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3.15 Examples We give examples of cartesian pre-closed categories which are not cartesian multi-closed. (a) The category [MATH] is not cartesian multi-closed (by Remark 2.10 ). It is cartesian pre-closed because it is infinitary extensive, thus Theorems 1 and 2 of |
apply. (b) [MATH] , the full subcategory of [MATH] spaces. The argument is the same. (c) The full subcategory [MATH] of [MATH] on all sets of cardinality other than continuum is cartesian pre-closed but not multi-closed. Indeed, this category has finite products. To prove that [MATH] is small, it is by Remark 3.4 suffi... |
[MATH] whose left-hand component is the evaluation provides a weakly initial element of [MATH] . In fact, given [MATH] in [MATH] there exists a (not necessarily unique) [MATH] with [MATH] |
[MATH] is not cartesian multi-closed because, having a strictly initial object, it would be cartesian closed, see Remark 2.10 . However, the sets [MATH] and [MATH] obviously fail to have an exponential object in [MATH] . (Since given [MATH] in [MATH] , its elements would bijectively correspond to maps [MATH] , hence [M... |
We are going to improve the above Theorem by moving to local cartesian closedness. For that we use the characterization of locally cartesian closed exact completions [MATH] due to Carboni and Rosolini |
, which we now recall. In that paper [MATH] was supposed to have weak finite limits. Recently, Emmenegger showed that the result is only correct if finite completeness is assumed, see |
. The following Definition and Theorem are from 3.16 Definition By a weak simple product of a morphism [MATH] over [MATH] is meant an object [MATH] and morphisms |
[MATH] and [MATH] forming a commutative triangle as follows [EQUATION] They are also requested to have the following weak universal property: given morphisms [MATH] and [MATH] |
with [MATH] , there exists a morphism [MATH] such that [MATH] and [MATH] 3.17 Theorem A finitely complete category has a locally cartesian closed exact completion if and only if it has weak simple products. |
3.18 Remark Weak simple products are also callled weak dependent products. The ”pre” version was called approximate dependent product in |
it is a set of objects [MATH] [MATH] and morphisms [MATH] [MATH] [MATH] such that [MATH] . The weak universal property states that given [MATH] and [MATH] |
with [MATH] , there exists [MATH] and a morphism [MATH] with [MATH] and [MATH] 3.19 Theorem For a complete category [MATH] the following conditions are equivalent: |
(i) [MATH] is locally cartesian closed, (ii) [MATH] is cartesian closed, (iii) [MATH] is cartesian pre-closed, (iv) [MATH] has weak simple products, and |
(v) [MATH] is cartesian pre-closed. Proof. iii [MATH] iv. In Lemma A1 of it is proved that a complete, cartesian pre-closed category [MATH] has approximate dependent products. We apply this to [MATH] to conclude that [MATH] has weak simple products. Indeed, given a morphism [MATH] , let |
[MATH] and [MATH] [MATH] form an approximate dependent product in [MATH] . Since [MATH] preserves coproducts (by Example 2.1 (e) and Remark 2.2 (2)), the following morphisms [MATH] |
and [MATH] are easily seen to form a weak simple product of [MATH] over [MATH] iv [MATH] i. [MATH] is locally cartesian closed by Theorem 3.17 |
[MATH] ii is clear. ii [MATH] iii. In Theorem 1 of every complete, infinitary extensive category [MATH] with [MATH] cartesian closed is proved to be cartesian pre-closed. Apply this to [MATH] , using Theorem 3.12 |
[MATH] ii follows from Theorem 3.12 3.20 Corollary Let [MATH] be a complete category which is a. cartesian closed, or b. additive, or |
c. dual to an extensive category. Then [MATH] is locally cartesian closed. Thus categories [MATH] [MATH] [MATH] and [MATH] (= accessible set functors) are locally cartesian closed. |
4. [MATH] is a pretopos but seldom a topos Recall that a pretopos is an exact and extensive category. That is, a category which is regular, has effective equivalence relations, and has finite coproducts which are disjoint and universal. |
4.1 Lemma Let [MATH] be a finitely pre-complete category. Then [MATH] is a pretopos. Proof. By Remark 3.7 and Corollary 3.8 [MATH] is closed under existing limits in [MATH] , and it is obviously closed under colimits. Thus [MATH] shares with [MATH] all the exactness properties of colimits and finite limits. In particul... |
4.2 Remark By the same argument we derive that if [MATH] is finitely pre-complete, then (a) [MATH] is infinitary extensive, see Remark 2.4 |
(b) every monomorphism in [MATH] is regular, and (c) every epimorphism in [MATH] is regular. For small categories [MATH] , we know that [MATH] is a topos. The converse ”almost” holds: |
4.3 Theorem Given a category [MATH] with copowers and finite intersections, then [MATH] is a topos iff [MATH] is essentially small. |
Proof. (1) We consider first the case that [MATH] is a preordered class. We are to prove that if [MATH] is a topos, then [MATH] is essentially small. Since [MATH] has a terminal object, [MATH] has a pre-terminal set of objects, and since [MATH] is wellpowered, so is [MATH] . It is easy to see that a wellpowered preorde... |
(2) Next consider [MATH] having an object [MATH] which is the domain of a parallel pair of distinct morphisms. Since for every cardinal [MATH] the coproduct injections of [MATH] are then pairwise distinct, [MATH] is not essentially small. We are going to prove that [MATH] is not topos. |
Assuming that [MATH] has a subobject classifier [MATH] , we derive a contradiction. Since [MATH] contains all hom-functors, Yoneda lemma implies that (up to natural isomorphism) [MATH] is given as follows: on objects [MATH] we have |
[EQUATION] and on morphisms [MATH] the map [MATH] forms preimages of subfunctors under [MATH] . We are going to prove that this presheaf [MATH] is not petty, which is the desired contradiction. |
For every set of elements of [MATH] [EQUATION] we are going to find an object [MATH] of [MATH] and an element of [MATH] [EQUATION] |
such that [MATH] for all [MATH] and all [MATH] . Our choice of [MATH] is a copower of the above object [MATH] [EQUATION] where [MATH] is a cardinal with |
[EQUATION] Denote by [MATH] [MATH] ) the coproduct injections. Choose arbitrary two elements [MATH] in [MATH] and form the intersection [MATH] |
[EQUATION] Given any distinct elements [MATH] in [MATH] , they yield the same intersection: [EQUATION] This follows from the (obvious) fact that there is an isomorphism [MATH] with [MATH] and [MATH] |
Our choice of the subfunctor [MATH] of [MATH] uses the above morphism [MATH] : put [MATH] and [MATH] for all [MATH] and define a small presheaf [MATH] as the wide pushout of the following morphisms |
[EQUATION] Up to natural isomorphism, [MATH] can be described as follows: on objects [MATH] put [EQUATION] where [MATH] is the least equivalence relation merging, for every morphism [MATH] , all [MATH] with [MATH] to one element (denoted by [MATH] ). The equivalence class of a morphism [MATH] not factorizing through [M... |
This small presheaf [MATH] is a subobject of [MATH] since the cocone [MATH] fulfills [MATH] (independent of [MATH] ). This yields a natural transformation |
[EQUATION] It assigns to [MATH] for [MATH] the value [EQUATION] and to [MATH] not factorizing through [MATH] the value [EQUATION] |
This morphism [MATH] is monic: (a) Given distinct [MATH] [MATH] , then [MATH] since [MATH] is monic: indeed, each [MATH] is a split monomorphism (the codiagonal splitting it). |
(b) Given distinct [MATH] and [MATH] not factorizing through [MATH] , then in case [MATH] we have [MATH] since [MATH] is a (split) monomorphism, and in case [MATH] we have [MATH] Otherwise [MATH] would factorize through [MATH] (and [MATH] through [MATH] ) due to the above pullback [MATH] |
(c) Given [MATH] and [MATH] not factorizing through [MATH] , then [MATH] : since [MATH] is a split monic and [MATH] , the equality [MATH] would yield a factorization [MATH] or [MATH] through [MATH] |
It remains to prove [EQUATION] Assuming to the contrary [MATH] , we prove that the morphisms [EQUATION] are pairwise distinct, contradicting [MATH] |
For elements [MATH] [MATH] of [MATH] with [MATH] we are going to verify that [MATH] . The equality [MATH] means that in [MATH] we have a pullback as follows: |
[EQUATION] Apply this to [MATH] in the component of [MATH] indexed by [MATH] : we get [EQUATION] with [EQUATION] since [MATH] . (Observe that [MATH] clearly does not factorize through [MATH] .) The codiagonal [MATH] yields |
[EQUATION] with [EQUATION] Indeed, naturality of [MATH] gives, since [MATH] [EQUATION] Thus, the above pullback applied to [MATH] provides |
[EQUATION] Since [MATH] does not factorize through [MATH] , neither does [MATH] , therefore [MATH] lies in some component [MATH] of [MATH] and [MATH] . Thus [MATH] factorizes through [MATH] , proving [MATH] , and [MATH] also factorizes through [MATH] , hence [MATH] , as desired. |
4.4 Example The categories [MATH] and [MATH] are locally cartesian closed (by Theorem 3.19 ) but not toposes. 4.5 Open problem Is there a category [MATH] which is not essentially small with [MATH] a topos? |
4.6 Remark (1) Menni characterized categories whose exact completion is a topos, see . He defined a generic proof as a map [MATH] such that for every map [MATH] there exists a map [MATH] for which [MATH] factorizes through [MATH] , and [MATH] factorizes through [MATH] . And he proved (in Theorem 1.2) that a category ha... |
(2) Consequently, the theorem just proved implies for complete, cartesian pre-preclosed categories [MATH] that the free colimit completion [MATH] is a topos iff the free coproduct completion [MATH] has a generic proof. |
4.7 Corollary If a complete, cartesian pre-closed category with copowers is not essentially small, then its free coproduct completion does not have a generic proof. |
See Theorem 3.19 The following examples are proved by Menni in a somewhat more technical manner (see , Propositions 5.7 and Lemma 5.5): |
4.8 Example (1) The category [MATH] does not have a generic proof. Indeed, it is equivalent to [MATH] (see Example 2.1 (c)). (2) For every small category [MATH] the free coproduct completion has a generic proof. The exact completion, which is the presheaf category for [MATH] , is namely a topos. |
5. [MATH] is sometimes wellpowered but not often This last section is devoted to the question whether [MATH] is wellpowered or cowellpowered. We first observe that the two problems are more or less equivalent. |
5.1 Proposition Every cowellpowered category [MATH] is also wellpowered. The converse holds for finitely pre-complete categories [MATH] |
Proof. (1) Let [MATH] be cowellpowered. Following Remark 4.2 , every monomorphism [MATH] in [MATH] is monic in [MATH] , too. Form the pushout in [MATH] |
[EQUATION] Since [MATH] and [MATH] are small, so are [MATH] and [MATH] . Thus we obtain a quotient [MATH] in [MATH] Distinct subobjects of [MATH] in [MATH] yield distinct quotients in [MATH] . Indeed, [MATH] certainly has the corresponding property, therefore, so does [MATH] . Since [MATH] has only a set of quotients, ... |
(2) Let [MATH] be finitely pre-complete, and [MATH] be wellpowered. Following Remark 4.2 , every epimorphism [MATH] is regular in [MATH] . Hence it is determined by its kernel pair [MATH] . Since [MATH] is a subobject of [MATH] |
[MATH] is cowellpowered. 5.2 Proposition If a wellpowered category [MATH] has all morphisms monic, then [MATH] is cowellpowered. |
Proof. (1) For every functor [MATH] denote by [MATH] a class of objects representing all [MATH] with [MATH] up to isomorphism. Then we verify that [MATH] is small iff [MATH] is a set. |
Indeed, let [MATH] be small. If [MATH] , then [MATH] is a set because [MATH] has only a set of subobjects. If [MATH] is a colimit of a diagram with objects [MATH] [MATH] , then clearly [MATH] . This is a set if [MATH] is small. |
Conversely, let [MATH] be a set. Recall that [MATH] is a colimit of the diagram of its elements. More precisely, we form the category [MATH] of all pairs [MATH] where [MATH] is an object of [MATH] and [MATH] . Morphisms [MATH] are those morphisms [MATH] of [MATH] with [MATH] The diagram |
[EQUATION] has canonical colimit [MATH] . Since [MATH] is a set, [MATH] is essentially a small category, thus, [MATH] is small. (2) The category [MATH] clearly has non-empty pre-limits. (Indeed, choose an object [MATH] of the given diagram [MATH] . Then every cone of [MATH] has a subobject of [MATH] as its domain.) Thu... |
5.3 Proposition [MATH] is cowellpowered for every finitely complete, wellpowered category [MATH] such that (1) all hom-sets of objects that are neither terminal nor initial are non-empty, and |
(2) morphisms with non-terminal codomains have (split epi, mono)-factorizations. Proof. (a) We first prove for every coproduct of representables |
[EQUATION] that it has only a set of subobjects in [MATH] . Since [MATH] is closed under coproducts in [MATH] and [MATH] is infinitary extensive, it is sufficient to prove that every representable functor has only a set of subobjects. |
To give a subobject [MATH] of [MATH] means to give, for every object [MATH] of [MATH] , a subset [MATH] such that for all morphisms |
[MATH] we have [EQUATION] In case [MATH] is a terminal object, then [MATH] is determined by a class [MATH] such that no morphism with domain in [MATH] has codomain in the complement [MATH] . Due to (1) either [MATH] or its complement consists of objects isomorphic to [MATH] (initial) or [MATH] (terminal), thus, the num... |
Suppose [MATH] is non-terminal. We prove that [MATH] is determined by its values at (i) 1 and (ii) subobjects of [MATH] . Since [MATH] is wellpowered, this proves that there is only a set of possibilities. |
For that, consider an object [MATH] . Given [MATH] factorize it as a split epic [MATH] followed by a monic [MATH] If [MATH] splits [MATH] then [MATH] and, since [MATH] , we have [MATH] . Conversely, since [MATH] [MATH] whenever |
[MATH] . This proves our claim that [MATH] is determined by [MATH] and all [MATH] (b) The full subcategory of [MATH] formed by all petty functors is cowellpowered. Indeeed, we only need to prove that coproducts of hom-functors [MATH] have sets of quotients in [MATH] . Observe that [MATH] is also a coproduct of hom-func... |
(c) Following Remark 3.4 , it is sufficient to prove that every subfunctor of a hom-functor is petty. 1. Consider first [MATH] for the terminal object [MATH] . To give a subfunctor [MATH] means to specify a class [MATH] of objects such that |
[MATH] and [MATH] implies [MATH] ; then the subfunctor assigns [MATH] to objects of [MATH] and [MATH] else. Due to (1), the only possibilities are [MATH] or [MATH] consisting of initial objects. In both cases every element [MATH] |
yields a weakly initial object of [MATH] , thus [MATH] is petty. 2. Next let [MATH] be a subfunctor of [MATH] where [MATH] is non-terminal. Without loss of generality assume that [MATH] for all [MATH] and that on morphisms [MATH] we have [MATH] . Since [MATH] is wellpowered, we have a set [MATH] |
[MATH] of subobjects representing the images of all morphisms [MATH] [MATH] , lying in [MATH] . As above, each [MATH] lies in [MATH] The elements [MATH] [MATH] , form a weakly initial set in [MATH] : for every [MATH] as above we have [MATH] |
5.4 Example [MATH] is wellpowered and cowellpowered for all of the following categories and their duals: (1) [MATH] (2) Vector spaces over every field. |
(3) Sets and partial functions. Indeed, [MATH] as well as [MATH] satisfy the assumptions of the above Proposition. ( [MATH] was the reason for the complicated condition (2).) |
Recall that a category [MATH] is concrete it there exists a faithful functor [MATH] 5.5 Remark (1) For a finitely pre-complete category [MATH] , the category [MATH] is concrete if and only if it is wellpowered. |
Indeed, [MATH] is finitely complete (Corollary 3.8 ) and monomorphisms are regular (Remark ( 4.2 ). Thus the result follows from |
Theorem 4.1(iii). (2) The limit completion of a category [MATH] is dual to [MATH] . Thus the limit completion of the categories in the above example are wellpowered and cowellpowered. |
5.6 Remark (1) The category [EQUATION] of all accessible set functors is wellpowered and cowellpowered. Indeed, recall from Remark 3.1 (e) that this is precisely the category [MATH] . Recall also that this category is a locally cartesian closed pretopos (see Corollary 3.20 and Lemma 4.1 ). |
(2) This category has also been studied by Barto who has also proved that it is concrete, and universal . That is, every concrete category can be fully embedded into [MATH] |
Recall that a clique on a set [MATH] is a graph whose arrows lead from every node to all distinct nodes. Graph homomorphisms between cliques are precisely the monic maps. Therefore, if [MATH] is a clique on |
[MATH] nodes, then (a) for all ordinals [MATH] infinitely many morphisms exist from [MATH] to [MATH] , and (b) no morphism from [MATH] to [MATH] exists if [MATH] |
Recall that a class [MATH] of morphisms of a category [MATH] is right-cancellative if [MATH] implies that [MATH] 5.7 Definition A class of objects [MATH] for [MATH] of a category is called clique-like provided that there exists a right-cancellative |
class [MATH] of morphisms such that (a) for every ordinal [MATH] more than one morphism from [MATH] to [MATH] exists in [MATH] , and |
(b) no morphism from [MATH] to [MATH] exists in [MATH] either (b1) for all pairs of ordinals [MATH] or (b2) for all pairs of ordinals [MATH] |
5.8 Proposition If a category [MATH] has a clique-like class of objects, then [MATH] is not cowellpowered. Proof. For every ordinal [MATH] denote by |
[EQUATION] the multiple coequalizer of the following morphisms of [MATH] (a full subcategory of [MATH] ): [EQUATION] for all morphisms [MATH] of [MATH] . These small functors [MATH] are quotients of |
[MATH] in [MATH] It remains to prove that they are pairwise non-isomorphic. Indeed, consider [MATH] Assume first that (b1) holds. Choose distinct morphisms [MATH] in [MATH] . They are merged by [MATH] because the element [MATH] of [MATH] is sent to [MATH] by [MATH] and to [MATH] by [MATH] . But [MATH] does not merge [M... |
[MATH] cannot be a composition [MATH] for any [MATH] in [MATH] and any [MATH] . Thus [MATH] cannot be merged with any distinct morphism from [MATH] to [MATH] by [MATH] |
In the case that (b2) holds the proof is the same, we just calculate the components [MATH] and [MATH] 5.9 Example In the category [MATH] of sets and monomorphisms we have a clique-like collection: choose a set [MATH] |
of power [MATH] and [MATH] all morphisms. Thus [MATH] is not cowellpowered. Compare this with the fact that by Proposition 5.2 [MATH] is. |
5.10 Example For the following categories [MATH] the colimit completions [MATH] and [MATH] are not cowellpowered: we present a clique-like class of objects in [MATH] as well as in its dual. Here [MATH] is always chosen to be the class of all morphisms of [MATH] |
(1) The category of graphs: let [MATH] be a clique on [MATH] vertices. This is a clique-like class in [MATH] . Now take the same class except changing [MATH] |
to the complete graph on 2 vertices (so that more than one homomorphism exists from every nonempty graph into [MATH] ). The result is a clique-like class in [MATH] |
(2) Any alg-universal category [MATH] , i.e, one admitting a full embedding [MATH] The monograph presents a number of examples of such categories, e.g., semigroups or rings with unit. |
If [MATH] is a clique-like collection in [MATH] or [MATH] , then [MATH] is one in [MATH] or [MATH] , resp. (3) The category [MATH] of accessible set functors is, as we have seen above, cowellpowered. By Remark 5.6 (2), this category is alg-universal, therefore [MATH] is not cowellpowered. |
(4) The category [MATH] of algebras on one unary operation. To present a clique-like class, we first construct unary algebras [MATH] for [MATH] with no homomorphisms from [MATH] to [MATH] whenever |
[MATH] Recall from the concept of the rank [MATH] of an element [MATH] of an algebra [MATH] which is either an ordinal or [MATH] , an element larger than all ordinals. We work with [MATH] given by a set [MATH] and its endomap [MATH] . The rank is defined to be [MATH] iff there exist elements [MATH] with [MATH] |
for all [MATH] and [MATH] . For all other elements [MATH] of [MATH] rank is defined by [EQUATION] Every homomorphism [MATH] is nondecreasing on ranks: |
[EQUATION] for all [MATH] . See , Proposition 5.2. Let us construct algebras [MATH] for [MATH] such that [MATH] contains an element |
[MATH] of rank [MATH] , but no element of rank at least [MATH] . We proceed by transfinite induction: For [MATH] put [MATH] [MATH] and [MATH] |
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