Semi-cubical tribes

Semi-cubical trib es El Mehdi Cherradi IRIF - CNRS - Université Paris Cité MINES ParisT e ch - Université PSL Abstract W e in tro duce a general notion of J -trib e, and construct the J -tribe of J -frames in a given trib e T , where J a suitable generalized direct category . This construction applies to semi-cubical diagrams for a category of semi-cubes with symmetries and rev ersals. Con ten ts In tro duction 2 1 F rames in a trib e 2 2 Generalized frames in a trib e 7 2.1 The trib e structure . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The enric hmen t . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Semi-cubical trib es 14 1 In tro duction Giv en a fibration category F and an ob ject x of F , the iterated path ob jects P n x of x define a semi-cubical ob ject in F , where the category of semi-cub e is taken to b e the free monoidal category □ ♯ generated b y tw o face maps δ i : I 0 → I 1 with domain the monoidal unit (note that, in particular, w e also get a semi-simplicial ob ject in F ). Ho w ev er, this do es not define a functor F → F □ op ♯ since the path ob ject are not canonical here. In the case of a trib e T , it is not difficult, using the lifting prop ert y of the ano dyne maps P n − 1 x → P n x against the fibrations P n x → P n − 1 x → P n − 1 x , to lift these semi-cubical ob jects to diagrams o v er the opp osite of a category of semi-cub e □ s ♯ that also has symmetries and rev ersals. A natural direc- tion is hence to make these diagrams canonical, by in tro ducing definitions of fibration categories/trib es enric hed in a suitable wa y ov er semi-cubical (or semi-simplicial) sets, and it is reasonable to exp ect the corresp onding categories to b e DK-equiv alent to their unenriched counterpart. In Section 3 of [KS19], the authors inv estigate such a notion of semi- simplicial trib e, based on the work of [Sch13] in the setting of (co)fibration categories. Their aim is to pro vide a “replacement” of the category of trib e T rb that enjo ys the structure of a fibration category , unlike the whole cat- egory of tribes. The key feature enjo yed by semi-simplicial tribes is the existence of a canonical factorization of the diagonals T → T × T through trib es P T of Reedy fibran t homotopical spans in T , defined b y considering the mapping x 7→ x ∆ 1 (where x ∆ 1 comes with t w o pro jections to x ). The goal of this document is to adapt this work to semi-cub es. T o do so, w e first develop a general framew ork for J -trib es and J -frames in a trib e, then rely on a construction in tro duced in [Che26] to carve out a trib e structure for semi-cubical frames in a giv en trib e T based on the usual Reedy trib e structure on a category of inv erse diagrams in T . In the last section, w e apply our result to a suitable category of semi-cub es. 1 F rames in a trib e W e first introduce a general notion J -trib e, in the spirit of [KS19, Defi- nitions 3.1 and 3.2]. Definition 1.1. Consider a small category J . A J -tribe T is a trib e enric hed o v er preshea v es on J and admitting cotensors by finite preshea v es suc h that the follo wing t w o prop erties hold: • If i : K → L is a monomorphism b et ween finite presheav es, and if p : a → b is a fibration in T , then the gap map a L → a K × b K b L is a fibration, and is moreo v er a trivial one whenever p is trivial. 2 • If f : i → j is a morphism in J , and a an ob ject in T , then a H om ( − ,f ) is a w eak equiv alence in T (where H om ( − , f ) : H om ( − , i ) → H om ( − , j ) is the represen table natural transformation represented by f ). • If K is a finite presheaf on J , and if p : a → b is ano dyne, then p K : a K → b K is also ano dyne. W e recall the follo wing imp ortant definition and result: Definition 1.2. A Reedy category J is an elegan t Reedy category when for ev ery monomorphism m : X → Y in Set J op and ev ery ob ject j of J , the relativ e latc hing map L j m is a monomorphism. Examples of elegant Reedy categories include the category ∆ as well as an y direct category . In what follows, we fix an elegant Reedy category J . Lemma 1.1. The c ate gory of pr eshe aves Set J op admits a c ofibr antly gen- er ate d we ak factorization system whose left class is the class of monomor- phisms. Mor e over, this class is gener ate d by the b or der inclusions ∂ J x → J x wher e J x is the r epr esentable pr eshe af r epr esente d by x and ∂ J x its sub- pr eshe af given as the “latching” c olimit (i.e, ∂ J x → J x is the latching map at x of the Y one da emb e dding J → Set J op ). Pr o of. This is Corollary 6.8 (and Example 6.9) in [R V13], by definition of elegan t Reedy categories. Lemma 1.2. A ny pr eshe af K ∈ Set J op c an b e de c omp ose d as a (p ossibly tr ansfinite) c omp osite of inclusions sk 0 K → ... → sk n K... wher e the suc c essive maps ar e obtaine d by pushouts, ⨿ S n ∂ J x ⨿ S n J x sk n − 1 K sk n K ⌜ for S n the set of maps J x → K with deg x = n . Pr o of. This is an instance of Prop osition 6.3 of [R V13]. W e no w supp ose that J is a direct category . 3 Definition 1.3. W e define a J -frame in a trib e T to b e a Reedy fibran t homotopical diagram J op → T where the direct category J is giv en the homotopical structure with all maps w eak equiv alences. W e write J -F r ( T ) for the category of frames in T . Consider a frame F : J op → T and an ob ject z of T . W e write T ( z , F ) for the presheaf defined as the comp osite: J op T Set F H om ( z , − ) F or K a finite presheaf on J , w e write F K for a represen ting ob ject of the functor T op → Set mapping z to H om Set J op ( K, T ( z , F )) , provided such an ob ject exists. The follo wing lemma is an analogue of Prop osition 3.3 in [Sch13]. Lemma 1.3. Consider a trib e T and a J -fr ame F in T . F or any finite pr eshe af K ∈ Set J op , F K exists in T . Mor e over, the functor F 7→ F K takes any (R e e dy) fibr ation (r esp. trivial fibr ation) b etwe en fr ames to a fibr ation (r esp. a trivial fibr ation), and it takes ano dyne maps to ano dyne maps. Pr o of. The existence of F K is tautological when K is the representable J x for x ∈ J (the represen ting ob ject is F ( x ) ). This is also true for the b oundaries ∂ J x , as, by definition of Reedy fibrancy , the matc hing ob ject M n F exists. The stated prop ert y of the functor F 7→ F K follo ws directly b y definition of Reedy (trivial) fibration and ano dyne maps (which are p oin t wise) in this case. The general case follows b y induction on the dimension of K , taking adv an tage of the existence of a skeletal filtration for K (Lemma 1.2). Ex- plicitly , when K is of dimension 0 , i.e., when it has only cells of degree 0 , the result is trivial: the representing ob ject is the terminal one. Assuming that the result holds for all dimensions up to n − 1 , w e can form the follow ing pullbac k square, P Π S n F J x F sk n − 1 K Π S n F ∂ J x ⌜ where S n is the set of cells J x → K of degree n , since the vertical map on the righ t is a fibration by the Reedy fibrancy assumption on F . The ob ject P is then the represen ting ob ject F K w e w ere looking for. 4 If p : F → F ′ is a fibration of J -frames, the induced map b et w een the pullbac ks in the follo wing diagram F K Π S n F J x F ′ K Π S n F ′ J x F sk n − 1 K Π S n F ∂ J x F ′ sk n − 1 K Π S n F ′ ∂ J x ⌜ ⌜ factors as id Π S n F J x × p sk n − 1 K : Π S n F J x × Π S n F ∂ J x F sk n − 1 K → Π S n F J x × Π S n F ∂ J x F ′ sk n − 1 K follo w ed b y Π S n p J x × id F ′ sk n − 1 K : Π S n F J x × Π S n F ∂ J x F ′ sk n − 1 K → Π S n F ′ J x × Π S n F ′ ∂ J x F sk n − 1 K where the first map is a fibration as a base c hange of the fibration p sk n − 1 K (b y our inductiv e hypothesis), and the second map is a base change of a finite pro duct of the fibration F J x → F ′ J x × F ′ ∂ J x F ∂ J x (b y the assumption that p is a Reedy fibration). Observ e that these tw o maps are also weak equiv alences whenever p is one (by induction or b y the characterization of Reedy trivial fibration). The argumen t applies to ano dyne maps w : F → F ′ since pullbacks of ano dyne maps along fibrations are ano dyne in a tribe. Lemma 1.4. F or any monomorphism i : K → L b etwe en finite pr eshe aves on J , and any fibr ation p : F → F ′ b etwe en J -fr ames, the gap map in the diagr am b elow F L F ′ L × F ′ K F K F K F ′ L F ′ K ⌜ is a fibr ation in T , that is mor e over a trivial fibr ation whenever p is so. 5 Pr o of. This is prov ed following the pattern of the pro of for Prop osition 3.5 of [Sc h13]. First note that F K → F ′ K is a fibration by Lemma 1.3, so the pullback indeed exists in the trib e T . W e pro ceed by induction on the n um b er of cells in L that are not in the image of i : K → L . If this num b er is 0 , it means that the monomorphism i is surjective in every dimension, so that i is in fact an isomorphism. Then, the pullbac k is isomorphic to F ′ L , and the gap map F L → F ′ L is a fibration by Lemma 1.3. If the n umber is strictly p ositiv e, we can choose a factorization j ◦ i ′ : K → L ′ → L of i suc h that K → L ′ is a monomorphism and L ′ has one cell less than L (say , J x → L ). W e observe that the gap map F L → F ′ L × F ′ K F K factors as sho wn in the diagram b elow, F J x F L F ′ J x × F ′ ∂ J x F ∂ J x F ′ L × F ′ L ′ F L ′ F L ′ F ′ L × F ′ K F K F ′ L ′ × F ′ K F K F K F ′ L F ′ L ′ F ′ K ⌜ ⌜ ⌜ ⌜ where F J x → F ′ J x × F ′ ∂ J x F ∂ J x is a fibration b y Reedy fibrancy of F → F ′ , and F L ′ → F ′ L ′ × F ′ K F K is a fibration b y our inductiv e hypothesis. If moreov er p is a trivial fibration, b oth of the previous t w o maps are trivial fibrations (by characterization of Reedy trivial fibrations and by the inductiv e h yp othesis resp ectiv ely), so we can conclude. In the rest of this section, we assume that J has a monoidal structure (and w e write ⊗ : J × J → J for the monoidal pro duct). Define the geometric pro duct of tw o presheav es K and L on J by the co end form ula b elo w, whic h is just the Day con v olution arising from the monoidal structure on J : K ⊗ L := Z i,j ∈ J K i × L j × J i ⊗ j Theorem 1.5. The c ate gory J -F r ( T ) of fr ames in T is enriche d over Set J op and admits c otensors by finite pr eshe aves on J . Mor e over, the c otensors satisfy the r e quir e d pr op erties for J -F r ( T ) to b e a J -trib e. 6 Pr o of. As in the semi-simplicial case ([KS19]), w e define the “cotensor” K ▷ F of a frame F b y a presheaf K ∈ Set J op b y the form ula ( K ▷ F ) x := F J x ⊗ K and define the enric hmen t by H om F r ( T ) ( F , F ′ ) x := H om ( F , J x ▷ F ′ ) With this definition, the pro of follo ws by analogy with the semi-simplicial case (Theorem 3.7 in [KS19]. Explicitly , the prop erties exp ected from the cotensor hold by Lemma 1.3n Lemma 1.4 and b y the fact that a frame is a homotopical diagram. 2 Generalized frames in a trib e In this section, R is a generalized direct category (in particular, an EZ- category) satisfying the follo wing condition, introduced in [Che26]: • There exists a functor c : R 0 → R from a (strict) direct category , suc h that ev ery arrow f : a → b in R lifts to an arrow k : x → y in R 0 up to isomorphisms, i.e., such that there are isomorphisms w : a ≃ c ( x ) and w ′ : b ≃ c ( y ) fitting in a commutativ e square: a b c ( x ) c ( y ) f w w ′ c ( k ) Recall the follo wing definition ([Che26, Definition 3.1]): Definition 2.1. W rite F for the free category comonad on Cat , then form the pushout as in the diagram b elo w: F ( R 0 ) F ( R ) R 0 F ≃ ( R ) R ⌜ p 0 7 Finally , write D R for the full subcategory of the twisted arrow category Tw ( F ≃ ( R )) spanned by ob jects consisting of an arrow x → y of the form x → z → y where x → z comes from a map in R 0 , and z → y corresp onds to a “free” isomorphism in R or is an identit y arro w (that is, an arrow obtained b y taking the image under F ( R ) → F ≃ ( R ) of a path of length one in R whose underlying arrow is an isomorphism). It comes with a pro jection p : D R → R . In [Che26], w e established the following: Lemma 2.1. D R inherits a dir e ct c ate gory structur e. Lemma 2.2. The c anonic al functor p : D R → R is absolutely dense (i.e., the pr e c omp osition functor p ∗ : Set R → Set D R is ful ly faithful). 2.1 The trib e structure In this subsection, w e fix a trib e T . Definition 2.2. W e define the class of p -fibrations as the class of morphisms m : F → F ′ in T R op suc h that p ∗ m is a Reedy fibration in T D op R . Definition 2.3. W e define a R -frame in the trib e T to b e a p -fibrant homo- topical diagram R op → T where the category R is giv en the homotopical structure with all maps weak equiv alences. W e write R -F r ( T ) for the full sub category of T R op spanned by the R -frames. W e define the fibrations b etw een t w o suc h diagrams to be the p -fibrations. W e write ( D R ) ≤ n for the full sub category of D R spanned b y the arro ws whose co domain’s degree is at most n . Corollary 2.3. Supp ose that ( D R ) ≤ n has a c ontr actible nerve for every natur al numb er n . Then, the functor p ∗ : T D op R f → T R op maps R e e dy fibr ant diagr ams that ar e homotopic al ly c onstant to homotopi- c al ly c onstant diagr ams. Pr o of. Let X b e a homotopically constant Reedy fibrant diagram of shap e D op R . It is enough to prov e that the restriction of p ∗ X to the full sub category R op ≤ n of R op spanned b y the ob jects of degree at most n is homotopically constan t, for ev ery natural n um b er n . Fix such a natural num b er n . W e ha v e the follo wing pullbac k square, that is also an exact square (as a pullbac k along a Grothendiec k fibration): 8 ( D R ) ≤ n D R R ≤ n R p n ⌜ p The restriction X n of X to ( D R ) ≤ n is Reedy fibran t b ecause X is Reedy fibran t and the inclusion ( D R ) ≤ n → D R is a discrete fibration, and it is also a finite diagram. Hence, it admits a limit l in T . Moreo v er, the comp onen ts of the cone ∆ ( D R ) op ≤ n ( l ) → X n are weak equiv alences since ( D R ) ≤ n has a con tractible nerv e b y assumption. But ( p n ) ∗ maps p oin t- wise w eak equiv alences to point wise w eak equiv alences, just lik e p ∗ , and maps the constan t diagram ∆ ( D R ) op ≤ n ( l ) = p ∗ n ∆ R op ≤ n ( l ) to the constant dia- gram ∆ R op ≤ n ( l ) = ( p n ) ∗ p ∗ n ∆ R op ≤ n ( l ) , so we can conclude. Theorem 2.4. R -F r ( T ) enjoys the structur e of a trib e. Pr o of. W e already kno w from [Che26, Theorem 2.3] that the sub category of p -fibran t diagrams in T R op admits a trib e structure with the fibrations the p -fibrations. W e still need to c hec k that this restricts to a trib e structure on R -F r ( T ) . The class of p -fibrations is stable under pullback (using Corollary 2.3) and comp osition. Moreo ver, p oin twise ano dyne maps are stable under pullbac k along p -fibration b ecause the p oin t wise ano dyne maps in T D op R f are the ano- dyne maps. Ov erall, this means that R -F r ( T ) also enjoys a trib e structure, with the fibrations the p -fibrations. Prop osition 2.5. If T is a π -trib e, then so is R -F r ( T ) . Pr o of. The corresp onding result for general diagram has b een established in [Che26, Prop osition 2.4]. W e now chec k that the restriction to homotopical diagrams still enjoys this structure. Given tw o p -fibrations f : A → B and g : B → C , we claim that p ∗ Π p ∗ g ( p ∗ f ) , which defines an internal pro duct Π g f of f along g , is homotopically constant: this is b ecause a diagram X in T R op is homotopically constant if and only if p ∗ X is homotopically constan t, since p is surjective on ob jects. But Π p ∗ g ( p ∗ f ) is homotopically constant since A , B and C are, so we can conclude by Corollary 2.3. In the following, w e assume that R 0 and R are endow ed with the struc- tures of a monoidal categories, and that the functor R 0 → R is (strong) monoidal. Definition 2.4. W e equip D R with a monoidal pro duct as follows. Given t w o ob jects of D R represen ted b y arrows x → y and x ′ → y ′ in R p ossibly 9 follo w ed by “free” isomorphisms w : y → z and w ′ : y ′ → z ′ , w e define their monoidal pro duct as the arro w x ⊗ x ′ → y ⊗ y ′ in R 0 follo w ed by the automorphism w ⊗ w ′ (unless b oth lists of isomorphisms are empt y , then w e tak e the empty list). The axioms for a monoidal category then follo w from those for the monoidal structure on R 0 and R . Note that the functor p is monoidal with this definition of monoidal structure on D R . Lemma 2.6. The functor p ! : Set D op R → Set ( R ) op is c anonic al ly a (str ong) monoidal functor. Pr o of. In terms of profunctors, preshea ves K and L on D R yield a profunctor K ⊗ L : D op R × D op R → ∗ (corresp onding to the external tensor product). The Day conv olution pro cess then corresp onds to precomp osition with the represen table profunctor D op R ( ⊗ , ∗ ) : D op R → D op R × D op R . In particular, w e ha v e that p ! ( K ⊗ L ) corresp onds to the top composite in the diagram b elo w, ( R ) op D op R D op R × D op R ∗ ( R ) op × ( R ) op H om ( p, ∗ ) H om ( ⊗ , ∗ ) H om ( ⊗ , ∗ ) K ⊗ L H om ( p × p, ∗ ) whic h, b y functorialit y , coincides with p ! ( K ) ⊗ p ! ( L ) . As a corollary , the adjunction p ! ⊢ p ∗ is a monoidal adjunction, where p ∗ inherits the structure of a lax monoidal functor. Lemma 2.7. L et T b e a R -trib e. Then T is also c anonic al ly enriche d over Set D op R , making it a D R -trib e. Pr o of. By Lemma 2.6, the left Kan extension functor p ! : Set D op R → Set R op is strong monoidal with resp ect to the monoidal structure obtained b y Day con v olution, hence the precomp osition functor p ∗ is lax monoidal. W e rely on the change-of-enric hment provided by p ∗ (i.e, we define the hom-ob ject of T to b e the image under p ∗ of the hom-ob ject for the original enric hmen t o ver Set ( R ) op ). Supp ose K is a finite presheaf ov er D R . It follows from the point wise for computing the left Kan extension functor p ! that p ! K is a finite presheaf o v er 10 R , and the follo wing natural isomorphisms prov e the existence of cotensors b y K : H om Set D op R ( K, p ∗ H om T ( x, y )) ≃ H om Set ( R ) op ( p ! K, H om T ( x, y )) ≃ H om T ( x, y p ! K ) Similarly , observe that an y monomorphism K → L b et w een finite presheav es o v er D R is mapp ed b y p ! to a monomorphism betw een finite preshea ves ov er R . Indeed, K → K is a relativ e cell complex built from b oundary of rep- resen table ∂ D x R → D x R b y Lemma 1.1, and these b oundaries inclusions are mapp ed to b oundary inclusions ∂ R px → R px , whic h are still monomorphisms since R is an EZ-category . Therefore, it follows that the cotensor by such morphisms is a fibration in T . Finally , since p ! maps representable presheav es to representable ones, the cotensors by any represen table monomorphism D x R → D y R is alwa ys a w eak equiv alence (hence a trivial fibration). This concludes the pro of that T is canonically a Set D op R -trib e. Corollary 2.8. If T is a R -trib e, then ther e is a “c anonic al fr ame” trib e morphism c : T → R -F r ( T ) mapping x ∈ T to r 7→ x R r . Pr o of. Observ e that c factors through the “canonical frame” functor c ′ : T → T D op R resulting from the D R -trib e structure on T (by Lemma 2.7). Precisely , w e ha v e a comm utative triangle T R -F r ( T ) D R -F r ( T ) c c ′ p ∗ and we conclude from the fact that c ′ is trib e morphism that c is also a trib e morphism. Lemma 2.9. Supp ose that ( D R ) ≤ n has a c ontr actible nerve for every natur al numb er n . Also assume that D R has only one obje ct, denote d by 0 , with c o domain an obje ct of R of de gr e e 0 (this is true if R has only one obje ct of de gr e e 0 admitting no automorphism). Consider a D R -fr ame F in T and a morphism x → F 0 . Then, the latter c an b e extende d to a c one ∆ x → F . 11 Pr o of. W e will inductiv ely extend the cone to the full sub categories D R, ≤ n of D R , whic h are finite categories with con tractible nerv es. F or n = 0 , D R, ≤ n has only one element 0 by assumption, and we already hav e a cone as part of the starting input data. No w, consider the functor q n from D R, ≤ n +1 to the arro w category → mapping D R, ≤ n to the source and mapping the rest ob ject in D R, ≤ n +1 but not in D R, ≤ n to the target. W e hav e the following comma square (which is also a pullbac k), D R, ≤ n D R, ≤ n +1 ∗ → q n s so w e can conclude, b ecause F is fibrant (hence so is its restriction to D R, ≤ n +1 ), that the diagram ( q n ) op ∗ F is a fibrant diagram (i.e., a fibration) with source the limit of the restriction of F to D R, ≤ n +1 , and with target the limit of its restriction to D R, ≤ n . This arro w is not only a fibration, but moreo v er also a weak equiv alence since b oth corresp onding diagrams are ho- motopically constan t diagrams of con tractible shap e. As a trivial fibration in a trib e, it admits a section b y [Jo y17, Corollary 3.4.7]. This section allows us to extend the cone from x at stage n to stage n + 1 , concluding the pro of. The follo wing result adapts Theorem 3.10 of [Sch13]: Prop osition 2.10. Under the assumptions of L emma 2.9, and assuming that R has only one obje ct 0 of de gr e e 0 . The functor ev 0 : R -F r ( T ) T → T mapping a fr ame F to its value at 0 is DK-e quivalenc e. Pr o of. Since F r ( T ) and T are, in particular, fibration categories, w e can use the characterization of DK-equiv alences b et w een fibration categories by means of the approximation prop ert y in tro duced b y Cisinski in [Cis10]. The first condition ( AP 1) is easily c hec k ed. F or the second, consider an arro w x → F 0 = ev 0 F , where F is a frame. W e use Lemma 2.9 to pro vide a cone ∆ x → p ∗ F . W e can now take a factorization of ∆ x → p ∗ F as weak equiv a- lence follo w ed by a Reedy fibration F ′ → p ∗ F . This provides a diagram x ev 0 F x ev 0 ( p ∗ F ′ ) id x ∼ 12 where the map x → ev 0 ( p ∗ F ′ ) is a weak equiv alence b ecause it is obtained by restriction: in the corresp onding computation for p oin twise Kan extensions, follo wing from the exact square b elow, p ↓ 0 D R ∗ R p 0 the category p ↓ 0 is the terminal category . This pro v es that ( AP 2) also holds. R emark 2.1 . The previous result states that a sub category of homotopi- cal diagrams of shap e R in the ( ∞ , 1) -category Ho ∞ ( T ) presented by T is equiv alent to Ho ∞ ( T ) . In the case of subcategory of all hom otopical diagrams, this should reasonably hold (for arbitrary T ) only when R has a con tractible nerv e. In particular, R -F r ( T ) need not present the ( ∞ , 1) - category Ho ∞ ( T ) R . This is the case for semi-cubical frames, discussed in the next section, since the category of semi-cub es w e consider do es not ha ve a con tractible nerv e. 2.2 The enric hment F or K a finite presheaf on R , and F an R -frame, w e also write F K for a represen ting ob ject of the functor T op → Set mapping z to H om Set R op ( K, T ( z , F )) , pro vided suc h an ob ject exists. Lemma 2.11. Consider a trib e T and an R -fr ame F in T . F or any finite pr eshe af K ∈ Set R op , F K exists in T . Mor e over, the functor F 7→ F K takes any p -fibr ation (r esp. trivial p -fibr ation) b etwe en fr ames to a fibr ation (r esp. a trivial fibr ation), and it takes ano dyne maps to ano dyne maps. Pr o of. W e ha v e a natural isomorphism H om Set R op ( K, T ( z , F )) ≃ H om Set R op ( p ∗ K, p ∗ T ( z , F )) so that the existence of the representing ob ject, whic h coincides with ( p ∗ F ) p ∗ K , follo ws from Lemma 1.3, as w ell as the stated prop erties (observing that p ∗ preserv es monomorphisms). Lemma 2.12. F or any monomorphism i : K → L b etwe en finite pr eshe aves on R , and any fibr ation q : F → F ′ b etwe en R -fr ames, the gap map in the diagr am b elow 13 F L F ′ L × F ′ K F K F K F ′ L F ′ K ⌜ is a fibr ation in T , that is mor e over a trivial fibr ation whenever q is so. Pr o of. Just like in the pro of of Lemma 2.11, this follows from the corresp ond- ing prop ert y for p ∗ F → p ∗ F ′ since p ∗ maps monomorphisms to monomor- phisms, whic h is pro vided by Lemma 1.4. Theorem 2.13. The c ate gory R -F r ( T ) of R -fr ames in T is enriche d over Set R op and admits c otensors by finite c ofibr ant pr eshe aves on R . Mor e over, the c otensors satisfy the r e quir e d pr op erties for R -F r ( T ) to b e an R -trib e. Pr o of. The enrichmen t is constructed as in Theorem 1.5: we define the “cotensor” K ▷ F of a frame F by a presheaf K ∈ Set R op b y the form ula ( K ▷ F ) x := F R x ⊗ K and define the enric hmen t by H om F r ( T ) ( F , F ′ ) x := H om ( F , R x ▷ F ′ ) Using this definition, the prop erties exp ected from the cotensor hold by Lemma 2.12, Lemma 2.11 and b y the fact that a frame is a homotopical diagram. This makes R -F r ( T ) an R -trib e, with the trib e structure pro vided by Theorem 2.4. 3 Semi-cubical trib es Consider the free monoidal category ( □ ♯ , ⊗ , I 0 ) generated by tw o face maps δ 0 , δ 1 : I 0 → I 1 with domain the monoidal unit. This is a subcat- egory of the cub e category I introduced in section 4 of [GM03], excluding degeneracies. Th us, the ob jects of □ ♯ are of the form I n := I 1 ⊗ ... ⊗ I 1 and can therefore b e identified with the natural num b ers (we may write [ n ] by analogy to the semi-simplex category). The morphisms are generated from the tw o face maps δ 0 and δ 1 (and the identit y maps) under the monoidal pro duct. W e will b e interested in a symmetric v ersion: the free symmetric 14 monoidal category ( □ s ♯ , ⊗ , I 0 ) generated b y tw o face maps δ 0 , δ 1 : I 0 → I 1 together with an in v olution of the interv al (a reversal) r : I 1 → I 1 . There is a monoidal functor r : ∆ a,♯ → □ s ♯ from the augmen ted semi- simplex category to the category of symmetric cub es we consider, defined by mapping [ n ] ∈ ∆ a,♯ to I n +1 , and the unique map [ − 1] → [0] to δ 0 . Thanks to the presence of reversals, this functor satisfies the condition assumed on R in Section 2, as prov ed in Lemma 3.1 b elo w. Therefore, we may apply the results from Section 2 after c hec king that D □ s ♯ has a con tractible nerv e. Lemma 3.1. The functor r : ∆ a,♯ → □ s ♯ induc es a functor r → : ∆ → a,♯ → ( □ s ♯ ) → b etwe en the arr ow c ate gories that is essential ly surje ctive on obje cts. Pr o of. First, observe every map f : I n → I m in □ s ♯ is essen tially generated b y the identit y map id I 0 : I 0 → I 0 and the face maps δ i : I 0 → I 1 in the sense that f factors as w ◦ f ′ , where w : I m → I m is an automorphism, and f ′ : I n → I m is a tensor pro duct of id I 0 , δ 0 and δ 1 . Recall that, b y definition, □ s ♯ is generated under the tensor product by id I 0 , δ i , the transp osition I 2 → I 2 and the reversal I 1 → I 1 . The observ ation can then b e prov ed inductiv ely on m . If m = 0 , then n = 0 and f is an iden tity arro w, so the result is obvious. Otherwise, either n < m , and f factors through I m − 1 , or n = m , and f is an isomorphism. In the latter case, the result is ob vious, so w e are only left with the former: b y assumption w e hav e a diagram I n I m − 1 I m I m − 1 f ′ 0 f 0 f ′ 1 ≃ f 1 where f ′ 0 is a tensor pro duct of the form w e wan t to obtain, by our inductiv e h yp othesis, and where f 1 (and f ′ 1 ) must b e a tensor of isomorphisms and exactly one face δ i . The face map can b e comp osed with an identit y arrow I 0 → I 0 in the decomp osition of f ′ 0 to pro vide an arrow f ′ : I n → I m of the form we wan t, and the remaining isomorphisms in the decomp osition of f ′ 1 define, together with the identit y arro w I 1 → I 1 an automorphism w of I m suc h that w ◦ f ′ is equal to f . This concludes the pro of of our observ ation. Since r is monoidal, and leveraging our previous observ ation, it will b e enough to chec k that the generators id I 0 , δ 0 and δ 1 are in the essential image of r → in order to conclude that every arrow of □ s ♯ is. This is obvious for id I 0 and δ 0 (whic h are in the exact image of r → ), and it is also the case of δ 1 whic h can b e written v ◦ δ 0 , where v : I 1 → I 1 is the rev ersal. 15 Lemma 3.2. The c ate gories D □ s ♯ has a c ontr actible nerve. Mor e over, the ful l sub c ate gories of D □ s ♯ sp anne d by obje cts whose c o domain is at most n also have c ontr actible nerve. Pr o of. W e define a functor d : D □ s ♯ → D □ s ♯ mapping an arro w [ m ] → [ n ] → [ n ] to [ − 1] → [ n ] → [ n ] (precomp osing it with [ − 1] → [ m ] ). There is a natural transformation id D □ s ♯ → d whose comp onents are given b y the t wisted squares b elo w: [ m ] [ − 1] [ n ] [ n ] W rite c : D □ s ♯ → D □ s ♯ for the constan t functor with v alue the arrow [ − 1] → [ − 1] . There is a natural transformation c → d defined by the squares of the follo wing form: [ − 1] [ − 1] [ n ] [ − 1] Ov erall, this pro vides a zig-zag b etw een the iden tit y functor on D □ s ♯ and a constan t functor, so that the nerve of D □ s ♯ is con tractible. F or the second assertion, it is enough to observ e that the functors and natural transformations constructed ab o v e restrict suitably (i.e., they give rise to functors betw een the considered full sub categories and corresp onding natural transformations). W rite scF r ( T ) for the category □ s ♯ -F r ( T ) . T o sum up, w e obtain the follo wing result b y sp ecializing those of Section 2: Theorem 3.3. The c ate gory scF r ( T ) of semi-cubic al fr ames in T enjoys the structur e of a □ s ♯ -trib e, and the evaluation functor ev 0 : scF r ( T ) → T establishes a DK-e quivalenc e. Mor e over, if T is a π -trib e, then so is scF r ( T ) . 16 References [Che26] El Mehdi Cherradi. “ Generalized Reedy diagrams in trib es”. In: arXiv pr eprint arXiv:2602.17355 (2026). [Cis10] Denis-Charles Cisinski. “ Catégories dériv ables”. In: Bul letin de la so ciété mathématique de F r anc e 138.3 (2010), pp. 317–393. [GM03] Marco Grandis and Luca Mauri. “ Cubical sets and their site.” In: The ory and Applic ations of Cate gories [ele ctr onic only] 11 (2003), pp. 185–211. [Jo y17] André Jo y al. “ Notes on clans and trib es”. In: arXiv pr eprint arXiv:1710.10238 (2017). [KS19] Krzysztof Kapulkin and Karol Szumiło. “ Internal languages of finitely complete ( ∞ , 1) -categories”. 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