aa r X i v : . [ m a t h . C T ] S e p ENRICHED LOCALLY GENERATED CATEGORIES
I. DI LIBERTI AND J. ROSICK ´Y
Abstract.
We introduce the notion of M -locally generated category for a factor-ization system ( E , M ) and study its properties. We offer a Gabriel-Ulmer dualityfor these categories, introducing the notion of nest. We develop this theory alsofrom an enriched point of view. We apply this technology to Banach spaces show-ing that it is equivalent to the category of models of the nest of finite-dimensionalBanach spaces. Contents
1. Introduction 12. Locally generated categories 33. Extended Gabriel-Ulmer duality 94. Enriched locally generated categories 125. Enriched Gabriel-Ulmer duality 166. Banach spaces 17References 211.
Introduction
Locally presentable categories were introduced by Gabriel and Ulmer [11] in 1971and, since then, their importance was steadily growing. Today, they form an estab-lished framework to do category theory in the daily practice of the working mathe-matician. The reason of their success is merely evidence-based. On the one hand,they are technically handy, allowing transfinite constructions (e.g. the small objectargument) and thus offering pleasing versions of relevant tools (e.g. the adjoint func-tor theorem). On the other hand, a vast majority of relevant categories happento be locally presentable (with some very disappointing exceptions, like the cate-gory of topological spaces). Gabriel and Ulmer [11] also introduced locally generatedcategories. While locally presentable categories are based on the classical algebraicconcept of a (finitely) presentable object, locally generated categories are based on
Date : September 23, 2020.The first author was supported by the Grant Agency of the Czech Republic project EXPRO20-31529X and RVO: 67985840. The second author was supported by the Grant Agency of theCzech Republic under the grant 19-00902S. (finitely) generated objects. This makes them easier to handle then locally pre-sentable ones. Although the both classes of categories coincide, a category can belocally finitely generated without being locally finitely presentable. Despite this, lo-cally generated categories were somewhat neglected. They were mentioned in [1] and,later, generalized in [2]. In a nutshell, a M -locally generated category is a cocompletecategory K , equipped with a factorization system ( E , M ), that is generated by a setof M -generated objects. In [11] they were only dealing with (Strong Epi, Mono), in[2], monomorphisms were replaced by any right part of a proper factorization system( E , M ).We slightly generalize [2] by taking any factorization system ( E , M ).A locally λ -presentable category is then a K → -locally λ -generated category. More importantly, weextend the (nearly forgotten) Gabriel-Ulmer duality for locally generated categoriesto our setting. While the famous Gabriel-Ulmer duality for locally finitely presentablecategories is based on the fact that they are sketched by a finite limit sketch, theirduality for locally finitely generated categories uses finite limits and multiple pullbacksof monomorphisms. This explains why locally finitely generated categories do notneed to be locally finitely presentable. In our situation, we use multiple pullbacks ofmorphisms from M .Our main goal is to apply these ideas to Banach spaces. For this, we have to makeour theory enriched. While enriched locally presentable categories were treated byKelly [16], enriched locally generated categories have been never considered. Ourmotivation was [3] where finite-dimensional Banach spaces, which are not finitelypresentable, were shown to be finitely generated w.r.t. isometries when Banach spacesare enriched over CMet , the category of complete metric spaces. We provide anappropriate framework for the observation [3] 7.8 that Banach spaces are locallyfinitely generated w.r.t. isometries. The relevant factorization system is (dense maps,isometries). Our main accomplishment is that Banach spaces can be sketched usingfinite-dimensional Banach spaces equipped with finite weighted limits and multiplepullbacks of isometries. This has turned out to be more delicate than expected,because
CMet is a quite ill-behaved enrichment base.In Section 2 we introduce the notion of a locally generated category, providingexamples and properties. For this, we need the concept of a λ -convenient factorizationsystem ( E , M ), which seems to be of its own interest. We discuss the main propertiesof λ -generated objects w.r.t. M . We provide a recognition theorem (2.22) for locally λ -generated categories which meets the spirit of the original definition of Gabriel andUlmer. In Section 3 we introduce the notion of a λ -nest, a sketch-like gadget thatwe use to recast a suitable version of Gabriel-Ulmer duality for M -locally generatedcategories. We introduce the notion of a model of a nest, and show that the categoryof models of a nest is a M -locally generated category. In Sections 4 and 5 we presentan enriched version of Sections 2 and 3, when the enrichment base V is locally λ -presentable as a closed category. Section 6 discusses the case of Banach spaces indetail, focusing on their axiomatizability via the ( ℵ , CMet )-nest
Ban fd . NRICHED LOCALLY GENERATED CATEGORIES 3 Locally generated categories
Locally λ -generated categories were introduced by Gabriel and Ulmer [11] as cocom-plete categories K having a strong generator consisting of λ -generated objects suchthat every λ -generated object has only a set of strong quotients. Here, an object A is λ -generated if its hom-functor K ( A, − ) : K →
Set preserves λ -directed colimitsof monomorphisms. A category is locally generated if it is locally λ -generated forsome regular cardinal λ . [11] showed that locally generated categories coincide withlocally presentable ones. Every locally λ -presentable category is locally λ -generatedbut a locally λ -generated category does not need to be locally λ -presentable. Thusthe passage to locally generated categories can lower the defining cardinal λ .The definition of a locally λ -generated category was reformulated in [1]: a cocom-plete category K is locally λ -generated if it has a set A of λ -generated objects suchthat every object is a λ -directed colimit of its subobjects from A . The fact that thesetwo definitions are equivalent follows from [1] 1.70 and [11] 9.2.A further step was made in [2] where a cocomplete category K with a properfactorization system ( E , M ) was called M -locally λ -generated if it has a set A of λ -generated objects w.r.t. M such that every object is a λ -directed colimit of A -objectsand M -morphisms. Here, an object A is λ -generated w.r.t. M if its hom-functor K ( A, − ) : K →
Set preserves λ -directed colimits of M -morphisms. Again, [2] provesthat a category is M -locally generated for some proper factorization system ( E , M )iff it is locally presentable. Moreover, locally λ -generated categories are M -locally λ -generated ones for the factorization system (strong epimorphisms, monomorphisms).We extend this definition a little bit. Definition 2.1.
Let K be a category with a factorization system ( E , M ) and λ aregular cardinal. We say that an object A is λ -generated w.r.t. M if its hom-functor K ( A, − ) : K →
Set preserves λ -directed colimits of M -morphisms. Examples 2.2. (1) For the factorization system (Iso , K → ), an object is λ -generatedw.r.t. K → iff it is λ -presentable.(2) For the factorization system ( K → , Iso), every object is λ -generated w.r.t. Iso.(3) For the factorization system (StrongEpi , Mono), an object is λ -generated w.r.t.Mono iff it is λ -generated. Definition 2.3.
Let λ be a regular cardinal. A factorization system ( E , M ) in acategory K will be called λ -convenient if(1) K is E -cowellpowered, i.e., if every object of K has only a set of E -quotients,and(2) M is closed under λ -directed colimits, i.e., every λ -directed colimit of M -morphisms has the property that(a) a colimit cocone consists of M -morphisms, and(b) for every cocone of M -morphisms, the factorizing morphism is in M .( E , M ) is convenient if it is λ -convenient for some λ . I. DI LIBERTI AND J. ROSICK ´Y
Remark 2.4.
If ( E , M ) is λ -convenient and λ ′ ≥ λ is regular then ( E , M ) is λ ′ -convenient. Definition 2.5.
Let K be a cocomplete category with a λ -convenient factorizationsystem ( E , M ) where λ a is regular cardinal. We say that K is M -locally λ -generated if it has a set A of λ -generated objects w.r.t. M such that every object is a λ -directedcolimit of objects from A and morphisms from M . K is called M -locally generated if it is M -locally λ -generated for some regularcardinal λ . Remark 2.6.
In the same way as in II. (i) of the proof of [1] 1.70, we show that A is dense in K . Hence the canonical functor E : K →
Set A op ( sending K to K ( − , K ) restricted on A op ) is fully faithful. Since K is cocomplete, E is a rightadjoint. Moreover, E preserves λ -directed colimits of M -morphisms. Consequently,in a M -locally λ -generated category, λ -small limits commute with λ -directed colimitsof M -morphisms. Notation 2.7.
In what follows, K λ will denote the full subcategory of K consistingof λ -generated objects w.r.t. M and by E λ the set E ∩ K → λ . Examples 2.8. (1) The factorization system (Iso , K → ) is convenient and K is locally λ -presentable iff it is K → -locally λ -generated.(2) The category K is Iso-locally generated iff it is small. In fact, every object K of K has a morphism ` A → K where the domain is the coproduct of all objects A from A . Thus ( K → , Iso) is convenient iff K is small.(3) A Mono-locally λ -generated category K in the sense of [11] and [1] satisfies2.5 for the factorization system (strong epi, mono). Then this factorization systemis λ -convenient. Indeed, following 2.6 the canonical functor E : K →
Set A op isfully faithful and preserves λ -directed colimits of monomorphisms. This implies thatmonomorphisms are closed in K under λ -directed colimits (in the sense of 2.3) becausethis holds in Set A op and E preserves and reflects monomorphisms. Now, we can showthat A is closed under strong quotients. Let e : A → B be a strong epimorphismwith A ∈ A , m i : K i → K a λ -directed colimit of monomorphisms and f : B → K .Since A is λ -generated, there is g : A → K i such that m i g = f e . Since m i is amonomorphism, the diagonalization property yields h : B → K i such that m i h = f .Hence B is λ -generated. Then II. in the proof of [1] 1.70 yields that K is locallypresentable, hence cowellpowered. This verifies 2.3 (1), hence the (strong epi, mono)factorization system is λ -convenient. We gave this argument in detail because [1] isnot accurate at this point.(4) For a factorization system (Epi , StrongMono), we get StrongMono-locally λ -generated categories. Here, we have to assume that this factorization system is λ -convenient because it does not seem that we could get it for free like in (3). NRICHED LOCALLY GENERATED CATEGORIES 5 (5) Any full reflective subcategory L of K determines a factorization system ( E , M )where E consists of morphisms sent by the reflector to an isomorphism (see [22]). Both(1) and (2) are special cases for L = K or L consisting just from a terminal objects. Notation 2.9.
We say that a category is locally λ -generated if it is M -locally λ -generated for some M . Locally λ -generated categories in the sense of [11] are thenMono-locally λ -generated. Lemma 2.10. λ -generated objects w.r.t. M are closed under E -quotients.Proof. Let e : A → B is in E and A ∈ A . Consider a λ -directed colimit m i : K i → K , i ∈ I of M -morphisms and f : B → K . Since A is λ -generated w.r.t. M , there is g : A → K i such that m i g = f e . Since m i is in M , the diagonalization propertyyields h : B → K i such that m i h = f . Assume that m i h ′ = f for h ′ : B → K i .Then m i he = m i h ′ e and, since A is λ -generated w.r.t. M , there is i ≤ j ∈ I suchthat m ij he = m ij h ′ e . Hence both m ij h and m ij h ′ are diagonals in the commutativesquare A e / / m ij he (cid:15) (cid:15) B f (cid:15) (cid:15) K j m j / / K and thus they are equal. Hence B is λ -generated. (cid:3) Lemma 2.11.
In an M -locally λ -generated category K , λ -generated objects w.r.t. M are closed under λ -small colimits and, up to isomorphism, form a set.Proof. Let colim A j be a λ -small colimit of λ -generated objects w.r.t. M and colim K i a λ -directed colimit of M -morphisms. Since λ -directed colimits commute in Set with λ -small limits, we have K (colim A j , colim K i ) ∼ = lim j K ( A j , colim K i ) ∼ = lim j colim i K ( A j , K i ) ∼ = colim i lim j K ( A j , K i ) ∼ = colim i K (colim j A, K i )Hence colim A j is λ -generated w.r.t. M .The second claim follows from the fact that every λ -generated object w.r.t. M isis a λ -directed colimit of A -objects and M -morphisms, which makes it a retract ofan A -object, i.e., a finite colimit of A -objects. Indeed, given a retract u : B → A with a section s : A → B , then s is a coequalizer of us and id A . (cid:3) I. DI LIBERTI AND J. ROSICK ´Y
Remark 2.12.
For the first claim, we do not need to assume that K is M -locally λ -generated. Notation 2.13.
Given a family of arrows E , we call E ⊥ the family of arrows thatare orthogonal to E . This means that an arrow m belongs to E ⊥ if it has the uniqueright lifting property with respect to every arrow in E . An object A is orthogonal to E if the unique morphism from A to the terminal object is in E ⊥ . Lemma 2.14. If K is M -locally λ -generated then M = ( E λ ) ⊥ .Proof. Every e : A → B in E is a λ -directed colimit ( a i , b i ) : f i → e in K → λ . Every f i : A i → B i has the factorization f i = m i e i with e i : A i → C i in E and m i : C i → B i in M . Then e = colim m i · colim e i where colim e i : A → C and colim m i : C → B . Since C = colim C i is a λ -directed colimit of M -morphisms and ( E , M ) is λ -convenient,colim m i is in M and thus it is an isomorphism. Hence e = colim e i . Consequently M = E ⊥ λ . (cid:3) Remark 2.15.
We have shown that, in a M -locally λ -generated category K , E isthe closure colim( E λ ) of E λ under all colimits in K → .Conversely, if M = E ⊥ λ then M satisfies 2.3(2).The proof of the next theorem follows [2] and, like [2], is based on [11]. However,[2] lacks the assumption that ( E , M ) is convenient. Theorem 2.16.
For every category K equivalent are: (1) K is locally presentable, (2) K is M -locally generated for some convenient factorization system ( E , M ) ,and (3) K is M -locally generated for every convenient factorization system ( E , M ) .Proof. Clearly, (3) ⇒ (1) ⇒ (2). The implication (1) ⇒ (3) is analogous to I. of theproof [1] 1.70.(2) ⇒ (1) follows [2]. In fact, let K be M -locally λ -generated. Then E : K →
Set A op makes K equivalent to a full reflective subcategory of L = Set A op closedunder λ -directed colimits of M -morphisms (see 2.6). Let P consist of reflections of L -objects in K which are either λ -small colimits in L of diagrams in A or codomainsof multiple pushouts in L of E -morphisms with a domain in A . Observe that P issmall: the case of λ -small colimits is clear, for the multiple pushout use 2.10 andthe fact that every λ -generated object w.r.t. M is a retract of an A -object. Thus,the class Ort( P ) of all L -objects orthogonal to P is a locally presentable category.Moreover, K is closed in Ort( P ) under(1) λ -small colimits of diagrams in A ,(2) multiple pushouts of E -morphisms with a domain in A , and(3) λ -directed colimits of M -morphisms. NRICHED LOCALLY GENERATED CATEGORIES 7
Consider a morphism f : A → L with A in A and L in Ort( P ). Let e f : A → K f be the cointersection in K of all E -morphisms through which f factors. Following(2) above, there is m f : K f → L such that f = m f e f . Clearly, e f is in E and m f is E -extremal in the sense that any E -morphism K f → K through which m f factors isan isomorphism. Moreover, this factorization is ”functorial”: given f ′ : A ′ → L and h : A → A ′ with f = f ′ h there exists h ∗ : K f → K f ′ in M with m f = m f ′ h ∗ . In fact,form a pushout A e f ′ h / / e f (cid:15) (cid:15) K f ′ ˜ e f (cid:15) (cid:15) K f h ∗ / / K There is g : K → L such that gh ∗ = m f and g ˜ e f = m f ′ . Since ˜ e f is in E and m f ′ is E -extremal, ˜ e f is an isomorphism.Every L in Ort( P ) is a λ -directed colimit d i : D i → L where D i are λ -small colimitsof diagrams in A . Form reflections r i : D i → D ′ i in K . Since r i ∈ P , d i factors through r i , say, d i = d ′ i r i . Since D ′ i is in A , we can take the factorization d ′ i = m i e i above with m i : D ′′ i → L . Since m i is E -extremal, d ′′ i is in M . Thus m i : D ′′ i → L is a λ -directedcolimit of M -morphisms, which implies that L is in K .We have proved that K = Ort( P ), hence K is locally presentable. (cid:3) Remark 2.17. (1) All implications, except (2) ⇒ (1), are valid for a given λ . In(2) ⇒ (1), λ can increase.(2) The previous theorem sheds a light on the notion of locally generated category.Indeed, the framework of locally generated categories is not more expressive than thatof locally presentable ones. Yet, as pointed out (1), it might happen that a categoryis locally finitely generated without being locally finitely presentable. From the pointof view of essentially algebraic theories (see 3.D in [1]), this means that some λ -aryessentially algebraic theories can be axiomatized by the data of a finite limit theoryand a factorization system. Thus, the complexity of infinitary (partial) operationscan be hidden under the carpet of the factorization system. From a less technicalperspective, the factorization system that makes the category locally generated isvery often an extremely natural one to consider. A sketch-oriented (see 2.F in [1])interpretation of this discussion is the core motivation for the forthcoming notion of λ -nest. Remark 2.18.
We do not know whether, given a locally λ -presentable category K ,one can find an ℵ -convenient factorization system ( E , M ) such that K is M -locallyfinitely generated. Following 2.6, finite limits should commute in K with directedcolimits of M -morphisms. Notation 2.19.
Gen λ ( K ) will denote the full subcategory of λ -generated objectsw.r.t. M in a M -locally λ -generated category K . I. DI LIBERTI AND J. ROSICK ´Y
Remark 2.20.
A multiple pullback P is a limit of a diagram consisting of morphisms f i : A i → A , i ∈ I . We can well-order I as { i , i , . . . , i j . . . } and form pullbacks P j as follows: P is the pullback P f / / ¯ f (cid:15) (cid:15) A f (cid:15) (cid:15) A f / / A Then P is the pullback P p / / (cid:15) (cid:15) P f ¯ f (cid:15) (cid:15) A f / / A We proceed by recursion and in limit steps we take limits. In this way, we transformmultiple pullbacks to limits of smooth well-ordered chains ( smooth means that inlimit steps we have limits).Conversely, a limit of a smooth chain p ij : P j → P i , i ≤ j < α is a multiplepullback of p j , j ∈ I . Moreover, if K is equipped with a factorization system ( E , M )then multiple pullbacks of M -morphisms correspond to limits of smooth well-orderedchains of M -morphisms. Indeed, M is stable under pullbacks. Corollary 2.21.
Let K be an M -locally λ -generated category. Then K is equivalentto the full subcategory of Set
Gen λ ( K ) op consisting of functors preserving λ -small limitsand sending multiple pushouts of E -morphisms to multiple pullbacks.Proof. Following [1] 1.33(8), this describes Ort( P ) from the proof of 2.16. (cid:3) The following theorem makes easier to identify M -locally generated categoriesamong cocomplete categories with a factorization system and goes back to [11]. Theorem 2.22.
A cocomplete category K equipped with a λ -convenient factorizationsystem ( E , M ) is M -locally λ -generated iff it has a strong generator formed by λ -generated objects w.r.t. M .Proof. Using 2.11 and 2.10, the closure A of our strong generator under λ -smallcolimits and E -quotients consists of λ -generated objects w.r.t. M . Following theproof of [1] 1.11, we show that every object in K is a λ -filtered colimit of A -objects.Using ( E , M )-factorizations, we get that every object of K is a λ -directed colimit of A -objects and M -morphisms. (cid:3) Theorem 2.23.
Let K be a M -locally λ -generated category where ( E , M ) is a proper λ -convenient factorization system and let T be a monad preserving λ -directed colim-its of M -morphisms. Then, assuming Vopˇenka’s principle, the category of algebras Alg ( T ) is locally λ -generated. NRICHED LOCALLY GENERATED CATEGORIES 9
Proof.
Following [4] 3.3, Alg( T ) is cocomplete. Consider the adjunction F : K ⇆ Alg ( T ) : U and put M ′ = U − ( M ). Using [5] 4.3.2, we get that M ′ is closed under λ -directed colimits. Following [20] 11.1.5, M ′ = F ( E ) ⊥ and F ( E ) = ⊥ M ′ . Since E = colim E λ (see 2.15), we have F ( E ) = colim( F ( E λ )). U is clearly conservative(and faithful), thus F maps the dense generator Gen λ ( K ) to a strong generator G in Alg ( T ). Let A be the closure of G under λ -small colimits. Following 2.12, A is densein K . Assuming Vopˇenka’s principle, K is locally presentable (see [1] 6.14). Hence,following [10] 2.2, ( F ( E ) , M ′ ) is a factorization system on Alg( T ). Since Alg( T ) iscowellpowered (see [1] 1.58) ad F ( E ) ⊆ Epi, this factorization system ( F ( E ) , M ′ ) is λ -convenient. We conclude the proof by the previous theorem and 3.11. (cid:3) Remark 2.24.
We do not know whether Vopˇenka’s principle is really needed. This isrelated to the Open Problem 3 in [1]. In fact, let L be a full reflective subcategory of alocally λ -presentable category K closed under λ -directed colimits of monomorphisms.Then the monad T = F G , where G : L → K is the inclusion and F its left adjoint,preserves λ -directed colimits of monomorphisms. Since L ∼ = Alg( T ), 2.23 withoutVopˇenka’ s principle would yield a positive solution of the Open Problem 3.3. Extended Gabriel-Ulmer duality
The Gabriel-Ulmer duality is a contravariant biequivalence between categories with λ -small limits and locally λ -presentable categories (see [11] 7.11 or [1] 1.45). We aregoing to to extend this duality to M -locally λ -generated categories. It will also coverthe Gabriel-Ulmer duality for Mono-locally λ -generated categories. In order to do so,we will introduce the notion of a nest. Definition 3.1. A λ -nest is a small category A equipped with a factorization system( E A , M A ) and having λ -small limits and multiple pullbacks of M -morphisms. Remark 3.2. λ -nests A with (Iso , A → ) are precisely small categories with λ -smalllimits. λ -nests with (StrongEpi , Mono) are precisely small ”echt” λ -complete cate-gories of [11]. Example 3.3.
Gen λ ( K ) op is a λ -nest for every M -locally λ -generated category K .This follows from 2.11, 2.10 and the fact that ( M A ∩ A → , E A ∩ A → ) is a factorizationsystem on Gen λ ( K ) op . Notation 3.4.
For a λ -nest A , Mod λ ( A ) denotes the category of all models , i.e., offunctors A →
Set preserving λ -small limits and multiple pullbacks of M -morphisms. Lemma 3.5.
Mod λ ( A ) is a locally presentable category equipped with a factorizationsystem ( E , M ) .Proof. Mod λ ( A ) is a full subcategory of Set A and the codomain restriction of theYoneda embedding Y : A op → Set A is a full embedding of A op to Set A . Following [1]1.51, Mod λ ( A ) is locally presentable. The factorization system on A op is ( M A , E A ) and the factorization system ( E , M ) is given, following [10] 2.2, as: E = colim( M A )and M = M ⊥A . (cid:3) Theorem 3.6.
Mod λ ( A ) is a M -locally λ -generated category for every λ -nest A .Proof. I. At first, we will show that K = Mod λ ( A ) is closed in L = Set A under λ -directed colimits of M -morphisms. Following [1] 1.33(8), K = Ort( P ) where P isdefined as in the proof of 2.16. This means that P consists of reflections of L -objectsin K which are either λ -small colimits of diagrams in A op or codomains of multiplepushouts of M A -morphisms in A op .Let k i : K i → K be a colimit in L of a λ -directed diagram of M -morphisms in K .Let r : colim A j → A be a reflection of a λ -small colimit in L of a diagram in A op ;its reflection in K lies in A op . It is easy to see that objects orthogonal to r are closedunder λ -directed colimits; in fact, these objects correspond to functors A →
Set preserving λ -small limits.Let r : P → A be a reflection of a multiple pushout of M A -morphisms. Alterna-tively, P can be seen as a colimit of a well ordered smooth chain P p −−−→ P p −−−→ P p −−−→ . . . of M A -morphisms. Let f : P → K . There exists i and g : P → K i such that k i g = f p . There exists i > i and g ′ : P → K i such that k i i g = g ′ p .Since k i i ∈ M ⊥A and p ∈ M A , there is g : P → K i such that g p = g and k i i g = g ′ . Continuing this procedure by taking colimits in limit steps, we get acocone g j : P j → K i inducing g : P → K i such that k i g = f . There is h : A → K i with hr = g . Hence k i hr = f . The uniqueness of this extension follows from A being in A op .II. From I., it follows that every object of A op is λ -generated w.r.t. M in K .Following 3.5 and 2.15, M satisfies 2.3(2). In the same way as in 2.10, we show thatevery E -quotient of an A op -object is λ -generated w.r.t. M . Since every λ -generatedobject w.r.t. M is a retract of an A op -object (cf. 2.11) and A op is closed underretracts, A op = Gen λ ( K ). Using ( E , M ) factorizations, we show that every object in K is a λ -directed colimit of A op -objects.It remains to show that K is E -cowellpowered. Let e : K → L be an E -quotientof K . Then e is a λ -directed colimit of E -morphisms e i : K i → L i where K i and L i are in A op . Since there is only set of expressions of K as a λ -directed colimit of A op -objects, there is only a set of E -quotients of K . (cid:3) Remark 3.7.
We have
A ≃ (Gen λ (Mod λ ( A ))) op for every λ -nest A , and K ≃
Mod λ (Gen λ ( K ) op )for every M -locally λ -generated category K . NRICHED LOCALLY GENERATED CATEGORIES 11
Note that these equivalences also include the corresponding factorization systems.In the first case, (colim( M A ) , M ⊥A ) restricts to ( M A , E A ) (see 3.5) and, in the secondcase, it follows from 2.15. Remark 3.8.
For a λ -nest A , Mod λ ( A ) is a full subcategory of the free comple-tion Ind λ ( A ) of A under λ -directed colimits. Indeed, the latter category consists offunctors A op → Set preserving λ -small limits. Definition 3.9. A morphism of locally λ -generated categories R : K → L is a rightadjoint preserving M -morphisms and λ -directed colimits of them. Remark 3.10. R preserves M if and only if its left adjoint L : L → K preserves E (see [20] 11.1.5). Lemma 3.11.
Let R : K → L be a morphism of locally λ -generated categories. Thenits left adjoint L preserves λ -generated objects w.r.t. M .Proof. Let A be λ -generated w.r.t. M in L . Consider a directed colimit colim K i of M -morphisms in K . Then K ( LA, colim K i ) ∼ = L ( A, R colim K i ) ∼ = L ( A, colim RK i ) ∼ = colim L ( A, K i ) ∼ = colim K ( LA, K i ) . (cid:3) Definition 3.12. A morphism of λ -nests F : A → B is a functor preserving λ -smalllimits, M -morphisms and multiple pullbacks of them. Notation 3.13.
Let LG λ be the 2-category of locally λ -generated categories and N λ be the 2-category of λ -nests, in the both cases 2-cells are natural transformations. Construction 3.14 (The functor Gen λ ) . Given a locally λ -generated category K ,we have defined Gen λ ( K ) op to be the opposite category of its λ -generated objects. Itis easy to see that this construction is (contravariantly) functorial. Indeed, given amorphism of locally λ -generated categories R : K → L , its left adjoint L restricts to λ -generated objects L : Gen λ ( L ) → Gen λ ( K )by 3.11, and passing to the opposite category, is a morphism of λ -nests because of2.10 and 2.11. Construction 3.15 (The functor Mod λ ) . Given a λ -nest A , we have seen thatthe category Mod λ ( A ) is locally λ -generated. We will extend this construction toa (contravariant) functor. Given a morphism of λ -nests F : A → B , the functorMod λ ( F ) sends H from Mod λ ( B ) to HF . This functor is the domain restriction of2 I. DI LIBERTI AND J. ROSICK ´Y
Let LG λ be the 2-category of locally λ -generated categories and N λ be the 2-category of λ -nests, in the both cases 2-cells are natural transformations. Construction 3.14 (The functor Gen λ ) . Given a locally λ -generated category K ,we have defined Gen λ ( K ) op to be the opposite category of its λ -generated objects. Itis easy to see that this construction is (contravariantly) functorial. Indeed, given amorphism of locally λ -generated categories R : K → L , its left adjoint L restricts to λ -generated objects L : Gen λ ( L ) → Gen λ ( K )by 3.11, and passing to the opposite category, is a morphism of λ -nests because of2.10 and 2.11. Construction 3.15 (The functor Mod λ ) . Given a λ -nest A , we have seen thatthe category Mod λ ( A ) is locally λ -generated. We will extend this construction toa (contravariant) functor. Given a morphism of λ -nests F : A → B , the functorMod λ ( F ) sends H from Mod λ ( B ) to HF . This functor is the domain restriction of2 I. DI LIBERTI AND J. ROSICK ´Y the functor Ind λ ( B ) → Ind λ ( A ) given, again, by precompositions with F . The latterfunctor has the left adjoint Ind λ ( F ) : Ind λ ( A ) → Ind λ ( B ). The domain restriction L ( F ) of this left adjoint is a left adjoint to Mod λ ( F ). The functor L ( F ) preserves E -morphisms because L ( F )( E Mod λ ( A ) ) = L ( F )(colim M A ) ⊆ colim( F ( M A )) ⊆ E Mod λ ( B ) . Thus Mod λ ( F ) preserves M -morphisms and, therefore, it is a morphism of λ -generatedcategories. Theorem 3.16.
Gen λ : LG λ ⇆ N op λ : Mod λ is a dual biequivalence between locally λ -generated categories and λ -nests.Proof. It follows from 3.7 and 3.15. (cid:3)
Remark 3.17. (1) Our duality restricts to the standard Gabriel-Ulmer duality be-tween locally λ -presentable categories and small categories with λ -small limits (for M = Iso) and, a little bit forgoten, Gabriel-Ulmer duality for locally λ -generatedcategories (see [11] 9.8). In the second case M = Mono and the dual is formed bysmall categories with λ -small limits and pullbacks of strong monomorphisms.On the other hand, our extended Gabriel-Ulmer duality is a restriction of thestandard one (see 3.15).(2) Like the Gabriel-Ulmer duality (see [18]), our duality is given by the category Set being both a large λ -nest and a locally λ -generated category. Clearly, models ofa λ -nest A are λ -nest morphisms A →
Set . Conversely, a morphism U : K →
Set oflocally λ -generated categories is uniquely determined by its left adjoint F : Set → K ,these restrictions uniquely corespond to objects in Gen λ ( K ).(3) [8] generalized the Gabriel-Ulmer duality to certain limit doctrines. This isbased on the commutation of certain limits and colimits in Set . But our duality(even that for Mono-locally λ -generated categories) does not fall under this scope.4. Enriched locally generated categories
In what follows, V will be a complete and cocomplete symmetric monoidal closedcategory. We will work with V -categories and under a λ -directed colimit we willmean a conical λ -directed one. V -factorization systems were introduced in [9] andtheir theory was later developed by Lucyshyn-Wright in [17]. We refer to [17] for themain definitions and notations. Definition 4.1.
Let K be a V -category with a V -factorization system ( E , M ) and λ aregular cardinal. We say that an object A is λ -generated w.r.t. M if its hom-functor K ( A, − ) : K → V preserves λ -directed colimits of M -morphisms. Remark 4.2.
In a tensored V -category K , V -factorization systems are precisely fac-torization systems ( E , M ) such that E is closed under tensors (see [17] 5.7). NRICHED LOCALLY GENERATED CATEGORIES 13
Definition 4.3. A V -factorization system ( E , M ) in a tensored V -category K is called λ -convenient if it is λ -convenient as an (ordinary) factorization system. Definition 4.4.
Let K be a cocomplete V -category with a λ -convenient V -factori-zation system ( E , M ) where λ a is regular cardinal. We say that K is M -locally λ -generated if it has a set A of λ -generated objects w.r.t. M such that every objectis a λ -directed colimit of objects from A and morphisms from M . K is called M -locally generated if it is M -locally λ -generated for some regularcardinal λ . Example 4.5.
For a cocomplete V -category K , (Iso , K → ) is a convenient V -factori-zation system. K → -locally λ -generated categories are locally presentable V -categoriesin the sense of [16]. Remark 4.6.
In 4.4, A is dense in K , i.e., the canonical V -functor E : K → V A op is fully faithful. In fact, E preserves λ -directed colimits of M -morphisms and, then,the result follows from [15], 5.19. Remark 4.7.
For a V -category K , the arrow category K → is a V -category with K ( f, g ) defined by the following pullback in VK → ( f, g ) / / (cid:15) (cid:15) K ( A, C ) K ( A,g ) (cid:15) (cid:15) K ( B, D ) K ( f,D ) / / K ( A, D )where f : A → B and g : C → D . Proposition 4.8.
Assume that pullbacks commute with λ -directed colimits in V .Then, for a M -locally λ -generated V -category K , the V -category K → is M → -locally λ -generated.Proof. Following [20] 13.1, K → is tensored and cotensored. Since K → has limits andcolimits (calculated pointwise), it is complete and cocomplete (see [5] 6.6.16). Thefactorization system ( E → , M → ) on K → is defined pointwise from that on K and isclearly λ -convenient. We will show that any morphism f : A → B with A and Bλ -generated w.r.t. M is λ -generated w.r.t. M → .Consider a λ -directed colimit g i → g on M → -morphisms in K → . We have pullbacks K → ( f, g g i ) / / (cid:15) (cid:15) K ( A, C i ) K ( A,g i ) (cid:15) (cid:15) K ( B, D i ) K ( f,D i ) / / K ( A, D i ) Since pullbacks commute with λ -directed colimits in V , K → ( f, g ) ∼ = colim K → ( f, g i ) . Clearly, every h in K → is a λ -directed colimit of λ -generated objcts w.r.t. M → and M → -morphisms. (cid:3) Lemma 4.9.
Assume that λ -small (conical) limits commute in V with λ -directedcolimits. Then, in an M -locally λ -generated V -category, λ -generated objects w.r.t. M are closed under λ -small (conical) colimits.Proof. The proof for λ -small conical colimits is the same as that of 2.11 and the prooffor λ -small weighted limits is analogous (cf. [6] 3.2). (cid:3) Remark 4.10.
For the first claim, we do not need to assume that K is M -locally λ -generated. Lemma 4.11.
Assume that finite conical limits commute with λ -directed colimits in V . Then, in an M -locally λ -generated V -category, λ -generated objects w.r.t. M areclosed under E -quotients.Proof. Let e : A → B is in E and A ∈ A . Express B as a λ -directed colimit m i : B i → B , i ∈ I , of A -objects and M -morphisms. Form pullbacks P i ¯ e / / ¯ m i (cid:15) (cid:15) B im i (cid:15) (cid:15) A e / / B Since pullbacks commute with λ -directed colimits in V , they do it in V A op as well.Since E preserves pullbacks and λ -directed colimits of M -morphisms, pullbacks com-mute with λ -directed colimits of M -morphisms in K . Hence ¯ m i : P i → A is a λ -directed colimit. Since M is λ -convenient, m i ∈ M for every i ∈ I . Hence ¯ m i ∈ M for every i ∈ I and, thus, ¯ m ij : P i → P j are in M for every i < j ∈ I . Since A is λ -generated w.r.t. M , m i splits for some i ∈ I . Thus there exists s : A → P i suchthat ¯ m i s = id A . We have m i ¯ es = e ¯ m i s = e. The commutative square A e / / ¯ es (cid:15) (cid:15) B id B (cid:15) (cid:15) B i m i / / B has the diagonal t : B → B i making B a retract of B i . Since t is a coequalizer ofid B i and tm i , 4.9 implies that B is λ -generated w.r.t. M . (cid:3) NRICHED LOCALLY GENERATED CATEGORIES 15
Lemma 4.12. If K is M -locally λ -generated V -category then M = ( E λ ) ⊥ V .Proof. It follows from 4.2, 2.14 and [17] 5.4. (cid:3)
Remark 4.13. (1) Given a small V -category, every V -functor H : A → V is aweighted colimit of representable V -functors. If V is locally λ -presentable as a closedcategory (see [16] 5.5) then, for a small category A with λ -small limits, any functor H : A → V preserving λ -small limits is a λ -filtered (and thus λ -directed) conicalcolimit of representable functors (see [6] 4.5). In such a V , λ -small limits commutewith λ -directed colimits (see [6] 2.4).(2) For a general V , let H : A → V preserve λ -small conical limits and consider thecategory Y ( A op ) ↓ H of representable functors over H . This category is λ -filteredand let H ∗ be its colimit in V A and γ : H ∗ → H be the comparison morphism. If A is a λ -nest then, due to ( E , M ) factorizations in A , the category Y ( A op ) ↓ H has acofinal subcategory D whose morphisms are Y ( m ) : Y ( A ) → Y ( B ) where m are in M . D is λ -directed and H ∗ is it colimit of the projection D : D → V A with a cocone ϕ d : Dd → H ∗ .Assume that V is equipped with a λ -convenient factorization system ( E V , M V ) andthat hom-functors A ( A, − ) : A → V send M -morphisms to M V -morphisms. Them,for every A in A , we have a λ -directed colimit ( ϕ d ) A : Dd ( A ) → H ∗ ( A ) of M V -morphisms in V . Since the factorization system ( E , M ) is λ -convenient, ( ϕ d ) A are in M .Assume that V is M V -locally λ -generated and that A has cotensors with λ -generated objects V w.r.t. M V . If H preserves these cotensors then the argumentfrom [6] 4.5 yields that γ is an isomorphism. Theorem 4.14.
Let V be locally λ -presentable as a closed category. Then, for every V -category K equivalent are: (1) K is locally presentable, (2) K is M -locally generated for some convenient V -factorization system ( E , M ) ,and (3) K is M -locally generated for every convenient V -factorization system ( E , M ) .Proof. We proceed like in 2.16. In the implication (2) ⇒ (1), we take Gen λ ( K ) for A .Following 4.9, P is closed under finite tensors. Analogously as in 4.12, we get that P ⊥ = P ⊥ V . Hence Ort( P ) consists of objects V -orthogonal to P . Finally, following4.13(1), every L in Ort( P ) is a λ -directed colimit of objects from Gen λ ( K ). (cid:3) Corollary 4.15.
Let V be locally λ -presentable as a closed category and K be an M -locally λ -generated V -category. Then K is equivalent to the full subcategory of V Gen λ ( K ) op consisting of V -functors preserving λ -small limits and sending multiplepushouts of E -morphisms to multiple pullbacks. Proof.
We proceed analogously as in 2.21. Note that [1] 1.33(8) is true in the enrichedsetting too but m i there should be m i : colim hom( Dd, − ) → hom(lim Dd, − ) . (cid:3) Notation 4.16.
Again, we say that a V -category is locally λ -generated if it is M -locally λ -generated for some M . Theorem 4.17.
Assume that λ -small conical limits commute in V with λ -directed col-imits. Then a cocomplete V -category K equipped with a λ -convenient V -factorizationsystem ( E , M ) is M -locally λ -generated iff its underlying category K has a stronggenerator formed by λ -generated objects w.r.t. M .Proof. Using 4.9 and 4.11, the closure A of our strong generator under λ -small conicalcolimits and E -quotients consists of λ -generated objects w.r.t. M . Then we followthe proof of 2.22. (cid:3) Enriched Gabriel-Ulmer duality
Definition 5.1.
A ( λ, V ) -nest is a small V -category A equipped with a V -factorizationsystem ( E A , M A ) and having λ -small limits and multiple pullbacks of M -morphisms. Example 5.2.
Gen λ ( K ) op is a ( λ, V )-nest for every M -locally λ -generated V -category K . This follows from 4.9, 4.11 and the fact that ( M A ∩ A → , E A ∩ A → ) is a V -factorization system on Gen λ ( K ) op . Remark 5.3.
Let V be locally λ -presentable as a closed category. Then a small V -category A is a ( λ, V )-nest if and only is it is a λ -nest and M A is closed under λ -small cotensors.We get this analogously to 4.2 applied to A op . Notation 5.4.
For a ( λ, V )-nest A , Mod λ ( A ) denotes the category of all V -functors A → V preserving λ -small limits and multiple pullbacks of M -morphisms. Lemma 5.5.
Let V be locally λ -presentable as a closed category. Then Mod λ ( A ) isa locally presentable V -category equipped with a V -factorization system ( E , M ) .Proof. We proceed like in 3.5 with [1] 1.51 replaced by [6] 7.3. We also use 4.2. (cid:3)
Theorem 5.6.
Let V be locally λ -presentable as a closed category. Then Mod λ ( A ) is a M -locally λ -generated V -category for every ( λ, V ) -nest A .Proof. It follows from 5.5, 3.6 and 4.2. (cid:3)
Definition 5.7. A morphism of λ -generated V -categories R : K → L is a right V -adjoint preserving M -morphisms and λ -directed colimits of them.A morphism of ( λ, V ) -nests F : A → B is a V -functor preserving λ -small limits, M -morphisms and multiple pullbacks of them. NRICHED LOCALLY GENERATED CATEGORIES 17
Theorem 5.8.
Let V be locally λ -presentable as a closed category. Then Gen λ : LG λ ⇆ N op λ : Mod λ is a dual biequivalence between locally λ -generated V -categories and ( λ, V ) -nests.Proof. It follows from 3.16, 5.2 and 5.6. (cid:3)
Remark 5.9. (1) Our duality restricts to the enriched Gabriel-Ulmer duality betweenlocally λ -presentable V -categories and small V -categories with λ -small limits given in[16].(2) 3.17 (2) applies to the enriched case as well.6. Banach spaces
Let
CMet be the category of (generalized) metric spaces and nonexpanding maps.This category is symmetric monoidal closed and locally ℵ -presentable as a closedcategory (see [3] 2.3(2) and 4.5(2)). But it is not locally ℵ -presentable. In fact, onlythe empty space is ℵ -presentable in CMet (see [3] 2.7 (1)).
CMet has a stronggenerator consisting of a one-point space 1 and of two-point spaces 2 ε where the twopoints have the distance ε > Ban of Banach spaces and linear maps of norm ≤ CMet . Moreover, it is locally ℵ -presentable CMet -category (see [3] 6.3).
Remark 6.1. (1) Epimorphisms in
CMet or Ban coincide with dense maps. See[19] 1.15 for
Ban and the argument for
CMet is analogous. Both in
CMet and
Ban , there is a factorization system ( E , M ) where E consists of dense maps and M of isometries (see [3] 3.16(2)). Hence isometries coincide, both in CMet and in
Ban with strong monomorphisms. Both
CMet and
Ban are E -cowellpowered and, fromthe description of directed colimits (see [3] 2.5), it follows that ( E , M ) is, in the bothcases, ℵ -convenient.(2) Both in CMet and
Ban , ℵ -generated objects w.r.t. M coincide with approx-imately ℵ -generated objects in the sense of [3] (see 5.11(3) in this paper). In CMet ,approximately ℵ -generated finite metric spaces are precisely discrete ones (see [3]5.18 and 5.19).(3) Ban is M -locally ℵ -generated (see [3] 7.8). Every finite-dimensional Banachspace is approximately ℵ -generated ([3] 7.6).(4) The concept of a finite weight does not make sense in CMet because only ∅ is ℵ -presentable. Thus, under finite limits in CMet , we understand finite conicallimits, cotensors with finite metric spaces, and their combinations. Here, cotensorswith finite metric spaces can be replaced by ε -pullbacks (see [3] 4.6). But in CMet , ℵ -generated objects w.r.t. M are not closed under these finite colimits (see [3]5.20). Hence, following 4.9 and [3] 5.20 and 4.1(4), finite limits do not commute withdirected colimits in CMet . Remark 6.2.
A metric space is called convex if for every points x and y there isa point z such that d ( x, z ) + d ( z, y ) = d ( x, y ). A subset S of a metric space A iscalled a metric segment if for every two distinct points a = b there is an isometry f : [0 , d ( a, b )] → A from a closed interval on R such that f (0) = a , f ( d ( a, b )) = b and f ([0 , d ( a, b )]) = S . A complete metric space is convex iff every distinct pointsare connected by a metric segment (see [7]). The proof consists in creating a denseset of points between a and b and taking its completion. Hence it also applies to d ( a, b ) = ∞ where the interval is [0 , ∞ ] in R with ∞ added. Lemma 6.3.
For every δ > , CMet (2 δ , − ) : CMet → CMet preserves directedcolimits of convex spaces and isometries.Proof.
Let k i : K i → K , i ∈ I be a directed colimit of convex complete metric spacesand isometries. Consider f : 2 δ → K and choose ε >
0. Denote the two points of 2 δ as x and y . There is i ∈ I and a, b ∈ K i such that d ( a, f x ) , d ( b, f y ) ≤ ε . Then d ( a, b ) ≤ d ( a, f x ) + d ( f x, f y ) + d ( f y, b ) ≤ ε + δ. Let S be the metric segment in K i connecting a and b . Choose a ′ , b ′ ∈ S such that d ( a, a ′ ) , d ( b, b ′ ) ≤ ε . Then d ( a ′ , b ′ ) ≤ δ . Let g : 2 δ → K i be given by f x = a ′ and f y = b ′ . Then d ( f, k i g ) because d ( a ′ , f x ) ≤ d ( a ′ , a ) + d ( a, f x ) ≤ ε and similarly d ( b ′ , f y ) ≤ ε . (cid:3) Example 6.4.
There is an approximately ℵ -generated Banach spaces which is notfinite-dimensional. Consider the complete metric space A = { } ∪ { n ; n = 1 . ., . . . } .Let A m = { n ; n = 1 , , . . . , m } and r m : A m → A be the inclusion. Let u m : A → A m be the identity on A m and send A \ A m to m . We have r m u m ∼ m id A m .Let F : CMet → Ban be the left adjoint to the unit ball functor U : Ban → CMet (see [3] 4.5(3)). This adjunction is enriched and thus F ( r m u m ) ∼ m id F ( A m ) .Since F (1) = C , F sends finite discrete metric spaces to finite-dimensional Banachspaces. We have surjective maps D m → A m where D m is discrete space of m points.Since U preserves isometries, F preserves E -maps. Thus F ( A m ) are finite-dimensionalBanach spaces. Following [3]7.7(1), F ( A ) is approximately ℵ -generated.Since u m are surjective, F ( u m ) are dense. Thus F ( A ) cannot be finite-dimensionalbecause it has dense maps to m -dimensional Banach spaces for every m . Lemma 6.5. An ε -pushout in Ban is A f / / g (cid:15) (cid:15) B g (cid:15) (cid:15) C f / / B ⊕ f,g,ε C NRICHED LOCALLY GENERATED CATEGORIES 19 where B ⊕ f,g,ε C is the coproduct B ⊕ C endowed with the norm k ( x, y ) k = inf {k b k + k c k + ε k a k \ x = b + f ( a ) , y = c − g ( a ) } . Proof.
In the special case of B ⊕ f, id A ,ε C , it is [12] 2.1. The general case is analogous. (cid:3) Corollary 6.6.
The dual of the full subcategory
Ban fd of Ban consisting of finite-dimensional Banach spaces is a ( ℵ , CMet )-nest.
Proof.
Finite-dimensional Banach spaces are closed under dense quotients. Following[3] 4.6 and 6.5, they are closed under finite colimits. (cid:3)
Theorem 6.7. Ban is equivalent to
Mod ℵ ( Ban op fd ) .Proof. Following 4.15,
Ban is equivalent to the full subcategory of Mod ℵ Ban opfd .Consider H in Mod ℵ Ban opfd and follow 4.13(2). For every Banach space X , H ( X ) isthe directed colimit colim d Dd ( X ) of complete metric spaces Dd ( X ) and isometries.Every Dd ( X ) is Ban ( A, X ) for some finite-dimensional Banach space A . Since thesecomplete metric spaces are convex, we have CMet (2 δ , colim d Dd ( X )) ∼ = colim d CMet (2 δ , Dd ( X )) . Since 1 and 2 δ , δ > CMet , γ : H ∗ → H is an isomor-phism (following the argument from [6] 4.5). (cid:3) Remark 6.8.
The category
Ban → does not seem to be M -locally ℵ -generated – 4.8does not apply to it. Hence [3] cannot be immediately applicable to the constructionof approximately ℵ -saturated objects in Ban → like in Ban . The existence of suchobjects in
Ban → was proved in [13]. Remark 6.9.
Let
CCMet be the category of convex complete metric spaces andnon-expansive maps.
CCMet is an injectivity class in
CMet given by 2 δ → [0 , δ ], δ >
0, sending the two points of 2 δ to the end-points of [0 , δ ]. Thus CCMet is a fullweakly reflective subcategory of
CMet . A weak reflection of a complete metric space A to CCMet is constructed as follows. If a, b ∈ A are not connected by a metricsegment of length d ( a, b ), we glue [0 , d ( a, b )] to A by identifying 0 with a and d ( a, b )with b . The distances of points of distinct added segments are ∞ . We repeat theprocedure and add metric segments [0 , ∞ ]. We proceed by induction and the union ∪ n A n , n = 1 , , . . . is a desired weak reflection. But CCMet is not reflective in
CMet because it is not closed under equalizers. Indeed, let C be the circle of radius1. Then the equalizer of the identity and the axial symmetry on C is 2 . Even, thisequalizer does not exist in CCMet at all.
CCMet does not have coproducts - forinstance 1 + 1 does not exist in
CCMet . CCMet is closed in
CMet under ℵ -directed colimits. Indeed, let k i : K i → K , i ∈ I be an ℵ -directed colimit of convex complete metric spaces in CMet . Let a, b ∈ K . Following [3] 2.5(1), there is i ∈ I and a ′ , b ′ ∈ K i such that d ( a ′ , b ′ ) = d ( a.b ), k i a ′ = a and k i b ′ = b . There is c ∈ K i such that d ( a ′ , c ) + d ( c, b ′ ) = d ( a ′ , b ′ ). Since d ( a, f c ) + d ( f c, b ) ≤ d ( a ′ , c ) + d ( c, b ′ ) = d ( a, b ), we have d ( a, f c ) + d ( f c, b ) = d ( a, b ).Thus K is convex. Analogously, using [3] 2.5(2), we prove that CCMet is closedunder directed colimits of isometries in
CMet . We do not know whether the segments[0 , δ ] are ℵ -generated w.r.t. isometries in CCMet .The tensor product A ⊗ B of convex complete metric spaces A and B is convexand complete. Recall that A ⊗ B is A × B with the +-metric d ( a, b ) , ( a ′ , b ′ )) = d ( a, a ′ ) + d ( b, b ′ ) . Indeed, let a ′′ ∈ A and b ′′ ∈ B satisfy d ( a, a ′′ ) + d ( a ′′ , a ′ ) = d ( a, a ′ ) and d ( b, b ′′ ) + d ( b ′′ , b ′ ) = d ( b, b ′ ) . Then d (( a, b ) , ( a ′′ , b ′′ )) + d (( a ′′ , b ′′ ) , ( a ′ , b ′ )) = d ( a, a ′′ ) + d ( b, b ′′ ) + d ( a ′′ , a ′ ) + d ( b ′′ , b ′ )= d ( a, a ′ ) + d ( b, b ′ ) = d (( a, b ) , ( a ′ , b ′ )) . We do not know whether
CCMet is symmetric monoidal closed.
Remark 6.10.
Let
CAlg be the category of C ∗ -algebras and CCAlg the category ofcommutative C ∗ -algebras. The forgetful functor U : CAlg → Ban preserves limits,isometries and ℵ -directed colimits. Thus it has a left adjoint F . The same holdsfor U : CCAlg → Ban . The unit η B : B → U F B is a linear isometry. Thus F isfaithful. In the commutative case, this left adjoint was described in [23] and calledthe Banach-Mazur functor.The forgetful functor U : CAlg → Ban even preserves directed colimits. In fact,directed colimits in
Ban are calculated like in
CMet , i.e., as a completion of adirected colimit in
Met . However, the same holds in
CAlg because one completesthe directed colimit of ∗ -algebras. Indeed, if x = lim n x n and y = lim m y m are in thiscompletion then both x · y = lim n,m ( x n · y m ) and x ∗ = lim n x ∗ n are there.Free (commutative) C ∗ -algebras over finite-dimensional Banach spaces are ℵ -ap-generated and, since U is conservative, they form a strong generator in CAlg . Thecategory of (commutative) C ∗ -algebras is locally ℵ -presentable and monadic over Set (see [21]). Since epimorphisms are surjective in
CCAlg (see [14]), (epi, strongmono)-factorization system on
CAlg is (surjective, dense) one and it is ℵ -convenient.Hence CAlg is a cocomplete
CMet -category with a ℵ -convenient V -factorizationsystem (surjective, isometry) and with a strong generator consisting of ℵ -generatedobjects w.r.t. isometries. But 4.17 does not apply to CAlg and we expect that thiscategory is not isometry-locally ℵ -generated. NRICHED LOCALLY GENERATED CATEGORIES 21
References [1] J. Ad´amek and J. Rosick´y,
Locally Presentable and Accessible Categories , Cambridge Uni-versity Press 1994.[2] J. Ad´amek and J. Rosick´y,
What are locally generated categories , Proc. Categ. Conf. Como1990, Lect. Notes in Math. 1488 (1991), 14-19.[3] J. Ad´amek and J. Rosick´y,
Approximate injectivity and smallness in metric-enriched cat-egories , arXiv:2006.01399.[4] M. Barr,
Coequalizers and free triples , Math. Z. (1970), 307-322.[5] F. Borceux,
Handbook of Categorical Algebra , Vol. 2, Cambridge University Press 1994.[6] F. Borceux, C. Quinteiro and J. Rosick´y,
A theory of enriched sketches , Th. Appl. Categ.4 (1998), 47-72.[7] V. W. Bryant,
The convexity of the subset of a metric space , Comp. Math. 22 (1970),383-385.[8] C. Centazzo and E. M. Vitale,
A duality relative to a limit doctrine , Th. Appl. Categ. 10(2002), 486-497.[9] B. Day,
On adjoint-functor factorisations , Lect. Notes in Math. 145 1970), 1-19.[10] L. Fajstrup and J. Rosick´y,
A convenient category for directed homotopy , Th. Appl. Cat.21 (2008), 7-20.[11] P. Gabriel and F. Ulmer,
Lokal pr¨asentierbare Kategorien , Lect. Notes in Math. 221,Springer 1971.[12] J. Garbuli´nska-W¸egrzyn,
Isometric uniqueness of a complementably universal Banach spacefor Schauder decompositions , Banach J. Math. Anal. 8 (2014), 211-220.[13] J. Garbuli´nska-W¸egrzyn and W. Kubi´s,
A universal operator on the Gurarii space , J.Operator Theory 73 (2015), 143-158.[14] K. H. Hofmann and K.-H. Neeb,
Epimorphisms of C ∗ -algebras are surjective ,arXiv:funct-an/9405003.[15] G. M. Kelly, Basic Concepts of Enriched Category Theory , Cambridge University Press1982.[16] G. M. Kelly,
Structures defined by finite limits in the enriched context , Cah. Top. G´eom.Diff. XXIII (1982), 3-42.[17] R. B. B. Lucyshyn-Wright,
Enriched factorization systems , Th. Appl. Cat. 29 (2014), 475-495.[18] M. Makkai and A. Pitts,
Some Results on Locally Finitely Presentable Categories , Trans.Amer. Math. Soc. 299 (1987), 473-496.[19] K. L. Pothoven,
A category of Banach spaces , Master thesis, Western Michigan Univ. 1968.[20] E. Riehl,
Categorial Homotopy Theory , Cambridge University Press 2014.[21] J. W. Pelletier and J. Rosick´y,
On the equational theory of C ∗ -algebras , Alg. Univ. 30(1993), 275-284.[22] J. Rosick´y and W. Tholen, Factorizations, fibrations and torsion, J. Hom. Rel. Str. 2 (2007),295-314.[23] Z. Semadeni, Some categorical characterizations of algebras of continuous functions , Symp.Math. XVII (1978), 97-112.2 I. DI LIBERTI AND J. ROSICK ´Y