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A category is concrete (see E. Krishnan [2000], §1̃.3) if there is a faithful functor . Since, by Proposition(1.9) of E. Krishnan [2000], small categories are concrete, the existence of such functors does not impose any additional restriction on the category under consideration. Here we shall consider a class of small subcategories of with images in which surjections are surjective maps (surjective morphisms in ). In the sequel, will be considered as a category with subobjects with respect to the set-theoretic inclusion. Unless otherwise stated, , , etc.will denote categories with images. Recall that, if is a subcategory of , then the inclusion is a category embedding of into .
Definition 3.1 (Definition of set-based categories). Let be a small category and let be a functor. We say that is set-based (-category for short) with respect to if the pair satisfy the following:
Notice that the condition (Sb:b) implies that the functor preserves imagefactorizations:
Lemma 3.1. Let be a category with images satisfying the condition (Sb:a). Then it satisfies (Sb:b) if and only if
If (Sb:b) (or (Sb:b)) hold, then
for any and .
Proof. If (Sb:b) holds for all then
and so, (Sb:b) holds. Conversely, assume that (Sb:b) is satisfied. Then, for any ,
Since is a monomorphism, (Sb:b) follows. The equality of Equation (4a) also follows from the proof above.
By definition for any . Since is an inclusion preserving functor, we have
This proves Equation (4b).
It is clear that there is a (large) category in which objects are -categories and morphisms are inclusion-preserving functors that preserve images.
The following is an immediate consequence of the definition of -category which will be useful in the sequal.
Corollary 3.2. Let be an -category with respect to the forgetful functor . Assume that is a subcategory of having images such that the inclusion preserve images of morphisms in . Then is an -category withrespect to .
Let be a subcategorty of the -category satisfying the requtrements of the corollary above. Ihen will be called an -subcategory of .
Let be a category with images and be an embedding. Let
(5) |
Let . Since is an embedding the composition exists in if and only if exists in and
Consequently is a subcategory of . Since is an embedding in if and only if in . Hence is a category with subobjects in which the subobject relation is the set-theoretic inclusion. Notice that is an embedding of into . It is routine to verify the following.
Proposition 3.3. Let be an embedding of a category with subobjects to . Let denote the category with subobjects defined above. Then is a subcategory of and the inclusion is an embedding. Moreover is an isomorphism of categories with subobjects and is a -category with respect to if and only if is an -category with respect to inclusion.
The result above shows that we may replace a -category by in which objects are sets, morphisms are mappings of sets and the image-factorizations of morphisms in are those factorizations in . Consequently it may be possible to replace categorical manipulations in by elementary set-theoretic manipulations. For example, we have the following computations (cf. Equation (6)) of the image and the surjection of the composite of a pair of composable morphisms which is useful in the sequel. One can see, by elementary set-theoretic arguements, that Equation (6) holds in the category . In the next proof we use this fact and the properties of -categories to extend the validity of Equation (6) to arbitrary -categories. Recall that two morphisms are parallel if they belong to the same home-set; that is, and .
Proposition 3.4. Suppose that is an -category and that is a pair of composable morphisms in . Then
(6) |
Moreover, if and are parallel and if then for all for which exists and for all for which exists.
Proof. The hypothesis implies that and for some . Then and
Thus . Since is an embedding, is injective and so .
Also by the first equality in (6),
and so . Since is faithful, we have . This proves Equation (6) for all -categories.
Now, to prove the remaining statements, let . Suppose that . We observe that for all , by definition
Hence if (or ) exists, then
by (6). Since and are parallel, exists if and only if exists. If this holds, we have and again by Equation (6)
This complets the proof.
Remark 3.1: Since epimorphisms are surjective in the categories , , , etc., they could regarded as (large) -categories. Notice that these categories also have unique factorization property. Also any partial order is a -category but don’t have unique factorization.
Example 3.1: Let denote the category with vertex set and morphism set . Setting identity relation on as the subobject relation, becomes a category with factorization in which is a surjection. Furthermore, has images. Assume that and are disjoint sets and is a map such that (). Then the assignments
is an embedding of into . However, is not an -category with respect to . On the other hand, if is a surjection, then the assignments
is an embedding of into such that is an -category with respect to . Similarly one can construct examples of distinct functors and such that is an -category with respect to both and .
It is clear from the remarks above that the structure of a -category depends both on the structure of as well as . However, the following proposition discuss a class of categories (with images) that are -categories with respect to functors where the functor is completely determined by the structure of itself. These are useful for discussing regular categories.
Proposition 3.5. Let be a small category with images. Define assignments and as follows: for each
Then is a functor satisfying the following:
Consequently is an -category if and only if every surjection in is a split-epimorphism.
Proof. For each , is a set and for each , is a well-defined map by Proposition 3.4. If and then for any ,
Hence is a functor. Taking in the above computation, we see that . Hence is an inclusion-preserving functor.
(a) If and then . Hence and so, . Further, by Proposition 3.4 (see Equation (6))
Hence . Conversely if , then and so . This gives . Consequently, is an order-embedding and so, is an embedding if it can be shown that is faithful. To this end assume that and . Then by Equation (7b)
Hence . This proves that is faithful and so (a) is proved.
(b) Let . Then, clearly, and and . Hence .
(c) Suppose that is a monomorphism. If then and so, . Since is a monomorphism, and so, . Therefore is injective. Conversely if is injective and if then
Since is injective, and so, . Hence is a monomorphism.
(d) Suppose that such that is surjective. Since , there exist such that . Since , we have and so, is a split epimorphism. On the other hand, if is a split epimorphism then for some . Then and so, is surjective. This complete the proof of (d).
If has images, statements (a) and (b) shows that satisfies axioms (Sb:a) and (Sb:c) of Definition 3.1. Suppose that every surjection in split. Then by (d), is surjective for all . Hence and so by Lemma 3.1, satisfies axiom (Sb:b). Therefore is a -category. Conversely, if is an -category then by Lemma 3.1, for all . Hence, by (d), every surjection split.
Given a category with images, we shall refer to the embedding constructed above as the natural embedding of into . is called a -category if it is a -category with respect to its natural embedding.
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