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(2) The embedding [MATH] clearly preserves existing limits. (3) Monomorphisms of [MATH] are precisely the morphisms [MATH] such that [MATH] is monic in [MATH] and each [MATH] is monic in [MATH] . We always represent them by collections with [MATH] where [MATH] is the inclusion map. |
(4) Analogously, epimorphisms in [MATH] are precisely the morphisms [MATH] with [MATH] epic in [MATH] and each [MATH] epic in [MATH] |
(5) [MATH] is wellpowered (or cowellpowered) iff [MATH] is. 2.3 Remark If [MATH] has pullbacks, then pullbacks of morphisms [EQUATION] |
in [MATH] are computed as follows: form a pullback of [MATH] and [MATH] in [MATH] [EQUATION] and for each [MATH] form a pullback in [MATH] |
[EQUATION] This defines an object [MATH] and morphisms [MATH] [MATH] of [MATH] (by the given functions [MATH] [MATH] and the given components [MATH] [MATH] ). It is easy to see that the following square |
[EQUATION] is a pullback in [MATH] 2.4 Remark Recall that a finitely complete category with coproducts is infinitary extensive if coproducts are universal (i.e., preserved by pulling back along a morphism) and disjoint (i.e., coproduct injections are monic and their pairwise intersections are formed by the initial obje... |
It is extensive if finite coproducts are universal and disjoint. 2.5 Proposition For every category [MATH] the following conditions are equivalent: |
(i) [MATH] is extensive, (ii) [MATH] is infinitary extensive and (iii) [MATH] is finitely multi-complete. Proof. [MATH] iii is clear: extensivity includes finite completeness, so use Remark 2.2 (1). |
iii [MATH] ii. We prove that coproducts in [MATH] are disjoint and universal. A coproduct of two objects [MATH] and [MATH] is obviously the collection [MATH] with [MATH] and [MATH] . The description of pullbacks in Remark 2.5 makes it clear that the pullback of the coproduct injections [MATH] and [MATH] is the empty co... |
For the verification that coproducts are universal in [MATH] we also use just a coproduct [MATH] of a pair of objects, the general proof is again completely analogous. Let |
[EQUATION] be an arbitrary morphism. Then [MATH] for [MATH] and [MATH] . Thus [EQUATION] where [MATH] and [MATH] . And the components of [MATH] define obvious morphisms [MATH] and [MATH] . It is easy to see that we get the following pullbacks |
[EQUATION] where [MATH] [MATH] are the coproduct injections of [MATH] ii [MATH] i is trivial. 2.6 Lemma Let [MATH] be objects of [MATH] for [MATH] . A product of this family exists in [MATH] iff for every function [MATH] a multi-product of [MATH] for [MATH] exists in [MATH] |
Proof. (1) We prove the necessity: Let [MATH] be a product [MATH] in [MATH] with projections [EQUATION] for [MATH] . Given [MATH] denote by [MATH] the set of all [MATH] with [MATH] for all [MATH] This defines cones in [MATH] as follows: for every [MATH] we have the [MATH] -component of [MATH] |
[EQUATION] The set of these cones (indexed by [MATH] ) is a multi-product of [MATH] for [MATH] Indeed, every cone in [MATH] [EQUATION] |
yields a cone [MATH] in [MATH] with [MATH] choosing [MATH] and the unique component of [MATH] being [MATH] . The fact that [MATH] has a unique factorization through [MATH] in [MATH] clearly implies that there exists a unique [MATH] and a unique factorization of [MATH] through [MATH] in [MATH] |
(2) Conversely, if all the above multi-products exist in [MATH] , then [MATH] is given in [MATH] as follows. For every [MATH] choose a multi-product, indexed by a set [MATH] , as follows: |
[EQUATION] This defines a morphism of [MATH] for every [MATH] : put [MATH] and let [EQUATION] be defined by [EQUATION] for all [MATH] with components |
[EQUATION] as above. For every object [MATH] of [MATH] and every cone [MATH] [MATH] , in [MATH] we prove that there is a unique factorization through [MATH] . It then follows immediately that the same holds for families of more that one object as well. |
The given cone chooses for every [MATH] a unique [MATH] , let [MATH] denote the resulting function. And [MATH] has a component [MATH] in [MATH] [MATH] . There exists a unique [MATH] and a unique factorization of [MATH] through [MATH] in [MATH] . This clearly implies that [MATH] factorizes uniquely through [MATH] in [MA... |
2.7 Remark Recall that a functor [MATH] is a left multi-adjoint if, for every object [MATH] , the category [MATH] has a multi-terminal set of objects. This is a set [MATH] [MATH] such that for every [MATH] there is a unique [MATH] and a morphism |
[MATH] such that [MATH] , and moreover [MATH] is also unique. 2.8 Definition A category [MATH] is called cartesian multi-closed if it has finite products and each endofunctor [MATH] is a left multi-adjoint. |
Explicitly, this means that for every pair of objects [MATH] and [MATH] there exists a set of objects [MATH] [MATH] and morphisms |
[EQUATION] multi-universal in the expected sense: For every morphism [MATH] there exists a unique [MATH] for which some morphism [MATH] makes the following triangle |
[EQUATION] commutative, and moreover, [MATH] is also unique. 2.9 Examples The following categories are cartesian multi-closed: (a) Every cartesian closed category. |
(b) Every category with binary products satisfying [MATH] (e.g. every additive category). Indeed, given [MATH] , the family [MATH] indexed by all [MATH] is obviously multi-universal: every morphism [MATH] has the form [MATH] for a unique pair [MATH] and [MATH] |
(c) Every dual of an extensive category (e.g. [MATH] ). Indeed, first observe that every object [MATH] has only a set of decompositions [MATH] in [MATH] (up to isomorphism). In fact, every such decomposition yields a morphism in [MATH] taking [MATH] to the left-hand summand and [MATH] to the right-hand one. And differe... |
Now, given [MATH] , the family [MATH] in [MATH] indexed by all decompositions [MATH] and all [MATH] is obviously multi-universal: every morphism [MATH] in [MATH] has the form [MATH] in [MATH] for unique [MATH] and |
[MATH] 2.10 Remark Let [MATH] have a strict initial object [MATH] , i.e., if a morphism [MATH] exists, then [MATH] is initial. Then [MATH] is cartesian closed iff it is cartesian multi-closed. Indeed, consider the above triangle for [MATH] : since [MATH] , we have a unique [MATH] , but for every [MATH] the unique [MATH... |
This implies that for example the category [MATH] of topological spaces is not cartesian multi-closed. 2.11 Theorem Let [MATH] have finite products. Then [MATH] is cartesian closed iff [MATH] is cartesian multi-closed and has multi-products. |
Proof. (1) Let [MATH] be cartesian closed. Given objects [MATH] [MATH] of [MATH] we denote the corresponding exponential object in [MATH] by |
[EQUATION] together with the counit having components [EQUATION] (1a) We prove that [MATH] is cartesian multi-closed. It is our task to prove that for every morphism [MATH] in [MATH] there exists a unique [MATH] with a commutative triangle |
[EQUATION] for some [MATH] , and moreover [MATH] is unique. Indeed, [MATH] is a morphism of [MATH] , thus we have a unique [MATH] with |
[EQUATION] in [MATH] . This morphism [MATH] is given by choosing [MATH] and a morphism [MATH] in [MATH] with [MATH] Conversely, suppose the triangle above commutes, then we must prove [MATH] and [MATH] is as above. The choice of [MATH] and [MATH] in that triangle defines a morphism [MATH] with [MATH] , and since [MATH]... |
(1b) [MATH] has multi-products. Indeed, for every collection [MATH] [MATH] in [MATH] let [MATH] be the object of [MATH] with components [MATH] , terminal in [MATH] . Since in [MATH] we have [MATH] and [MATH] has the exponential object [MATH] , it has the following product |
[EQUATION] By Lemma 2.6 for every [MATH] a multi-product of [MATH] [MATH] exists in [MATH] . This proves our claim: put [MATH] (2) Let [MATH] be cartesian multi-closed and have multi-products. Then [MATH] has for every pair [MATH] [MATH] of objects a multi-universal collection of morphism [MATH] [MATH] w.r.t [MATH] . W... |
[MATH] and [MATH] . The exponential object [MATH] in [MATH] is formed as follows: let for every [MATH] a multi-universal collection be given |
[EQUATION] This defines a morphism [EQUATION] whose index function [MATH] is [MATH] and whose components are [MATH] . We prove that in [MATH] we have |
[EQUATION] with the above counit [MATH] . Indeed, to give a morphism [MATH] means to choose for every [MATH] (a) indices [MATH] and [MATH] , and |
(b) a morphism [MATH] And to give a morphism [MATH] means to choose for every [MATH] an index [MATH] and a morphism [MATH] , where the latter is equivalent to choosing [MATH] and a morphism from [MATH] to [MATH] , due to the universal property of [MATH] |
Turning to a general pair of objects [MATH] and [MATH] , the existence of exponential objects [MATH] and the fact that [MATH] is a coproduct in [MATH] of [MATH] imply |
[EQUATION] the above product exists because [MATH] has multi-products (see Lemma 2.6 ). 2.12 Corollary [MATH] is cartesian closed for every complete, additive category. |
2.13 Definition A category [MATH] is called locally cartesian multi-closed if every slice [MATH] is cartesian multi-closed. 2.14 Theorem |
Let [MATH] be a complete category. Then [MATH] is locally cartesian closed iff [MATH] is locally cartesian multi-closed. Proof. Let [MATH] be locally cartesian multi-closed. For every object [MATH] of [MATH] we verify that [MATH] is cartesian closed. This category is isomorphic to the product of categories [MATH] [MATH... |
Since [MATH] has products, from its cartesian multi-closedness follows that [MATH] is cartesian closed (see Theorem 2.11 ). And a product of cartesian closed categories is cartesian closed (with exponential objects defined componentwise). |
Conversely, let [MATH] be locally cartesian closed. For every object [MATH] of [MATH] the category [MATH] is isomorphic to [MATH] and is thus cartesian closed. Therefore [MATH] is cartesian multi-closed (see Theorem 2.11 ), proving that |
[MATH] is locally cartesian multi-closed. 2.15 Corollary Let [MATH] be a complete category with a strict initial object. Then [MATH] is (locally) cartesian closed if and only if [MATH] is (locally) cartesian closed. |
This follows from Remark 2.10 , Theorem 2.11 and Theorem 2.14 2.16 Examples (i) For the topos [MATH] of finite sets, [MATH] is not cartesian closed: [MATH] does not have multi-products (see Remark 2.2 (1)). Analogously, [MATH] is not cartesian closed where [MATH] is the category of finite abelian groups. |
(ii) In contrast, [MATH] is locally cartesian closed, indeed a topos, see Example 2.1 (c). (iii) [MATH] is cartesian closed by Corollary 2.12 |
(iv) The category [MATH] of small categories is cartesian closed but not locally so. Since it is complete and has a strict initial object, we conclude that [MATH] is cartesian closed but not locally so. |
2.17 Open problem Is [MATH] locally cartesian closed? 2.18 Definition By a multi-topos is meant a finitely complete, cartesian multi-closed category with a subobject classifier. |
2.19 Theorem For a complete category [MATH] the completion [MATH] is a topos iff [MATH] is a multi-topos. Proof. (1) Let [MATH] be a multi-topos with a subobject classifier [MATH] . Then the morphism |
[EQUATION] corresponding to [MATH] is a subobject classifier of [MATH] , which by Theorem 2.11 proves that [MATH] is a topos. Indeed, given a subobject [MATH] with [MATH] (see Remark 2.2 (2)), the corresponding pullbacks in [MATH] |
[EQUATION] yield the following morphism [EQUATION] The function [MATH] assign to every [MATH] the left-hand object of [MATH] , (with the component [MATH] above) and to every [MATH] the right-hand one. The square below |
[EQUATION] is a pullback in [MATH] . This follows easily from Remark 2.3 It remains to prove that the pulback ( 2.1 ) determines [MATH] uniquely. Suppose [MATH] makes the corresponding square a pullback in [MATH] , too. For every [MATH] the commutativity of that square proves that |
[MATH] is the left-hand object, [MATH] , iff [MATH] . Thus, [MATH] . For every [MATH] we are to prove that the following square [EQUATION] |
is a pullback in [MATH] – then [MATH] and the proof is complete. Indeed, given a commutative square in [MATH] [EQUATION] define [MATH] by [MATH] [MATH] and [MATH] for all [MATH] Then the last square and the pullback ( 2.1 ) with [MATH] in place of [MATH] imply that the following square |
[EQUATION] commutes in [MATH] . Thus, it factorizes uniquely through the modified square ( 2.1 ), which proves that [MATH] factorizes uniquely through ( 2.2 ), as required. |
(2) Let [MATH] be a topos. By Theorem 2.11 [MATH] is cartesian multi-closed. Denote by [MATH] the subobject classifier of [MATH] Let [MATH] be the element given by the indexing function of [MATH] , and let |
[EQUATION] be the unique component of [MATH] . For every subobject [MATH] in [MATH] , since [MATH] is also monic in [MATH] , we have a morphism [MATH] forming a pullback as follows: |
[EQUATION] Thus [MATH] chooses [MATH] (since [MATH] does) and the unique component [MATH] defines a pullback in [MATH] as follows: |
[EQUATION] Conversely, given [MATH] making the last square a pullback in [MATH] , then the corresponding morphism [MATH] makes the above square a pullback. Therefore, [MATH] is unique. This proves that [MATH] is a subobject classifier in [MATH] |
2.20 Corollary For a complete category [MATH] with a strict initial object [MATH] is a topos if and only if [MATH] is a topos. This follows from the above Theorem and Remark 2.10 |
However, we have seen in Example 2.16 (i) that for the topos [MATH] the category [MATH] is not even cartesian closed. 3. [MATH] is often locally cartesian closed |
We now turn to the free completion [MATH] of a category [MATH] under colimits. In the present section we concentrate on the question whether [MATH] is (locally) cartesian closed, in the next ones we ask whether it is a topos and study the (co)wellpoweredness of it. |
Recall the concept of a free completion [EQUATION] under colimits: the category [MATH] is cocomplete, and there is a full embedding [MATH] such that every functor [MATH] with [MATH] cocomplete has an extension to a colimit-preserving functor [MATH] , unique up to natural isomorphism. |
3.1 Example (a) [MATH] is the free colimit completion of the terminal category. (b) [MATH] is the free colimit completion of the two-element chain. |
(c) For small categories [MATH] is the presheaf category [MATH] and [MATH] is the Yoneda embeddding. (d) For general categories [MATH] can be described as the full subcategory of [MATH] on all small functors , i.e., small colimits of hom-functors (see |
2.29). And [MATH] is the codomain restriction of the Yoneda embedding, we denote it by [MATH] (e) [MATH] is the category of all accessible set functors. Indeed, for every locally presentable category [MATH] a functor into [MATH] is accessible iff it is a small presheaf on [MATH] (see |
). Thus [MATH] is the category of all accessible set-valued functors on [MATH] The completions [MATH] and [MATH] are closely related: the former one is an exact completion of the latter. Recall that a category is exact if it has finite limits, regular factorizations with regular epimorphisms stable under pullbacks, and... |
, is an exact category [EQUATION] with a full embedding [MATH] such that every functor [MATH] with [MATH] exact has an extension to an exact functor [MATH] unique up to natural isomorphism. The following is Lemma 3 in |
3.2 Lemma For every finitely complete category [MATH] we have [EQUATION] Freyd introduced the following concepts in 3.3 Definition |
(1) By a pre-limit of a diagram [MATH] is meant a set of cones such that every cone of [MATH] factorizes through some of them (not uniquely in general). A category is pre-complete if every diagram has a pre-limit. Dual concept: pre-colimit and pre-cocomplete category. |
(2) A functor [MATH] is petty if it is a quotient of a coproduct of representable functors. It is lucid if, moreover, for every pair [MATH] of natural transformations the equalizer of [MATH] has a petty domain. |
3.4 Remark (a) In the definition of a lucid functor we can restrict [MATH] to representable functors. Thus, in categories where all subfunctors of representable functors are petty, any petty functor is lucid. |
(b) A functor [MATH] is petty iff its category [MATH] of elements has a weakly initial set. That is, we have a set of objects [MATH] |
[MATH] of [MATH] and elements [MATH] such that every element [MATH] has, for some [MATH] and some morphism [MATH] of [MATH] , the form |
[MATH] 3.5 Examples (a) Complete categories are pre-complete. (b) Small categories are pre-complete. (c) Accessible categories are pre-cocomplete. Indeed, if [MATH] is a diagram in a [MATH] -accessible category and if [MATH] is chosen so that |
(i) every object of [MATH] is [MATH] -presentable, and (ii) [MATH] has less than [MATH] morphisms, then the (essentially small) set of all cocones with [MATH] -presentable codomains is a pre-colimit of [MATH] |
The following is essentially Lemma 1 in 3.6 Lemma Every lucid functor is small. If [MATH] has finite pre-limits, then every small functor [MATH] is lucid. |
Proof. (1) Every lucid functor [MATH] is small. Indeed, since [MATH] is petty, we have a quotient [MATH] [MATH] [MATH] a set. Thus [MATH] is the coequalizer of its kernel pair |
[EQUATION] which yields a petty functor [MATH] (since [MATH] is the equalizer of [MATH] and [MATH] and [MATH] is petty). Therefore, [MATH] is also a quotient [MATH] [MATH] [MATH] a set. Consequently, [MATH] is the coequalizer of [MATH] and [MATH] , hence [MATH] is small. |
(2) If [MATH] has finite pre-limits, every small functor [MATH] is lucid. Indeed, by the usual reduction of colimits to coproducts and coequalizers we have a coequalizer in [MATH] as follows |
[EQUATION] where [MATH] and [MATH] are sets. By 1.4 and 1.2 in , both of the coproducts above are lucid, and by 1.9 in , applied to [MATH] , it follows that [MATH] is lucid. |
3.7 Remark Whenever a limit of a diagram in [MATH] exists, it is formed objectwise. That is, [MATH] is closed under existing limits in [MATH] . This follows easily from Yoneda Lemma. |
3.8 Corollary [MATH] is (finitely) complete iff [MATH] is (finitely) pre-complete. Proof. Indeed, from (finite) pre-completeness of [MATH] it follows that (finite) limits of lucid functors are lucid, see 10 , (1.7) 1.12] . Thus 3.6 implies that [MATH] is closed under (finite) limits in [MATH] |
Conversely, let [MATH] be (finitely) complete. For every (finite) diagram [MATH] in [MATH] its composite with the Yoneda embedding |
[MATH] has, by 3.7 , a petty pointwise limit in [MATH] . This is clearly equivalent to [MATH] having a pre-limit in [MATH] 3.9 Remark |
Analogously, [MATH] has finite products iff [MATH] has finite pre-products. 3.10 Lemma Given objects [MATH] [MATH] of [MATH] for which [MATH] has the following product |
[EQUATION] it follows that a pre-product of [MATH] [MATH] exists in [MATH] More generally, given a diagram [MATH] for which [EQUATION] |
exists in [MATH] , then [MATH] has a pre-limit of the diagram [MATH] Proof. We prove the first statement, the latter is analogous. The functor [MATH] assigns to every object [MATH] the collection of all [MATH] where [MATH] is a morphism of [MATH] for some [MATH] . This follows from [MATH] being closed under colimits an... |
[EQUATION] for every [MATH] . Consequently, if [MATH] is the set of all [MATH] with [MATH] for every [MATH] , then the cones [EQUATION] |
for [MATH] form the desired pre-product. 3.11 Definition (see A category [MATH] is called cartesian pre-closed if it has finite products and for every pair of objects [MATH] and [MATH] |
the functor [MATH] is small. The second author proved in Proposition 1 that for pre-complete categories [MATH] cartesian closedness of [MATH] is equivalent to cartesian pre-closedness of [MATH] . We will show that this also implies pre-completeness of [MATH] |
3.12 Theorem Let [MATH] have finite products. Then [MATH] is cartesian closed if and only if [MATH] is pre-complete and cartesian pre-closed. |
Proof. Let [MATH] be cartesian closed. Then [MATH] is pre-complete. Indeed, for every diagram [MATH] we form the constant diagram [MATH] with value [MATH] , a terminal object, and use the exponential object [MATH] where [MATH] . The existence of [MATH] implies (due to colimits preserved by functor [MATH] in [MATH] ) th... |
[EQUATION] exists in [MATH] . Apply Lemma 3.10 to [EQUATION] We deduce that a pre-limit of [MATH] exists. The rest follows from Proposition 1. |
3.13 Remark In a cartesian pre-closed category every pair [MATH] and [MATH] of objects posseses a set of morphisms [EQUATION] with the universal property of Definition 2.8 except that neither [MATH] nor [MATH] are required to be unique. Indeed, by Remark |
3.4 (b) this is precisely the fact that the functor [MATH] is petty. It is an open question whether (in analogy to the free coproduct completion) the pettiness of the above functors implies that they are small, i.e., that our category is cartesian pre-closed. |
3.14 Example Every cartesian multi-closed category is cartesian pre-closed. Indeed, given the multi-universal morphism [MATH] [MATH] of Definition 2.8 , then in [MATH] we have |
[EQUATION] This is clear since to give a morphism [MATH] means precisely to give a (unique) [MATH] and a (unique) moprhism [MATH] . Thus, [MATH] is small. |
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