Algebraic $K$-theory of stably compact spaces

ALGEBRAIC K -THEOR Y OF ST ABL Y COMP A CT SP A CES GEORG LEHNER Abstract. W e compute the v alue of ﬁnitary localizing inv ariants, including algebraic K -theory , on cate- gories of sheav es ov er stably lo cally compact spaces X . Our formula sim ultaneously generalizes the cases of locally compact Hausdorﬀ and coheren t (sp ectral) spaces and recov ers several smaller K -theory calculations as sp ecial instances. Ernst Haeckel, K unstformen der Natur , 1904, plate 71: Stephoidea, Public domain, via Wikimedia Commons 2020 Mathematics Subje ct Classiﬁcation. Primary: 19D99, Secondary: 06D22, 54B40, 18F20, 18F70. K ey words and phr ases. Stably compact spaces, V erdier duality , patch topology , algebraic K -theory , ∞ -categories of sheav es on a space. 1 2 GEORG LEHNER Contents 1. In tro duction 2 1.1. Motiv ation 4 1.2. The pro of strategy 8 1.3. A ckno wledgements 9 2. Preliminaries 9 2.1. Compactly assembled ∞ -categories 10 2.2. Dualizable stable ∞ -categories and lo calizing inv arian ts 12 2.3. T op oi 13 2.4. Lo cales 14 2.5. Sublo cales 17 3. Stable lo cal compactness 21 3.1. Stably lo cally compact top oi 21 3.2. Stably lo cally compact spaces 23 3.3. Shea ves on stably lo cally compact spaces 30 3.4. de Gro ot and V erdier duality for stably compact spaces 32 3.5. The patch top ology 34 4. Nisnevic h-type descent for lo calizing inv ariants 38 4.1. The sheaf condition on D 3 40 4.2. Extending excisiveness 42 5. Descen t on stably compact spaces 44 6. The main theorem 50 App endix A. The sheaf condition ov er free distributive lattices of arbitrary ﬁnite rank 51 References 52 1. Introduction Recen t adv ances in the K -theory of large categories as laid out b y the articles [Eﬁ25a], [Eﬁ25b], [Eﬁ25c] and [KNP24], see also the lecture [Nik23] b y Nikolaus, hav e raised the exciting p ossibility of providing a conceptual framework for assembly conjectures, such as the F arrell-Jones, No viko v, and Borel conjectures using the language of sheaf theory . It also seems p ossible to ﬁnally bridge the gap betw een the (algebraic) F arrell-Jones conjecture and its op erator-theoretic counterpart, the Baum-Connes conjecture. The central insigh t is that localizing in v ariants, suc h as algebraic K -theory , can b e extended from small stable ∞ - categories to the class of dualizable stable ∞ -categories. This includes ∞ -categories of sp ectral sheav es Sh( X , Sp) on a lo cally compact Hausdorﬀ space X , which play an analogous role to C ∗ -algebras of the form C 0 ( X, C ) in the con text of op erator theory . Whilst the general picture is still v ery muc h work in progress, an imp ortant computation that motiv ated this pro ject is the following. Theorem 1.1 ([Eﬁ25a] Theorem 6.11, see also [KNP24] Theorem 3.6.1) . L et X b e a lo c al ly c omp act Haus- dorﬀ sp ac e. Then K cont (Sh( X , Sp)) ≃ H cs ( X ; K ( S )) wher e Sp is the ∞ -c ate gory of sp e ctr a and the right-hand side r efers to c omp actly supp orte d she af c ohomolo gy of X with r esp e ct to the lo c al system given by the K -the ory of the spher e sp e ctrum. There are some extensions that are p ossible for Theorem 1.1. F or one, the theorem holds not just for K -theory , but for arbitrary ﬁnitary lo calizing inv arian ts, as long as the target of the lo calizing inv arian t ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 3 is a dualizable ∞ -category . It also generalizes to non-constant coeﬃcients, in the sense that an analogous form ula holds when one considers a presheaf C on X v alued in dualizable categories. W e prop ose a generalization of Theorem 1.1 b y enlarging the class of spaces X considerably . As w as already observed by Eﬁmov in [Eﬁ25a, Remark 6.2], Hausdorﬀness of a space X is not essential to guarantee that Sh( X , Sp) is dualizable, rather a w eak er notion suﬃces. Recall the notion of the wa y-below relation b et w een op en subsets of a top ological space X : T wo open sets U, V satisfy the relation U ≪ V iﬀ for all directed sets of op ens W i , i ∈ I it holds that [ i ∈ I W i ⊃ V implies ∃ i ∈ I : W i ⊃ U. (In other words, the inclusion U ⊂ V is a compact morphism in the p oset O ( X ) of op en sets of X .) Deﬁnition 1.2. A top ological space X is called stably lo c al ly c omp act if it is sob er, and the wa y-b elow relation ≪ satisﬁes the t w o conditions: • F or all op en U , we hav e U = [ V ≪ U V . • The wa y-below relation is stable under in tersection, in the sense that U ≪ V 1 and U ≪ V 2 implies U ≪ V 1 ∩ V 2 . A con tinuous map f : X → Y will be called p erfe ct if f − 1 preserv es the wa y-b elow relation. A partial map f : X → Y will b e called p artial p erfe ct , if its domain of deﬁnition is op en, and f restricted to its domain is a p erfect map. W e deﬁne the category SLC p of stably lo cally compact spaces and partial p erfect maps. An y lo cally compact Hausdorﬀ space X is in particular an example of a stably lo cally compact space. Moreo ver, the inclusion of the full sub category of lo cally compact Hausdorﬀ spaces in to stably locally compact spaces has a right adjoin t LCH p SLC p patch ⊣ whic h equips a stably lo cally compact space X with its p atch top olo gy . (See Section 3.5) If X is stably lo cally compact, the category of sheav es Sh ( X, Sp) is dualizable (Corollary 3.52), and hence it makes sense to apply lo calizing inv arian ts, such as K -theory . W e hav e obtained the following generalization of Theorem 1.1. Theorem 1.3 (See Theorem 6.2) . L et X b e stably lo c al ly c omp act, let C b e a dualizable stable ∞ -c ate gory and let F b e a ﬁnitary lo c alizing invariant Cat perf → E with values in a dualizable stable ∞ -c ate gory E . Then F cont (Sh( X , C )) ≃ H cs ( X patch ; F cont ( C )) . Here, the right-hand side refers to the c omp actly supp orte d c ohomolo gy of X patch . W e also obtain the follo wing p erhaps surprising dualit y statement. Giv en X and C as ab ov e, deﬁne the ∞ -category of cosheav es on X as Cosh( X , C ) = F un L (Sh( X , Sp) , C ) . Corollary 1.4 (See Corollary 6.3) . Let X b e stably lo cally compact, let C b e a dualizable stable ∞ -category and let F b e a ﬁnitary lo calizing inv ariant Cat perf → E with v alues in a presentable stable ∞ -category E . Then F cont (Sh( X , C )) ≃ F cont (Cosh( X , C )) . In the case of X b eing locally compact Hausdorﬀ the ab ov e statement easily follo ws from V erdier dual- it y . How ev er, in the absence of Hausdorﬀness, there is no a priori existing functor comparing shea ves and coshea ves on X directly . (There is a version of V erdier duality for stably compact spaces, which inv olv es the de Gr o ot dual X ∨ of X , a statement which w e will discuss in Section 3.4.) 4 GEORG LEHNER Remark 1.5. By Eﬁmov [Eﬁ25c], the target of the universal ﬁnitary lo calizing inv ariant U : Cat perf → Mot is a dualizable stable ∞ -category . Since every ﬁnitary lo calizing in v ariant F factors through U , this means that at least in principle one knows the v alue of Sh( X , C ) for any ﬁnitary lo calizing inv arian t F : Cat perf → E , ev en when E is not dualizable. 1.1. Motiv ation. W e should say that the purp ose of this article is t wo-fold. F or one, a simple motiv ation for Theorem 1.3 is that it is widely applicable to many diﬀeren t examples. Let us elab orate on some of them. • In the case of X b eing Hausdorﬀ, we hav e X = X patch and recov er the known fact that K cont (Sh( X , C )) ≃ H cs ( X ; K cont ( C )) giv en by Theorem 1.1. • The sp ecial case of X = S b eing the Sierpinski space, i.e. the top ological space giv en by a t w o element set 2 with a top ology with a unique op en p oint, can be enlightening. W e hav e S patch = 2 disc is the t wo p oint space equipp ed with the discrete top ology . There is a natural equiv alence Sh( S , C ) ≃ C ∆ 1 and the contin uous map p : 2 disc → S induces the functor p ∗ : C × C → C ∆ 1 ( c, d ) 7→ ( c × d → c ) . The statemen t that p ∗ induces an equiv alence on lo calizing inv ariants reco v ers the well-kno wn ad- ditivity the or em . W e thus w ould like to think of Theorem 1.3 as an analytic generalization of the additivit y theorem. • The case X = − − − → [0 , ∞ ) , with topology given by op ens of the form [0 , a ) for 0 ≤ a ≤ ∞ , is also quite imp ortan t. There is an equiv alence of ∞ -categories Sh( − − − → [0 , ∞ ) , C ) ≃ Sh( R , C ) ≥ 0 , using the terminology of Eﬁmo v [Eﬁ25a, p. 76]. In the case C = Sp , the dualizable ∞ -category Sh( R , Sp) ≥ 0 corepresen ts the functor that sends a dualizable ∞ -category C to Q -indexed diagrams in C with compact transition morphisms whenever a < b ∈ Q . Eﬁmo v shows that Sh( R , Sp) ≥ 0 is an ω 1 -compact generator of Pr L dual , [Eﬁ25a, Theorem D.1], and computes that F cont (Sh( R , C ) ≥ 0 ) = 0 for any ﬁnitary lo calizing inv arian t F , [Eﬁ25a, Prop osition 4.21]. W e recov er this result as a sp ecial case, as − − − → [0 , ∞ ) patch = [0 , ∞ ) with the ordinary metric topology , whic h has a con tractible one-p oint compactiﬁcation. • In the case of a c oher ent sp ac e X , also sometimes called sp e ctr al sp ac e , the patch top ology on X agrees with the constructible top ology . The statemen t of Theorem 1.3 in this case was veriﬁed in the author’s previous article [Leh25a]. • If P is a lo c al ly ﬁnite p oset , which means that the do wn-sets P ≤ p are ﬁnite for each p ∈ P , we can consider X = P Alex to b e the set P equipp ed with the Alexandrov top ology , which means that op en sets are deﬁned to b e do wnw ard closed sets. This giv es a lo cally coherent space ([Leh25a, Section 3.1]), and one can sho w that there is a natural equiv alence Sh( X , C ) ≃ F un( P op , C ) , see [Aok23a, Prop osition 5.18]. 1 One can see that P patch Alex = P disc is just P equipp ed with the discrete top ology , hence we obtain K cont (F un( P op , C )) ≃ K cont (Sh( X , C )) ≃ M p ∈ P K cont ( C ) , 1 W arning: W e are using the opp osite conv en tion from Aoki for the deﬁnition of a lo cally ﬁnite p oset. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 5 a result that can of course also b e veriﬁed indep endently using the fact that F un ( P op , C ) admits a P - indexed semiorthogonal decomp osition, [Eﬁ25a, Prop osition 4.14]. Examples of lo cally ﬁnite p osets are face p osets of ﬁnite dimensional simplicial or p olyhedral complexes, or v ariations thereof. They arise naturally as gadgets for homotop y theory (the case of ﬁnite p osets mo dels the homotopy theory of ﬁnite CW-complexes), or when considering stratiﬁcations of spaces, and sheav es on these spaces pla y a crucial role in combinatorial areas of geometry , e.g. Com binatorial In tersection Cohomology (See [Bar+00], [Kar04] and [Bra06]), or the theory of T ropical (Co)-Homology , [Bru+15]. The abilit y to ha v e these examples in the same framew ork as the inﬁnitary analytic examples is one of the strengths of considering the general framework of sheav es on stably lo cally compact spaces, and not just lo cally compact Hausdorﬀ spaces. • Let V b e a ﬁnite-dimensional real v ector space and γ a non-zero proper conv ex closed cone. Kashiw ara- Sc hapira [KS02] deﬁne the γ -top ology V γ on V , whic h is giv en by the set of op en subsets U of V whic h satisfy U + γ = U . They sho w that Sh V × γ ( V , C ) ≃ Sh( V γ , C ) , where Sh V × γ ( V , C ) is the ∞ -category of sheav es on V with microsupport V × γ . (See Theorem 1.5 in [KS17].) Zhang pro ved in [Zha25] that the v alue of any ﬁnitary lo calizing in v ariant on this ∞ -category v anishes. The top ological space V γ is not sob er, how ever its sobriﬁcation adds b oundary p oin ts tow ards the direction of the cone γ . This results in a stably lo cally compact space. (W e remark that ∞ -categories of sheav es cannot distinguish b et ween a space and its sobriﬁcation.) It w ould b e an interesting av enue for future work to reprov e Zhang’s result using purely top ological means; by geometrically understanding the patch top ology of ( V γ ) sob and applying Theorem 1.3. The secondary purp ose of this article is to illustrate that the apparent usefulness of the class of stably lo cally compact spaces given by the ab ov e list of examples is by no means an accident. One can think of it as a partial answer to the following question. What is the largest (con v enient) class of spaces X such that Sh ( X, Sp) is dualizable? Since it is the case that Sh( X , Sp) ≃ Sh( X, An) ⊗ Sp , where An denotes the ∞ -top os of anima/ ∞ - group oids, one might ask instead for the sligh tly stronger condition that the ∞ -topos Sh( X , An) is a com- pactly assembled ∞ -category . Let us mention the follo wing facts. • Giv en any Grothendieck top ology τ on a p oset P , which is closed under binary meets, there exists a natural frame F = F ( P, τ ) such that Sh( P , τ , An)  Sh ( F, An) . Such Grothendieck top ologies app ear naturally in man y contexts. • If for a given frame F the ∞ -topos Sh( F, An ) is a compactly assem bled ∞ -category , then F is isomorphic to the set of op ens of a sob er and locally compact space X . (See Theorem 3.49 and Theorem 3.16.) • Ho wev er, it is not true that for every sob er and lo cally compact space X the ∞ -top os Sh( X , An) is also compactly assem bled. An example that already fails in the case of 1 -top oi is given in [JJ82, Example 5.5]. • Under the additional assumption that X is in fact stably lo cally compact this issue disapp ears, and Sh( X , An) is compactly assembled. Stabilit y of X corresponds precisely to the additional prop erty of Sh ( X, An ) that compact morphisms are closed under pro ducts. While it can still happ en that for a top ological space X (or frame, for that matter) the category of sheav es Sh( X , C ) is dualizable for a given dualizable C , without X itself b eing stably lo cally compact, the category 6 GEORG LEHNER of stably lo cally compact spaces is thus in a sense an optimal candidate for a category of spaces for which w e get dualizable categories of sheav es. Remark 1.6. The condition of an ∞ -top os X to b e compactly assembled as an ∞ -category is equiv alen t to X being exp onentiable in the ∞ -category R T op of top oi, a fact prov en indep endently by Anel-Leja y [AL18], [AL], as well as Lurie [Lur18, Section 21.1.6], and generalizes the analogous statement for 1 -top oi due to Johnstone-Jo yal [JJ82]. On a deep er level, there are man y close parallels betw een the theory of stably lo cally compact spaces and that of dualizable categories. Eﬁmov originally observed shadows of this picture as several analogies b et w een dualizable categories and compact Hausdorﬀ spaces, see [Eﬁ25a, App endix F]. Among these are for example versions of Urysohn’s lemma (in the form as stating that [0 , 1] and Sh ( R , Sp) ≥ 0 are generators of the categories CH op and Pr L dual resp ectiv ely), and the Tyc honoﬀ Theorem (stating that the inclusion functors CH op → T op op and Pr L dual → Pr L preserv e colimits). Ho wev er, this analogy remains somewhat v ague. When one switches the p erspective to stably lo cally compact spaces, one can observ e the following tw o similarities. • A space X is stably lo cally compact iﬀ X is sob er and there exists a diagram of adjoints O ( X ) Ind( O ( X )) y ˆ y k ⊣ ⊣ in ternal to the 2 -category of low er b ounded distributive lattices. • An accessible stable ∞ -category C is dualizable iﬀ there exists a diagram of adjoints C Ind( C ) y ˆ y k ⊣ ⊣ in ternal to the ( ∞ , 2) -category of (large) stable ∞ -categories. Similarly , partial perfect maps betw een stably lo cally compact spaces, as w ell as strongly contin uous functors can also b e characterized in purely 2 -categorical terms. The formal setup for this is that b oth cases are instances of c ontinuous algebr as for a lax-idemp otent monad on an ( ∞ , 2) -category . 2 The corresp onding 2 -categorical version is sometimes also referred to as a KZ-monad. The requirement of the sobriety condition on X is an indicator that truly , one should rather consider the ambien t setting of frame/locales (of which O ( X ) is an example), rather than top ological spaces, as the right formal analogue to the ∞ -category Pr L st . F rames are in fact the algebras for the Ind -monad on distributive lattices, the same wa y presentable stable ∞ -categories are (up to size issues) the algebras for the Ind -monad on stable ∞ -categories. The functor whic h assigns a frame F to its ∞ -category of shea ves Sh( F , Sp) provides a bridge b et w een the tw o cases, and descends to a well-deﬁned functor from (SLC p ) op to Pr L dual , which do es in fact send the generator − − − → [0 , ∞ ) (see Corollary 3.46) to the generator Sh( R , Sp) ≥ 0 . A oki has observed that the relationship b etw een frames and presentable categories b ecomes quite tight, whic h he formalized in his sheav es-sp ectrum adjunction. 2 The general theory of lax-idemp oten t monads on ( ∞ , 2) -categories is still w ork in progress, with forthcoming results by Abellán–Blom [AB26]. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 7 Theorem 1.7 ([Aok25a]) . Ther e exists an adjunction F rm CAlg (Pr L ) Sh( − , An) cIdem ⊣ b etwe en the c ate gory of fr ames F rm and the ∞ -c ate gory of pr esentably symmetric monoidal ∞ -c ate gories, wher e Sh( F , An ) is the ∞ -c ate gory of she aves on a fr ame F , and cIdem( C ) is the sp e ctrum of c oidemp otents of a pr esentably symmetric monoidal ∞ -c ate gory C . He ﬂeshed this adjunction out further in the article [A ok25b], in a manner whic h w e summarize b y the follo wing commutativ e diagram of adjunctions. Let us call a space X stably c omp act if it is compact and stably lo cally compact. Denote b y SC the category of stably compact spaces and p erfect maps, and b y CH the full subcategory of compact Hausdorﬀ spaces. (W e note that any contin uous map b etw een compact Hausdorﬀ spaces is automatically p erfect.) Rig CAlg(Pr L dual ) CAlg(Pr L st ) CAlg(Cat perf ) CH op SC op F rm DLatt bd Sm rig “add duals” f or get Sm con Ind f or get Sm Ind Idemp Sh Sh patch f or get Sh Ind f or get Sh( − ,f in ) ω Ind ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ ⊣ A few comments. • The functors Sh and Sh( − , f in ) in this diagram refer to sheav es with v alues in sp ectra. • The category CAlg (Pr L dual ) refers to presentably symmetric monoidal stable ∞ -categories, whose underlying category is dualizable, with compact 1 and such that ⊗ preserves compact morphisms in b oth v ariables. • The category Rig refers to rigid presen tably symmetric monoidal categories. F or a reference, see [KNP24] and [Ram24b]. • All blue arrows in the diagram are left adjoints, and all squares comp osed of them commute. The red arrows are all righ t adjoints to the blue ones, and the green arrows are all left adjoints to the blue ones. • The central columns are given as algebras and contin uous algebras, resp ectiv ely for the Ind -monad. • The images of the right most columns under Ind give the case of coherent spaces and compactly generated stable ∞ -categories. These can b e understo o d as fr e e contin uous algebras, which generate the rest under retracts. • The relationship of CH and Rig to the rest of the picture will not b e further discussed in this pap er, with the exception of the patch-topology functor, but is highly interesting. Remark 1.8. On a more philosophical persp ective on wh y one could expect Theorem 1.3 to be true, w e remark that K -theory can b e considered as a form of universal me asur e - In fact, there are several close analogies b etw een measure theory and the K -theory of categories of sheav es ov er certain classes of lo cales, as in v estigated to some degree in the author’s previous articles [Leh25a, 7.2] and [Leh25b, Remark 1.20], motiv ated by the observ ation that b oth areas of math are morally sp eaking just formalizations of the inclusion-exclusion principle together with the principle of exhaustion from b elow. W e can deﬁne a me asur e on a lo cale L to b e a contin uous v aluation µ : O ( L ) → [0 , ∞ ] . If L corresp onds to a lo cally compact Hausdorﬀ space X , then lo c al ly ﬁnite measures on L agree with the classical notion of R adon me asur e on X , see [Leh25b, Theorem 9.4 and Section 12.2]. It is in fact true that lo cally ﬁnite measures on a stably lo cally 8 GEORG LEHNER compact space X extend uniquely to lo cally ﬁnite measures on X patch , as pro v en in [KL05, Theorem 8.3]. Con versely , any lo cally ﬁnite measure on X patch pro duces a lo cally ﬁnite measure on X via pushforw ard along the p erfect map X patch → X , resulting in an isomorphism Meas lf ( X )  Meas lf ( X patch ) , natural in partially deﬁned p erfect maps of stably lo cally compact spaces with op en supp ort. Analogously , one could exp ect K -theory to behav e in the same w ay . This w as one of the motiv ations for the author to in vestigate the formula given in Theorem 1.3. 1.2. The pro of strategy. The pro of of Theorem 1.3 rests on a meta theorem which is a generalization of an argument due to Clausen in the case of compact Hausdorﬀ spaces, see [KNP24, Theorem 3.6.11]. Theorem 1.9 (See Theorem 5.2) . L et D b e a c omp actly assemble d pr esentable ∞ -c ate gory, and let F : SC op → D b e a c ontr avariant functor on the c ate gory of stably c omp act sp ac es such that: (1) F ( ∅ ) = 1 . (2) P erfect descent: Whenever K , L ⊂ X ar e two p erfe ct emb e ddings, then F ( K ∪ L ) F ( K ) F ( L ) F ( K ∩ L ) ⌟ is a pul lb ack. (3) Coﬁltered descent: Whenever X i , i ∈ I , is a c oﬁlter e d system in SC , then F (lim i ∈ I X i ) ≃ colim i ∈ I F ( X i ) . Then ther e is a natur al e quivalenc e of functors F ≃ Γ(( − ) patch ; F (pt)) . The pro of works formally almost the same as the one given by Clausen, along with one crucial observ ation: The set of p erfect embeddings K ⊂ X is simply isomorphic to the set of closed subsets C ⊂ X patch . Hence for each given stably compact space X , the restriction of F to the set of p erfect embeddings of X pro duces a K -sheaf on X patch . The corresp onding sheaf F X under the equiv alence of K -sheav es with sheav es (Theorem 3.61) satisﬁes that its compactly supp orted sections are computed as F X ( X ) = F ( X ) . No w one needs to sho w that this sheaf is lo cally constant. W e verify this in detail in Section 5. It is not to o hard to v erify that K -theory , or any ﬁnitary lo calizing inv arian t for that matter, satisﬁes (1) and (3). (See Corollary 2.34.) How ever, showing p erfect descent for ﬁnitary lo calizing in v ariants needs careful analysis. W e reduce it to tw o diﬀerent descen t conditions: Close d and Satur ate d c omp act descent, and then use an inductive argument to extend it to all p erfect embeddings in Section 4. This rests on a concrete description of the closed sets of the patch top ology as generated under intersections by pairwise unions of closed and saturated compact sets, see Section 3.5. Remark 1.10. As the reader will observe, the pro of of Theorem 1.3 needs little in the sense of actual K -theory computations. Most of the diﬃculty relies in understanding the correct “p oin t-set top ological” features of stably lo cally compact spaces and their ∞ -categories of sheav es. Remark 1.11. W e originally hop ed that the pro of of Theorem 1.3 could b e reduced to the following claim: ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 9 Conjecture 1.12. The universal ﬁnitary lo calizing in v ariant U : CAlg (Pr L dual ) → CAlg (Mot) is right Kan extended from its restriction to CAlg (Pr L st ,ω ) . The analogous statement without reference to symmetric monoidality is true by Eﬁmov [Eﬁ25a, Theorem 4.16]. If Conjecture 1.12 were in fact true, one could use the adjointabilit y of the square of adjoints (CohSp c) op CAlg(Pr L st ,ω ) (SC) op CAlg(Pr L dual ) Sh Sm con Sh Sm con ⊣ ⊣ ⊣ ⊣ to show that U (Sh( − )) is right Kan extended along its restriction to (CohSp c) op , where CohSp c is the category of c oher ent sp ac es , also called sp ectral spaces. F rom here, we ha ve the existence of the adjointable square ProFin CohSp c CH SC const patch ⊣ ⊣ whic h reduces the statement to the result that U (Sh( X ; Sp)) ≃ H • ( X const , U (Sp)) , for a coherent space X , as prov en b y the author in [Leh25a], together with a statement ab out whether the sheaf cohomology functor is the right Kan extension of its restriction along (ProFin ) op → (CH) op , a question that is dealt with using the newly developed setup of condensed cohomology , see [Sch19]. As it stands, we are not sure whether Conjecture 1.12 is correct, which is why we are left with the more granular pro of strategy as presented in this article. Remark 1.13. In the case of X being locally compact Hausdorﬀ, Eﬁmo v [Eﬁ25a, Theorem 6.11] prov es a more general statement, namely that for an y presheaf C : O ( X ) op → Pr L dual on X with v alues in dualizable ∞ -categories it holds that U (Sh( X , C )) ≃ H • c ( X, U ( C ) # ) . It is plausible that Theorem 1.3 holds in this greater generality as well for stably lo cally compact spaces, ho wev er w e hav e not attempted this during the work on this article. Perhaps the main issue w ould b e in understanding why U ( C ) should lift to a presheaf on X patch , where U denotes the univ ersal lo calizing in v ariant. This p oint is something to b e addressed in future work. 1.3. A c kno wledgemen ts. W e w ant to thank Thomas Nikolaus, Maxime Ramzi, Thorger Geiß, Phil Pützstück, Dustin Clausen, Bingyu Zhang, Mathieu Anel and Bastiaan Cnossen for helpful comments and discussions. F urthermore, the author was funded by the Deutsc he F orsch ungsgemeinschaft (DF G, German Researc h F oun- dation) – Pro ject-ID 427320536–SFB 1442, as w ell as under German y’s Excellence Strategy EX C 2044/2 - 390685587, Mathematics Münster: Dynamics-Geometry-Structure. 2. Preliminaries W e will largely follow the conv en tions on ∞ -category theory as laid out in [Lur12], [Lur17] and [Lur18]. The sym b ol [1] refers to the p oset { 0 ≤ 1 } . The sym b ol An denotes the ∞ -category of anima (equiv alently sp ac es , or ∞ -group oids). W e denote the ∞ -category of small ∞ -categories by Cat ∞ , and the ∞ -category of 10 GEORG LEHNER large ∞ -categories as [ Cat ∞ . If C is an ∞ -category , then C op refers to the ∞ -category obtained by reversing the 1 -morphisms. The sym b ol Pr L refers to the ∞ -category of presen table ∞ -categories together with left adjoin t functors as morphisms. This ∞ -category is symmetric monoidal when equipped with the Lurie tensor pro duct C ⊗ D = F un R ( C op , D ) , with unit An and which preserv es colimits in b oth v ariables. The corresp onding internal hom is given by F un L ( C , D ) . Commutativ e algebra ob jects in (Pr L , ⊗ ) are giv en by presen table symmetric monoidal ∞ -categories ( C , ⊗ ) such that the given tensor pro duct preserv es colimits in b oth v ariables. A functor F : C → D in Pr L with fully faithful right adjoint is called a Bousﬁeld lo c alization . W e remark that F is a Bousﬁeld lo calization iﬀ for every other presentable ∞ -category E precomp osition with F induces a fully faithful functor F un L ( D , E ) → F un L ( C , E ) , see [Lur12, Prop osition 5.5.4.20]. Using the tensor-hom adjunction, it is clear that Bousﬁeld lo calizations are closed under tensor pro ducts in Pr L . W e will use light 2 -categorical techniques o ccasionally . If C is an ( ∞ , 2) -category , it makes sense to sp eak of an internal adjunction b et w een ob jects c, d , c d g f ⊣ deﬁned analogously to how one deﬁnes an adjunction b etw een ∞ -categories, see e.g. [GR19, Ch. 12]. An in ternal left adjoin t f is called an internal emb e dding if the unit id d ⇒ g f is an equiv alence. It is called an internal quotient if the counit f g ⇒ id c is an equiv alence. Note that these notions only depend on the underlying homotopy 2 -category of C . The following lemma will b e useful on o ccasion. Lemma 2.1. If F : C → D is a 2 -functor b et w een ( ∞ , 2) -categories it preserv es internal adjunctions, internal em b eddings and internal quotients. The pro of of this lemma reduces to the observ ation that in ternal adjunctions, internal em b eddings and in ternal quotients are c haracterized as b eing equiv alent to 2 -functors out of basic 2 -categorical diagram shap es, and comp osition of 2 -functors gives 2 -functors. W e will occasionally use the following standard lemma. Lemma 2.2. Supp ose C is an ( ∞ , 2) -category and the diagram c 1 d 1 c 0 d 0 f 1 i L c f L 1 i L d i c f 0 i d ⊣ ⊣ ⊣ is a diagram in C with i L c and i L d b eing in ternal quotien ts and the square in v olving i c , i d , f 0 and f 1 b eing comm utative. Then f 0 has a left adjoint giv en by i L c f L 1 i d : d 0 → c 0 . The pro of follows directly from the same statement in the situation of an ordinary 2 -category , whic h is a standard fact. Of course a dual version, demanding that i L c and i L d are in ternal em beddings, holds true as w ell. 2.1. Compactly assem bled ∞ -categories. The notion of a compactly assembled ∞ -category is central to this article. W e note that if sp ecialized to 1-category theory , a compactly assembled category corresp onds to what is also kno wn as a c ontinuous category . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 11 Deﬁnition 2.3 (See also [Lur18] Section 21.1.2, [Eﬁ25a] Deﬁnition 1.9 and 1.10) . Let C b e an accessible ∞ -category . W e call C c omp actly assemble d if the canonical inclusion y : C → Ind( C ) admits tw o further left adjoin ts ˆ y ⊣ k ⊣ y , C Ind( C ) . y ˆ y k ⊣ ⊣ A functor F : C → D will b e called str ongly c ontinuous if the commuting square C Ind( C ) D Ind( D ) y F Ind( F ) y is twice left-adjointable, i.e. the resulting natural transformations k Ind( F ) → F k and ˆ yF → Ind( F ) ˆ y are equiv alences. The resulting sub category of [ Cat ∞ of large ∞ -categories given b y compactly assem bled ∞ -categories and strongly contin uous functors will b e denoted by Cat ca . F urther, we denote by Pr L ca the ∞ -category of presen table and compactly assembled ∞ -categories, and strongly contin uous left adjoin t functors b et w een them, as a (non-full) sub category of [ Cat ∞ . Note that y : C → Ind( C ) is fully faithful, and therefore there is alwa ys a natural transformation ˆ y → y . Deﬁnition 2.4. Let C b e a compactly assembled ∞ -category . A map f : c → d in C is called c omp act if y ( f ) : y ( c ) → y ( d ) factors through the natural map ˆ y ( d ) → y ( d ) . Remark 2.5. Left adjointabilit y of the square in Deﬁnition 2.3 for a functor F : C → D simply expresses that F preserves ﬁltered colimits. If F itself has a further righ t adjoin t F R , then the statement that F is strongly contin uous is equiv alent to F R preserving ﬁltered colimits. Example 2.6. If C is an ∞ -category with ﬁltered colimits, denote b y C ω the full sub category of c omp act obje cts , i.e. those c ∈ C suc h that Map C ( c, − ) preserves ﬁltered colimits. Then C is called c omp actly gener ate d , if C ≃ Ind( C ω ) . If this is the case, C is in particular compactly assembled with ˆ y obtained b y applying Ind to the inclusion C ω  → C . Theorem 2.7 ([Lur18], Section 21.1.2) . A n ∞ -c ate gory C admitting ﬁlter e d c olimits is c omp actly assemble d iﬀ it is a r etr act via ﬁlter e d c olimit pr eserving functors of a c omp actly gener ate d ∞ -c ate gory. In p articular, if C is a r etr act of a c omp actly assemble d ∞ -c ate gory D via ﬁlter e d c olimit pr eserving functors, then C is also c omp actly assemble d. Let us p oint out tw o sp ecial cases of Theorem 2.7. • If we hav e an adjunction C D R L ⊣ with R b eing fully faithful, and b oth L, R preserving ﬁltered colimits, then ˆ y C is obtained as the comp osite Ind( L ) ˆ y D R , as a direct application of Lemma 2.2. • If we hav e an adjunction C D L R ⊣ 12 GEORG LEHNER with L b eing fully faithful, and both L, R preserving ﬁltered colimits, then the composite ˆ y D L factors through Ind( L ) , giving the functor ˆ y C . Concretely , this means that for eac h c ∈ C , the formal diagram ˆ y D L ( c ) = “colim” i ∈ I d i has a coﬁnal system of ob jects L ( c j ) , j ∈ J in the essential image of L , resulting in the diagram ˆ y C ( c ) = “colim” j ∈ J c j . This is typically inexplicit to compute in practice, but we remark that a map f : c → c ′ in C is compact iﬀ L ( f ) is compact in D , see [Aok25b, Lemma 3.26]. Compactly assembled presentable ∞ -categories are furthermore closed under tensor pro ducts. Theorem 2.8 ([KNP24] Prop osition 2.12.2.) . The Lurie tensor pr o duct ⊗ r estricts to a symmetric monoidal structur e on Pr L ca , which is close d. Lastly , the follo wing lemma will b e useful for detecting strongly contin uous functors. Lemma 2.9 ([KNP24], Lemma 2.12.1) . Suppose F : C → D is a left adjoint functor in Pr L . Then F is strongly contin uous iﬀ there exists a commutativ e square in Pr L C D C ′ D ′ F L C L D F ′ suc h that F ′ : C ′ → D ′ is a left-adjoint functor preserving compact ob jects b etw een compactly generated presen table ∞ -categories C ′ , D ′ , and L C and L D are internal embeddings in Pr L , i.e. they are b oth fully faithful and hav e right adjoints preserving colimits. 2.2. Dualizable stable ∞ -categories and lo calizing in v arian ts. Let us recall some basic prop erties of dualizable ∞ -categories. The standard references are [Eﬁ25a], [KNP24] and [Ram24a]. Deﬁnition 2.10. A compactly assem bled presentable ∞ -category C is called dualizable if it is also stable. Denote by Pr L dual ⊂ Pr L ca the full sub category spanned by dualizable ∞ -categories. A sequence C F − → D G − → E of dualizable ∞ -categories C , D , E and strongly contin uous left adjoints F , G is called exact , or also a V er dier se quenc e , if it is a coﬁber sequence in Pr L dual , and F is fully faithful. A functor F : Pr L dual → E , where E is an accessible stable ∞ -category , is called a lo c alizing invariant , if F (0) = 0 and F maps exact sequences to ﬁb er sequences in E . If E is furthermore presentable, then F will b e called ﬁnitary if F preserv es ﬁltered colimits. Theorem 2.11 (See [Eﬁ25a] Prop osition 1.65, and [KNP24] Prop osition 2.12.9) . The inclusion Pr L dual → Pr L st pr eserves and cr e ates c olimits. Mor e over, the Lurie tensor pr o duct ⊗ r estricts to a symmetric monoidal structur e on Pr L dual , which is close d, and the stabilization functor − ⊗ Sp : Pr L ca → Pr L dual is str ong monoidal and left adjoint to the inclusion of Pr L dual into Pr L ca . Let Cat perf denote the ∞ -category of stable, idempotent complete ∞ -categories. The essential image of the functor Ind : Cat perf → Pr L dual are called c omp actly gener ate d stable ∞ -categories. There already exists a notion of lo calizing inv ariant on Cat perf , see [BGT10], which includes for example the algebraic K - theory functor, or top ological Ho c hsc hild homology THH . Eﬁmo v observed that the v alues of any lo calizing in v ariant on Pr L dual are determined uniquely by their restriction along Ind . Theorem 2.12 (Eﬁmov, see [Eﬁ25a] Theorem 0.1) . R estriction along Ind F un(Pr L dual , E ) → F un (Cat perf , E ) induc es an e quivalenc e on the ful l sub c ate gories of ﬁnitary lo c alizing invariants. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 13 W e will denote the inv erse under this functor of a lo calizing in v ariant F deﬁned on Cat perf b y F cont . The follo wing is a simple, but sometimes useful, observ ation. Lemma 2.13. Let F cont : Pr L dual → E b e a ﬁnitary lo calizing in v ariant, and D be a dualizable ∞ -category . Then F cont ( − ⊗ D ) is again a ﬁnitary lo calizing inv ariant. Pr o of. The operation of − ⊗ D preserves the zero category and ﬁltered colimits. It furthermore preserves coﬁb er sequences. It remains to argue that if F : C  → E is a fully faithful strongly contin uous left adjoint, then also F ⊗ id D is fully faithful. This follows from the observ ation that b etw een dualizable ∞ -categories, the right adjoint F R of a strongly contin uous left adjoint has a further right adjoint F RR . In other words, F is an in ternal embedding in Pr L , and will b e mapp ed to an in ternal embedding under the image of the 2 -functor − ⊗ D . □ 2.3. T op oi. Let us introduce some general notions ab out higher top oi. The reference for (higher) top oi that we follow is [Lur12]. F or the sake of this article, the word top os will denote what is usually called an ∞ -top os, by whic h we mean a left exact accessible lo calization of an ∞ -category of the form F un( C op , An) , where C is some small ∞ -category . If X , Y are t wo ∞ -top oi, a geometric morphism f : X → Y is deﬁned via its pul lb ack p art f ∗ : X ← Y , which is required to b e a left adjoin t functor preserving ﬁnite limits. Its right adjoin t will be denoted as f ∗ : X → Y and called the dir e ct image part of f . The ∞ -category of top oi and geometric morphisms will b e denoted by R T op , where R refers to the fact that we consider maps as directed along the right adjoint direct image functors. If X is a top os, there exists a unique geometric morphism X → An , whose direct and inv erse image parts will b e denoted as X An . X ∗ X ∗ ⊣ If X = Sh( X , An) is the ∞ -category of shea ves on some top ological space or lo cale X , w e also use the notation X ∗ ⊣ X ∗ . Deﬁnition 2.14. Let X a top os and E a presentable ∞ -category . W e call the ∞ -category X ⊗ E = F un R ( X op , E ) the ∞ -category of E -v alued sheav es ov er X . (Compare with e.g. [V ol25, Section 2.2].) Ob jects in the essential image of E X ∗ ⊗ id E − − − − − → X ⊗ E are referred to as c onstant E -v alued shea v es. If E ∈ E , and the con text is clear, we will use the notation E for the asso ciated constan t sheaf with v alue E . W e denote the right adjoint to X ∗ ⊗ id E b y Γ( X ; − ) : X ⊗ E → E and call it the glob al se ctions functor . Deﬁnition 2.15. Let f : X → Y b e a geometric morphism. W e say: • f is c ontr actible if f ∗ is fully faithful. 3 • f is essential if f ∗ admits a further left adjoint f ! . The functor f ! is referred to as r elative shap e of X ov er Y . A top os X is called c ontr actible iﬀ the unique geometric morphism X → An is contractible, and lo c al ly c ontr actible iﬀ X → An is essen tial. F or a lo cally contractible top os X , the morphism X ! is sometimes also written as Π X ∞ . The v alue Π ∞ ( X ) = Π X ∞ (1) is called the shap e of X . 3 This is equiv alent to the statement that the relative shap e of X ov er Y is trivial in the sense of [V ol25, Deﬁnition 3.1]. 14 GEORG LEHNER F or more information ab out shap e theory of top oi, we refer the reader to [Lur17, App endix A] as well as [Ho y18]. W e will need the following simple lemma ab out contractible and essen tial geometric morphisms. Lemma 2.16. Let f : X → Y b e a geometric morphism and E a presentable ∞ -category . Then f ∗ ⊗ id E preserv es E -v alued constant sheav es. Moreov er, if f is a contractible and essential geometric morphism then the right adjoint to f ∗ ⊗ id E also preserves constant sheav es. Pr o of. The statement ab out f ∗ ⊗ id E is immediate. Now assume that f is contractible and essential, in other w ords we hav e an adjunction of the form X Y f ! f ∗ f ∗ ⊣ ⊣ with f ∗ fully faithful. Since − ⊗ E is a 2-functor on Pr L , the same holds for the diagram X ⊗ E Y ⊗ E f ! ⊗ id E ( f ∗ ⊗ id E ) ∗ f ∗ ⊗ id E ⊣ ⊣ where ( f ∗ ⊗ id E ) ∗ denotes the right adjoint to f ∗ ⊗ id E , and f ∗ ⊗ id E is fully faithful. Therefore w e see that ( f ∗ ⊗ id E ) ∗ ( X ∗ ⊗ id E ) ≃ ( f ∗ ⊗ id E ) ∗ ( f ∗ ⊗ id E )( Y ∗ ⊗ id E ) ≃ ( Y ∗ ⊗ id E ) , in other words ( f ∗ ⊗ id E ) ∗ preserv es constant ob jects. □ Example 2.17. If X is a paracompact top ological space with the homotopy type of a CW-complex, then X is lo cally contractible, and its shap e Π ∞ ( X ) agrees with the homotop y type of X , [Lur17, Remark A.1.4]. In particular, if I = [0 , 1] is the closed in terv al and X = I S is a Hilb ert cub e for some set S , then Sh( I S ) is an example of a con tractible and lo cally contractible top os. If furthermore N is a subset of S , then the canonical pro jection I S → I N arises as the pro duct of the canonical map I S \ N → pt with I N , and hence the induced geometric morphism Sh ( I S ) → Sh( I N ) is con tractible and essential. 2.4. Lo cales. W e will use the notion of frames and lo cales as developed for example in [PP12] and [Joh82]. Lo cales should b e thought of as a suitable generalization of the notion of top ological spaces, 4 that ﬁx several pathological shortcomings that the notion of top ological spaces has. Lo cales can b e understo od as the 0 - categorical analogue of top oi. (In fact lo cales embed fully faithfully in to top oi by taking shea ves.) Let us recap some basic notions and ﬁx notation. A frame is a complete lattice ( F , ≤ ) suc h that binary meets distribute ov er arbitrary joins. 5 A left adjoint f ∗ : F → F ′ is called a p artial fr ame homomorphism if f ∗ comm utes with binary meets. It is further called a fr ame homomorphism if f ∗ also preserves the top element, equiv alently all ﬁnite meets. Analogously to geometric morphisms, we will denote the corresp onding right adjoint by f ∗ . W e denote by F rm part and F rm the categories of frames with partial frame homomorphism, respectively frame homomorphisms, and deﬁne Lo c part = (F rm part ) op and Lo c = F rm op as the categories of lo cales and (partial) contin uous maps b etw een them. There exists an adjunction T op Lo c O pts ⊣ 4 While it is not technically true that all top ological spaces embed fully faithfully in to lo cales, typically in practice no mathematical content is lost by only considering sob er spaces, which do embed. 5 Or in the language of category theory , a cartesian closed presentable category that is a p oset. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 15 where the left adjoint sends a top ological space X to its frame O ( X ) , and a lo cale L to the space pts( L ) with underlying set Map Loc (pt , L ) where pt is the lo cale obtained from the one p oin t space, with frame [1] . This adjunction is idemp otent, and identiﬁes the sub category of sob er topological spaces with the full sub category of spatial lo cales. Sob erness is a mild separation condition, whic h implies T0 and is implied b y T2 , see [PP12, Chapter I.1]. F or the sake of notation, we sometimes also write O ( L ) for the asso ciated frame representing a lo cale, and call it the fr ame of op ens of L . The notion of partial frame homomorphism corresp onds to the notion of a p artial c ontinuous map f : X → Y , whose domain U is op en in X . W e note that F rm part and F rm are naturally (1 , 2) -categories, since their hom-sets are naturally p osets. Example 2.18. If F is a frame and U ∈ F an op en, then F /U = { V ∈ F | V ≤ U } is again a frame, called the op en sublo c ale asso ciated with U . The inclusion F /U ⊂ F is a partial frame homomorphism. It has an righ t adjoint − ∧ U : F → F /U , which is frame homomorphism (and hence an in ternal right adjoint in F rm part ), corresp onding top ologically to the inclusion of the op en sublo cale. An y other partial frame homomorphism f ∗ : F → F ′ naturally factors as F f ∗ − → F ′ /f ∗ (1) → F ′ where the left functor is a frame homomorphism. T op ologically , this means that every partial map of lo cales has an op en domain f ∗ (1) , on which it is an actual map of lo cales. Construction 2.19 (The non-Hausdorﬀ one-p oin t compactiﬁcation) . The inclusion of the (wide) sub cate- gory F rm  → F rm part admits a left adjoin t, whic h is given by sending a frame F to the frame F ⊤ giv en b y adding a single new top elemen t. T op ologically , this adds a new point + given by the frame homomorphism + ∗ : F ⊤ → [1] with + ∗ ( U ) = 0 for all U ∈ F . This point is a c o-generic close d p oint, in the sense that its only op en neighborho o d is the top element 1 . The adjunction F rm F rm part ( − ) + ⊣ lifts for formal reasons to an adjunction F rm / [1] F rm part coim ( − ) + ⊣ where the right adjoin t sends a frame homomorphism f ∗ : F → [1] to the frame coim( f ∗ ) = ( f ∗ ) − 1 (0) ⊂ F , and the left adjoint ( − ) ⊤ b ecomes fully faithful. T op ologically , this corresp onds to a fully faithful embedding L 7→ L + of Lo c part in to p oin ted lo cales Lo c pt / . Its left adjoin t sends a p oin ted lo cale to the (maximal) op en complemen t of the chosen basep oint. W e note that for the case of L + , the newly added p oint + is closed in L + , with op en complemen t given by the original top element of O ( L ) . Giv en a lo cale L , its ∞ -category of shea v es Sh( L ) = Sh( L, An) is given b y the full subcategory of F un( O ( L ) op , An) consisting of those functors F suc h that: • F (0) ≃ 1 . • F or all U, V ∈ O ( L ) it holds that F ( U ∨ V ) F ( U ) F ( V ) F ( U ∧ V ) is a pullback square. 16 GEORG LEHNER • F or all directed sets U i , i ∈ I , the canonical map F ( _ i ∈ I U i ) → lim i ∈ I F ( U i ) is an equiv alence. The ∞ -category Sh( L ) is a left exact and accessible localization of F un( O ( L ) op , An) and hence a top os, [Lur12, Corollary 6.2.1.7 and Lemma 6.2.2.7]. Moreo ver, if f : L → L ′ is a partial map of lo cales, it is straightforw ard to see that the precomposition F 7→ ( U 7→ F ( f ∗ ( U ))) preserves the sheaf condition, and hence gives a well-deﬁned functor f ∗ : Sh( L ) → Sh ( L ′ ) . It follo ws that f ∗ has a left adjoin t f ∗ , whic h is given by left Kan extension, follow ed by sheaﬁﬁcation. In case f is an actual map of locales, i.e. f ∗ : O ( L ′ ) → O ( L ) preserv es the top element, the induced left adjoint f ∗ on sheaf ∞ -categories preserves ﬁnite limits, i.e. f ∗ : Sh ( L ) → Sh( L ′ ) is a geometric morphism. F or a partial map of lo cales f : L → L ′ , we ha ve the decomp osition L ⊃ f ∗ (1) f − → L ′ , which means that the left adjoint f ∗ : Sh( L ′ ) → Sh ( L ) factors as Sh( L ′ ) → Sh ( L ) /y f ∗ (1) → Sh( L ) hence do es not preserve the terminal ob ject, how ever pullbacks and binary pro ducts. 6 W e record this b y the statemen t that there exist functors Sh : (Lo c part ) op → Pr L and Sh : Lo c → R T op  → (Pr L ) op . T op oi in the essential image of Lo c in R T op are called lo c alic top oi . The functor Sh has a left adjoint. Giv en an y top os X , the sub category X ≤ 0 = Sub(1 X ) of sub ob jects of 1 is a frame, [Lur12, Prop osition 6.4.5.4.]. The assignmen t X 7→ X ≤ 0 is in fact left adjoint to the shea v es functor, [Lur12, Prop osition 6.4.5.7.], i.e. we hav e the adjunction 7 Lo c R T op . Sh ( − ) ≤ 0 ⊣ The unit of this adjunction is giv en b y the geometric morphism X → Sh( X ≤ 0 ) , whose pullback part is given b y left Kan extending the inclusion y : X ≤ 0  → X . The functor y has a left adjoint τ ≤− 1 , whose main prop erties we summarize here. Prop osition 2.20. Let X b e a top os. Then there exists an adjunction X ≤ 0 X y τ ≤− 1 ⊣ where the left adjoint is giv en by sending X ∈ X to its supp ort , given b y the canonical eﬀective epimorphism- monomorphism factorization X 1 τ ≤− 1 ( X ) . ∃ ! The inclusion y preserves ﬁltered colimits and τ ≤− 1 preserv es ﬁnite pro ducts. 6 The statement that the forget functor from an ov ercategory preserv es con tractible limits is given as Lemma 2.2.7. in [GHK20], see also [Lur12, Prop osition 4.4.2.9]. The fact that in this case also binary pro ducts are preserv ed is because y U ∈ Sh( L ) is a sub ob ject of 1 . 7 W e remark that this adjunction arises via restriction from the more general sheaves-coidempotents adjunction due to Aoki, given in Theorem 1.7. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 17 Pr o of. W e cite the relev an t parts from [Lur12]. The existence of the left adjoint is provided by Prop osition 5.5.6.18. The statement that τ ≤− 1 preserv es ﬁnite pro ducts is giv en by [Lur12, Lemma 6.5.1.2]. The claim that y preserves ﬁltered colimits follo ws from the statement that ﬁltered colimits are exact in ∞ -top oi (Example 7.3.4.7.), and hence monomorphisms are closed under ﬁltered colimits. □ Let ( P , τ ) b e a Grothendiec k top ology τ on a p oset P . Deﬁne the p oset F ( P , τ ) as the set of all U ⊂ P that are down ward closed and compatible with sieves, in the sense that if S ⊂ { q ∈ P | q ≤ p } is a cov ering siev e of p ∈ P , we ha ve p ∈ U iﬀ q ∈ U for all q ∈ S . Prop osition 2.21. Let ( P , τ ) b e a site with P a p oset. Then F ( P, τ ) is a frame. If P is closed under binary meets then it holds that Sh( P , τ ) ≃ Sh( F ( P, τ )) . Pr o of. The p oset F ( P, τ ) can be seen to agree with the lo calic reﬂection of Sh( P, τ ) , whic h is giv en by Sub(1) ≃ Sh( P , τ ; [1]) , b y sending F : P op → [ 1] to the down w ard closed set F − 1 (1) . The rest of the claim is a sp ecial case of Lemma 6.4.5.6 in [Lur12], which states that Sh ( P, τ ) is lo calic if P is closed under binary meets. □ W arning 2.22. The requiremen t that P is closed under binary meets cannot b e dropp ed in general. As an example, the trivial Grothendieck top ology on P leads to the frame F un ( P op , [1]) , which is isomorphic to the set of op en sets of the Alexandroﬀ topology . Ho wev er, it is not the case in general that F un ( P op , An) is equiv alen t to Sh( P Alex ) . Rather, the former agrees with the latter after hypercompletion, see [Aok23b, App endix A] for a discussion. 2.5. Sublo cales. Let us collect some basic notions on sublo cales. Deﬁnition 2.23. Let f : L → M b e a contin uous map of lo cales. W e call f a quotient map , if f ∗ is injective. W e call f an emb e dding if f ∗ is surjective. In this case L will also b e called a sublo c ale of M . Remark 2.24. It is sensible to extend Deﬁnition 2.23 to the case where f : L → M is a partial map, and talk ab out partial quotients and partial embeddings. The notion of partial embedding how ever do es not add an ything new: A partial frame homomorphism f ∗ suc h that f ∗ is surjective automatically preserves 1 , since 1 is obtained the suprem um of all op ens in M . W e remark that since frames are in particular p osets, it holds that f is an embedding iﬀ the right adjoint f ∗ is injective. The notion of sublo cale can b e expressed in multiple diﬀerent forms. Prop osition 2.25 ([PP12], Chapter I II, also [Joh82], Chapter I I.2) . Let L b e a lo cale. The following p osets are isomorphic. (1) The set of isomorphism classes of embeddings { S → L } . (2) The set of isomorphism classes of regular monomorphisms { S → L } , that is, maps S → L that arise as equalizers of pairs of maps f , g : L → M . (3) The set of subsets S ⊂ O ( L ) that are closed under arbitrary meets and suc h that for every V ∈ S, U ∈ O ( L ) it holds that the Heyting implication U → V ∈ S . (4) The set of fr ame c ongruenc es of L , that is, equiv alence relations of O ( L ) that are subframes of O ( L ) × O ( L ) = O ( L ⨿ L ) . (5) The set of nuclei of L , that is functors N : O ( L ) → O ( L ) such that for all U, V ∈ O ( L ) it holds U ≤ N ( U ) , N 2 = N and N ( U ∧ V ) = N ( U ) ∧ N ( V ) . Moreo ver, the p oset of em b eddings is in fact a co-frame, or dually the p oset of nuclei (ordered p oint wise) is a frame. 18 GEORG LEHNER Let us men tion just a few k ey facts ab out ho w to c hange b et w een these descriptions. An embedding i : S  → L leads to the nucleus N = i ∗ i ∗ . The corresp onding congruence can b e understo o d as the quotient p : L ⨿ L → L ⨿ S L . Here the frame of L ⨿ S L is given as { ( U, V ) ∈ O ( L ) × O ( L ) | i ∗ ( U ) = i ∗ ( V ) } ⊂ O ( L ) × O ( L ) , and S is recov ered as the equalizer of the natural maps i 1 , i 2 : L → L ⨿ S L , with ( i 1 ) ∗ and ( i 2 ) ∗ b eing giv en b y the pro jection onto the ﬁrst or second v ariable, and ( i 1 ) ∗ ( U ) = ( U, N ( U )) . W e remark here as well, that the pushforward p ∗ is given by the form ula p ∗ ( U, V ) = ( U ∧ N ( V ) , V ∧ N ( U )) , a fact that will pro v e useful later on. Lemma 2.26. Let i : S  → L b e the em b edding of a sublo cale, and C b e a presentable ∞ -category . Then the induced functor i ∗ : Sh( S, C ) → Sh( L, C ) is fully faithful. Pr o of. T aking presheav es is a 2 -functor, hence sends the internal quotient map O ( S ) O ( L ) i ∗ i ∗ ⊣ to the adjunction F un( O ( S ) op , C ) F un( O ( L ) op , C ) i ∗ i ∗ ⊣ with fully faithful right adjoint. Then i ∗ on the level of sheav es is simply obtained by restriction to sub cat- egories, hence still fully faithful. □ Let us highlight tw o main examples of sublo cales. Example 2.27. Let L b e a lo cale, and U an op en of L . Then i ∗ : − ∧ U : O ( L ) → O ( L ) /U is a surjective frame homomorphism. The corresp onding sublo cale is called the op en sublo c ale of L given by U . The functor i ∗ has a further left adjoint i ! : O ( L ) /U → O ( L ) which is giv en by inclusion. It is a partial frame homomorphism. On the level of ∞ -categories of sheav es, we get induced adjunctions Sh( U, C ) Sh( L, C ) i ! i ∗ i ∗ ⊣ ⊣ with i ∗ giv en by precomp osition with i ! , and i ∗ giv en by precomp osition with i ∗ . F urthermore, i ! is fully faithful. W e remark that Sh( U, An ) can b e iden tiﬁed with Sh( L ; An ) /y U and i ! corresp onds to the forget functor, [Lur12, Section 6.3.5]. Example 2.28. Let L b e a lo cale, and U an open of L . Then j ∗ : − ∨ U : O ( L ) → O ( L ) U / is a surjective frame homomorphism. The corresp onding sublo cale is called the close d sublo c ale of L complementary to U . The following simple statement will b e very useful later on, hence we w an t to highlight it. W e remark that meets of sublo cales are computed as pullbacks, and open / closed sublo cales are closed under pullback [PP12, Section 6.3]. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 19 Prop osition 2.29 (Second Isomorphism Theorem for Lo cales) . Let L b e a lo cale, S  → L a sublocale and C  → L a closed sublo cale. Then the inclusions S ∧ C  → S and C  → S ∨ C are closed, and the inclusion S  → S ∨ C induces a homeomorphism on their resp ective op en complements. Pr o of. The inclusions S ∧ C  → S and C  → S ∨ C are obtained from pullback along the closed inclusion C  → L , hence are themselves closed. No w let U denote the op en complement of C , and denote the nucleus corresp onding to S by N . The nucleus corresp onding to S ∨ C is then given by N ∧ ( − ∨ U ) . W e can verify that the left adjoint to the inclusion S  → S ∨ C is simply given by N , as for V op en we hav e N ( N ( V ) ∧ ( V ∨ U )) = N ( V ) ∧ N ( V ∨ U ) = N ( V ) , as N ( V ∨ U ) ≥ N ( V ) . This means the induced adjunction on the op en complements is given as O ( S ) / N ( U ) O ( S ∨ C ) /U −∧ U N ⊣ W e simply need to verify that these tw o adjoin ts are in fact inv erse to each other. T o see this, we verify the follo wing: • Let V ∈ O ( S ) / N ( U ) , i.e. V = N ( V ) and V ≤ N ( U ) . Then N ( V ∧ U ) = N ( V ) ∧ N ( U ) = N ( V ) = V since V ≤ N ( U ) implies N ( V ) ≤ N ( U ) . • Let W ∈ O ( S ∨ C ) /U , i.e. W = N ( W ) ∧ ( W ∨ U ) , and W ≤ U . Then W = N ( W ) ∧ ( W ∨ U ) = ( N ( W ) ∧ W ) ∨ ( N ( W ) ∧ U ) = W ∨ ( N ( W ) ∧ U ) hence N ( W ) ∧ U ≤ W . But W ≤ U and W ≤ N ( W ) also imply W ≤ N ( W ) ∧ U , hence W = N ( W ) ∧ U . □ A core prop erty of sheav es on a lo cale is given by the existence of op en-closed decomp ositions. 8 Prop osition 2.30 (Op en-closed decomp osition) . Let L b e a lo cale, i : U  → L an op en sublo cale, and j : U c  → L its closed complement. Let C b e a presentable ∞ -category . Then the sequence Sh( U, C ) Sh( L, C ) Sh( U c , C ) i ! i ∗ j ∗ j ∗ ⊣ ⊣ is a coﬁb er sequence in Pr L . If C is furthermore stable, it is an exact sequence in Pr L st . Pr o of. The functor i ∗ is obtained b y precomp osition with the inclusion O ( L ) /U , and the functor j ∗ is obtained by precomp osition with − ∨ U : O ( L ) → O ( L ) U / , hence it is clear that the comp osite i ∗ j ∗ , given by precomp osition with the constant partial frame homomorphism O ( L ) /U → O ( L ) U / agrees with the constant functor giv en b y the terminal ob ject. W e need to sho w that the essential image of j ∗ actually agrees with the kernel of i ∗ . 8 See also [Lur17, App endix A.8] for more information. 20 GEORG LEHNER Assume that F is a sheaf on L such that i ∗ F ≃ 1 , or in other words, suc h that for all V ≤ U we hav e F ( V ) ≃ 1 . Deﬁne F ′ to b e the restriction of F to O ( L ) U / . W e claim this is a sheaf. Since the inclusion i : O ( L ) U /  → O ( L ) preserves non-empty suprema, the sheaf condition for non-empt y cov erings is satisﬁed. Moreo ver, the initial ob ject is giv en by U ∈ O ( L ) U / , and by assumption F ( U ) ≃ 1 , hence also in this case the sheaf condition holds. Next we argue that there is a natural equiv alence F ≃ j ∗ F ′ , or in other words, that for all W ∈ O ( L ) we hav e that the restriction maps F ( W ∨ U ) → F ( W ) are equiv alences. But observe that we hav e the pullback squares F ( W ∨ U ) F ( W ) F ( U ) F ( W ∧ U ) . ⌟ By assumption, b oth of the b ottom terms are the terminal ob ject, hence the top map is an equiv alence. □ Corollary 2.31. Let j : C  → L b e a closed inclusion of lo cales, and let C b e a presentable ∞ -category . Then j ∗ : Sh( C, C ) → Sh( L, C ) preserves ﬁltered colimits. Pr o of. This is immediate since Sh( C, C ) can b e identiﬁed with the full sub category of shea v es F on L such that i ∗ ( F ) ≃ 1 . The functor i ∗ preserv es colimits, and ﬁltered colimits with constant v alue the terminal ob ject remain terminal. □ Corollary 2.32. Let L b e a lo cale, and assume that C is a presen table stable ∞ -category . If Sh ( L, C ) is a dualizable ∞ -category it then follows that for any op en U of L the sequence given in Prop osition 2.30 is an exact sequence of dualizable ∞ -categories. Giv en any contin uous map of lo cales f : L → M , w e can factor f uniquely (up to the obvious notion of isomorphism) into L ↠ im( f )  → M where the map f : L ↠ im( f ) is a quotient map, and the map im ( f )  → M is the inclusion of a sublocale corresp onding to the nucleus f ∗ f ∗ , see [PP12, IV.1.4.]. The existence of this image factorization has the follo wing pleasant consequence. Lemma 2.33 (See [Joh02] A.1.3.1.) . Let L b e a lo cale. The inclusion Sub( L )  → Lo c /L has a left adjoint giv en by ( f : M → L ) 7→ im( f ) . In particular, directed intersections of sublocales agree with limits computed in the category Lo c , a fact that is useful for the following corollary . Corollary 2.34. Let L b e a lo cale, S i , i ∈ I a coﬁltered family of sublo cales of L , and C a presen table ∞ -category . Denote by S = ^ i ∈ I S i the intersection of the sublo cales S i . Then Sh( S, C ) ≃ colim i ∈ I Sh( S i , C ) , where the colimit is tak en in the category Pr L . Pr o of. Since the Lurie tensor pro duct preserves Pr L -colimits in eac h v ariable, we can assume w.l.o.g. that C = An . Now the statement follows directly from Lemma 2.33. The functor Sh( − , An) : (Lo c op ) = F rm → CAlg(Pr L ) ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 21 is a left adjoint by Theorem 1.7, and ﬁltered colimits in CAlg(Pr L ) are computed as ﬁltered colimits in Pr L . □ 3. St able local comp actness In the following section we will discuss stably lo cally compact spaces, as well as sheav es on them. It will b e conv enient to introduce the more general notion of a stably lo cally compact top os ﬁrst. 3.1. Stably lo cally compact top oi. Deﬁnition 3.1. Let X b e a top os. W e call X c omp act if 1 ∈ X is a compact ob ject. W e call X lo c al ly c omp act if X is a compactly assembled ∞ -category . A geometric morphism f : X → Y betw een lo cally compact top oi is called p erfe ct if f ∗ is strongly contin uous. W e call X stably lo c al ly c omp act or also quasi- sep ar ate d if it is lo cally compact and the diagonal ∆ : X → X ⊗ X is a p erfect geometric morphism. W e call X stably c omp act if it is stably lo cally compact and compact. 9 Lo cal compactness for a top os X is a pleasant prop ert y to hav e. It was shown indep endently by Anel and Leja y [AL18], [AL], as well as Lurie [Lur18, Section 21.1.6 ] that a top os X is locally compact iﬀ X is an exp onen tiable object in the ∞ -category R T op , i.e. the functor X ⊗ − has a right adjoint. Moreov er, it is clear that if a top os X is lo cally compact, and C is a dualizable stable ∞ -category , the ∞ -category of shea ves Sh( X , C ) = X ⊗ C is dualizable, hence lo cally compact top oi give us a health y supply of dualizable stable ∞ -categories to work with. The prop erty of stability should be though t of as a separation condition analogous to quasi-separatedness of sc hemes, and can b e phrased in diﬀerent w ays. Prop osition 3.2. Let X b e a lo cally compact top os. Then the following are equiv alent. (1) X is stably lo cally compact. (2) The functor ˆ y : X → Ind( X ) preserves binary pro ducts. (3) If f , g are tw o compact maps in X , then so is their pro duct f × g . It will b e useful to hav e the following lemma. Lemma 3.3 ([A ok25b], Prop osition 3.32) . Suppose C , D , E ∈ Pr L ca and F : C × D → E is a functor preserving colimits in b oth v ariables. Then the induced left adjoint functor C ⊗ D → E is contin uous iﬀ for any t w o compact maps f in C and g in D the map F ( f , g ) is a compact map in E . W e will add a slight alteration to Lemma 3.3. Note that Ind( C ) ⊗ Ind( D ) ≃ Ind( C ⊗ rex D ) where C ⊗ rex D is the tensor pro duct of ∞ -categories with ﬁnite colimits, [Lur17, Section 4.8.1]. Since C ⊗ rex D  → C ⊗ D embeds full faithfully , we see that Ind( C ) ⊗ Ind( D )  → Ind( C ⊗ D ) 9 W e remark that quasi-separation is muc h weak er than the notion of a top os X b eing sep ar ate d in the sense of [Aok23a, Deﬁnition 6.1.], which is the requirement that the diagonal is prop er. As an example, the topos of sheav es on the Sierpinski space is quasi-separated but not separated. 22 GEORG LEHNER is a fully faithful, strongly con tin uous left adjoint. No w, if F : C × D → E is as ab o v e, we can consider the diagram C ⊗ D E Ind( C ) ⊗ Ind( D ) Ind( E ) Ind( C ⊗ D ) F ˆ y ⊗ ˆ y ˆ y ˜ F Ind( F ) where ˜ F is deﬁned as the comp osite Ind( F ) i . W e obtain the following as a trivial consequence. Lemma 3.4. The functor F is strongly contin uous iﬀ ˜ F ( ˆ y ⊗ ˆ y ) ≃ ˆ y F . Pr o of of Pr op osition 3.2. This is a sp ecial case of Lemma 3.3 and Lemma 3.4 applied to × : X ⊗ X → X . □ Example 3.5. The ∞ -top os An is stably compact, since ˆ y : An → Ind(An) is obtained by applying Ind to the inclusion An ω  → An , which is closed under ﬁnite pro ducts. More generally , if X ∈ An , then the ∞ -top os An /X ≃ F un( X , An) is alwa ys lo cally compact. It is compact iﬀ X ∈ An ω . Ho we ver, stability can often fail: Consider X = S 1 . Then ˆ y is given b y applying Ind to the inclusion An ω /S 1  → An /S 1 . This inclusion is not closed under binary pro ducts: The pullback of the inclusion of a p oint pt → S 1 against itself in An is Z , whic h is not a compact anima. Example 3.6. Let D b e a small ∞ -category admitting binary pro ducts. Then PSh( D ) = F un( D op , An) is stably lo cally compact. Under the identiﬁcation PSh( D ) ⊗ PSh( D ) ≃ PSh( D × D ) , the diagonal ∆ ∗ is given b y left Kan extension of the pro duct functor × : D × D → D , and therefore is strongly contin uous. Deﬁnition 3.7. A geometric morphism i : Y → X is called a subtopos of X if i ∗ is fully faithful. It is furthermore called a p erfe ct subtop os if i ∗ preserv es ﬁltered colimits. Prop osition 3.8. Let X b e a top os and i : Y → X a p erfect subtop os. • If X is compact, then so is Y . • If X is lo cally compact, then so is Y . • If X is stably lo cally compact, then so is Y . Pr o of. If 1 ∈ X is compact, then so is i ∗ (1) = 1 ∈ Y , proving the ﬁrst claim. The second claim follows by Theorem 2.7. In particular ˆ y Y exists and is giv en by Ind( i ∗ ) ˆ y X i ∗ . This implies the last claim, since if X is stably lo cally compact, the functor ˆ y Y is given as a comp osite of functors that preserve binary pro ducts. □ Stably lo cally compact top oi are also closed under passage to op en subtop oi. Prop osition 3.9. Let X b e a top os and U ∈ X b e a sub ob ject of 1 . • If X is lo cally compact, then so is X /U . • If X is stably lo cally compact, then so is X /U . • If X is compact, and U is a compact ob ject, then X /U is compact. Pr o of. This is a direct corollary of Theorem 2.7, by noting that the forget functor X /U  → X is fully faithful, since U is assumed to b e a sub ob ject of 1 , and has the colimit preserving left adjoint − × U . Note that the remark follo wing Theorem 2.7 also states that an ob ject/morphism in X /U is compact iﬀ it is compact when considered as an ob ject/morphism. The forget functor preserves binary pro ducts, hence the statement ab out stable lo cal compactness follows. The statement about compactness of X /U is clear b y the same reasoning. □ ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 23 Remark 3.10. Compactness is of course not inherited b y passing to arbitrary op en subtop oi. 3.2. Stably lo cally compact spaces. Let us deﬁne the main class of top ological spaces of interest for the purp oses of this article: The so-called stably lo c al ly c omp act sp ac es. Standard References for the theory of stably lo cally compact spaces are [Gie+03, Section VI-6], [Gou13, Section 9] and [Law11]. 10 Deﬁnition 3.11. Let F b e a frame. W e say that F is • lo c al ly c omp act , if F is compactly assembled when considered as an ∞ -category , • stably lo c al ly c omp act , if furthermore ˆ y : F → Ind( F ) is a partial frame homomorphism, and • stably c omp act , if ˆ y : F → Ind( F ) is a frame homomorphism. A partial frame homomorphism f ∗ : F → F ′ b et w een lo cally compact frames is called p erfe ct if it is strongly con tinuous. W e deﬁne the categories: • SLCF rm p of stably lo cally compact frames and partial p erfect frame homomorphisms. • SCF rm of stably compact frames and p erfect frame homomorphisms. Deﬁnition 3.12. A sob er top ological space X is called (stably) (lo cally) compact if its corresp onding frame O ( X ) is (stably) (lo cally) compact. 11 A partially deﬁned con tinuous map f with op en supp ort is called p erfe ct , if the corresp onding partial frame homomorphism f − 1 is. W e deﬁne the categories: • SLC p of stably lo cally compact spaces and p erfect partially deﬁned contin uous maps. • SC of stably compact spaces and p erfect con tinuous maps. W e can rephrase Deﬁnitions 3.11 and 3.12 into the following more classically known terms. Recall the notion of the way b elow relation b etw een opens of a frame F : If U ≤ V ∈ F are t w o op ens, w e say that U is way b elow V , in symbols U ≪ V , if for every directed family of op ens W i , i ∈ I , it holds that V ≤ _ i ∈ I W i implies ∃ i ∈ I : U ≤ W i . 12 W e call an op en U c omp act op en if U ≪ U . A frame / space is compact iﬀ 1 is a compact op en. Prop osition 3.13. Let F b e a frame. • F is lo cally compact iﬀ for all op ens U ∈ F , we ha ve U = _ V ≪ U V . • F is stably lo cally compact iﬀ it is lo cally compact, and whenever U ≪ V 1 and U ≪ V 2 it follo ws that U ≪ V 1 ∧ V 2 . • F is stably compact iﬀ F is stably lo cally compact and 1 ∈ F is a compact op en. Let f ∗ : F → F ′ b e a partial frame homomorphism. Then f ∗ is p erfect iﬀ it preserves the wa y-b elow relation ≪ , or equiv alently its right adjoin t f ∗ preserv es directed joins. The pro ofs for these facts are straightforw ard, and can b e found (with slight v ariation) in the pro ofs given in [Aok25b, Prop osition 4.4 and 4.6]. Remark 3.14. It is tempting to refer to p erfect maps as pr op er , how ev er, the terminology of pr op er map often additionally assumes closedness in the literature. 10 W e warn the reader ab out a conﬂict of terminology . The term c oher ent as a prop erty of spaces in the references [Gou13] and [Gie+03] refers to the prop erty that the wa y-b elo w relation is stable under intersection, whereas coher ent sp ac e used in [Leh25a] refers to what is otherwise also known as a spe ctr al sp ac e , and is in accordance with the term c oher ent lo c ale , as used in [Joh82]. This terminology matches the more general notion of a coherent top os, following [Lur18, App endix A]. 11 W e note that our notion of lo cal compactness for a space is sometimes referred to as c or e-c omp actness in the literature and is a stronger notion than just the naive requirement that every p oin t has a compact neighborho o d. 12 This is just saying that U ≤ V is a c ompact morphism when considering F as a category . 24 GEORG LEHNER Prop osition 3.15. A sob er, lo cally compact space X is stably lo cally compact iﬀ the diagonal ∆ : X → X × X is a p erfect map. F urthermore, X is compact iﬀ the canonical map X → pt is p erfect. Pr o of. The inv erse image part ∆ ∗ : O ( X ) ⊗ O ( X ) → O ( X ) is given by sending U ⊗ V to U ∧ V . With this understo od, w e see that the ﬁrst claim is a special case of Lemma 3.3. The statement ab out compactness follo ws by observing that O (pt) = {∅ ⊂ { pt }} has a single non-trivial compact elemen t and thus p : X → pt is p erfect iﬀ p − 1 ( { pt } ) is compact op en. □ By deﬁnition, the notions of stably (locally) compact spaces make reference only to their corresp onding frames. Since we also assumed sobriety , it follows that the opp osite of the category of stably (locally) compact spaces embeds fully faithfully into stably (lo cally) compact frames. A v arian t of Birkhoﬀ ’s completeness theorem guarantees that this is an equiv alence of categories. Theorem 3.16 ([Joh82], page 311) . A lo c al ly c omp act fr ame is sp atial. Corollary 3.17. There exist equiv alences of categories (SLC p ) op ≃ SLCF rm p and (SC) op ≃ SCF rm , giv en on ob jects by X 7→ O ( X ) . There are several inheritance properties we wan t to discuss. Stably lo cally compact spaces are closed under p erfect subspaces, partial p erfect quotients and disjoint unions. Deﬁnition 3.18. Let L b e a lo cale. A p artial quotient of L is given by a partial map p : L → S such that p ∗ is fully faithful. W e call a partial quotient p erfe ct , if furthermore f ∗ preserv es directed suprema. A p artial quotient p : L → S is simply called a quotient , if p is a (globally deﬁned) map of lo cales. Prop osition 3.19. Let X b e a lo cale, and p : X → S a partial quotient. • If X is lo cally compact, then so is S . • If X is stably lo cally compact, then so is S . • If X is lo cally compact, and the domain p ∗ (1) is compact op en, then S is also compact. Pr o of. This is a direct application of Theorem 2.7. □ Example 3.20. A particular example is given by considering an op en U of a lo cally compact frame F . The inclusion F /U  → F is in fact the inv erse image part of a partial p erfect quotient, as it has the colimit preserving left adjoint − ∧ U . It follo ws that the op en sublo cale F /U is lo cally compact, and the inclusion F /U  → F is a partial p erfect frame homomorphism. If F is stably lo cally compact, then so is F /U , how ever compactness is only given if U is a compact op en of F . If f ∗ : F → F ′ is a partial p erfect frame homomorphism, then in the natural factorization F f ∗ − → F ′ /f ∗ (1) → F ′ the left functor is also p erfect, as its right adjoin t is given by f ∗ : F ′ /f ∗ (1) → F , and suprema in the ov er frame are computed as suprema in F ′ . Hence, analogously to the situation for partial maps of lo cales, we can think of partial p erfect maps as p erfect maps deﬁned on some op en domain f ∗ (1) . Deﬁnition 3.21. Let L b e a lo cale. A sublo cale i : S  → L is called p erfe ct , if i ∗ preserv es directed suprema. Prop osition 3.22. Let X b e a lo cale, and i : S  → L a p erfect sublo cale. • If X is lo cally compact, then so is S . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 25 • If X is stably lo cally compact, then so is S . • If X is compact, then S is also compact. Pr o of. The pro of of this is completely analogous to the pro of of Prop osition 3.8. □ Example 3.23. Let X b e a locale, and U an op en. The sublo cale C corresp onding to the closed complement of U is given b y i ∗ : O ( X ) U /  → O ( X ) , and is eviden tly p erfect. (The inclusion preserves non-empty suprema, in particular directed ones.) W e now come to closure under copro ducts. Lemma 3.24. Let X b e a lo cale, and U i , i ∈ I and op en cov er such that all U i are lo cally compact. Then X is lo cally compact. Pr o of. The inclusions O ( X ) /U i  → O ( X ) are strongly contin uous partial frame homomorphisms, hence preserv e the wa y-below relation. Let V b e an op en of X . Then V ∧ U i = _ W ≪ V ∧ U i W since U i is lo cally compact. Hence V = _ V ∧ U i = _ i ∈ I ,W ≪ V ∧ U i W is a join of w ay b elow elements of V . □ Remark 3.25. W e cannot exp ect the analogous statement for stable lo cal compactness to work. Consider the lo cale L obtained as [0 , 1] ⨿ (0 , 1) [0 , 1] . Then L admits a co v er by tw o stably lo cally compact op ens, hence is lo cally compact. How ever, L is not stably lo cally compact. Prop osition 3.26. Let X i , i ∈ I , b e a collection of lo cales and let X = a i ∈ I X i . • If all X i are lo cally compact, then so is X . • If all X i are stably lo cally compact, then so is X . • If I is ﬁnite, and all X i are compact, then so is X . W e remark that the frame O ( X ) is obtained as the pr o duct of the frames O ( X i ) . The statement applied to ﬁnite index sets can also b e found in [Gou13, Prop osition 9.2.1]. Pr o of. The canonical inclusions j i : X i  → X are open, and corresp ond to the frame homomorphism giv en b y the pro jection Y i ∈ I O ( X i ) → O ( X i ) Its left adjoint sends an op en U of X i to the sequence ( j i ) ! ( U ) which is U at v alue i , and 0 otherwise. The lo cale X admits an op en cov er given by the images of the X i under j i . Hence lo cal compactness of X follo ws from lo cal compactness of X i b y Lemma 3.24. T o see stable lo cal compactness, note that V ≪ V ′ holds in X only if V is contained in the image of ﬁnitely man y inclusions j i , since V ′ can b e exhausted b y the op ens V ′ ∧ X i . Hence if V ≪ V 1 and V ≪ V 2 holds, we can restrict ourself to b eing supported on some ﬁnite union of X i , where the statemen t is clear. The same reasoning applies to the last p oin t ab out compactness. □ Let us give some explicit examples of stably lo cally compact lo cales. 26 GEORG LEHNER Example 3.27. Ev ery lo cally compact Hausdorﬀ space X is stably lo cally compact. Moreo v er, b etw een lo cally compact Hausdorﬀ spaces, one sees that (partial) p erfect maps are exactly the (partial) prop er maps. Example 3.28 (The directed interv al) . A crucial example of a stably compact space is giv en by the following. The dir e cte d interval I ≤ is given by equipping the real in terv al I = [0 , 1] with the top ology consisting of op en sets of the form [0 , a ) for some a ∈ [ 0 , 1] , together with the op en set I itself. Another example of a stably locally compact space is given by the open subspace [0 , 1) ≤ , which is homeomorphic to the space − − − → [0 , ∞ ) men tioned in the introduction. Remark 3.29. One ma y equip the reals R with the top ology given b y op en sets of the form ( −∞ , a ) for some a ∈ {−∞} ⊔ R ⊔ { + ∞} to obtain a space R ≤ . The frame ( {−∞} ⊔ R ⊔ { + ∞} , ≤ ) is in fact stably lo cally compact, how ever R ≤ is not sober, hence not a stably lo cally compact space. Intersections of non-empty op ens alw ays remain non-empty , so the sobriﬁcation adds the additional p oint {−∞} , which is contained in ev ery non-empty op en set. The resulting space is of course homeomorphic to [0 , 1) ≤ . Example 3.30 (Finite spaces) . Any ﬁnite frame F is automatically stably compact, as then Ind( F ) ≃ F , and hence all structure maps are just equal to the identit y . Finite frames, or equiv alently ﬁnite distributive lattices alw a ys corresp ond to ﬁnite p osets equipp ed with the Alexandroﬀ topology , and will b e referred to as ﬁnite sp ac es . (See [Leh25a, Section 3.4] for a summary .) They are equiv alently describ ed as top ological spaces that are ﬁnite and T0. Special cases worth mentioning are the p oint pt and the Sierpinski space 2 , i.e. the space obtained from the p oset [1] = { 0 ≤ 1 } . Moreo ver, any contin uous map f : X → Y with X b eing a ﬁnite space and Y stably lo cally compact is automatically p erfect. W e note that the same do es not hold for maps into a ﬁnite space, as we hav e just seen in Prop osition 3.15. Example 3.31 (Lo cally coherent spaces) . Let us elab orate on the case when X is stably lo cally compact, and the frame O ( X ) is compactly generated. In this case X is called a lo c al ly c oher ent sp ac e . In case X is furthermore compact, X is also called a c oher ent sp ac e , which is equiv alent to X arising as the sp ectrum of a ring. (See [Leh25a, Section 3] for a summary .) A space X is lo cally coheren t if it is sob er, the top ology of X is generated by compact op ens, and the subset of compact op ens forms a low er b ounded distributiv e lattice. In fact, Stone duality gives an equiv alence of categories Lo cCohSp op p ≃ DLatt lb where Lo cCohSp p is the full sub category of SLC p spanned by lo cally coherent spaces, and a lo cally coherent space is mapped to its lo wer b ounded distributiv e lattice of compact opens. The i nv erse sends a low er b ounded distributive lattice D to the stably lo cally compact frame Ind( D ) . This equiv alence restricts to the classical Stone duality b et w een coherent spaces and bounded distributive lattices. The imp ortance of the category of lo cally coherent spaces for the category of stably lo cally compact spaces is the following. The natural adjunction F Ind( F ) ˆ y y ⊣ ⊣ for any stably lo cally compact frame F means that every stably locally compact space arises as a (partial) retract of a coherent space, and every partial p erfect map of stably lo cally compact spaces arises as a retract of a partial p erfect map betw een coherent spaces. W e also remark that coherent spaces arise exactly as in verse limits of ﬁnite p osets equipp ed with the Alexandroﬀ top ology , tying in with the previous example. The additional prop ert y of Hausdorﬀness for lo cally coherent spaces is also worth elab orating on. A lo cally coheren t space is Hausdorﬀ iﬀ it is totally disconnected. This is the case iﬀ the lattice of compact op ens forms a Bo olean ring. T otally disconnected lo cally compact Hausdorﬀ spaces are also called td -sp ac es . The ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 27 additional assumption on X to b e compact corresp onds to the requirement that the compact opens form a Bo olean algebra. In this case X is called a pr oﬁnite sp ac e . Proﬁnite spaces arise exactly as inv erse limits of ﬁnite and discrete sets. Let us summarize this picture in the following table. Spaces Compact Op ens Compactly Generated Case Lo cally compact Join-semilattice Stably lo cally compact Low er b ounded distributiv e lattice Lo cally coherent Stably compact Bounded distributive lattice Coheren t Lo cally compact Hausdorﬀ Bo olean ring td -space Compact Hausdorﬀ Bo olean algebra Proﬁnite space The deﬁnition of a stably (locally) compact frame is no acciden t. Observ e that for a frame F , the poset Ind( F ) is again a frame. In fact, Ind giv es an endofunctor F rm part → F rm part . W e observe that there are natural transformations k F : Ind( F ) → F and Ind( y ) F : Ind( F ) → Ind 2 ( F ) , which giv e Ind the structure of a comonad on F rm part → F rm part . Moreo ver, one perhaps un usual feature is that Ind( y ) F is left adjoint to k Ind( F ) . The same holds for Ind deﬁned on F rm instead of F rm part as w ell. This structure is known as a lax-idemp otent c omonad , also called KZ -comonad. The theory of lax-idemp otent (co)-monads on 2 -categories is very rich, and was originally developed b y K o c k [K o c95] and Zöb erlein [Zöb76]. 13 The unique feature of a lax-idemp otent comonad T on a 2 -category C is that for a given ob ject c ∈ C there is essentially a unique wa y for c to hav e the structure of a coalgebra, giv en by the existence of a left adjoint to the counit T ( c ) → c . Similarly , homomorphisms of algebras are giv en by maps f : c → d satisfying a left adjoin tabilit y prop erty . The wa y we phrased the deﬁnition of a stably (lo cally) compact frame and p erfect (partial) frame homomorphism was inten tional, as it highlights that they are just special cases of the deﬁnition of a coalgebra and a morphism of coalgebras for a lax- idemp oten t comonad. (Compare [Koc95] Deﬁnition 2.1. and 2.4.) F rom this p ersp ective, the following prop osition is purely formal, as it is a sp ecial case of the statement that for a comonad T on C there is an induced adjunction CoAlg( C , T ) C f or get T ⊣ Prop osition 3.32 ([Koc95] Prop osition 3.3) . Consider the comonad Ind : F rm part → F rm part . There is an equiv alence of 2 -categories CoAlg(F rm part , Ind) ≃ SLCF rm p . The analogous statement is true with F rm and SCF rm as well. Corollary 3.33 (Tyc honoﬀ ’s Theorem for stably (lo cally) compact spaces) . The forget functor SLC p → Lo c part admits a left adjoint γ , giv en on frames by F 7→ Ind( F ) . In particular, SLC p admits all limits, and the forget functor to Lo c part preserv es them. The analogous statement is true with F rm and SCF rm as well. Remark 3.34. Analogously , also the deﬁnition of a compactly assem bled ∞ -category , compactly assembled presen table ∞ -category , as well as a dualizable ∞ -category ﬁt into the lax-comonadic picture by considering Ind as deﬁned on Cat ω (the ∞ -category of accessible ∞ -categories with ﬁltered colimits and ﬁltered colimit preserving functors), Pr L and Pr L st , ignoring the obvious set theoretic issues. In fact, all of these examples arise from simpler lax-idemp otent monads, deﬁned on Cat , Cat rex and Cat st , resp ectively . The 2 -categories 13 See also [Joh02, Section B1.1] 28 GEORG LEHNER F rm part and F rm arise from considering DLatt lb and DLatt bd , the 2 -categories of low er b ounded (resp ectiv ely b ounded) distributive lattices. A t presen t the ( ∞ , 2) -categorical version of the theory of lax-idemp oten t monads is only work in progress, with up coming results due to Ab ellán–Blom [AB26]. Nonetheless, inv estigating these analogies has b een a ma jor driver for this article. Example 3.35 (The directed Hilbert cube) . If S is a set, we call the product I S ≤ the dir e cte d Hilb ert cub e of cardinality S , which is a stably compact space. W e will need to talk ab out satur ate d c omp act subsets . Let us give a further deﬁnition. Deﬁnition 3.36. Let F b e a frame. A nonempty subset F ⊂ F is called Sc ott op en ﬁlter if: • F is upw ard closed, • F is closed under ﬁnite intersections, and • F is Sc ott op en , which means that if U i , i ∈ I is a directed family of op ens and _ i ∈ I U i ∈ F , then there exists i ∈ I such that U i ∈ F . Note that the deﬁnition of Scott op enness of course abstractly mimics the prop erty of the op en neighbor- ho od ﬁlter of a compact subset. This is substantiated by the following theorem. If X is a top ological space, and K ⊂ X a subset, we say that K is satur ate d , if K arises as an intersection of op en sets. W rite Q ( X ) for the set of satur ate d c omp act subsets of X . Theorem 3.37 (Hofmann-Mislov e, see [Gie+03] Theorem I I-1.20.) . L et X b e a sob er sp ac e. Ther e is an isomorphism of p osets Q ( X ) op  OFilt( O ( X )) , wher e a satur ate d c omp act subset K ⊂ X is sent to the Sc ott op en ﬁlter of op en sets of X c ontaining K . This allows us to switc h b etw een the top ological and frame-theoretic viewp oin t. Example 3.38. Let X b e a sob er space, and x ∈ X a p oint. Then the op en neigh b orho od ﬁlter U x = { V ⊂ X op en | x ∈ V } is a Scott op en ﬁlter. It corresponds to a saturated compact set S x , which con tains x and is con tained in every op en neigh b orho od of x , which is the satur ate d c omp act closur e of { x } . W e note that this implies that for every op en U and p oin t x ∈ U there exists a saturated compact set S suc h that x ∈ S ⊂ U . If X is Hausdorﬀ, then S x = { x } . Theorem 3.39 (Urysohn’s Lemma for stably compact spaces, see [Gou13] Theorem 9.4.11) . L et X b e a stably c omp act sp ac e. F or every c omp act satur ate d subset S and close d subset C of X such that S ∩ C = ∅ , ther e exists a pr op er map f : X → I ≤ such that f has c onstant value 0 on S and c onstant value 1 on C . Let us phrase Urysohn’s Lemma in a more categorical wa y . Theorem 3.40. L et X b e a stably c omp act sp ac e. Ther e exists a p erfe ct emb e dding i : X  → I S ≤ for some set S . Pr o of. Let S b e the set of p erfect maps X → I ≤ . Then, by Tyc honoﬀ ’s Theorem, there exists a canonically induced p erfect con tin uous map i : X → I S ≤ . F or each f ∈ S , denote by p f : I S ≤ → I ≤ the asso ciated pro jection. T o see that i is an em b edding we need to show that i ∗ is surjective. Let U be an op en of X . Denote by C the closed complement. F or each p oint x ∈ U pic k a saturated compact neighborho o d x ∈ S x ⊂ U , together with a p erfect map f x : X → I ≤ whic h is 0 on S x and 1 on C . Then [0 , 1) is op en in ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 29 I ≤ and hence f ∗ x ([0 , 1)) = i ∗ ( p f x ) ∗ ([0 , 1)) ⊂ U is in the image of i ∗ , and moreo v er contains the p oint x . The image of i ∗ is closed under suprema, and thus we see that U = _ x ∈ U f ∗ x ([0 , 1)) is also in the image of i ∗ . □ Remark 3.41. A curious observ ation ab out the pro of of Theorem 3.40 is that, when applied to the case when X = P Alex for a ﬁnite p oset P the pro of constructs a monotone embedding of ( P , ≤ ) into a ﬁnite-dimensional cub e ( I n , ≤ ) . Corollary 3.42 (Urysohn’s Lemma for stably compact spaces, categorical form ulation) . The directed in- terv al I ≤ generates the category SC under limits. 14 Pr o of. Let X b e stably compact. Pick a p erfect embedding i : X  → I S ≤ for some S . Then X is given as the equalizer of its congruence relation I S ≤ ⇒ I S ≤ ⨿ X I S ≤ , which is a diagram in stably compact spaces. Perfectly em b ed the latter again into I S ′ ≤ for some set S ′ . Then X is equal to the equalizer of I S ≤ ⇒ I S ′ ≤ . □ Remark 3.43. One can show that I ≤ is ω 1 -compact, when considered as an ob ject of SCF rm , where ω 1 is an y c hoice of uncountable regular cardinal. It follo ws that SCF rm is a presentable category . This is p erhaps surprising, as the category of frames F rm is not presentable. The pro of given here is completely analogous to the pro of of Ramzi’s theorem given by Eﬁmov in [Eﬁ25a, App endix C], which states that the ∞ -category Pr L dual is presentable. Let us now discuss one-p oin t compactiﬁcations of stably lo cally compact spaces. Recall the adjunction Lo c part Lo c pt / ( − ) + coim ⊣ discussed in Construction 2.19. Prop osition 3.44. The giv en adjunction b etw een Lo c part and Lo c pt / descends to an adjunction SLC p SC pt / ( − ) + coim ⊣ with fully faithful right adjoin t. Pr o of. W e observe the follo wing facts. • If F is a stably lo cally compact frame, then so is F ⊤ . Moreov er, the newly added top elemen t is unreac hable from below by arbitrary joins, and so in particular compact, hence F ⊤ is stably compact. (See also [Gou13, Prop osition 9.1.10].) • The left adjoint coim assigns to a p ointed stably compact space x : pt → X the interior U of the complemen t of { x } (given as the join of ( x ∗ ) − 1 (0) ⊂ O ( X ) , or equiv alen tly x ∗ (0) ). The space U is, as an op en subspace of a stably compact space, again stably lo cally compact. • Giv en a partial frame homomorphism f ∗ : F → F ′ the asso ciated pushforward on one-p oin t com- pactiﬁcations is giv en by ( f ∗ ) ⊤ : ( F ′ ) ⊤ → ( F ) ⊤ , which agrees with f ∗ on F ′ and preserves the newly added top elements. If f ∗ preserv es directed suprema, so do es ( f ∗ ) ⊤ , as can b e chec ked b y verifying t wo cases. 14 W e warn the reader that this notion is strictly weaker than the statement that the full sub category spanned by I ≤ in SC op is dense in the sense that SC op → F un( { I ≤ } , Set) is fully faithful, see [Lan98, Chapter X.6]. 30 GEORG LEHNER • If ( F , x ) and ( F ′ , x ′ ) are tw o pointed frames with coimages U = x ∗ (0) and U ′ = x ′ ∗ (0) , then a p oin ted frame homomorphism f ∗ : F ′ → F b et w een p ointed frames satisﬁes f ∗ ( x ∗ (0)) = x ′ ∗ (0) , and therefore the induced righ t adjoint coim( f ) ∗ : F /U → F ′ /U ′ is simply giv en by f ∗ . If f ∗ preserv es directed suprema, so do es coim ( f ) ∗ . □ Example 3.45. Assume that X is stably lo cally compact and consider the one-p oin t compactiﬁcation X + . Then an y Scott op en ﬁlter F ⊂ O ( X ) pro duces a Scott open ﬁlter of O ( X + ) , by adding the top element. All Scott op en ﬁlters of O ( X + ) are of this form, with the exception of { 1 } . Hence we see that Q ( X + ) arises from Q ( X ) by fr e ely adding a b ottom element . Corollary 3.46 (Urysohn’s Lemma for stably lo cally compact spaces) . Let X b e stably lo cally compact. • Let S ⊂ X be saturated compact, and C ⊂ X closed, such that S ∩ C = ∅ . Then there exists a partial p erfect map f : X → [0 , 1) ≤ , which is iden tically 0 on S and has op en domain of deﬁnition con tained in C c . • There exists a p erfect embedding X  → [0 , 1) S ≤ for some set S . In particular, the category SLC p is generated under limits b y [ 0 , 1) ≤ . Pr o of. Let S ⊂ X b e saturated compact, and C ⊂ X closed b e giv en, such that S ∩ C = ∅ . Then S ⊂ X ⊂ X + remains saturated compact. The set C ′ = C ∪ {∞} is closed in X + , and it holds that S ∩ C ′ = ∅ as subsets of X + . Applying the Urysohn lemma for stably compact spaces pro duces a p erfect map f : X + → I ≤ , which is iden tically 0 on S and identically 1 on C ′ . In particular, it is a pointed map and thus corresp onds to a partial p erfect map f : X → [0 , 1) ≤ with op en domain giv en by f ∗ ([0 , 1)) , whic h is contained in C c . The remaining tw o claims now follow completely analogously to the same statements given for stably compact spaces. □ Remark 3.47. As an application of the patch topology functor, which we will in tro duce later in Section 3.5, we recov er the classical statement that any compact Hausdorﬀ space embeds as a closed subset of some Hilb ert cub e [0 , 1] S for some set S , when equipp ed with the usual (euclidean) top ology . W e also get as a direct corollary the slightly lesser known fact that any lo cally compact Hausdorﬀ space X embeds as a close d subset in some half-op en cub e [0 , 1) S . Remark 3.48. Whereas the use of stably compact spaces requires the use of the axiom of choice at several p oin ts for the dev elopmen t of the theory , the same is not necessary for the notion of stably compact frame/lo- cale, which has a robust constructive theory , a fact that has b een observ ed early on during the developmen t of p ointfree top ology , [Joh83]. This comment is not merely philosophical: Any constructively v alid theorem giv es statements that are true internally to any 1-top os - Imp ortant sp ecial cases w ould b e given by working o ver a given base space, or b y working equiv arian tly with resp ect to some ﬁxed group G . These examples are crucial in algebraic and geometric top ology . As an example of how far one can go with applying constructiv e tec hniques when working within algebraic geometry , see the work of Ingo Blechsc hmidt [Ble21]. Ho wev er, we will not insist on this distinction during this article for purely pragmatic reasons. First of all, neither the theory of higher categories, nor the theory of K -theory of large categories has a work ed out constructiv e version at the momen t of writing. Second of all, the proof strategy for Theorem 6.1 that we emplo y in this article is highly dep endent on the existence of enough p oints, and thus only classically v alid. 3.3. Shea v es on stably lo cally compact spaces. Let us no w come to the question of how the notion of stable lo cal compactness of spaces interacts with the notion of stable lo cal compactness of top oi. Recall that an y top os X has an asso ciated frame X ≤ 0 , given by the set of sub ob jects of 1 . Theorem 3.49. L et X b e a top os. Consider the fr ame X ≤ 0 . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 31 • If X is lo c al ly c omp act, then X ≤ 0 is lo c al ly c omp act. • If X is stably lo c al ly c omp act, then X ≤ 0 is stably lo c al ly c omp act. • If X is c omp act, then X ≤ 0 is c omp act. Pr o of. By Prop osition 2.20 we hav e an adjunction X ≤ 0 X i τ ≤− 1 ⊣ where the inclusion i preserves ﬁltered colimits. Hence the left adjoint τ ≤− 1 preserv es compact ob jects, pro ving (3). P oin ts (1) and (2) follo w by an application of Lemma 2.2: If the adjoin t ˆ y X exists, then ˆ y X ≤ 0 exists as w ell and is giv en b y Ind( τ ≤− 1 ) ˆ y X i , which, assuming that ˆ y commutes with binary pro ducts, also comm utes with binary pro ducts. □ Let us now concern ourselv es with the conv erse question. Theorem 3.50. L et X , Y b e a stably lo c al ly c omp act sp ac es and f : X → Y a p artial p erfe ct map. • The top os Sh( X ) is a stably lo c al ly c omp act. • If X is c omp act, so is Sh( X ) . • The induc e d functor f ∗ : Sh( Y ) → Sh( X ) is str ongly c ontinuous. Pr o of. Applying Sh( − ) to the commuting squares of partial frame homomorphisms O ( X ) Ind( O ( X )) O ( Y ) Ind( O ( Y )) ˆ y k f ∗ ˆ y Ind( f ∗ ) k ⊣ ⊣ sho ws that f ∗ : Sh( Y ) → Sh( X ) is a Pr L -retract of the induced functor Sh (Ind( O ( Y )) → Sh(Ind( O ( X ))) of Ind( f ∗ ) . The frame Ind( O ( X )) is a coherent frame and thus Sh (Ind( O ( X ))) is a compactly generated ∞ -category , see Example 3.31 and [Leh25a, Corollary 3.17]. Similarly for Ind( O ( Y )) . The left adjoint functor induced by Ind( f ∗ ) on sheav es preserves compact generators by construction and is thus strongly con tinuous. Applying Lemma 2.9, we see that f ∗ : Sh( Y ) → Sh( X ) is a strongly contin uous functor b etw een compactly assembled ∞ -categories. In particular, Sh( − ) sends p erfect maps b etw een stably lo cally compact spaces to p erfect geometric morphisms. W e can directly apply this to get the remaining t wo op en claims: By Prop osition 3.15, the diagonal ∆ : X → X × X is a p erfect map, and hence ∆ ∗ : Sh( X ) → Sh( X ) ⊗ Sh( X ) is a p erfect geometric morphism, which shows stability . The statement ab out compactness follo ws the same w ay by considering the map X → pt instead. □ Remark 3.51. The corresponding statemen t that for locally compact sob er spaces X , the top os Sh( X ) is also lo cally compact is unfortunately not true. This failure is already presen t at the level of 1 -top oi, see [JJ82, Example 5.5]. Johnstone-Jo y al characterize those spaces X such that Sh ( X, Set) is compactly assem bled as those X that are metastably lo c al ly c omp act . It is an interesting question whether an analogous criterion exists for the analogous questions with Sh( X , An) instead. Corollary 3.52. The functor Sh : F rm part → Pr L descends to a well-deﬁned functor Sh : SLCF rm p → Pr L ca . Remark 3.53. The restriction of this claim to stably compact frames was done b y Aoki in [A ok25b, Theorem 4.29]. The pro of given here is completely analogous to the argument given by Aoki. 32 GEORG LEHNER In the case of shea v es on stably compact spaces, there exists further structure. Theorem 3.54 ([Aok25b], Corollary 5.13) . Ther e exists an adjunction SCF rm CAlg (Pr L ca ) . Sh Sm con ⊣ 3.4. de Gro ot and V erdier duality for stably compact spaces. There is a very useful dualit y for stably compact spaces, that can b e understoo d as exchanging the roles of op enness with that of (co)-compactness. Note that the p oset Q ( X ) op  OFilt( O ( X )) of saturated compact sets, or equiv alen tly Scott op en ﬁlters, alw ays has directed colimits (computed just as unions of their resp ective ﬁlters), and ﬁnite in tersections. In case X is stably compact ho w ever, more holds: It b ecomes a frame itself. Theorem 3.55 (de Gro ot duality , see [Gou13] Theorem 9.1.38) . L et X b e a stably c omp act sp ac e. Then the set of c omplements of satur ate d c omp act subsets gives a top olo gy on X . Denote the r esulting top olo gic al sp ac e by X ∨ . The sp ac e X ∨ is again stably c omp act, and it holds that ( X ∨ ) ∨ = X . Mor e over, a p erfe ct map f : X → Y is again a p erfe ct map f : X ∨ → Y ∨ when c onsidering the c o c omp act top olo gies. In other words, we obtain an inv olution ( − ) ∨ : SC → SC . W e note that ( − ) ∨ is cov ariant in 1 -morphisms, but reverses 2 -morphisms. Example 3.56. In the case of X = I ≤ b eing the directed in terv al, w e ha v e ( I ≤ ) ∨ = I ≥ is the down wards directed interv al, with reversed top ology . Example 3.57. In the case of a coherent space X corresp onding to a b ounded distributive lattice D under Stone dualit y , the dual X ∨ is again coherent and corresp onds to D op . The restriction of de Gro ot duality to the sub category of coherent spaces is also referred to as Ho chster duality . 15 Example 3.58. If X is compact Hausdorﬀ, it is a standard fact that a subset K ⊂ X is compact saturated iﬀ it is closed. W e thus see that for compact Hausdorﬀ spaces it holds that X ∨ = X . Crucial will b e the follo wing theorem, which explains the interaction of de Gro ot duality with the corre- sp onding ∞ -categories of shea ves. Theorem 3.59 (See [Aok25b], Theorem 4.46.) . L et X b e a stably c omp act sp ac e and let C b e a pr esentable stable ∞ -c ate gory. Then ther e is a c anonic al e quivalenc e VD : Sh( X , C ) → Cosh( X ∨ , C ) p ointwise given by VD( F )( X \ K ) = colim U ⊃ K op en ﬁb( F ( X ) → F ( U )) wher e K is a c omp act satur ate d subset of X . There is a related version of Theorem 3.59, whic h works without the assumption of stabilit y of C , or the requirement that X is compact, whic h is known in the Hausdorﬀ case under the statement “Shea v es = K -sheav es”, see [Lur17, Section 7.3.4]. Deﬁnition 3.60. Let X b e a stably locally compact space. A K - she af on X with v alues in a presentable ∞ -category C is a functor F : Q ( X ) op → C such that • F ( ∅ ) = 1 15 de Gro ot duality can actually b e understo o d as a formal extension of Hochster duality , using the p ersp ectiv e of lax- idempotent monads. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 33 • F or each C, C ′ ∈ Q ( X ) the square F ( C ∪ C ′ ) F ( C ′ ) F ( C ) F ( C ∩ C ′ ) is a pullback square. • Whenev er C = \ i ∈ I C i is a ﬁltered intersection of saturated compact sets, the natural map colim i ∈ I F ( C i ) ∼ − → F ( C ) is an equiv alence. W e deﬁne Sh K ( X, C ) as the full sub category of F un( Q ( X ) op , C ) spanned by K -sheav es. Theorem 3.61 ([Aok25b], Prop osition 4.36) . L et X b e a stably c omp act sp ac e, and let C b e a pr esentable ∞ -c ate gory such that ﬁlter e d c olimits ar e exact. Then ther e is an e quivalenc e of ∞ -c ate gories ( − ) K : Sh( X , C ) ∼ − → Sh K ( X, C ) that is obtaine d on obje cts by G K ( K ) = colim U ⊃ K op en G ( U ) . Remark 3.62. Theorem 3.61 remains true even if X is only stably lo cally compact, with essentially the same pro of as giv en by Aoki. Ho w ev er, we will not need this greater generality here. Observ ation 3.63. Let X b e stably compact, and G a sheaf on X . • If x ∈ X is a saturated p oint, then the v alue of G K at the saturated compact set { x } agrees by deﬁnition with the stalk G x . Note that if X is Hausdorﬀ, any p oint is saturated. • The v alue G K ( X ) = G ( X ) = Γ( X ; G ) is just the v alue of the global sections of G . Remark 3.64. Theorem 3.59 is in fact directly obtained from Theorem 3.61, by noting that if F is a K -sheaf on X and the target C is stable, the ﬁb er of F ( X ) const → F is on the nose a cosheaf on X ∨ . The pro cess of taking these ﬁb ers is an equiv alence, again b ecause C is assumed to b e stable. Here, F ( X ) const refers to the constan t functor Q ( X ) op → C with v alue F ( X ) . Remark 3.65. Plugging in C = [1] in Theorem 3.61 reco vers the statemen t that X = ( X ∨ ) ∨ on the level of frames for X stably compact. F or stably lo cally compact spaces this giv es a generalization of de Groot dualit y . Note that in the absence of compactness Q ( X ) op is not a frame, as it is lacking a b ottom element, ho wev er, that is the only defect. W e will need a sligh t strengthening of Theorems 3.59 and Theorems 3.61, namely that the equiv alences men tioned in b oth theorems are in fact natural in perfect maps. W e ﬁrst remark that de Gro ot dualit y implies the following lemma, b y dualizing the deﬁnition of a compact subset. Lemma 3.66. Let X b e stably compact, and K = \ i ∈ I K i a ﬁltered in tersection of saturated compact sets K i . If U is an op en subset of X such that K ⊂ U , then there exists i ∈ I such that K i ⊂ U . If f : X → Y is p erfect, w e see that for K ⊂ Y a saturated compact subset, f − 1 ( K ) ⊂ U ⊂ X is again saturated compact. Hence we get a functor f − 1 : Q ( Y ) → Q ( X ) preserving arbitrary intersections and ﬁnite unions. 34 GEORG LEHNER Lemma 3.67. Let f : X → Y b e a p erfect map, and C ⊂ Y a saturated compact subset. Then for all op en U ⊃ f − 1 ( C ) in X there exists an op en V ⊂ C in Y such that U ⊃ f − 1 ( V ) ⊃ f − 1 ( C ) . Pr o of. W rite C = \ C ≪ C ′ C ′ , for saturated compact C ′ , where C ≪ C ′ means that there exists an op en V suc h that C ⊂ V ⊂ C ′ . Applying f − 1 w e get f − 1 ( C ) = \ C ≪ C ′ f − 1 ( C ′ ) . If U ⊃ f − 1 ( C ) by Lemma 3.66 there thus exists C ⊂ V ⊂ C ′ with V op en, C ′ saturated compact, such that U ⊃ f − 1 ( C ′ ) ⊃ f − 1 ( V ) ⊃ f − 1 ( C ) . □ Using precomp osition, we see that we hav e an induced functor f + : Sh K ( X, C ) → Sh K ( Y , C ) , giv en by f ∗ ( G )( K ) = G ( f − 1 ( K )) . Prop osition 3.68. Let f : X → Y b e a p erfect map b etw een stably compact spaces. The equiv alence pro vided in Theorem 3.61 is natural in f , in the sense that the square Sh( X , C ) Sh K ( X, C ) Sh( Y , C ) Sh K ( Y , C ) ∼ f ∗ f + ∼ is commutativ e. W e note that the analogous statement for V erdier dualit y provided by Theorem 3.59 follows immediately as well. Pr o of. T racing through deﬁnitions, we see that this amoun ts to the question whether for a given sheaf F on X , and C ⊂ Y compact saturated the natural map colim V ⊃ C op en in Y F ( f − 1 ( V )) → colim U ⊃ f − 1 ( C ) op en in X F ( U ) is an equiv alence. But w e see that the diagram on the left maps coﬁnally in to the diagram on the right by Lemma 3.67. □ Remark 3.69. The statement of Prop osition 3.67 is very particular to p erfect maps and fails for more general contin uous maps. In the con text of the six-functor formalism on lo cally compact Hausdorﬀ spaces, see e.g. [V ol25], it corresp onds to the fact that for prop er maps f there is an equiv alence f ! ≃ f ∗ . Remark 3.70. A more top down p erspective on V erdier duality is to observe that it is ab out the statement that the ( ∞ , 2) -categories SC and Pr L dual come equipped with dualities ( − ) ∨ , and these dualities commute with the functor that assigns to a stably compact space X its ∞ -category of shea v es with v alues in sp ectra. Both dualities are 2-functors, cov ariant in 1-morphisms, con trav ariant in 2-morphisms, and b oth are in v o- lutions. The pro of of V erdier dualit y can b e reduced via the theory of lax-idemp oten t monads to a simple computation for distributive lattices. 3.5. The patch top ology. There is a universal wa y of adding op en sets to the top ology of a stably lo cally compact space X to make it Hausdorﬀ, while retaining lo cal compactness. This is called the p atch top olo gy of X . Let us give a short summary of how to work with this top ology . The general reference will b e the article by Escardó [Esc01], how ever information can also b e found in [Gie+03] and [Gou13]. Since Escardó’s w ork is done purely in the language of lo cales, whereas older literature uses top ological spaces, we will need to discuss quickly how to translate b etw een those pictures. Deﬁnition 3.71. Let L be a lo cale. W e call a n ucleus N on L p erfe ct if it preserves directed suprema. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 35 Lemma 3.72. Let S  → X b e a sublocale of a stably locally compact space X . Then the following are equiv alent: (1) S is a p erfect sublo cale. (2) The induced nucleus N = i ∗ i ∗ on X is p erfect. (3) S is obtained by a subset S ′ of X , such that the induced subspace top ology is stably lo cally compact, and the embedding S ′  → X is a p erfect map. (4) The asso ciated quotient map p : X ⨿ X ↠ E corresp onding to the frame congruence of S is p erfect. In particular, the asso ciated frame congruence E is itself stably lo cally compact. Pr o of. W e ﬁrst remark that if S is a p erfect sublo cale of a stably lo cally compact space X , it is itself stably lo cally compact by Prop osition 3.22. The equiv alence of (1) and (3) is now immediate from the equiv alence of the category of stably lo cally compact spaces and stably lo cally compact frames giv en by Corollary 3.17. T o see the equiv alence of (1) and (2) is also straightforw ard. If i ∗ preserv es ﬁltered suprema, then so do es i ∗ i ∗ , as i ∗ is left adjoint. Conv ersely , if the nucleus N of X is given, the asso ciated sublo cale S is obtained as the frame of op ens { U ∈ O ( X ) | N ( U ) = U } ⊂ O ( X ) , and i ∗ is simply giv en by the inclusion. W e only need to show that N preserving directed suprema implies that this subset is closed under directed suprema. Let U i , i ∈ I , b e a directed set of op ens such that U i = N ( U i ) , and U = _ i ∈ I U i . Then N ( U ) = N ( _ i ∈ I U i ) = _ i ∈ I N ( U i ) = _ i ∈ I U i = U, and hence we are done. T o see how (4) is equiv alent to the rest, ﬁrst observe that if (4) holds, then S  → L is obtained as an equalizer of stably lo cally compact spaces and p erfect maps, hence itself p erfect. Conv ersely , if S is a sublo cale, the quotient map p : X ⨿ X ↠ E is given by the formula p ∗ ( U, V ) = ( U ∧ N ( V ) , V ∧ N ( U )) . Hence if the nucleus N is p erfect, this map preserves directed suprema, and hence p is p erfect. □ W e no w come to the cen tral notion of this section. The patch top ology asso ciated to a stably lo cally compact space. Denote by LCH p the full sub category of SLC p spanned by lo cally compact Hausdorﬀ spaces. W e will simply write LCH and SLC for the corresp onding wide sub categories given by only considering globally deﬁned p erfect maps. F urthermore, for a given stably lo cally compact space X , let us denote by K ( X ) the set of subsets of X obtained via p erfect embeddings. Note that K ( X ) is isomorphic to the opp osite of the p oset of p erfect nuclei on X . Theorem 3.73 (The patc h top ology , [Esc01], Theorem 5.8) . The inclusion of the ful l sub c ate gory of lo c al ly c omp act Hausdorﬀ sp ac es into stably lo c al ly c omp act sp ac es has a right adjoint, LCH SLC patch ⊣ given by sending a stably lo c al ly c omp act sp ac e X to the stably lo c al ly c omp act fr ame K ( X ) op of p erfe ct nuclei on X . Remark 3.74. Since the space pt is a ﬁnite space, every contin uous map pt → X is automatically p erfect. (See Example 3.30.) It is also Hausdorﬀ, hence we see that X = Map(pt , X )  Map(pt , X patch ) , 36 GEORG LEHNER and therefore the underlying set of p oints of X patch agrees with that of X . The new top ology on X is referred to as the p atch top olo gy of X . In light of Lemma 3.72 it is also immediate what this top ology is. The closed subsets of X patch are given exactly by the set K ( X ) of images of p erfect embeddings. Remark 3.75. The patc h top ology on a stably compact space giv es a compact Hausdorﬀ space. This is seen by observing that the p erfect map X → pt is sent to a p erfect map X patch → pt . Example 3.76. In the case of the directed in terv al I ≤ , it holds that I patch ≤ = I is the interv al equipp ed with the ordinary metric top ology on I , see [Gou13, Example 9.1.33]. More generally , since patch preserves limits, if S is a set we also see that for the directed Hilb ert cub e I S ≤ it holds that ( I S ≤ ) patch = I S . Ev en more generally , any p erfect subspace K  → I S ≤ is obtained as an equalizer of p erfect maps b etw een stably compact spaces, and hence K patch agrees with the induced subspace top ology when considering K as a subset of I S . Example 3.77. If X is a coherent space, corresp onding to a b ounded distributive lattice D , then X patch is a proﬁnite set, corresp onding to the Bo olean algebra Bool ( D ) given by Bo oleanization, [Esc01, Section 3]. The patch top ology in this case is also referred to as the c onstructible top olo gy on X , see [Leh25a, Section 3.2]. Example 3.78. Let P b e a p oset which is compactly assembled when considered as an ∞ -category . This notion is referred to as a domain in [Gie+03], and is also called c ontinuous p oset by Eﬁmov [Eﬁ25a]. There are tw o notable top ologies on P , one b eing the Scott top ology , where op en sets are Scott op en upw ards closed subsets of P , [Gie+03, Chapter I I], as well as the Lawson top ology , which is generated from the Scott top ology by adding the lo w er top ology , [Gie+03, Chapter I II]. If P is compact in the Lawson top ology , then P equipp ed with the Scott top ology is stably compact, with its de Gro ot dual giv en by the low er top ology , and its patch topology giv en by the La wson top ology , [Gie+03, Prop osition VI-6.24]. The condition for a domain P to b e compact can b e understo o d in multiple diﬀeren t wa ys, see [Gie+03, Chapter I II-5]. While the description of the patch top ology in terms of p erfect em b eddings is elegan t, we will need a concrete wa y on how to build it. Let us highlight tw o sp ecial classes of p erfect embeddings for a given stably lo cally compact space X . • An y close d subset C ⊂ X is p erfect. • An y saturated compact subset S ⊂ X is p erfect. Example 3.79. In the case of the directed interv al I ≤ , saturated compact sets are of the form [0 , a ] for a ∈ I , whereas closed sets are of the form [ b, 1] , for b ∈ I . Deﬁnition 3.80. Let X b e a stably locally compact space. A subset E ⊂ X is called elementary c omp act if it is of the form C ∪ S for C ⊂ X closed, and S ⊂ X saturated compact. W rite E ( X ) ⊂ K ( X ) for the set of elementary compact subsets. Elemen tary compact subspaces are closed under ﬁnite unions. Moreov er, they generate the patch top ology in the following sense. Lemma 3.81 ([Esc01], Lemma 5.4) . Let X b e stably lo cally compact. The set of elemen tary compact subspaces E ( X ) generates K ( X ) under intersections. In other w ords, when considering reverse inclusions, elemen tary compact sets form a basis of the frame K ( X ) op  O ( X patch ) . F rom the p ersp ective of sheaf theory , this means that—at least in principle—working with shea ves on the patch top ology only requires an understanding of the p oset E ( X ) and its induced Grothendiec k top ology . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 37 Remark 3.82. In the case that X is stably compact it is th us clear that the patch top ology applied to the de Gro ot dual X ∨ also in results in X patch . This is since de Groot dualit y simply interc hanges the roles of closed and saturated compact subsets, result in the same sets of elementary compact subspaces. Example 3.83. F or X = I ≤ the directed in terv al, it is straightforw ard to see that ( I ≤ ) patch = I is the standard real interv al. Since taking the patc h top ology preserves limits we therefore see that for any directed Hilb ert cub e I S ≤ w e hav e ( I S ≤ ) patch = I S . Using this w e see that the stably compact versions of the Urysohn lemma pro duce classical v ersions of the Urysohn lemma for compact Hausdorﬀ spaces. As the last part of this section, observ e that the P atch top ology is even functorial in partial p erfect maps. This is easiest to see on the level of n uclei. If N is a p erfect nucleus on X , and f : X → Y a partial p erfect map, then f ∗ N f ∗ is a p erfect n ucleus on Y . W e claim that the adjunction given in Theorem 3.73 generalizes to this functorialit y as well. One could in principle retread the pro of given by Escardó by analyzing what happ ens to partial frame homomorphisms, how ev er, w e will circumv en t the trouble by using one-p oin t compactiﬁcations. T o start, let us recall the notion of the (classical) one-p oin t compactiﬁcation X ∗ of a lo cally compact Hausdorﬀ space, giv en by taking the top ology on X ⨿ {∗} which consists of opens U ⊂ X and complements of compact subsets K ⊂ X . The following is a standard fact ab out lo cally compact Hausdorﬀ spaces. Prop osition 3.84 ([Bun21], Lemma 5.2) . There exists an equiv alence of categories LCH p ≃ CH pt / , giv en on ob jects by sending a lo cally compact Hausdorﬀ space X to its one-p oint compactiﬁcation. Corollary 3.85. There exists an adjunction LCH p CH ( − ) ∗ ⊣ with left adjoint given b y the inclusion of compact Hausdorﬀ space and contin uous maps into LCH p , and the right adjoint by one-p oin t compactiﬁcation. Lemma 3.86. Let X be stably lo cally compact. Then the inclusion X patch  → ( X + ) patch is op en. Pr o of. First note that { + } is closed in ( X + ) patch , and hence X as the complementary set, equipp ed with the induced subspace top ology , is op en. Recall that by Example 3.45, the set of saturated compact sets of X + is given by the set of saturated compact sets of X ⊂ X ⨿ { + } together with the en tire set X ⨿ { + } . Since the top ology on ( X + ) patch is generated from intersections of op ens on X + and complements of saturated compact sets, w e see that the induced subspace topology on the set X is generated from intersections of op ens in X and complements of saturated compact sets in X . But this simply agrees with the patch top ology on X . □ Corollary 3.87. Let X b e stably lo cally compact. Then there is a natural equiv alence ( X patch ) ∗  ( X + ) patch . Pr o of. The partial p erfect map ( X + ) patch → X patch giv en by the identit y on the op en domain X patch giv es, b y applying the adjunction from Corollary 3.85 the con tinuous map ( X + ) patch → ( X patch ) ∗ . This is clearly a bijection. But a contin uous bijection b etw een compact Hausdorﬀ spaces is a homeomorphism, therefore proving the claim. □ 38 GEORG LEHNER Corollary 3.88. The inclusion of the full sub category of lo cally compact Hausdorﬀ spaces into stably lo cally compact spaces and partial p erfect has a righ t adjoint, LCH p SLC p patch ⊣ giv en on ob jects by X 7→ X patch . Pr o of. First note that the adjunction from Prop osition 3.44 applied to stably compact spaces and compact Hausdorﬀ spaces lifts to an adjunction CH pt / SC pt / . ( − ) patch ⊣ The commutativ e square of left adjoints SLC p SC pt / LCH p CH pt / giv es, as an application of Lemma 2.2, the statement that the inclusion LCH p  → SLC p , has a righ t adjoin t giv en b y sending a stably lo cally compact X to the op en complement of { + } in ( X + ) patch . But this agrees with X patch b y Lemma 3.86. □ Remark 3.89. Note that the sp e cialization or der equips the underlying set of a coherent space X with the structure of a p oset. (This is nothing more than an instance of the category of lo cales b eing naturally a 2 -category via its enric hmen t in p osets.) In case X is a stably compact space, the relation ≤ is closed as a subset of X patch × X patch . A compact Hausdorﬀ space equipp ed with such a closed p oset structure is called a c omp act p osp ac e . Given a compact p ospace ( Y , ≤ ) , the set of down w ard closed opens actually forms a stably compact top ology on Y . These tw o pro cedures result in an equiv alence of categories SC ≃ CompPO , see e.g. [Gou13, Prop osition 9.4.10]. W e hav e the follo wing table of corresp ondences. Stably compact space Compact p ospace P erfect subspace Closed subset Saturated compact Closed and down w ard closed Closed Closed and upw ard closed de Gro ot dual Opp osite order ≤ op P atch top ology F orget order 4. Nisnevich-type descent for localizing inv ariants In this section we are concerned with statements ab out descent for ﬁnitary lo calizing inv ariants applied to shea ves on stably (lo cally) compact spaces. This is somewhat inspired by the well-kno wn notion of Nisnevich- Descen t in the con text of K -theory of schemes. Let us ﬁx throughout the conv en tion that F : Cat perf → E denotes a lo calizing inv arian t. W e note that all statements made ab out Sh( X , Sp) in this section generalize immediately when Sp is replaced by an arbitrary dualizable ∞ -category C , in light of Lemma 2.13, whic h states that F cont ( − ⊗ C ) is again a lo calizing in v ariant. Before we b egin, we wan t to record the follo wing simple lemma ab out pullbac k squares in stable categories. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 39 Lemma 4.1. Let E b e a stable ∞ -category . Then a square A 0 A 1 B 0 B 1 f A f B is a pullback square iﬀ the induced map on horizontal ﬁb ers ﬁb( f A ) → ﬁb( f B ) is an equiv alence. Pr o of. Consider the diagram ﬁb( f A ) A 0 B 0 0 A 1 B 1 . ⌟ f A f B If the righ t hand square is a pullback, then so is the total square, and hence ﬁb( f A ) → ﬁb( f B ) is an equiv alence. Conv ersely , if the map ﬁ b( f A ) → ﬁb( f B ) is an equiv alence, w e see that the left and total squares are pullbacks. Using stability of E , b oth squares are also pushouts, and hence w e can use the pasting lemma for pushouts, [Lur12, Lemma 4.4.2.1], to conclude that the right square is a pushout. □ Prop osition 4.2 (Perfect Nisnevich Descent) . Let f : Y → X b e a p erfect map of stably lo cally compact spaces, and C ⊂ X a closed subspace. Supp ose that f induces a homeomorphism on the op en complements of f − 1 ( C ) and C . Then the square F cont (Sh( X , Sp)) F cont (Sh( C, Sp)) F cont (Sh( Y , Sp)) F cont (Sh( f − 1 ( C ) , Sp)) f ∗ f ∗ is a pullback. Pr o of. Straigh tforward: Observe that the horizon tal ﬁb ers of b oth rows are equiv alent by Prop osition 2.30, then apply Lemma 4.1. □ The main application is the simple case of a prop er inclusion, whic h follo ws directly from the Second Isomorphism Theorem (Prop osition 2.29). Corollary 4.3. Let X b e stably lo cally compact, K ⊂ X b e a p erfect inclusion, and C ⊂ X b e a closed inclusion. Then w e hav e a pullback square F cont (Sh( K ∪ C, Sp)) F cont (Sh( C, Sp)) F cont (Sh( K, Sp)) F cont (Sh( K ∩ C, Sp)) . ⌟ Our goal is to generalize the statement of Corollary 4.3 to the case where C is allow ed to b e an arbitrary p erfect subspace K ′ . In fact, if the formula of Theorem 1.3 is true, it must necessarily b e the case that this form of p erfect descent holds. W e will do this in steps. The ﬁrst will b e the use of V erdier duality . Note that the fact that ( − ) ∨ = F un L ( − , Sp) is an auto equiv alence on Pr L dual implies that it preserves V erdier sequences. This will b e critical in the following observ ation. 40 GEORG LEHNER Prop osition 4.4. Let X b e stably compact, K ⊂ X b e a p erfect inclusion, and C ⊂ X b e a saturated compact inclusion. Then we hav e a pullback square F cont (Sh( K ∪ C, Sp)) F cont (Sh( C, Sp)) F cont (Sh( K, Sp)) F cont (Sh( K ∩ C, Sp)) . ⌟ Pr o of. The prop osition follows analogously to Corollary 4.3 by applying V erdier dualit y (Theorem 3.59). The square of sheaf categories in question is equiv alent to Cosh( K ∨ ∪ C ∨ ; Sp) Cosh( C ∨ ; Sp) Cosh( K ∨ ; Sp) Cosh( K ∨ ∩ C ∨ ; Sp) This square is obtained from the corresp onding one inv olving shea ves of dual spaces b y applying ( − ) ∨ . The saturated compact inclusion C → X is sent to a closed inclusion C ∨ → X ∨ under duality , and we can simply v erify that horizontal ﬁb ers of the square of sheaf categories are equiv alent by Prop osition 2.29. □ Remark 4.5. T o b e more concrete, the ﬁb er of Sh( K ∪ C, Sp) → Sh( C , Sp) is given b y Cosh( U, Sp) , where U is the op en subset of X ∨ corresp onding to the Scott op en ﬁlter of the compact saturated set C . Since U is not in general stably compact, but only stably lo cally compact, V erdier dualit y is a bit more subtle. Note that this is exactly the same issue as mentioned in Remark 3.65. W arning 4.6. It is not true that for p erfect embeddings K , L of X the square Sh( K ∪ L ; Sp) Sh( L ; Sp) Sh( K ; Sp) Sh( K ∩ L ; Sp) is necessarily a pullback square in Pr L st , unlik e in the case when X is Hausdorﬀ. As a coun terexample, consider K = [0 , a ] ≤ , L = [ b, 1] ≤ for a < b in I ≤ . Their in tersection is empty , how ever Sh (([0 , a ] ∪ [ b, 1]) ≤ ; Sp) do es not split as a pro duct. T o sum up, we hav e excision for lo calizing in v ariants for p erfect embeddings if either of the embeddings is closed or saturated compact. W e are left with showing that this implies prop er descent. Before we do so, w e will introduce a tec hnical lemma to simplify induction of descent-t ype conditions. 4.1. The sheaf condition on D 3 . W e will need a tec hnical lemma that allows us a reduction of a sheaf condition on three generators to a statement ab out pullbacks. The conceptual reason for this lemma is that the space X = S 3 giv en by the pro duct of three copies of the Sierpinski space represents the functor on lo cales that chooses three op ens. It is somewhat tautological that the sheaf condition for the case of three op ens on an arbitrary lo cale can alwa ys b e chec k ed b y pushforward to S 3 . The space S 3 is simply a combinatorial cub e equipp ed with the Alexandroﬀ top ology , and its frame of op ens is D 3 , the free distributive lattice generated b y three elements. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 41 Lemma 4.7. Let D b e an ∞ -category with ﬁnite limits. Then A A 3 A 2 A 23 A 1 A 13 A 12 A 123 is a limit diagram, i.e. A is the limit ov er the b ottom part of the diagram, iﬀ the square A A 3 A 1 × A 12 A 2 A 13 × A 123 A 23 is a pullback square. Remark 4.8. Cubical diagrams as in Lemma 4.7 are referred to as c artesian cub es in the con text of Go odwillie calculus, see e.g. [Ane+18]. W e exp ect Lemma 4.7 to b e well-kno wn, but were not aw are of a direct reference, hence included it. The ob vious generalization to n -dimensional cubes is included in App endix A. Pr o of. Let us deﬁne tw o diagram shap es, namely D 1 = • • • • • • • and D 2 = • • • • • • • • • . Both are given as realizations of p osets. The p oset of D 1 em b eds fully faithfully in to D 2 as the set of black v ertices. Now, if we ha v e the starting datum of the following diagram A • , indexed b y D 1 , in D as represented b y A 2 A 3 A 1 A 12 A 23 A 13 A 123 42 GEORG LEHNER then the right Kan extension to D 2 is given by A 1 × A 12 A 2 A 2 A 3 A 1 A 12 A 13 × A 123 A 23 A 23 A 13 A 123 b y the general formula for right Kan extension. The limits o v er bot h diagrams agree. Finally , observe that the inclusion of the spine S • • • is ﬁnal in D 2 , hence we obtain the formula for the limit lim D 1 A • as the pullback lim F A • A 1 × A 12 A 2 A 3 A 13 × A 123 A 23 . ⌟ □ 4.2. Extending excisiveness. With Lemma 4.7 under our b elt, we can extend from closed and saturated compact descent to descent for elementary compact sets. Prop osition 4.9 (Elementary compact descent) . Supp ose F : SC op → D is a functor, with D an ∞ -category with ﬁnite limits, such that for all stably compact X , K a compact subset of X , C a closed subset of X , and S a saturated compact subset of X the squares F ( K ∪ C ) F ( C ) F ( K ∪ S ) F ( S ) F ( K ) F ( K ∩ C ) F ( K ) F ( K ∩ S ) . ⌟ ⌟ are pullback squares. Then it also holds that for any compact subset E ⊂ X of the form E = S ∪ C , with S b eing saturated compact and C b eing closed, that the square F ( K ∪ E ) F ( K ) F ( E ) F ( K ∩ E ) ⌟ is a pullback square. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 43 Pr o of. The pro of is just an iterated application of Lemma 4.7. Observe that by assumption we hav e the pullbac ks F ( S ∪ C ) F ( C ) F (( K ∩ S ) ∪ ( K ∩ C )) F ( K ∩ C ) F ( S ) F ( S ∩ C ) F ( K ∩ S ) F ( K ∩ S ∩ C ) ⌟ ⌟ since S ⊂ S ∪ C and K ∩ S ⊂ ( K ∩ S ) ∪ ( K ∩ C ) = K ∩ ( S ∪ C ) are saturated compact subsets. Applying Lemma 4.7, this means that the square F ( K ∪ S ∪ C ) F ( K ) F ( S ∪ C ) F ( K ∩ ( S ∪ C )) is a pullback square iﬀ F ( K ∪ S ∪ C ) is the limit of the corresp onding cub e built from F ( K ) , F ( S ) and F ( C ) , whic h, again by Lemma 4.7, is the case if the squares F ( K ∪ C ∪ S ) F ( S ) F ( K ∪ C ) F ( C ) F ( K ∪ C ) F (( K ∩ S ) ∪ ( C ∩ S )) F ( K ) F ( K ∩ C ) ⌟ ⌟ and F (( K ∩ S ) ∪ ( C ∩ S )) F ( C ∩ S ) F ( K ∩ S ) F ( K ∩ C ∩ S ) ⌟ are pullbacks. This is the case for all three squares since: • S ⊂ K ∪ C ∪ S is saturated compact. • C ⊂ K ∪ C is closed. • C ∩ S ⊂ ( K ∩ S ) ∪ ( C ∩ S ) = ( K ∪ C ) ∩ S is closed. Therefore in all of these cases, w e can use excision for either closed or saturated compact inclusions, hence w e conclude the claim. □ No w assume we ha v e a functor F : SC op → D as in Prop osition 4.9, and that D is furthermore stable. Since crucially in this context pushout squares agree with pullback squares, we see that F restricted to elementary compact sets satisﬁes a c oshe af condition on elemen tary compacts (with cov ers given b y intersections!). W e claim that this, together with compatibility with ﬁltered in tersections, is enough to pro duce a cosheaf on the frame O ( X patch )  K ( X ) op of all p erfect subspaces. The only cav eat is that F (1) , 0 in general, an issue that can b e solv ed by taking coﬁb ers. Prop osition 4.10. Let X b e a stably compact space, let D b e stable and closed under colimits, and let F : O ( X patch )  K ( X ) op → D b e a functor satisfying the tw o conditions: (1) F or all K ⊂ X compact and E ⊂ X elemen tary compact we hav e a pushout square F ( K ∪ E ) F ( K ) F ( E ) F ( E ∩ K ) . ⌟ 44 GEORG LEHNER (2) Whenev er K = \ i ∈ I K i is a directed intersection of compacts, we hav e F ( K ) ≃ colim i ∈ I F ( K i ) . Then the functor F # deﬁned via F # ( K ) = coﬁb( F (1) → F ( K )) is a cosheaf on X patch . Before we b egin the pro of, let us cite a standard lemma on the use of Grothendieck top ologies. Lemma 4.11 (Comparison lemma, see [Hoy14], Lemma C.3) . Let ( P , τ ) b e a poset equipped with a Grothendiec k top ology and P 0 ⊂ P a subset of P such that: • Ev ery ob ject in P can b e cov ered by ob jects in P 0 . • P 0 is closed under meets in P . Let τ 0 b e the induced Grothendiec k top ology on P 0 b y restriction. Then the restriction Sh( P , τ ) → Sh( P 0 , τ 0 ) is an equiv alence of ∞ -categories, with inv erse given by right Kan extension. Let us remark that the same statement thus follows for cosheav es, how ever with in v erse given by left Kan extension. Pr o of of Pr op osition 4.10. It is immediate that F # restricted to the set of elemen tary compacts E ( X ) op is a cosheaf for the induced Grothendieck top ology giv en b y restriction from O ( X patch ) . Using Lemma 4.11, w e are left to sho w that the natural map η : LanRes F # → F # is an equiv alence, which is a statement we only need to verify ob ject wise for all compact K ⊂ X . W e pro ceed in tw o steps: (a) W e ﬁrst show that η K is an equiv alence for K of the form K = E 1 ∩ · · · ∩ E n with E i elemen tary compact, by induction. The induction start is true since the inclusion of elementary compacts into all compacts is fully faithful. Now assume we know that η is an equiv alence for intersections of n − 1 man y elementary compacts. Since the left Kan extension of Res F # is a cosheaf on O ( X patch ) , the v alue of Lan Res F # on K is given as the pushout F # (( E 1 ∩ · · · ∩ E n − 1 ) ∪ E n ) F # ( E n ) F # ( E 1 ∩ · · · ∩ E n − 1 ) . LanRes F # ( E 1 ∩ · · · ∩ E n − 1 ∩ E n ) ⌟ But this pushout agrees with F # ( E 1 ∩ · · · ∩ E n − 1 ∩ E n ) by assumption from p oin t (1). (b) F or arbitrary K , we simply write K as a directed intersection of ﬁnite intersections of elemen tary compacts, and see from using (a) together with assumption (2) that η K is also an equiv alence. □ 5. Descent on st abl y comp act sp aces In this section we presen t an extension of the following theorem due to Clausen. Theorem 5.1 (Clausen, see [KNP24] Theorem 3.6.15) . L et D b e a c omp actly assemble d pr esentable ∞ - c ate gory and let F : CH op → D b e a functor such that: (1) F ( ∅ ) = 1 . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 45 (2) Closed descent: Whenever K , L ⊂ X ar e two close d emb e ddings, then F ( K ∪ L ) F ( K ) F ( L ) F ( K ∩ L ) ⌟ is a pul lb ack. (3) Proﬁnite descent: Whenever X i , i ∈ I , is a c oﬁlter e d system in CH , then F (lim i ∈ I X i ) ≃ colim i ∈ I F ( X i ) . Then ther e exists a natur al e quivalenc e of functors F ≃ Γ( − , F (pt)) , wher e for a given c omp act Hausdorﬀ sp ac e X , the expr ession Γ( X , F (pt)) r efers to glob al se ctions of the c onstant she af with value F (pt) ∈ D . This extension for stably compact spaces is obtained by switching closed descen t with prop er descent. Theorem 5.2. L et D b e a c omp actly assemble d ∞ -c ate gory, and let F : SC op → D b e a functor such that: (1) F ( ∅ ) = 1 . (2) P erfect descent: Whenever K , L ⊂ X ar e two p erfe ct emb e ddings, then F ( K ∪ L ) F ( K ) F ( L ) F ( K ∩ L ) ⌟ is a pul lb ack. (3) Coﬁltered descent: Whenever X i , i ∈ I , is a c oﬁlter e d system in SC , then F (lim i ∈ I X i ) ≃ colim i ∈ I F ( X i ) . Then ther e is a natur al e quivalenc e of functors F ≃ Γ(( − ) patch ; F (pt)) . Let us provide the setup for the pro of of Theorem 5.2. F or the remainder of the section, ﬁx a functor F : SC op → D satisfying the conditions of Theorem 5.2. F or a given stably compact space X , restricting F to the set of p erfect embeddings K ⊂ X pro duces a K -sheaf F X : K ( X ) op → D . on X patch , since the set of p erfect subsets of X agrees with the set of compact subsets of X patch . Under the equiv alence of K -sheav es with sheav es, F X corresp onds to a sheaf F X ∈ Sh( X patch , D ) . W e observe the follo wing, using the facts giv en in Observ ation 3.63. • W e hav e equiv alences Γ( X patch , F X ) = F X ( X ) ≃ F X ( X ) = F ( X ) by construction. • F or each p oin t x ∈ X , the stalks compute as F X x ≃ F ( { x } ) ≃ F (pt) . • The unique map X → pt produces a canonical map F (pt) → F ( X ) , whic h using the adjunction giv es a natural transformation α X : F (pt) → F X of sheav es in Sh( X patch , D ) . 46 GEORG LEHNER • The natural transformation α X is a stalkwise equiv alence for all p oints x ∈ X . Therefore, we arrive at the following simple conclusion. Observ ation 5.3. Let X b e a stably compact space. If α X is an equiv alence, then the statemen t that F ( X ) ≃ Γ( X patch ; F (pt)) holds. Recall that a top os X is said to hav e enough p oints , if the family of stalk functors x ∗ : X → An for all p oin ts x of X is join tly conserv ative, i.e. if equiv alences can b e detected stalkwise. T o hav e the same statemen t when considering as target the compactly assembled ∞ -category D , w e cite the follo wing lemma. Lemma 5.4 ([Hai22], Lemma 2.12) . Let { p ∗ i : T → S i } i ∈ I b e a jointly conserv ative family of ﬁnite limit preserving left adjoin t functors b et w een presen table ∞ -categories, and let E b e a compactly assembled presen table ∞ -category . Then the family of left adjoints { p ∗ i ⊗ E : T ⊗ E → S i ⊗ E } i ∈ I is jointly conserv ative. The following is immediate. Lemma 5.5. The statemen t F ( X ) ≃ Γ( X patch ; F (pt)) holds for all stably compact spaces X such that Sh( X patch , An) has enough p oints. W e note that the condition that Sh ( X patch , An) has enough p oints fails for example for X = I S ≤ a directed Hilb ert cub e of inﬁnite dimension S . Nonetheless, we now hav e a large class of examples for which Theorem 5.2 holds, due to the following tw o theorems. Theorem 5.6 ([Lur12] Corollary 7.2.1.17) . L et X b e a top olo gic al sp ac e. Supp ose that Sh ( X, An ) is lo c al ly of homotopy dimension ≤ n for some inte ger n . Then Sh( X, An ) has enough p oints. Theorem 5.7 ([Lur12] Corollary 7.2.3.7.) . L et X b e a p ar ac omp act top olo gic al sp ac e. The fol lowing c ondi- tions ar e e quivalent. (1) The c overing dimension of X is ≤ n . (2) The homotopy dimension of Sh ( X, An ) is ≤ n . (3) F or every close d subset A ⊂ X , every m ≥ n , and every c ontinuous map f 0 : A → S m , ther e exists f : X → S m extending f 0 . It follows that Theorem 5.2 holds for stably compact X such that X patch em b eds as a closed subset of I N for some ﬁnite N , as I N is paracompact and has cov ering dimension N . At this p oint it might seem that we are stuck, since we cannot argue via stalks for arbitrary stably compact X . In the case of compact Hausdorﬀ spaces, the pro of argumen t given by Clausen pro ceeds by using the fact that every compact Hausdorﬀ space is obtained as an inv erse limit of ﬁnite-dimensional p olyhedra, and then applying proﬁnite descent. Our own pro of will slightly diverge here from this argumen t, as we will instead argue directly that α X is an equiv alence for every stably compact space X . Before w e do so, we should add some comments on the naturalit y of the assignmen t X 7→ α X . Construction 5.8. Given a p erfect map f : X → Y , we see that there is a natural transformation of K -shea ves on Y , F Y → f + F X giv en for K ⊂ Y perfect as the map F ( K ) F ( f | f − 1 ( K ) ) − − − − − − − − → F ( f − 1 ( K )) . Under the naturality provided b y Prop osition 3.68, this corresponds to a natural transformation of sheav es F Y → f ∗ F X . The extreme case, X → pt , giv es the map F (pt) → X ∗ F X that is adjoint to α X . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 47 If X f − → Y g − → Z are tw o p erfect maps, it is clear that we hav e a commutativ e triangle of natural transformations of sheav es ov er Z , F Z g ∗ F Y ( g f ) ∗ F X . A djoining g ∗ o ver gives the commutativ e triangle g ∗ F Z F Y f ∗ F X . Moreo ver, the map g ∗ F Z → f ∗ F X naturally factors as the comm utativ e triangle g ∗ F Z f ∗ f ∗ g ∗ F Z f ∗ F X . By pasting triangles, we arriv e at the natural comm utative square of sheav es on Y , g ∗ F Z F Y f ∗ f ∗ g ∗ F X f ∗ F X . The sp ecial case of Z = pt gives a map of arrows from α Y → f ∗ α X . With this understo o d, we can get rid of the ﬁnite dimensionality assumption b y the observ ation that an arbitrary directed Hilb ert cub e I S ≤  lim N ⊂ S ﬁnite I N ≤ is obtained as an inv erse limit of stably compact spaces with ﬁnite dimensional patch top ology . This will follow from the (strong) homotopy inv ariance provided by the following lemma. Lemma 5.9. Let X b e stably compact such that X patch em b eds as a closed subset of [0 , 1] M for some ﬁnite M , and let S b e a set of arbitrary cardinality . Then F ( X ) ≃ F ( X × I S ≤ ) via the canonical pro jection X × I S ≤ → X . Pr o of. Using coﬁltered descent, we see that F ( X × I S ≤ ) ≃ colim N ⊂ S ﬁnite F ( X × I N ≤ ) . The spaces X × I N ≤ also satisfy that ( X × I N ≤ ) patch  X patch × I N ⊂ I M ⨿ N is paracompact with ﬁnite co vering dimension, hence Lemma 5.5 together with con tractibility of I N sho ws that the colimit diagram is in fact the constant diagram with v alue F ( X ) . □ As a direct application w e see the following. 48 GEORG LEHNER Lemma 5.10. Let N ⊂ S b e a ﬁnite subset, and denote by p N : I S ≤ → I N ≤ the asso ciated pro jection b etw een directed Hilb ert cub es. Then there exists a natural equiv alence of sheav es F I N ≤ ≃ p N ∗ F I S ≤ on I N . Pr o of. The corresp onding K -sheaf of F I N ≤ is given by F I N ≤ , and the corresp onding K -sheaf of p N ∗ F I S ≤ is given b y ( K ⊂ I N ≤ ) 7→ F I S ≤ ( K × I S \ N → ) = F ( K × I S \ N → ) . W e can see that their v alues are naturally equiv alent, as b y Lemma 5.9, w e ha v e F ( K × I S \ N → ) ≃ F ( K ) ≃ F X ( K ) . □ Prop osition 5.11. The natural transformation α X is an equiv alence for X = I S ≤ a directed Hilb ert cube for an arbitrary set S . Pr o of. Since I S ≤ = lim N ⊂ S ﬁnite I N ≤ , taking patch top ology and using the fact that taking sheav es with v alues in D sends coﬁltered limits of stably compact spaces to ﬁltered colimits in Pr L , we hav e that Sh( I S , D ) ≃ colim N ⊂ S ﬁnite Sh( I N , D ) , with the colimit taken in Pr L , i.e. as a limit of ∞ -categories along righ t adjoints. Th us, the map α I S ≤ : F (pt) → F I S ≤ corresp onds to the family of maps p N ∗ ( α I S ≤ ) for N ⊂ S ﬁnite. W e claim that p N ∗ ( α I S ≤ ) is equiv alent as a natural transformation to α I N ≤ . W e ha ve seen that F I N ≤ ≃ p N ∗ F I S ≤ b y Lemma 5.10. W e hav e also seen that the map p N induces a contractible and essential geometric morphism after applying the patch top ology functor (see Example 2.17), hence p N ∗ preserv es constant ob jects by Lemma 2.16. □ Lemma 5.12. If i : K  → X is a p erfect em b edding of stably compact spaces, and α X is an equiv alence, then α K is an equiv alence as well. Pr o of. The map of natural transformations α X → i ∗ α K pro duces by adjunction a map i ∗ α X → α K . W e claim that this map is an equiv alence of natural transformations, which is equiv alent to saying that the natural transformations i ∗ X ∗ ( F (pt)) → Y ∗ ( F (pt)) and i ∗ ( F X ) → F K are equiv alences. The former is clear, as i ∗ preserv es constant sheav es. The latter follows from the observ ation that the corresp onding K -sheaf of i ∗ ( F X ) is obtained b y restricting F X to K ( K ) . This restriction is giv en by precomp osing with the inclusion K ( K )  K ( X ) /K  → K ( X ) whic h preserves non-empty inﬁma and arbitrary suprema, hence precomp osition preserves K -sheav es. □ W e can no w ﬁnish the pro of of Theorem 5.2. Pr o of of The or em 5.2. If X is a stably compact space, embed X into a directed Hilb ert cub e, using Theorem 3.40. The com bination of Prop osition 5.11 and Lemma 5.12 implies that α X is an equiv alence. □ Remark 5.13. The only part where an argument via stalks was used during this pro of was to show that α X is an equiv alence for X = I N ≤ a directed, ﬁnite-dimensional cub e. Joint surjectivity of the stalk functors in this case unfortunately fails if D is not assumed to b e compactly assembled, see the discussion in [KNP24], Section 3.6, in particular Corollary 3.6.16. ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 49 Remark 5.14. W e hav e b een somewhat imprecise in the actual construction of a natural transformation Γ(( − ) patch , F (pt)) → F . Let us address p oten tial concerns here. 16 Consider the functor Sh(( − ) patch , D ) : SC → d Cat ∞ , with functoriality given by f : X → Y b eing mapp ed to f ∗ : Sh ( X patch , D ) → Sh( Y patch , D ) . The inclusion of the p oint pt → SC is terminal, hence there is an induced natural transformation Sh(( − ) patch , D ) ⇒ D , where D : SC op → d Cat ∞ is the constant functor with v alue D . V aluewise it is just given by forming global sections. Up on applying unstraightening, w e get an induced functor b etw een cartesian ﬁbrations Z SC op Sh Z SC op D ≃ SC op × D SC op Γ p q where the ∞ -category Z SC op Sh is giv en on objects as pairs ( X ∈ SC , F ∈ Sh( X patch , D )) , and with morphisms ( Y , G ) → ( X, F ) given b y a pair of a p erfect map f : X → Y and a natural transformation G ⇒ f ∗ ( F ) in Sh( Y patch , D ) . The functor Γ is given on ﬁb ers ov er a given stably compact space X as X ∗ : Sh( X patch , D ) → D . The functor Γ has a left adjoin t relativ e SC op , as can b e chec ked directly via [Lur17, Prop osition 7.3.2.6], with the lab els E = SC op , C = SC op × D , D = Z SC op Sh and G = Γ . Up on taking sections, this leads to the adjunction F un / SC op (SC op , Z SC op Sh) F un / SC op (SC op , SC op × D ) F un(SC op , D ) Γ ≃ ⊣ The p oin t of Construction 5.8 is that it provides a lift F − of F : SC op → D along Γ . The counit of the ab ov e adjunction applied to F − reco vers v aluewise for each stably compact space X the natural transformation α X : F (pt) → F X . When applying Γ , this provides the natural transformation Γ(( − ) patch , F (pt)) ⇒ F . Remark 5.15. Theorem 5.1 and Theorem 5.2 can b e phrased more structurally as stating that there are equiv alences of ∞ -categories F un desc (SC op , D ) F un desc (CH op , D ) D res patch ∗ Γ( − ; − ) ∼ ∼ where with F un desc w e mean functors satisfying the descent conditions of either Theorem 5.1 or Theorem 5.2. In this sense, Theorem 5.2 actually contains Theorem 5.1 as a sp ecial case, as precomp osing a functor on CH that satisﬁes closed and proﬁnite descent with the patch functor patch : SC → CH gives a functor that satisﬁes p erfect and coﬁltered descent. 16 W e thank Maxime Ramzi for suggesting this argument. 50 GEORG LEHNER 6. The main theorem Theorem 6.1. L et X b e a stably c omp act sp ac e, C a dualizable ∞ -c ate gory and F : Cat perf → E a ﬁnitary lo c alizing invariant with values in a dualizable ∞ -c ate gory E . Ther e exists a natur al e quivalenc e F cont (Sh( X , C )) ≃ H • ( X patch , F cont ( C )) , wher e H • ( X patch , F cont ( C )) = Γ( X patch , F cont ( C ) sh ) is the E -value d she af c ohomolo gy of X patch with value in the c onstant she af asso ciate d to F cont ( C ) . Pr o of. W e apply Theorem 5.2 to the functor F C = F cont (Sh( − , C )) : SC op → E . T o do so we need to chec k the three conditions: (1) The statement F cont (Sh( ∅ , C )) ≃ F cont (0) ≃ 0 is clear. (2) Coﬁltered descen t for F C follo ws by using that F is assumed to b e a ﬁnitary inv ariant together with Corollary 2.34. (3) P erfect descent for F C follo ws by stacking the follo wing results: • Corollary 4.3 shows that closed excision holds. • Prop osition 4.4 shows that saturated compact excision holds. • These tw o results together imply elementary compact descent using Prop osition 4.9. • Finally elementary compact descent together with coﬁltered descent implies p erfect descent by Prop osition 4.10. □ The extension of the result of Theorem 6.1 to the case of stably lo cally compact spaces is straightforw ard. W e recall that for a lo cally compact Hausdorﬀ space X , and E some dualizable ∞ -category the c omp actly supp orte d c ohomolo gy H • cs ( X, E ) of X with v alue some E ∈ E is deﬁned as the ﬁb er of the map H • ( X ∗ , E ) → H • (pt , E ) ≃ E , induced by the inclusion of the p oint at inﬁnity in to the one-p oint-compactiﬁcation X ∗ of X . (See e.g. [V ol25, Deﬁnition 5.6.] for more information.) Theorem 6.2. L et X b e a stably lo c al ly c omp act sp ac e, C a dualizable ∞ -c ate gory and F : Cat perf → E a ﬁnitary lo c alizing invariant with values in a dualizable ∞ -c ate gory E . Ther e exists a natur al e quivalenc e F cont (Sh( X , C )) ≃ H • cs ( X patch , F cont ( C )) . Pr o of. The op en-closed decomp osition of X + in to the op en subspace X and its closed complement {∞}  pt giv es the ﬁb er sequence F cont (Sh( X , C )) F cont (Sh( X + , C )) F cont ( C ) . No w, Theorem 6.1 together with Corollary 3.87 imply that the middle term computes as F cont (Sh( X + , C )) ≃ H • (( X patch ) ∗ , F cont ( C )) , and hence we conclude that F cont (Sh( X , C )) ≃ H • cs ( X patch , F cont ( C )) . □ Corollary 6.3. Let X b e stably lo cally compact, let F b e a ﬁnitary lo calizing inv ariant Cat perf → E with E presentable stable, and let C b e a dualizable ∞ -category . Then F cont (Sh( X , C )) ≃ F cont (Cosh( X , C )) . ALGEBRAIC K -THEOR Y OF ST ABL Y COMP ACT SP ACES 51 Pr o of. The statemen t for E b eing dualizable follows directly for stably compact spaces X b y using V erdier dualit y (Theorem 3.59), as w e ha v e the zigzag Sh( X , C ) → Sh( X patch , C ) ← Sh( X ∨ , C ) ≃ Cosh( X , C ) , where all maps are equiv alences on ﬁnitary lo calizing inv arian ts by Theorem 6.1. (The patch top ology of the de Groot dual X ∨ agrees with X patch , see Remark 3.82.) Note that this zig-zag itself is natural in prop er maps of stably compact spaces. Therefore for X only stably lo cally compact, we hav e F cont (Sh( X , C )) ≃ ﬁb( F cont (Sh( X + , C )) → F cont ( C )) ≃ ﬁb( F cont (Cosh( X + , C )) → F cont ( C )) ≃ F cont (Cosh( X , C )) , ﬁnishing the claim. F or E not necessarily dualizable, we can use the fact that F factors through the universal ﬁnitary lo calizing in v ariant U : Cat perf → Mot , using the result by Eﬁmo v that states that Mot is dualizable [Eﬁ25c]. □ Appendix A. The sheaf condition over free distributive la ttices of arbitrar y finite rank While ultimately not needed for the purp oses of this article, it can b e useful to ha v e a generalization of Lemma 4.7 to the case of cubical diagrams of arbitrary dimension. The main argument is exactly the same, but we need to set up a few deﬁnitions. Let us deﬁne the follo wing p osets: • W e deﬁne the spine S = [1] ⨿ [0] [1] to b e the shap e b a c obtained by glueing tw o copies of [1] along the top elemen t. W e denote its elemen ts by a, b, c as lab elled in the diagram. • Let N b e a set with n elemen ts. Denote b y C ( N ) = P ( N ) \ {∅} , a cub e with the b ottom corner remo ved, ordered via inclusion of subsets. Observe that [1] n  P ( N ) is the cone of C ( N ) . Fix an index i ∈ N . Then there exist t wo copies of C ( N \ { i } ) in C ( N ) given as C 0 = { A ⊂ N | A , ∅ and A ∩ { i } = ∅} C 1 = { A ⊂ N | A , ∅ , A , { i } and A ∩ { i } = { i }} . The full subp oset given b y the union of C 0 and C 1 is isomorphic to C ( N \ { i } ) × [1] , and agrees with to C ( N ) \{ i } . • F urthermore, recall that the join P  P ′ of tw o p osets ( P , ≤ P ) and ( P ′ , ≤ P ′ ) is given by the set P ⨿ P ′ equipp ed with the order ≤ ⋆ , whic h is the smallest order which agrees with the orders ≤ P and ≤ P ′ when restricted to P and P ′ resp ectiv ely , and satisﬁes p ≤ ⋆ p ′ for all p ∈ P and p ′ ∈ P ′ . 17 Lemma A.1. Let D b e a category with ﬁnite limits, and let F : [1] n → D be a diagram. Then F is a limit diagram iﬀ the square F ( ∅ ) F ( { i } ) lim C 0 F | C 0 lim C 1 F | C 1 is a pullback square. 17 This agrees with the notion of join of simplicial sets in the sense of Lurie [Lur12, 1.2.8], in the sense that the simplicial join of the nerves of P and P ′ is given by the nerve of the p osetal join P ⋆ P ′ . 52 GEORG LEHNER Pr o of. Consider the inclusion of C ( N ) into S  ( C ( N ) \{ i } ) , by sending { i } to the element b ∈ S . W e no w righ t Kan extend F from C ( N ) along this inclusion. The entries at S are then given by F ( { i } ) lim C 0 F | C 0 lim C 1 F | C 1 Since right Kan extension preserves limits, the limit of the righ t Kan extended diagram agrees with lim C ( N ) F . 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