Notions of Lawvere theory

Notions of La wv ere theory ∗ Stephen Lac k Sc ho ol of Computing and Mathematics Univ ersit y o f W estern Sydney Lo c k ed Bag 1797 P enrith South DC NSW 1797 Australia s.lack@uws. edu.au Ji ˇ r ´ ı Rosic k´ y Departmen t of Mathematics and Statistics Masaryk Univ ersit y Kotl´ a ˇ rsk´ a 2 60000 Brno Czec h Republic rosicky@mat h.muni.cz Octob er 29, 2018 Abstract Categoric a l universal alg e br a can be developed either using Lawv ere theor ies (single- sorted ﬁnite product theories) or using monads, and the category of Lawv ere theories is equiv alent to the categor y of ﬁnitary monads on Set . W e show ho w this equiv a lence, a nd the ba sic results o f universal algebra, can b e g eneralized in three wa ys: r eplacing Set by another ca tegory , working in an enric hed setting, and by working with another c la ss of limits than ﬁnite pro ducts. Classical universal alg ebra b egins with structures of the follo wing t yp e: a set X equipp ed with op erations X n → X for v arious natural num b ers n , su b ject to equations b etw een in duced op erations. F or example the structure of group can b e encoded usin g a binary op eration m : X 2 → X (m ultiplication), a unary op eration ( ) − 1 : X → X (in v erse) and a n ullary one i : 1 → X (un it). Then there is an equation enco ding asso ciativit y , t wo equations for th e t w o unit la ws, and tw o equations for the inv erses. There are t w o main wa ys for treating suc h structures categorical ly . In eac h case, one ends up giving all the op erations generated, in a suitable sense, by a presen tation as in the pr evious para- graph. Thus the structure of group (as opp osed to some particular group ) b ecomes a mathematical ob j ect in its o wn right. On the one h an d , for eac h such t yp e of structure th ere is a small category T with ﬁ nite pro du cts, suc h that a particular instance of th e str u cture is a ﬁnite-pro duct-pr eservin g fu nctor from T to Set . T he ob jects of T will b e exactl y the p o wers T n of a ﬁxed ob ject T . Su c h a cat egory is calle d ∗ Both authors gratefully acknow ledge t he sup port of th e Au stralia n Researc h Council; the second-named author also gratefully ackno wledges the supp ort of the Ministry of Education of the Czech R epublic by the pro ject MSM 0021622 4409 and of the Grant Agency of th e Czech Rep ublic by the pro ject 201/06/ 0664. 1 a L awver e the ory . In this case, the op erations in the structure are seen as the morp hism in the theory T . On the other hand, for eac h suc h t yp e of structure there is a monad T on Set , such that a particular ins tance of the structure is an algebra f or the monad. The basic idea this time is to describ e the stru ctur e in qu estion by sa ying w hat th e f ree algebras are. Since we ha v e b een considering only ﬁnitary op erations (op erations X n → X where n is ﬁ nite) the resulting monad T will also b e ﬁnitary . Th is means that it is determined by its b eha viour on ﬁnite sets; formally , this means that the end ofunctor T p reserv es ﬁltered colimits. There is a v ery well develo p ed theory of Lawv ere theories and of ﬁnitary monads, and of the equiv alence b et w een the t w o notions. In this pap er w e are interested in wh at happ ens when we mo v e b eyond the con text of str uctures b orne b y a set, to consider structures b orne b y other ob jects. La wv ere theories are single-sorted theories; m uc h of the time we sh all sp end on th e many-so rted case, or rather th e “unsorted” case, wh ere a theory consists of an arbitrary sm all categ ory with ﬁnite pro ducts, and a mo del is j ust a ﬁnite-pro duct-preserving functor. W e shall consider thr ee diﬀeren t wa ys in whic h th e classical setting could b e adapted; th e three w a ys are to some exten t in dep enden t of eac h other. Replace the ba se category Set b y some other suitable cate gory K W e could consider structures b orne n ot by sets but b y ob jects of other categories. F or example we could tak e K to b e the category Gph of (directed) graph s, and consider structur es b orne by graphs. An ob vious example is a category: this is a graph equipp ed with a comp osition la w and iden tities. F rom the p oint of view of monads, this change is straigh tforw ard enough: one can simply con- sider monads on Gph rather than monads on Set . As for theories, there is no problem co nsid er in g mo dels of a L a wvere theory T in Gph , one s im p ly tak es ﬁn ite-prod uct-preserving functors from T to Gph rather th an from T to Set . Bu t often this is to o restrictive ; for example it would not include the case of categ ories. One needs to allo w more general arities than natural n umb ers; in fact it is (certain) ob jects of Gph which will serv e as arities. Deal not wit h ordinary categories but w ith categories enric hed ov er some suitable monoidal category V A monoidal category can clea rly b e thought of as a categ ory with extra structure, of an algebraic sort. When we come to think of the morp h isms b et we en suc h structures, this naiv e algebraic p oin t of view w ou ld lead us to tak e as morphisms the strict monoidal functors (those which preserve the unit and tensor pro du ct in the strict sense of equalit y: F ( X ⊗ Y ) = F X ⊗ F Y ). Although suc h strict monoidal fu nctors hav e an imp ortan t role to pla y , muc h more imp ortan t are v arious “w eak” notions, wh ere the monoidal structur e is preserve d up to coheren t isomorphism, or p erh ap s only u p to a coheren t non-inv ertible comparison map. These w eak er notions of homomorphism can b e reco v ered by working not o ver the category of small categories and functors, bu t o v er the 2-ca tegory of small categories, functors, and natural transformations. In other wo rd s , w e w ork with categories enric hed in Cat . More generally one could consider man y other monoidal categ ories V , suc h as ab elian group s , c hain complexes, simp licial sets, and many m ore: see [11]. When we are w orking o v er V , the “default” c hoice of K b ecomes V rather th an Set , bu t man y other V -categories ma y also b e suitable. 2 Replace ﬁnite pro ducts b y some suitable class Φ of limits. This is probably the most imp ortan t asp ect for this p ap er. Some stru ctures cann ot b e deﬁn ed w ith ﬁ nite pro ducts alone. Once again the structure of catego ries is a goo d example. An in ternal category consists of an ob ject-of-ob jects C 0 , an ob ject-of- morphisms C 1 , with domain and co domain maps d, c : C 1 → C 0 , together with iden tities i : C 0 → C 1 and comp osition m : C 2 → C 1 satisfying v arious equations, where C 2 is ob ject-of-comp osable-pairs, giv en by the pullbac k C 2 / /   C 1 c   C 1 d / / C 0 . So to deﬁne internal cate gories one needs a class Φ of limits whic h cont ains pullbac ks. Once again, the limits in Φ should b e w eigh ted limits in the V -enric hed sense. Our m ain new example of su c h a class is where Φ consists of what we call stron gly ﬁnite limits. This is a class int ermediate b et ween ﬁnite pro ducts and ﬁnite limits. An ob ject X ∈ V is str ongly ﬁnite if the in ternal hom f unctor [ X , − ] : V → V p r eserv es sifted colimits (see [1 ] or Section 4.2 b elo w). Th e strongly ﬁnite limits are those whic h can b e constructed u sing ﬁn ite pro du cts and p o w ers (cotensors) by s trongly ﬁ nite ob jects. Th ese corresp ond to monads whic h pr eserv e sif ted colimits. As observ ed for example in [21], ev ery monad on a co complete sym m etric monoidal closed catego ry arising from an op erad pr eserves ﬁltered colimits and reﬂexiv e co equalizers; th us, us in g Prop osition 3.2, it preserves sifted colimits. It is wo rth observing that as w ell as b eing the free completion of the categ ory Set f of ﬁnite sets under ﬁltered colimits, Set is also the free completio n of Set f under sifted colimits. Thus an endofunctor of Set pr eserves ﬁltered colimits if and only if it preserves sifted colimits. Thus the w ell-kno wn equiv alence b et w een La wv ere theories and ﬁnitary monads on Set could also b e seen as an equiv alence b et ween L a wvere theories and sifted-colimit-preserving monads on Set . This is a v ery sp ecial prop erty of Se t ; for other b ase ca tegories, one therefore p oten tially has (at lea st) t w o w a ys to generaliz e the equiv alence. So altoge ther we ha v e three changes (A) replace the base category Set b y some other suitable category K (B) deal not with ordinary catego ries but with categories enriched o v er some suitable m onoidal catego ry V (C) r eplace ﬁnite pro ducts b y some suitable class Φ of limits and these app ear to b e indep endent to some exten t, apart f rom the obvio us restrictions th at K should b e a V -category and the limits in Φ should b e weigh ted V -limits. But the real r elati onship b et w een these conditions app ears when w e try to say what “suitable” should mean in eac h case; doing this will b e one of the main aims of the pap er. W e’ll see that a k ey asp ect is that it giv es a sort of d ecomposition or factorizati on of all limits in to t wo p arts. In fact it’s easier to think in terms of colimits. F or example if Φ consists of the ﬁnite limits, then we hav e the decomp osition of arbitrary colimits as ﬁltered colimit s of ﬁ nite ones. 3 Then again, if Φ consists of the ﬁ n ite pro ducts w e h a ve a corresp ond in g decomposition in to sifted colimits of ﬁnite copro du cts (see Section 3 for more ab out sifted colimits). There is a trade-oﬀ b et w een d iﬀeren t lev els of generalit y of theory . As usual, the more general the notion of theory , the more general the catego ries of mo dels that can b e describ ed, but the less that ca n b e pr o ved ab out them. Ther e are also more su b tle eﬀects. F or example, consider the case V = Set of unen r ic h ed categories and compare ﬁnite limit th eories with ﬁnite pro du ct theories. Finite limit theories are of course more general. On the other h and, in the ﬁrst instance one needs a category with ﬁn ite limits as the category K in which mo dels are tak en; whereas for a ﬁn ite pro duct theory , K n eed only h a ve ﬁ nite p ro d ucts. As one goes further , how eve r, it turns out that for the strongest results, K should itself b e the category of mo dels for a theory of the giv en t yp e, and this is a stronger condition on K wh en we use ﬁnite pro ducts than it is when w e use ﬁnite limits. The main results of u niv ersal algebra that we should lik e to obtain in our general setting are: Φ -algebraic functors ha v e left adjoin ts. If G : S → T is morphism of theories, then there is an in duced map G ∗ : Mo d ( T ) → Mo d ( S ), and s uc h a G ∗ w e call Φ -alge braic (or just alge braic if Φ is understo o d). Our ﬁr st basic fact is that such a G ∗ has a left adjoin t; or rather, w e giv e an explicit construction of a left adjoin t in terms of colimits in the base category K — the existence of the left adjoint is kno wn for general r easons. This includes the existence (and construction) of free algebras for single-sorted theories. The reﬂectiveness of mo dels. F or an y theory T , the categ ory Mo d ( T , K ) of algebras in K is reﬂectiv e in [ T , K ]. Th is means that Mo d ( T , K ) will b e complete and cocomplete pro vided that K is so, as w e shall usu ally supp ose to b e the case. But w e shall do more than just p ro ve the reﬂectivit y , we shall construct a r eﬂection in terms of colimits in K , and so obtain a d escription of colimits in Mo d ( T , K ) in terms of colimits in K . In the ﬁ nal c hapter of [11], the reﬂectiv en ess was p ro v ed und er extremely general conditions, but w ith ou t an explicit construction. Ou r setting w ill b e muc h more restrictiv e, but will allo w an explicit construction (as explicit as is our kno wledge of colimits in K ). Although w e can obtain an explicit construction of all colimits in Mo d ( T , K ), it is particularly simple f or the class of colimits whic h commute in K with Φ-limits, since these colimits are formed in Mo d ( T , K ) as in [ T , K ], and so no reﬂ ection is required. Once again w e see the trade-oﬀ: the smaller the class of limits in Φ, the clo ser Mod ( T , K ) is to [ T , K ], and so the greater our knowledge of colimits in Mo d ( T , K ). F or example, if we tak e K = V = Set and compare the case where Φ consists of all ﬁnite limits with that w here it consists only of ﬁn ite pro ducts, in th e ﬁrst case the inclusion Mo d ( T , Set ) → [ T , Set ] pr eserv es ﬁltered colimits, wh ile in the second case it also pr eserv es reﬂexive co equalizers. It is p erhaps w orth noting that the reﬂectivit y of m od els is a sp ecial case of the existence of left adjoin ts to algebraic fun ctors. Let F T b e obtained b y freely adding Φ-limits to T . Since T already has Φ-limits, the canonical inclusion T → F T has a righ t adjoint R , and now the induced algebraic functor R ∗ : Mod ( T , K ) → Mo d ( FT , K ) ma y b e iden tiﬁed with the inclusion Mo d ( T , K ) → [ T , K ], and so the left adjoin t to R ∗ giv es the desir ed reﬂection. On the other hand, we also ha v e a s ort of con v erse: the left adjoin t to G ∗ : Mo d ( T ) → Mo d ( S ) can b e obtained by ﬁ rst taking th e left Kan extension along G and then r eﬂecting in to the sub category of mo dels. 4 The corresp ondence b etw een (single-sorted) theories and monads. F or su itable V and Φ, we consider the class of all colimits w hic h comm ute in V with Φ-limits; these will b e called Φ -ﬂat . W e shall say that a V -fu n ctor is Φ -ac c essible if it p reserv es Φ-ﬂat colimits, and that a V -monad is Φ -acc essible if its und erlying V -functor is so. F or suitable V -categories K , we shall describ e a notion of K -b ase d Φ - the ory , or Φ -the ory in K , and prov e that the category of these is equiv alen t to th e cate gory of Φ-accessible V -monads on K , and further m ore that the algebras for a Φ-accessible monad are the same as the mo dels for the corresp onding Φ-theory . This generalizes the classical equ iv alence b et w een La wve re theories and ﬁnitary monads on Set , and is one of the main results of the pap er. Outline of pap er W e b egin in S ectio n 1 with a review of the en r ic h ed category theory that will b e needed in the pap er. In S ection 2 w e describ e the basic assumptions we shall m ake connecting our monoidal catego ry V , our base V -category K , and the class of limits Φ to b e considered. Section 3 is largely a review of v arious id eas relating to sifted colimits: these are the colimits whic h comm ute in Set with ﬁnite p ro ducts. In Section 4 w e describ e the v arious p ossible classes of limits, and the corresp onding requiremen ts on V (and on K ). The last t w o sections cont ain our main results; w e ha v e divided these into those w hic h are indep end en t of the sorts, in S ectio n 5, and those w hic h relate sp eciﬁcally to single-sorted theories, in Section 6. 1 Review of enric hed category theory W e w ork ov er a s y m metric m on oidal closed category V = ( V 0 , ⊗ , I ) wh ose u n derlying ord inary catego ry V 0 is complete and cocomplete. The general results in [11] on reﬂectivit y of mo dels, referred to ab o v e, used the f urther assump tion that V is lo call y b ounded , in the sense of [11, Chapter 6]. This includes all th e ke y examples of [11 ], including the categories of sets, p oint ed sets, ab elian group s , mo dules o v er a commutati ve ring, c h ain complexes, cate gories, grou p oids, simplicial sets, compactly generated spaces (Hausdorﬀ or otherwise, p ointed or otherw ise), Banac h spaces, sheav es on a site, truth v alues, and La wv ere’s p oset of extended n on-negati ve real n umbers. W e shall often mak e stronger assumptions on V . If the base V is clear, we generally omit the preﬁ x “ V -” and sp eak simply of a category , functor, or natural transform ation. A weight will b e a pr esheaf F : A op → V w hic h is a small colimit of represent ables. If A is small, an y pr esheaf on A is a small colimit of rep resen tables, so there is no restriction. W e sometimes sa y that F is small to mean that it is a small colimit of representa bles. See [6] for more on small functors. F or a V -functor S : A op → K , the limit { F , S } of S weig hted b y F is deﬁ n ed by a natural isomorphism K ( X , { F , S } ) ∼ = [ A op , V ]( F , K ( X, S )) while for a V -functor R : A → K , the colimit F ∗ R of R weigh ted b y F is deﬁned b y a n atural isomorphism K ( F ∗ R , X ) ∼ = [ A op , V ]( F , K ( R, X )) . 5 Note that the pr evious t wo displa ye d equations app ear to inv olv e hom-ob jects in [ A op , V ]. If A is large th en [ A op , V ] d oes not exist as a V -category; nonetheless, the d esired hom-ob ject will exist as an ob ject of V , since F is small (see [6] for example). In p articular, if F and G are b oth small, then [ A op , V ]( F , G ) exists as an ob ject of V , and so w e do ha v e a V -category P A of all small preshea ve s on A . T h is is the free completion of A under colimits [17]. A class Φ of limits means a class of w eigh ts; th en Φ-completeness or Φ -con tinuit y means the existence or the existence and preserv ation of all limits with w eigh ts in Φ. An imp ortant sp ecial case is th e w eigh t C : I op → V , where I is the u nit V -catego ry consisting of a single ob ject ∗ w ith hom I ( ∗ , ∗ ) = I . Then to give the weig ht is just to giv e an ob ject C ∈ V . The C -w eigh ted limit of a V -functor S : I op → K (that is, of an ob ject S ∈ K is deﬁn ed by a natural isomorphism K ( X, C ⋔ S ) ∼ = V ( C , [ X , S ]) has traditionally b een called a c otensor , b u t we shall simply call a p ower , or C -p ow er where nec- essary . The corresp ond in g colimit, written C · S , used to b e called a tensor , but we shall call a c op ower . The ordinary , un w eigh ted n otion of limit can b e seen as a sp ecial case. Let D b e an ordin ary catego ry , and let F D b e the free V -cate gory on D . Th en V -functors F D → K corresp ond to functors D → K 0 , and w e d eﬁne the limit in K of a f unctor S : D → K 0 to b e the limit of the corresp onding R : F D → K w eigh ted by the terminal w eigh t ∆ 1 : F D → K . Limits of this form are called c onic al . The universal prop ert y of { ∆1 , R } inv olv es an isomorphism in V , and is strictly stronger than the u niv ersal p rop ert y of lim S , wh ic h in vo lve s on ly a bijection of sets. Nonetheless, the u niv ersal prop ert y of lim S d o es serv e to iden tify { ∆1 , R } if the latter is kn o wn to exist. Thus if K and L h a ve the relev an t limits, then to sa y that F : K → L pr ese rve s a particular conical limit is equiv alen t to sa ying that F 0 : K 0 → L 0 do es so. W e shall need the follo wing basic result: Prop osition 1.1 L et F : A op → V b e a weig ht, and J : A → B and S : B → C functors. Then F ∗ S J ∼ = Lan J F ∗ S either side existing if the other do es. Pr oof : Here L an J F denotes the left Kan extension of F : A op → V along J : A → B op . The result follo ws from the calculat ion C ( F ∗ S J, C ) ∼ = [ A op , V ]( F , C ( S J, C )) ∼ = [ B op , V ](Lan J F , C ( S, C )) ∼ = C (Lan J F ∗ S, C ) .  Deﬁnition 1.2 A class Φ of weig hts is said to b e s atur ated if for any diagr am S : D → [ A op , V ] in a pr eshe af c ate gory, with e ach S D : A op → V in Φ , and for any F : D op → V in Φ , the c olimit F ∗ S ∈ [ A op , V ] is also in Φ . 6 The original referen ce [4] used the word “clo sed” in place of “saturated”, but the latter is no w standard. The basic result ab out a saturated class Φ is that the full su b catego ry of [ A op , V ] consisting of the preshea v es in Φ is the free completion of A un der Φ-colimits (pro vided that A is small). A V -functor F : A → B with small domain is said to b e dense [11, Chapter 5] if the in d uced V -fu nctor B ( F , 1) : B → [ A op , V ] sending B ∈ B to B ( F − , B ) : A op → V is fu lly faithful. Let F : A op → V and G : B op → V b e w eigh ts, and K a category w ith F -limits and G - colimits. F or any S : A op ⊗ B → K , and an y B ∈ B , w e can form the limit { F , S ( − , B ) } in K , and this d eﬁnes the ob j ect part of a functor { F, S } : B → K : B 7→ { F , S ( − , B ) } , to whic h we can no w apply G ∗ − to obtain G ∗ { F , S } ∈ K . Similar we can form a functor G ∗ S : A op → K and then { F , G ∗ S } ∈ K , and there is a canonical comparison map G ∗ { F , S } → { F , G ∗ S } in K . If th is is in v ertible for all S , we sa y th at F -limits comm ute in K with G -colimit s. T he follo w ing observ ation w as made in [14] in the case K = V : Prop osition 1.3 L et the V -c ate gory K have al l F -limits and al l G -c olimits. Then the fol lowing ar e e quiv alent: 1. F -limits c ommute in K with G -c olimits; 2. { F , −} : [ A , K ] → K is G -c o c ontinuous; 3. G ∗ − : [ B op , K ] → K is F -c ontinuous. More generally , if Φ and Ψ are classes of wei ghts, w e sa y that Φ-limits comm ute in K with Ψ-colimits if this is so for all F ∈ Φ and all G ∈ Ψ. If G commutes in V w ith Φ-limits we follo w [14] in calling G is Φ -ﬂat , or just F -ﬂat if Φ = { F } . This is b y analogy with the case wh ere V = Ab and Φ consists of th e ﬁ nite conical limits. T hen a one-ob ject Ab -categ ory B is a r ing, and a we igh t G : B op → Ab is a B -mo dule, while G ∗ − corresp onds to tensoring o v er B . A mo dule is ﬂat exactly wh en tensorin g with that m od ule pr eserv es ﬁnite limits. In [14, Pr op ositio n 5.4 ], the class of Φ-ﬂat w eigh ts is sh o wn to b e satur ated. An imp ortan t part of this is the f ollo wing: Prop osition 1.4 If F : A op → V i f Φ -ﬂat and G : A op → B op is arbitr ary, then Lan G F : B op → V is Φ -ﬂat. Pr oof : T o sa y that Lan G F is Φ-ﬂ at is to say that Lan G F ∗ − is Φ-con tin uous. But b y Prop osi- tion 1.1, this Lan G F ∗ − is giv en b y the comp osite [ B , V ] [ J, V ] / / [ A , V ] F ∗− / / V and [ J, V ] pr eserv es all limits, since limits in p resheaf categories are calculated p oin t wise, wh ile F ∗ − is Φ-con tin uous since F is Φ-ﬂat.  7 2 Key requiremen ts The ﬁr st requiremen t in v olv es Φ and V . It is con v enient to s upp ose that Φ is lo c al ly smal l [14] in the sen se that for an y small A , th e closure of the representa bles in [ A op , V ] under Φ-colimits is again s m all; typica lly th is will happ en b ecause V 0 is lo cally pr esen table and all weigh ts in Φ are α -presen table for some regular cardinal α . Axiom A. If A is a smal l V -c ate gory with Φ -limits, and F : A → V is Φ -c ontinuous, then so is F ∗ − : [ A op , V ] → V . Note that F ∗ − is th e left Kan extension Lan Y F of F along the Y o neda em b edd ing. This condition has b een considered b y many diﬀerent authors in v arious sp ecial cases, and some of these are listed b elo w when w e turn to examples. In p articular, it holds in the case V = Set if Φ consists of either ﬁnite pro du cts or ﬁnite limits. It was considered, still in the case V = Set , f or a general class of conical limits in [2], and in full generalit y in [14]. It could equiv alen tly b e stated as Φ-limits comm ute in V with colimits that ha v e Φ-con tinuous we ight s or All Φ-con tin uous w eigh ts are Φ -ﬂat. It is this condition whic h allo ws us to “decompose” colimits, in analo gy with th e ﬁn ite/ﬁlte red decomp osition, wh ere no w Φ-colimits pla y th e role of “ﬁnite”, and Φ-ﬂat colimits p la y th e r ole of “ﬁltered”. This can b e done thank s to the follo wing, whic h is a restatemen t of parts of [14, Theorems 8.9, 8.11]: Theorem 2.1 The fol lowing c ondition on the class Φ of weights ar e e quiv alent: 1. If F : A → V is Φ -c ontinuous then so is F ∗ − : [ A op , V ] → V (Axiom A) 2. The c ate gory Φ -C t s ( A , V ) of Φ - c ontinuous pr eshe aves is the fr e e c ompletion of A op under Φ -ﬂat c olimits 3. Any pr eshe af F : A → V is a Φ - ﬂat c olimit of pr eshe aves in Φ wher e in e ach c ase A is al lowe d to b e any smal l V - c ate gory, Φ - c omplete i n the ﬁrst two c ases. Prop osition 2.2 L et A and B b e smal l V -c ate gories with Φ -limits, and G : A → B an arbitr ary V - functor. If M : A → V pr ese rve s Φ -limits then so do es Lan G M : B → V . Pr oof : There is a functor B ( G, 1) : B → [ A op , V ] s en ding B ∈ B to B ( G − , B ) : A op → V whic h in turn sends an ob ject A ∈ A to the h om-ob ject B ( GA, B ). This fu nctor pr eserv es all existing limits, and its comp osite w ith Lan Y M : [ A op , V ] → V is Lan G M .  W e no w tur n to the requiremen ts on K . It is p ossible to deﬁne mo dels in any V -category with Φ-limits, but in order to dev elop th e theory , s omewhat more is required. W e shall consider t wo lev els of generalit y (more precise cond itions will b e giv en later). 8 Axiom B1. K is lo c al ly Φ - pr esentable: this is e quivalent to saying that K itself has the form Φ -Cts ( T , V ) for some smal l V -c ate gory T with Φ -limits. It fol lows that K is r eﬂe ctive in [ T , V ] , and so is c omplete and c o c omplete. Axiom B2. K has Φ - limits, and the inclusion y : K → P K has a Φ -c ontinuous left adjoint. By Theorem 2.1, the Axiom B1 is equiv alent to saying that K is the free completion under Φ-ﬂat colimits of a small Φ-co complete V -category . Axiom B2 implies in p articular that K is co complete ; note that P K h as Φ-limits by [6, Prop osition 4.3] and Axiom A. Axiom B2 is a strong exactness condition, related to lex-totalit y [23 ]: when V is Set , and Φ consists of the ﬁnite limits, it holds in any Grothendiec k top os. Remark 2.3 Both axioms imply that Φ-limits comm u te in K with Φ-ﬂat col imits, sin ce this is true in V , an d so in b oth [ T , V ] and P K , since the the limits and colimits are computed p oin t wise there. No w Φ -Cts ( T , V ) is closed in [ T , V ] under limits and Φ-ﬂ at colimits, so th e desired comm utativit y remains true there. In th e case of Axiom B2, b oth Φ-limits and arbitrary colimits m a y b e computed in K by passin g to P K (where they comm ute) and th en r eﬂecting bac k into K . Prop osition 2.4 A pr eshe af c ate gory K = [ C , V ] satisﬁes Axioms B1 and B2. Pr oof : Axiom B1 is easy: if T is the free V -category with Φ-limits on C , then Φ -Ct s ( T , V ) ≃ [ C , V ]. As for Axiom B2, since [ C , V ] is co complete, the Y o neda functor Y : [ C , V ] → P [ C , V ] certainly has a left adjoin t L ⊣ Y . Explicitly , for a small presh eaf G : [ C , V ] op → V , the reﬂection LG ∈ [ C , V ] is the functor sending C ∈ C to G ( C ( C, − )). W riting ev C for the functor [ C , V ] giv en b y ev aluation at C , and y C for the represen table C ( C, − ), we ma y therefore c haracterize L b y the isomorphism s ev C L ∼ = ev y C , natural in C . Let F : D → V b e in Φ , and S : D → P [ C op , V ]. W e m ust sh o w that L preserve s the limit { F , S } ; but this will b e true if and only if ev C L preserve s the limit for eac h C ∈ C . This w e show as follo ws: ev C L { F , S } ∼ = ev y C { F , S } ∼ = { F , ev y C S } ∼ = { F , ev C LS } using the fact that limits are preserved by ev aluation fu nctors.  3 Sifted colimits and lo cally strongly ﬁnitely presen table cate- gories A k ey notion will b e that of sifte d c olimit [1, 16]. A small cat egory D is said to b e sifte d if D -colimits comm ute in Set with ﬁnite pro du cts; equiv alen tly , if D is non-empty and for all ob jects A, B ∈ D , the category of cospans from A to B is connected. Reﬂexive co equalizers and ﬁltered colimits are b oth sifted, bu t Ad´ amek [3] h as giv en an example of a category with reﬂexiv e co equalizers and ﬁltered colimits bu t not all sifted colimits. 9 Muc h of th e theory of ﬁltered colimits, inv olving ﬁ n itely p resen table ob jects and locally ﬁnitely present able categ ories, has analogues in v olving sifted colimits. T his w as d ev elop ed in [1], and put in to a more general setting in [2]. F or example, a functor F : A → Set is said to b e sifte d-ﬂat [1] if the left Kan extension Lan Y F : [ A op , Set ] → Set preserve s ﬁnite pr od ucts. Since Lan Y F is also th e functor F ∗ − : [ A op , Set ] → Set calculating the F -weig hted colimit of a functor A op → Set , to say that F is sifted-ﬂat is equiv alen tly to sa y that F -weigh ted colimits commute in Set w ith ﬁn ite pro ducts. In particular, if A is sifted then ∆1 : A op → Set is sifted-ﬂat. The sifted-ﬂat functors were c haracterized in [1, Theorem 2.6] as the pr eshea ves whic h are sifted colimits of represent ables. An ob ject X of a cate gory K with sifted colimits is calle d str ongly ﬁnitely pr esentable [1] if the represent able fu nctor K ( X , − ) : K → Set p reserv es sifted colimits. In the follo w in g theorem, the equiv alence b et w een (i) and (ii) is a sp ecial case of the general c haracterizatio n of free completions und er colimits [11, Prop osition 5.62], while the equiv alence b et w een (ii) and (iii) is [1, Th eorem 3.1 0], b ut is essen tially already in [7, Prop osition 5.52]. Theorem 3.1 F or a c ate gory K the fol lowing c onditions ar e e quivalent: (i) K is c o c omplete and is the f r e e c ompletion under sifte d c olimits of a smal l c ate gory G (ii) K is c o c omplete and has a smal l ful l sub c ate g ory G c onsisting of str ongly ﬁnitely pr esentable obje cts, such that every obje ct of K is a sifte d c olimit of obje cts in G (iii) K is e quivalent to the c ate gory FPP ( G op , Set ) of ﬁnite-pr o duct-pr eservi ng functors fr om a smal l c ate gory G with ﬁnite c opr o ducts to Set . The str ongly ﬁnitely pr esentable obje cts wil l b e the closur e under r etr acts of the c ate gory G in e ach c ase. Suc h a category K is called locally strongly ﬁnitely presentable. T he locally strongly ﬁnitely present able categories are the (p ossibly m ultisorted) v arieties. W e s a w ab o v e that reﬂexiv e co equalizers and ﬁltered colimits are n ot enough to guaran tee all sifted colimits. On th e other han d , the f ollo wing pr op ositio n sho ws that pr eservation of reﬂexiv e co equalize rs and ﬁltered colimits is enough to guarantee p reserv ation of sifted colimits, p ro vided that all colimits actually exist. The h istory of this result is sligh tly complicated. The ﬁrst-named author knew the result and its pro of from s oon after the time of the ﬁrst pap ers on sifted colimits, but did not kno w that it w as regarded as an imp ortan t op en p roblem, and did not pub lish it. An analogue in the con text of quasicateg ories was recen tly pro v ed b y Joy al [9], and inspired by this, the result itself was pro v ed b y Ad´ amek [3]. Prop osition 3.2 L et K b e a c o c omplete c ate gory, and F : K → L a f unctor. Then F pr eserves sifte d c olimits if and only if it pr eserves r eﬂexive c o e qu alizers and ﬁlter e d c olimits. Pr oof : Since reﬂexiv e co equalizers and ﬁltered colimits are b oth sifted colimits, one direction is immediate. F or the con ve rse, supp ose that F preserve s reﬂexive co equalizers and ﬁltered colimits. Let D b e a sifted category , and S : D → K a d iagram. W e must sh o w that F p reserv es the colimit of S . Let F am D b e the free completion of D under ﬁnite coprod ucts, and J : D op → (F am D ) op the canonical inclusion. Let G = Lan J ∆1 b e the left Kan extension of the terminal fun ctor ∆1 : D op → 10 Set along J . Sin ce D is sifted, ∆1 : D op → Set is sifted-ﬂat, hence s o by Prop osition 1.4 is its left Kan extension G , and so G is a sifted colimit of repr esentables. But F am D has ﬁnite copro ducts, and so sifted colimits can b e constructed u sing reﬂexiv e co equalizers and ﬁltered colimits by [1 , Example 2.3(2 )]; thus G can b e constructed f rom the represent ables us in g reﬂexive co equalize rs and ﬁltered colimits. W e conclude that an y functor preserving reﬂexiv e co equalizers and ﬁltered colimits also preserves G -w eigh ted colimits. F or a functor R with domain F am D , w e hav e (see P r op osition 1.1) canonical isomorphisms G ∗ R = Lan J ∆1 ∗ R ∼ = ∆1 ∗ RJ ∼ = colim( RJ ) with all terms existing if an y one of th em d oes. If R : F am D → K is the ﬁn ite-co pro du ct- preserving fun ctor extending S , then we get G ∗ R ∼ = colim( RJ ) ∼ = colim( S ). Since F pr eserv es reﬂexiv e co equalizers and ﬁltered colimits, it preserves G -w eighte d colimits, and n o w F colim( S ) ∼ = F ( G ∗ R ) ∼ = G ∗ F R ∼ = ∆1 ∗ F RJ ∼ = colim( F R J ) ∼ = colim( F S ) .  4 P ossible c hoices for the class of limits 4.1 Finite limits Fix a symm etric monoidal closed catego ry V = ( V 0 , ⊗ , I ) with un derlying catego ry V 0 complete and complete. As u s ual, an ob j ect x ∈ V 0 is called ﬁnitely presen table if the rep resen table functor V 0 ( x, − ) : V 0 → Set p reserv es ﬁltered colimits. Kelly [10] d eﬁnes V to b e lo c al ly ﬁnitely pr esentable as a close d c ate gory if V 0 is locally ﬁn itely present able in the usual sense, and th e ﬁ nitely p resen table ob jects are closed under the monoidal structure: the unit I is ﬁn itely p resen table, and the tensor p ro d uct of an y t wo ﬁnitely presen table ob j ects is ﬁnitely presenta ble. Remark 4.1 In f act all the key results of [10] remain true if we drop the assump tion that I is ﬁnitely presen table: see Remark 4.5 b elo w. What is lost is the fact that ﬁn ite presentabilit y in V is th e same as ﬁ nite pr esen tability in V 0 : one only kno ws that ev ery ﬁnitely pr esen table ob ject in V 0 is ﬁnitely p resen table in V , n ot the con v erse. This p ossible generalization seems to b e of limited interest — we kno w of no imp ortan t new examples — but we men tion it here, b ecause a similar generali zation will b e imp ortant when w e mo ve fr om the locally ﬁnitely presen table case to the lo cally strongly ﬁnitely p resen table case. This now giv es a go o d notion of ﬁnite limit in the V -enric hed sense. First of all an ob ject X of a co complete V -cate gory K is said to b e ﬁnitely presen table if the h om-fu nctor K ( X, − ) : K → V preserve s ﬁltered colimits. F or an ob ject X of a co complete V -catego ry K , there is in general no relation b et w een the prop ert y of b eing ﬁnitely presen table in K and the pr op ert y of b eing ﬁnitely presentable in the underlying ordinary categ ory K 0 of K . But if V is lo cally ﬁnitely presen table as a closed category , then these t w o notions agree for K = V (and more generally for an y lo cally ﬁnitely pr esen table V -category K ). 11 The ﬁnite limits are no w those in the saturation of the class of ﬁn ite conical limits and V f - p o w ers, where V f -p o w ers are p o wers (cotensors) b y ﬁnitely p resen table ob jects of V . W e no w tak e these ﬁn ite limits to b e our class Φ. The fact that Axiom A holds is Theorem 6.12 of [10]. F u r thermore, Axiom B will h old if K is any lo cally ﬁn itely presenta ble V -category , in the sense of [10]; in other words, if K is a full reﬂectiv e sub category of a pr esheaf category [ C , V ] wh ic h is closed in [ C , V ] und er ﬁltered colimits; or, equiv alen tly , if K is the category Lex ( T , V ) of mo dels of a small V -categ ory T with ﬁn ite limits. Man y examples of lo cally ﬁn itely presen table V w ere giv en in [10]; they in clude th e closed catego ries of sets, p oin ted sets, ab elian groups, mod u les o v er a commutati ve rin g, chain complexes, catego ries group oids, and simplicial sets. Most of the general results we sh all prov e ab out mon ad s and theories for K = V were obtained in th is case b y Kelly in [10], but for the part in v olving monads on, and single-sorted theories in, V see [19]. Th e treatmen t of monads and single-sorte d theories for more general K (still with Φ the class of ﬁn ite limits) app eared in [18]. 4.2 Finite pro ducts In [5] the notion of π -category was int ro du ced as a suitable setting for un iv ersal algebra. E xplicitly , this is a complete and co complete symmetric monoidal category V , such th at Axiom A holds for Φ the class of ﬁ nite pro ducts, and furthermore, the functors − × X : V → V and X × − : V → V preserve reﬂexiv e coequalizers and ﬁltered colimits for all ob jects X . By Pr op osition 3.2, this condition on − × X and X × − is equiv alen t to sa ying that they pr eserve sifte d c olimits . Prop osition 4.2 L et K b e a c o c omplete c ate gory with ﬁnite pr o ducts. The fol lowing c onditions ar e e quiv alent: (i) X × − : K → K pr eserves sifte d c olimits for al l X (ii) − × X : K → K pr eserve s sifte d c olimits for al l X (iii) × : K × K → K pr eserves sif te d c olimits (iv) sifte d c olimits c ommute in K with ﬁnite pr o ducts In the setting of [5], the V -cate gory w e are calling K is alw ays V itself. Under these assump- tions, the v arious r esults w e h a ve considered w ere p r o ved in [5] (for the case K = V ): left adjoin ts to algebraic fu nctors, reﬂectiv eness of mo dels, corresp ondence b etw een monads and theories, and so on. No w ﬁ n ite p ro ducts commute with sifted colimits in Set , and m ore generally in an y lo cally strongly ﬁnitely presentable cate gory . If V 0 is lo cally strongly ﬁn itely presen table then − × X and X × − will preserve sifted colimits; and no w if Axiom A holds, V will b e a π -cate gory . In the case V = C at , the Φ-accessible monads on Cat will b e the str ongly ﬁnitary 2-monads of [12]. These corresp ond to (a ﬁn itary v ersion of ) the discr ete L awver e the ories of [20]. In the follo win g section these sifted colimits will play a still more central role, and w e shall see ho w they allo w a more expressiv e notion of theory than that of [5], although with somewhat greater restrictions on V . 12 4.3 Strongly ﬁnite limits In this section, w h ic h is one of the main original contributions of the pap er, w e adapt the setting of [10] using sifted colimit s in place of ﬁltered ones. Th e notion of strongly ﬁnite limit, in tro duced b elo w, redu ces to that of ﬁnite pro duct in the case V = Set , b ut not in general. Supp ose as usual that V = ( V 0 , ⊗ , I ) is a complete and cocomplete symmetric monoidal clo sed catego ry . This time w e supp ose that V 0 is lo cally strongly ﬁn itely presen table, and so h as the form FPP ( G op , Set ) for a category G with ﬁnite copr od ucts, which we ma y tak e to b e the category of strongly ﬁnitely presentable ob jects of V 0 . The d irectly analogous approac h to [10] w ould b e to sup p ose that G was closed under the monoidal str u cture; unfortu nately this is not true for key examples suc h as V = Gph (see Ex- ample 4.14). W e weak en the assu mption sligh tly by supp osing that G is closed under the tensor pro duct, but not that it con tains the un it I . W e then say that V is lo c al ly str ongly ﬁnitely pr e- sentable as a ⊗ - c ate gory . Giv en su c h a V , we can no w dev elop the theory of lo cally strongly ﬁ n itely presen table catego ries in the V -enric hed conte xt. W e say th at an ob j ect X of a co complete V -categ ory K is str ongly ﬁnitely pr ese ntable if the hom-functor K ( X , − ) : K → V preserve s sifted colimits, and write K sf for the full su b categ ory of K consisting of suc h ob jects. Just as in [10], it is imp ortant to d istinguish b et w een X b eing strongly ﬁnitely presenta ble in K , in the sense that K ( X, − ) : K → V p reserv es sifted colimits, and X b eing strongly ﬁnitely present able in K 0 , in the sens e that K 0 ( X, − ) : K 0 → Set preserves su c h sif ted colimits. (On the other h and, s ince K and V do hav e sifted colimits, to say that the V -functor K ( X, − ) : K → V p r eserv es sifted colimits is n o diﬀerent to sa ying that the underlying ordinary functor K ( X , − ) 0 : K 0 → V 0 preserve s sifted colimits.) If I ∈ V 0 w ere strongly ﬁn itely pr esen table, th en ev ery strongly ﬁn itely presen table ob ject in K would b e strongly ﬁnitely presen table in K 0 , but w e are n ot assuming this. What w e do ha ve is: Lemma 4.3 If X ∈ V is str ongly ﬁnitely pr esentable as an obje ct of V 0 , in the sense that V 0 ( X, − ) : V 0 → Set pr e serves sifte d c olimits, then it is str ongly ﬁnitely pr esentable in V . Pr oof : Supp ose that X is strongly ﬁn itely presentable in V 0 . Then it is a retract of a ﬁnite copro duct of ob jects in G . If we no w tensor X by an arbitrary G ∈ G , the resulting G ⊗ X is again a retract of a ﬁ nite copro duct of ob jects in G , since G is closed un der tensorin g. T h us G ⊗ X is again s tr ongly ﬁnitely presen table in V 0 , and so V 0 ( G ⊗ X , − ) preserv es sifted colimits. But this means that V 0 ( G, V ( X , − )) preserves sifted colimits. Since the V 0 ( G, − ) preserve and join tly reﬂect sifted colimit s, it follo ws that V ( X , − ) pr eserv es sifted colimits, and so that X is strongly ﬁnitely present able in V .  The con ve rse is f alse: we sh all see in Example 4.14 b elo w that if V is the cartesian closed catego ry Gph of graphs, then the terminal ob ject is strongly ﬁnitely presenta ble in V but not in V 0 . As a furth er indication of the distinction b et w een the p rop erties of b eing strongly ﬁ nitely present able in V or V 0 , notice that although we had to assume that the str ongly ﬁnitely presenta ble ob j ects of V 0 w ere closed und er tensoring, th is is automatic for the strongly ﬁnitely presenta ble ob j ects of V , sin ce V ( X , V ( Y , − )) ∼ = V ( X ⊗ Y , − ) and sifted-colimit- preservin g fu nctors are closed under comp osition. 13 An imp ortan t tec hn ical result is: Prop osition 4.4 V sf is (e quivalent to) a smal l V -c ate gory. Pr oof : Since V 0 is lo cally strongly ﬁn itely presen table, it is also lo cally ﬁnitely presentable. Thus there is a regular cardinal α for whic h I is α -presen table; and V 0 is still lo cally α -presen table. The α -presen table ob jects of V 0 are the α -colimits of th e ﬁ nitely presen table ones, and these are closed under tensorin g, and b y assumption they con tain the un it ob ject I . It no w follo ws , j u st as in [10, 5.2, 5.3] that an ob j ect is α -p r esen table in V if and only if it is α -present able in V 0 . An ob ject of V sf is certainly α -presen table in V ; th us ( V sf ) 0 is a full sub category of ( V 0 ) α , which is small, and so V sf to o is small.  Remark 4.5 All the k ey resu lts of [10] remain true if V 0 is locally ﬁnitely presen table and the tensor pro duct of t wo ﬁ n itely p r esen table ob jects is ﬁnitely present able. Th e reason for assu ming that the un it I is ﬁn itely presenta ble is that th en the notions of ﬁnite presenta bilit y in V and in V 0 agree. But th is is only used at one p oin t: in the p ro of of 7.1, in order to pro v e that the full sub catego ry V f of ﬁnitely pr esentable ob jects in V is small. Ho w ev er this can b e obtained alternativ ely as follo ws. As observed in [13], if V 0 is lo cally ﬁ nitely present able, then V is locally α -presen table as a closed category for s ome regular cardinal α — b y the s ame argument that was used in the previous p rop osition. Then the f u ll sub category V α of V consisting of the α -pr esen table ob j ects is small, b y the same argument as in [10]; b ut V f is clearly conta ined in V α and so is also small. Ha ving ﬁxed our monoidal categ ory V , we n ow tu rn to the class Φ of w eigh ts. W e tak e for Φ the saturation of the class of ﬁnite pro ducts and V sf -p o w ers (p o w ers by ob jects of V sf ). Prop osition 4.6 If K is c o c omplete, the str ongly ﬁnitely pr ese ntable obje cts of K ar e close d under Φ -c olimits. Pr oof : It suﬃces to show that they are closed un d er ﬁ n ite copro du cts and under V sf -cop o wers. If X 1 , . . . , X n are strongly ﬁn itely presen table, then K ( X 1 + . . . + X n , − ) ∼ = K ( X 1 , − ) × . . . × K ( X n , − ) and eac h K ( X i , − ) preserve s sifted colimits since X i is strongly ﬁ nitely present able, while ﬁ- nite pro du cts of sifted-colimit-preserving functors in to V still preserve sifted colimit s, since ﬁ nite pro ducts commute with sifted colimits in V . This pro ve s that X 1 + . . . + X n is s tr ongly ﬁ n itely present able. Similarly if X ∈ K is strongly ﬁnitely pr esentable, and G ∈ V is strongly ﬁ nitely presen table, then K ( G · X , − ) ∼ = V ( G, K ( X, − )) whic h preserv es sifted colimits since K ( X, − ) and V ( G, − ) do, thus G · X is strongly ﬁnitely present able.  Corollary 4.7 If F : A op → V is in Φ then it is str ongly ﬁnitely pr esentable as an obje ct of [ A op , V ] . 14 Pr oof : This is immediate from the case K = [ A op , V ] of the pr op ositio n, give n that repr e- sen tables are strongly ﬁn itely pr esen table, and F is a Φ-colimit of represent ables, via th e Y oneda isomorphism F ∼ = F ∗ Y .  The follo wing theorem, adapted f rom [10, Theorem 6.11], imp lies in particular that Axiom A holds: Theorem 4.8 L et T b e a smal l V -c ate gory with Φ -limits. F or a V -fu nctor F : T → V the fol lowing ar e e quivalent: (1) F is a sifte d c olimit of r epr esentables; (2) F is Φ -ﬂat; (3) F is Φ -c ontinuous. Pr oof : Represent ables are Φ-ﬂat, and sifted colimits comm ute in V w ith Φ -limits, thus (1) implies (2). T o see th at (2) implies (3), observ e that if F is Φ-ﬂat then Lan Y F is Φ-con tin uous, but Y is Φ-con tin uous, hence so is F = (Lan Y F ) Y . So it r emains to p ro v e that (3) implies (1). Supp ose then th at F is Φ-con tin uous. Consider the underlying ordinary functor F 0 : T 0 → V 0 , and the indu ced V 0 ( I , F 0 ) : T 0 → Set . Like any Set -v alued fu nctor, this is canonically a colimit of representables, W e form the category of elemen ts E and the induced P : E → T op 0 . Exp licit ly , an ob ject of E consists of an ob ject T ∈ T equipp ed with a V -natur al T ( T , − ) → F . Th en V 0 ( I , F 0 ) is the colimit of the comp osite E P / / T op 0 Y / / [ T 0 , Set ] Since F preserves ﬁnite p ro d ucts, s o do es V 0 ( I , F 0 ); it follo w s that E h as ﬁn ite copro ducts, and so is sifted. Thus w e ha v e expr essed V 0 ( I , F 0 ) as a sifted colimit of represen tables. The idea is to adapt this to obtain F and not just V 0 ( I , F 0 ). Consider no w the comp osite E P / / T op 0 Y 0 / / [ T , V ] 0 whic h sends an ob ject ( T , x : T ( T , − ) → F ) of E to T ( T , − ). W e shall show that it h as colimit F , and so that F is a sifted colimit of representables. Th er e is an evident co cone γ : Y 0 P → ∆ F , whose copro j ection at ( T , x : T ( T , − ) → F ) is just x . W e m ust show that this is a colimit. It will b e a colimit if and only if it giv es a colimit after ev aluating at all S ∈ T ; in other words, if the E -indexed d iagram T ( T , S ) / / F S is a colimit (in V , or equiv alen tly in V 0 ) for all S . No w the hom-functors V 0 ( G, − ) : V 0 → Set for G ∈ G jointly reﬂect colimits, since G is a strong generator, and they p reserv e sifted colimits, by assumption on G . Thus w e are reduced to showing that the E -indexed d iagram V 0 ( G, T ( T , S )) / / V 0 ( G, F S ) is a colimit in Set , f or all S ∈ T and all G ∈ G . But by the universal p rop ert y of p ow ers , and the fact that F preserve s G -p o wers, this diagram is equiv alen tly V 0 ( I , T ( T , G ⋔ S )) / / V 0 ( I , F ( G ⋔ S )) 15 and this is a colimit s in ce V 0 ( I , T ( T , − )) / / V 0 ( I , F ) is one.  W e n o w turn to examples of V which are lo cally strongly ﬁnitely pr esen table as ⊗ -categories. An imp ortan t sp ecial case is where V is a p resheaf category , equipp ed with the cartesian closed structure. First we pro v e: Lemma 4.9 In a pr eshe af c ate gory [ C op , Set ] , an obje ct is a r etr act of a ﬁnite c opr o duct of r epr e- sentables if and only if it is a ﬁnite c opr o duct of r etr acts of r epr esentables. Pr oof : If R i is a retract of the r epresen table y C i for i = 1 , . . . , n then P i R i is a retract of P i y C i . T h us a ﬁnite copro duct of retracts of repr esentables is a retract of a ﬁ nite copro duct of represent ables. Con v ersely , supp ose that R is a retract of a ﬁnite copro duct P i y C i of repr esen tables. By extensivit y of [ C op , Set ], we can write the inclusion R → P i y C i as a copro duct P i R i → P i y C i where eac h R i → y C i is the inclusion of a retract.  Corollary 4.10 If idemp otents split in C , then the r etr acts of ﬁnite c opr o ducts of r epr esentables ar e just the ﬁnite c opr o ducts of r epr esentables. Corollary 4.11 A r etr act of a ﬁnite c opr o duct of r etr acts of ﬁnite c opr o ducts of r epr esentables is a r etr act of a ﬁnite c opr o duct of r epr esentables. Prop osition 4.12 If V = [ C op , Set ] , e quipp e d with the c artesian close d structur e, then V is lo c al ly str ongly ﬁnitely pr esentable as a ⊗ -c ate gory if and only if the pr o duct of any two r epr esentables is a r etr act of a ﬁnite c opr o duct of r epr esentables. Pr oof : In this case ( V 0 ) sf is obtained from C b y freely adjoining ﬁ nite copro ducts and then splitting idemp oten ts; in other words it ma y b e identiﬁed with the fu ll sub category of [ C op , Set ] consisting of the retracts of ﬁnite copro ducts of repr esen tables. Then V will b e lo cally strongly ﬁnitely p resen table as a ⊗ -categ ory if and only if this sub catego ry is closed under binary prod ucts. This certainly implies that the pr od uct of any t wo r epresen tables is in the sub category , but in fact it is equiv alen t: if R and S are retracts of ﬁ nite copro ducts P i C i and P j D j of repr esen tables, then R × S is a r etract of the ﬁnite sum P i P j C i × D j of p ro ducts of representa bles. If eac h C i × D j is a retract of a ﬁnite copro duct of represent ables, then so is R × S by Corolla ry 4.11.  Corollary 4.13 [ C op , Set ] i s lo c al ly str ongly ﬁnitely pr esentable as a ⊗ -c ate gory if C has binary pr o ducts. Example 4.14 The cartesia n closed catego ry Gph of (directed) graphs is locally strongly ﬁnitely present able as a ⊗ -categ ory . Th ere are tw o representables: the free-living edge E , and the free- living v ertex V . It is ea sy to c hec k that V × V ∼ = V , V × E ∼ = E × V ∼ = V + V , and E × E ∼ = V + E + V , so that the p ro d uct of an y tw o r epresen tables is a ﬁ n ite co pr od uct of representa bles. On the other hand, the terminal graph 1 (whic h is of course th e unit for the cartesian m onoidal structur e) is not a ﬁnite copr od uct of representa bles, and so Gph is n ot locally strongly ﬁn itely presen table as a close d category . 16 W e can also see this m ore directly b y exhibiting a s ifted colimit in Gph wh ic h Gph (1 , − ) : Gph → Set fails to pr eserv e. Now Gph (1 , − ) sends a grap h G to a vertex v ∈ G with a c hosen lo op. C on s ider the parallel pair E + V f / / g / / E where f and g act as th e iden tit y on the E comp onent , while on the V comp onen t f pic ks out the source of the edge in E , and g pic ks out th e target. Th e coequalizer is formed b y identifying the sou r ce and target, whic h no w giv es a lo op: in other w ords, the co equ alize r is 1. This pair is clearly r eﬂexiv e, and so its co equalizer is a sifted colimit. But it is n ot preserved by Gph (1 , − ), since Gph (1 , 1) has one elemen t, while Gph (1 , E ) is empty . Example 4.15 The cartesian closed category RG ph of r eﬂexiv e graph s is not lo cally strongly ﬁnitely presentable as a ⊗ -category . In th is case there are t wo represen tables: the terminal graph 1 with a single lo op, and the reﬂexive graph E generated by a single edge. It f ollo ws that an y s trongly ﬁnitely p resen table r eﬂexiv e graph can only hav e these graph s as its connected co mp onents. But if w e tak e the pro duct of t w o copies of E w e obtain the reﬂ exive graph • / /   @ @ @ @ @ @ @   •   • / / • whic h, b y Corollary 4.1 0, is not strongly ﬁn itely pr esentable. (This time the terminal ob ject is not just strongly ﬁ n itely presen table, but representable.) Example 4.16 Let I b e a sk eletal category of ﬁ n ite sets and injections, and V the cartesian closed catego ry [ I , Set ]. This is locally strongly ﬁnitely presen table as a ⊗ -category: giv en representa bles I ( m, − ) and I ( n, − ) w e h av e the form ula I ( m, − ) × I ( n, − ) ∼ = m + n X k =max( m,n )  m m + n − k  ×  n m + n − k  · I ( k , − ) exhibiting I ( m, − ) × I ( n, − ) as a ﬁn ite coprod uct of represen tables. The category [ I ,Set] is one of mo dels giving a d enotatio nal semantic s for the π -calculus [22]. Since all op erations used there ha v e s trongly ﬁ nitely p resen table arities, our formalism might b e useful there. It is no w p ossible to dev elop a theory of lo cally strongly ﬁ nitely presentable V -cate gories as one migh t exp ect. W e sa y that a V -catego ry K is lo cally strongly ﬁnitely presen table if (i) K is co complete (ii) K has a small f ull sub category G consisting of strongly ﬁnitely pr esen table ob j ects (iii) every ob ject of K is a sifted colimit of ob jects in G W e shall pro vide v arious c haracterizations b elo w ; in the mean time we pro ve : 17 Theorem 4.17 Any lo c al ly str ongly ﬁnitely pr esentable V -c ate gory K is e quivalent to the c ate g ory of Φ -c ontinuous V - functors fr om T to V for some smal l Φ -c omplete V -c ate gory T . Pr oof : Let G b e the closure of G in K under Φ-colimits; that is, under ﬁn ite copro d ucts and V sf -cop o wers. Since V sf is s m all, it follo ws that G is still small. By assumption it has Φ-colimits, so T = G op is a small V -category with Φ-limits. The inclusion of G → K ind uces a V -functor W : K → [ G op , V ] = [ T , V ]. S ince Φ-coli mits of strongly ﬁ nitely presen table ob jects are strongly ﬁnitely presen table, the ob jects of G are strongly ﬁ n itely presentable, and so th e fu n ctor W : K → [ G op , V ] p reserv es sifted colimits. Since G is con tained in K sf it follo ws by Prop osition 4.6 that G is so too, and so that W actually lands in the catego ry Φ -Cts ( T , V ) of Φ-con tinuous functors. Since G is d ense [11, Theorem 5.35 ] it follo ws that G is d ense and so that W is fu lly faithful [11, Theorem 5.13]. It r emains to sho w that Φ -Cts ( T , V ) is the imag e of W ; in other w ords th at every Φ-con tinuous F : T → V has the form W A for some A ∈ K . Supp ose then that F : G op → V is Φ-con tin uous; b y T heorem 4.8 it is a sif ted colimit of represent ables, sa y F = colim i y C i . W rite J : G → K for the inclusion. Since K is co complete, w e ma y form the colimit F ∗ J = (colim i y C i ) ∗ J ∼ = colim i ( y C i ∗ J ) ∼ = colim i ( J C i ) wh ic h will b e preserve d by W , since W preserve s sifted colimits. T hen W ( F ∗ J ) ∼ = F ∗ W J ∼ = F ∗ Y ∼ = F , and so F is indeed in the image of W .  W e sh all see b elo w th at th e conv erse is also tru e, but we sh all prov e this u sing some of the more general theory devel op ed in Section 5 b elo w. 4.4 Sound do ct rines W e no w sho w the case of ﬁnite limits and of strongly ﬁnite limits ﬁt into a common fr amew ork. The treatmen t follo ws that of Section 4.3 ve ry closely , so w e lea v e out some of the details. Recall from [2] the n otion of soun d do ctrine. In that p ap er a do ctrine consisted of a s m all collect ion D o f small categories D . Then a cate gory C was said to b e D -ﬁltered if C -colimits comm ute in Set with D -limits. It follo ws that for an y D ∈ D and an y diagram S : D op → C , the catego ry of co cones is connected. If con ve rsely , the connectedness of the category of cocones of a diagram D op → C implies that C is D -ﬁltered, then the do ctrine D is said to b e sound . The ﬁrst main theorem ab out suc h do ctrines [2, Theorem 2.4] includ es in particular: Theorem 4.18 F or a sound do ctrine D and a functor F : A → Set with A smal l, the fol lowing ar e e quiv alent: 1. Lan Y F is D -c ontinuous 2. F is a D -ﬁlter e d c olimit of r epr esentables and if A has D -limits then these ar e further e quivalent to 3. F is D -c ontinuous. W e reco v er the setting of Sections 4.1 and 4.3 by taking for D , resp ectiv ely , all ﬁnite categories and all ﬁnite discrete categories. 18 A category is s aid to b e lo c al ly D -pr e sentable [2] if it is equiv alen t to the categ ory of D -con tin uous functors f rom A to Set , for a small category A w ith D -limits. T h ere are analogues for all the main results ab out lo cally ﬁ n itely presen table categories, and these results are then reco v ered on taking D to b e the ﬁ nite cat egories. Supp ose that D is sound, and let V 0 b e lo cally D -presen table, with the sub catego ry ( V 0 ) D of D -presenta ble ob jects closed under the tensor p ro duct in V . W e then sa y that V is lo c al ly D -pr esentable as a ⊗ -c ate g ory . Let V D b e the full sub- V -category of V consisting of the D -p resen table ob jects of V : those ob j ects G ∈ V for w hic h [ G, − ] : V → V pr eserv es D -ﬁltered colimits. Lemma 4.19 If G is D -pr esentable in V 0 then it is D -pr esentable in V . Pr oof : W e kno w that V 0 is (equiv alen t to) the cate gory of D -con tinuous fun ctors from T to Set for some small category T with D -limits. If D -limits do not already include the splittings of idemp oten ts th en we ma y split the idemp oten ts of T without c h anging the D -con tin uous functors. A D -presentable ob ject of V 0 is then a r epresen table fu nctor y T = T ( T , − ). W e m ust sho w that the internal hom [ y T , − ] : V → V preserves D -ﬁltered colimits, or equiv alen tly that the underlying functor [ y T , − ] 0 : V 0 → V 0 do es. But the V 0 ( y S, − ) : V 0 → Set preserve and detect D - ﬁltered colimits, s o it suﬃces to show that the V 0 ( y S, [ y T , − ]) preserv e D -ﬁltered colimits. Finally V 0 ( y S, [ y T , − ]) ∼ = V 0 ( y S ⊗ y T , − ), bu t V 0 ( y S ⊗ y T , − ) pr eserv es D -ﬁltered colimits since y S ⊗ y T is D -present able by assumption.  Prop osition 4.20 V D is (e quivalent to) a smal l V -c ate gory. Pr oof : Any lo cally D -presen table category is lo cally α -present able for some regular cardinal α (see [2, Theorem 5.5]): w e may tak e α to b e larger than the cardinalit y of an y D ∈ D and suc h that the unit I is α -presen table. It then follo ws that V is lo cally α -presenta ble as a closed category , and so that V α is small; but V D is con tained in V α .  Let Φ b e the saturation of the class of all (conical) D -limits and V D -p o w ers. These limits certainly commute in V with D -ﬁltered colimits. Prop osition 4.21 If K is a c o c omplete V -c ate gory, the Φ -pr e sentable obje cts of K ar e close d under Φ -c olimits. In p articular, if F : A op → V is in Φ , then it is Φ -pr esentable in [ A op , V ] . W e no w p ro v e follo wing analogue of Theorem 4.8, whic h shows in particular that Axiom A is satisﬁed. Theorem 4.22 L et T b e a smal l V - c ate gory with Φ -limits. F or a V -functor F : T → V the fol lowing ar e e quivalent: (1) F is a D - ﬁlter e d c olimit of r epr esentables; (2) F is Φ -ﬂat; (3) F is Φ -c ontinuous. 19 Pr oof : Since D -ﬁltered colimits comm ute in V with Φ-limits, the Φ-ﬂat w eigh ts are closed under D -ﬁltered colimits. Since repr esen tables are certainly Φ-ﬂat, we d educe that (1) implies (2). Of course (2) implies (3) since the Y oneda embed ding preserves all existing limits. So it remains to sho w that (3) implies (1). Supp ose th en that T is a small V -cate gory with Φ-limits and that F : T → V is Φ-con tin uous. Just as in the p ro of of Th eorem 4.8 , w e consider the ordinary functor T 0 F 0 / / V 0 V 0 ( I , − ) / / Set its categ ory of elemen ts E and th e the induced P : E → T op 0 , and observe that V 0 ( I , F 0 ) is canonically the colimit of E P / / T op 0 Y / / [ T , Set ] . Since F preserv es D -limits, so do es V 0 ( I , F 0 ) : T 0 → Set ; it f ollo ws that E is D -ﬁltered. If the colimit of E P / / T op 0 Y 0 / / [ T , V ] 0 is F , then F will b e a D -ﬁltered colimit of r epresen tables, as required. The v eriﬁcation goes exactly as in the pro of of Theorem 4.8.  4.5 Finite connected limits This is the case where D consists of the ﬁnite connected categories. A cate gory K is lo cally D - present able if and only if it is locally ﬁ nitely p resen table and fu r thermore its category of ﬁnitely present able ob j ects is itself the fr ee completion und er ﬁ n ite colimits of a full su b catego ry . The D -present able ob jects are those whic h are b oth ﬁnitely present able and connected. W e s upp ose that V 0 is suc h a category and that the ﬁ nitely p resen table connected ob jects are closed und er tensorin g. This time we tak e Φ to b e the saturation of the class of all ﬁnite connected conical limits and all G -p ow ers. Example 4.23 The cartesian closed categories Gph , R Gph , Cat , Gp d , SSet , and CGT op of graphs, reﬂexiv e graphs, categories, group oids, simplicial sets, and compactly ge nerated spaces are all examples; so is an y presheaf top os (su ch as Gph , RGph , and SSet ), and so, of course, is Set . A monad will b e Φ-accessible if and only if it is ﬁnitary and preserv es copro ducts. 5 Man y-sorted theories Let V and Φ b e giv en, satisfying Axiom A. T o s tart with, w e allo w K to b e an arb itrary V - catego ry w ith Φ-limits, but b efore long we shall supp ose that it satisﬁes Axiom B1 or B2. T h e most imp ortant case is K = V , w hic h satisﬁes b oth Axiom B1 and B2. 5.1 Theories and mo dels A small V -category T w ith Φ-limits is called a Φ - the ory in V , or just a theory when Φ and V are understo o d. A mo del of T in K is a Φ-conti nuous V -fun ctor from T to K . 20 The V -category of mo dels of T in K is the fu ll sub category Φ -C ts ( T , K ) of the fu nctor catego ry [ T , K ] consisting of the mo dels. When K = V , we write simply Φ -Mo d ( T ) for Φ -Cts ( T , V ). 5.2 Left adjoin ts to algebraic functors A morphism of the ories is a Φ-cont inuous V -fun ctor G : S → T . Comp osition with G induces a V -fun ctor G ∗ : Φ -Mo d ( T ) → Φ -Mod ( S ); s u c h a V -functor is called Φ-algebraic, or just algebraic. Suc h algebraic fu nctors hav e left adjoints: given a mo del M : S → V we ma y form the left Kan extension Lan G M : T → V of M along G and b y Prop osition 2.2 this is Φ-con tin uous, and so is a mo del of T ; it is easy to see that it h as the required univ ersal p rop ert y . (In fact the existence of a left adjoin t holds m uc h more generally; the p oint here is that it can b e constructed via left Kan extension.) W e no w turn to the case of a general K . Once again G induces a V -functor G ∗ : Φ -Cts ( T , K ) → Φ -Cts ( S , K ); such a G migh t b e called “Φ-algebraic relativ e to K ”. Prop osition 5.1 L et A and B b e smal l V -c ate gories with Φ -limits, and G : A → B an arbitr ary V - functor. If K satisﬁes Axiom B2 and M : S → K is Φ -c ontinuous, then Lan G M : T → K is also Φ - c ontinuous. Pr oof : Let Y : K → P K b e th e Y on ed a em b edding and L ⊣ Y its Φ -con tinuous left adj oin t. O f course Y p reserv es Φ-limits (and an y other existing limits). Since L is co con tin uous, it preserves left Kan extensions, and so Lan G M ∼ = Lan G LY M ∼ = L Lan G Y M . Now L is Φ-con tin uous by assumption, thus it will suﬃce to sho w th at Lan G Y M is. In other wo rds , we can w ork with P K rather than K . But in P K b oth the left Kan extensions and the Φ-limits are computed p oint wise, an d so w e actually need only consider th e case K = V ; this is Prop osition 2.2.  Th us when Axiom B2 holds w e can once agai n constru ct left adjoint s to algebraic f unctors by Kan extension. Once again, the existence of the left adjoint holds muc h more generally , certainly whenev er K is lo cally presenta ble. W e shall s ee b elo w that left adjoin ts to algebraic fun ctors include in p articular free mo dels for single-sorted theories. 5.3 Reﬂectiv eness of mo dels Let F T b e the fr ee completion of T und er Φ-limits. S ince T has Φ-limits, the canonical inclusion J : T → FT has a righ t adjoint R , and the algebraic fu nctor R ∗ : Φ -Mo d ( T ) → Φ -Mo d ( F T ) ≃ [ T , V ] has a left adjoin t by the previous result. But this is just the full inclusion Φ -Mo d ( T ) → [ T , V ]. Thus th e mo dels form a full reﬂ ectiv e s u b catego ry of the fu n ctor category , and in particular Φ -Mo d ( T , V ) is complete and cocomplete. Theorem 5.2 Φ -Mo d ( T ) is r eﬂe ctive in [ T , V ] and so is c omplete and c o c omplete. It is close d in [ T , V ] under al l limits and under Φ -ﬂat c olimits. 21 In fact the mod els will b e r eﬂectiv e m uch m ore generally (see [11, Chapter 6]), but our f ramew ork giv es a simp le construction, whic h can b e computed in pr actic e (pro vided that colimits in the base catego ry V can b e computed). Note also that wheneve r the mo dels are r eﬂectiv e we also ha v e adjoin ts to algebraic fun ctors: giv en a morphism of th eories G : S → T and a mo del M : S → K , ﬁ rst take the left K an extension Lan G M : T → V and then reﬂ ect into mo dels. More generally , Φ -Mo d ( T , K ) will b e r eﬂ ectiv e in [ T , K ] if K satisﬁes Axiom B2. 5.4 Characterization In this section we c haracterize V -categories of mo dels in V . Let M b e a V -category with Φ -ﬂat col imits. W e deﬁn e an ob ject M ∈ M to b e Φ -pr esentable if the hom-fun ctor M ( M , − ) : M → V pr eserv es Φ-ﬂat colimits. W e deﬁne a V -category M to b e lo c al ly Φ -pr esentable if it is cocomplete and has a small full sub category G consisting of Φ-present able ob jects such that ev er y ob ject of M is a Φ -ﬂat colimit of ob jects in G . It follo ws that M is the free completion of G un der Φ-ﬂat colimits. Let G b e the closure of G in M un der Φ-colimits. This is a small dense full sub category consisting of Φ-presen table ob jects, and it is Φ -co complete b y construction. T h e inclus ion J : G → M ind uces a V -fun ctor W : M → [ G op , V ] whic h is fully faithful sin ce G is d ense. It has a left adjoin t sending F : G op → V to the colimit F ∗ J ∈ M . The comp osite W J is the Y oneda em b edding. Explicitly , W sen ds an ob ject M ∈ M to M ( J − , M ) which is Φ -con tinuous since the inclusion J : G → M is Φ-co con tin uous. Thus W lands in Φ -Mo d ( G op ). F ur th ermore, W preserv es Φ-ﬂat colimits, since G consists of Φ-present able ob jects. Prop osition 5.3 The V - functor W : M → Φ -Mo d ( G op ) is an e quivalenc e. Pr oof : W e already kn o w that W is fully faithful, so it will su ﬃ ce to show that it is essentia lly surjectiv e on ob jects. Su pp ose then that F : G op → V is Φ-con tin uous. By Axiom A it is also Φ-ﬂat, and so W p r eserv es F -weig hte d colimits. Now ev ery presheaf is a colimit of represen tables, w eigh ted b y itself, and so we ha v e F ∼ = F ∗ Y ∼ = F ∗ W J ∼ = W ( F ∗ J ) whic h completes the pro of.  Th us any lo cally Φ-presenta ble V -category is the categ ory of m od els in V for a small theory . Con v ersely , let T be a theory and consid er Φ -Mo d ( T ). This is reﬂectiv e in [ T , V ] and so co complete . The represen tables pr o vid e a small dense sub category Since Φ-limits commute in V with Φ-ﬂ at colimits, the inclusion Φ -Mo d ( T ) → [ T , V ] p re- serv es Φ-ﬂat colimits, which is equ iv alen t to sa ying that the r epresen tables are Φ-pr esen table in Φ -Mo d ( T ). Finally if F : T → V is Φ-con tin uous th en it is Φ-ﬂat by Axiom A, and s o once again F ∼ = F ∗ Y ∼ = F ∗ W J ∼ = W ( F ∗ J ) sho w s that F is a Φ -ﬂ at colimit in Φ -Mo d ( T ) of represen tables. This prov es that Φ -Mo d ( T ) is locally Φ-presen table, and so giv es: Theorem 5.4 A c ate gory M is lo c al ly Φ -pr esentable if and only if it is e quivalent to a c ate gory of Φ -c ontinuous V -functors fr om T to V for a smal l Φ - c omplete V -c ate gory T . 22 As observed ab o v e, if Φ consists of the ﬁnite limits, then this was pro v ed in [10]; and , in the case V = Set , is of course due to Gabriel-Ulmer [8]. W e no w return to the sp ecial case of Section 4.3. Theorem 5.5 L et V b e lo c al ly str ongly ﬁnitely pr esentable as a ⊗ -c ate gory and let Φ b e the satu- r ation of the ﬁnite pr o ducts and V sf -p owers. F or a V -c ate gory M the fol lowing ar e e quivalent: (i) M is lo c al ly str ongly ﬁnitely pr esentable (ii) M i s lo c al ly Φ -pr esentable (iii) Ther e is a smal l V -c ate gory T with ﬁnite pr o ducts and V sf -p owers for which M is e quivalent to the ful l sub c ate gory of [ T , V ] c onsisting of the V -functors which pr eserve these limits. Pr oof : The equiv alence of (ii) and (iii) is a sp ecial case of the p revious theorem. Th e f act that (i) implies (iii) is Th eorem 4.17. T o see th at (iii) implies (i), observe that a M satisfying (iii) is co complete , since (iii) implies (ii), and no w consider the small f ull sub category consisting of the represent ables T → V . Certainly these are strongly ﬁnitely presen table. I t remains to sho w that ev ery Φ-cont inuous F : T → V is a sifted colimit of representables. But this is T heorem 4.8.  This reduces to Theorem 3.1 in the case V = Set ; see the discussion b efore that theorem for the history of the r esult in that case. 6 La wve re theories In this section we turn to theories which can b e thought of as single-so rted, and see ho w to extend the classical corresp onden ce b et w een such theories and monads. A ma j or diﬀerence is that the resulting theories n eed n ot ha ve all the limits und er consideratio n. This w as observed in [18] in the case wh ere Φ consists of the ﬁ nite limits, an d our appr oac h is mo delled on that of [18]. As a consequence, in this more general s etting, a La wve re theory need not b e a theory; we sh all s ee, ho wev er, ho w a La wve re theory do es generate a theory w ith the same mo dels. Let A and B b e V -categories with Φ-ﬂat colimits. W e sh all s ay that a V -functor F : A → B is Φ -ac c essible if it p reserv es Φ-ﬂat colimits. A monad on A will b e ca lled Φ-accessible if its underlying endofun ctor is so. W e write Mnd Φ ( A ) for the cate gory of Φ-accessible monads on A . 6.1 La wv ere Φ -theories Let K b e a V -category satisfying Axiom B1. Let J : K Φ → K b e the fu ll sub category of K consisting of th e Φ-presentable ob jects; then K Φ has Φ-colimits and J preserves them. F urthermore, K is equiv alen t to the catego ry of Φ-cont inuous V -fun ctors from K op Φ to V . Let T = ( T , m, i ) b e a Φ-accessible V -monad on K , write K T for the Eilen b erg-Mo ore V - catego ry , with forgetful functor U T : K T → K and left adjoint F T ⊣ U T . W e ma y factorize the comp osite F T J : K Φ → K T as an identit y-on-ob jects V -fu n ctor E : K Φ → G follo wed b y a fully faithful H : G → K T . The opp osite of the r esulting V -category G will b ecome the La wvere theory L corresp onding to T . No w K Φ has Φ-colimits, preserved b y J , while F T preserve s all colimits, so the comp osite H E = F T J : V Φ → V T preserve s Φ-colimits. It f ollo ws that E p reserv es Φ -col imits, but it do es 23 not follo w that G has all Φ-colimits. It do es ha v e Φ-colimits of diagrams in the image of E , but need not hav e Φ -col imits in general. Note, h ow ever, that G will hav e F -w eigh ted colimits for any F ∈ Φ whic h is a w eigh t for copro ducts or cop o we rs, since these in v olv e only th e ob jects of G , and so the resulting d iagrams will b e in the image of the iden tit y-on-ob ject V -functor E . Remark 6.1 Thus in the sp ecial case of Section 4.3 wh ere all weig hts in Φ are of this type, G will ha v e Φ-colimits. Th e same is true if Φ consists of j ust the ﬁnite pro ducts. But in general, we mak e the follo wing d eﬁnition, give n in [18] for the case where Φ is the ﬁnite limits. Deﬁnition 6.2 A L awver e Φ -the ory in V is an identity-on-obje ct V -functor E : V op Φ → L which pr eserves Φ - limits. A morphism of L awver e Φ -the ories is a c ommutative triangle of (identity-on- obje ct) V -functors; we write Law Φ ( K ) for the r esulting c ate gory of L awver e Φ -the ories on K . W e cannot in general simply d eﬁne a m od el to b e a Φ -con tinuous V -functor with domain L , since L m a y not h a ve all Φ-limits. In stead w e deﬁne the V -category of mo dels of L by th e follo w ing pu llb ac k in V -Cat Φ -Cts ( L , V ) / / U   [ L , V ] [ E , V ]   K K ( J , 1) / / [ K op Φ , V ] Remark 6.3 As obs erv ed in [15] in the case of ﬁn ite limits, since K is equiv alent to the V -categ ory of Φ-cont inuous V -functors fr om K op Φ to V , u p to an equiv alence, a mo del of L is just a V -functor M : L → V whose restriction along E is Φ -con tinuous. Remark 6.4 If Φ consists only of ﬁnite pro du cts and/or p o w ers, then Φ-limits in L are determined b y those in K op ϕ , and so th e restriction of M along E is Φ-contin uou s if and only if M is Φ- con tin uous. Prop osition 6.5 U is monadic via a Φ -ac c essible monad. Pr oof : [ L , V ] has all coequ alizers, [ E , V ] has b oth adjoints (giv en by left and right Kan exten- sion), and [ E , V ] is also conserv ativ e b ecause E is the bijectiv e on ob jects. An easy ap p licati on of Bec k’s theorem sh o ws that [ E , V ] is monadic. The pullbac k U of [ E , V ] will still satisfy the conditions of Bec k’s th eorem, and so b e monadic, p r o vid ed th at it has a left adjoint. No w K ( J, 1) : K → [ K op Φ , V ] is fully faithful and has a left adjoin t, [ E , V ] has a left adjoint, and the inclus ion Φ -Cts ( L , V ) → [ L , V ] has a left adjoin t, th us U do es indeed h av e a left adjoin t and so is monadic. It remains to sho w that th e monad is Φ-accessible, or equiv alen tly that U pr eserv es Φ -ﬂat colimits. But this follo ws b ecause th e other three functors in th e deﬁnition of Φ -Cts ( L , V ) preserv e Φ-ﬂat colimits, and K ( J, 1) is fu lly faithfu l so also reﬂects them.  Th us to ev ery La wvere theory we hav e asso ciated a Φ-acc essible monad. This giv es the ob ject- part of a fu nctor mnd : La w Φ ( K ) → Mnd Φ ( K ). 24 Con v ersely , for a Φ -accessible monad T on K , the inclusion H : L op = G → K T induces a V -fu nctor K T ( H , 1) : K T → [ L , V ]. Theorem 6.6 The V - functor K T ( H , 1) : K T → [ L , V ] r estricts to an isomorphism of V - c ate gories K T ≃ Φ - Cts ( L , V ) . Pr oof : Comp osition with E : K op Φ → G op induces a V -functor [ G op , V ] [ E , V ] / / [ K op Φ , V ] whic h has b oth adjoin ts, give n by left and right Kan extension. Since E is bijectiv e on ob jects, [ E , V ] is conserv ativ e, and n o w by the Bec k theorem, it is monadic. W rite S for the induced monad. Consider what happ ens when w e restrict the ind u ced monad S along the fully f aithful K ( J , 1) : K → [ K op Φ , V ]. W e ha v e (Lan E K ( J , X )) E ∼ = Z c ∈ K Φ G ( E − , E c ) · K ( J c, X ) ∼ = Z c K T ( H E − , H E c ) · K ( J c, X ) (b ecause H is fully faithful) = Z c K T ( F T J − , F T J c ) · K ( J c, X ) ∼ = Z c K ( J − , T J c ) · K ( J c, X ) (b y adjoint ness) ∼ = K ( J , X ) ∗  K ( J , 1) T  (b y the co end form ula for w eigh ted colimits) ∼ = K ( J , 1)  K ( J , X ) ∗ T  (b ecause K ( J, 1) preserv es Φ-ﬂat colimits and K ( J, X ) is Φ-ﬂat) ∼ = K ( J , 1)  T X  (b ecause T is Φ-accessible) = K ( J, T X ) Th us the functor part of S restricts to T . In fact the monad itself restricts, and so we conclude that a T -algebra is an S -algebra (an ob ject of [ G op , V ]) whose underlying ob ject (restriction along E ) is in the image of K ( J, 1). But this is exactly the deﬁnition of mo dels of L . Similarly , a morph ism of T -algebras is the same as a morphism of the corresp onding S -algebras, whence the result.  W e ha ve asso ciated a Lawv ere Φ-theory L to ev ery Φ -acc essible V -monad on K . This pro cess is clearly fun ctorial , giving a functor th : Mnd Φ ( K ) → L aw Φ ( K ). Theorem 6.7 The functors mnd and th form an e quivalenc e of c ate gories Mnd Φ ( K ) ≃ La w Φ ( K ) . Pr oof : The previous theorem giv es an isomorphism mnd ◦ th ∼ = 1. F or th e other isomorphism th ◦ mnd ∼ = 1, let E : K op Φ → L b e a La wvere th eory , and T th e induced monad mnd ( L ). Then th ( mnd ( L )) can b e obtained by factorizing F T J : K Φ → Φ -Cts ( L , V ) as an id en tit y- on-ob ject f unctor follo wed by a fully faithful one. No w to form F T J c , for c ∈ K Φ , we send J c to 25 K ( J, J c ) : K op Φ → V , then to its left Kan extension Lan E K ( J, J c ) : L → V , and then reﬂect this in to Φ - C ts ( L , V ). But K ( J, J c ) ∼ = K Φ ( − , c ), whose left Kan extension alo ng E is L ( − , E c ), and this is already in Φ -Cts ( L , V ). But since F T J send s c to L ( − , E c ), its identit y-on-ob ject/fully- faithful factorizatio n give s just L .  W e n o w tur n to the Φ-theory generated by a La wv ere Φ-theory L . Eve ry represen table fu nctor L ( L, − ) : L → V is a m o del of L , and so w e get a fully faithful em b edding Y : L op → Φ -Cts ( L , V ). F orm the closure of the representables in Φ - Cts ( L , V ) und er Φ-colimits. This giv es fully faithful K : L → T and P : T → Φ -Cts ( L , V ) op . Clearly H pr eserv es Φ-limits, while P p reserv es those Φ-limits in the image of E . No w T is a small V -category with Φ-limits; that is, a Φ-theory . F urthermore, it has the same mo dels as L . This is really a sp ecial case of [11, Prop osition 6.23 ], but w e outline here the argumen t. First of all th e comp osite L op K / / T op P / / Φ -Cts ( L , V ) is dense, and b oth K and P are fully f aithfu l. It follo ws by [11, Theorem 5.13 ] that b oth K and P are dense, and that P ∼ = Lan K ( P K ). Since P is dense, the indu ced fun ctor Φ -Cts ( L , V )( P, 1) : Φ -Cts ( L , V ) → [ T , V ] is fully faithful. W e m ust sho w that its imag e is exactly the Φ-con tinuous functors. No w P : T op → Φ -Cts ( L , V ) is Φ-cocontin u ou s , by constru ction of T op , and so the in- duced Φ -Cts ( L , V )( P − , M ) : T → V will b e Φ-con tin uous for all mo dels M . Th is pro v es that Φ -Cts ( L , V )( P, 1) tak es v alues among the Φ-con tinuous fun ctors. Con versely , let G : T → V b e Φ-con tin uous. Th en GK E is Φ-con tinuous, and so GK is (isomorphic to) a mo del; and indeed the mo del can b e calculated as GK ∗ P . Thus GK ∼ = Φ -Cts ( L , V )( P K − , GK ∗ P ) . 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