3. Set based categories

A category 𝒞 is concrete (see E. Krishnan [2000], §1̃.3) if there is a faithful functor U : 𝒞Set. 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 𝒮𝒞 of small subcategories of Set with images in which surjections are surjective maps (surjective morphisms in Set). In the sequel, Set will be considered as a category with subobjects with respect to the set-theoretic inclusion. Unless otherwise stated, 𝒞, 𝒟, etc.will denote categories with images. Recall that, if 𝒞 is a subcategory of Set, then the inclusion ȷSet𝒞 : 𝒞Set is a category embedding of 𝒞 into Set.

Definition 3.1 (Definition of set-based categories). Let 𝒞 be a small category and let U : 𝒞Set be a functor. We say that 𝒞 is set-based (𝒮-category for short) with respect to U if the pair (𝒞,U) satisfy the following:

(Sb:a)
U is an embedding.
(Sb:b)
U(imf) = imU(f) for all f 𝒞.
(Sb:c)
The functor U has the following property: for c,c 𝖛𝒞 and x U(c) U(c) there is d 𝖛𝒞 such that
d c,d c andx U(d).

Notice that the condition (Sb:b) implies that the functor U preserves imagefactorizations:

Lemma 3.1. Let 𝒞 be a category with images satisfying the condition (Sb:a). Then it satisfies (Sb:b) if and only if

(Sb:b)
U(f) = U(f) for all f 𝒞.

If (Sb:b) (or (Sb:b)) hold, then

U fȷcod f im f = U(f)ȷcod U(f) im U(f) (4a)

for all f 𝒞. Moreover, we have

U(fc) = U(f)U(c) andU f(c) = U(f) U(c) (4b)

for any f 𝒞 and c domf.

Proof. If (Sb:b) holds for all f 𝒞 then

imU(f) = codU(f) = codU(f) = U(codf) = U(imf)

and so, (Sb:b) holds. Conversely, assume that (Sb:b) is satisfied. Then, for any f 𝒞,

U(f) = U fȷcod f im f = U fȷU(cod f) U(im f)  by (Sb:a) = U fȷcod U(f) im U(f)  using (Sb:b) and the fact that U is a functor. The image factorization of U(f) in Set gives U(f) = U(f)ȷcod U(f) im U(f) .

Since ȷcod U(f) im U(f) is a monomorphism, (Sb:b) follows. The equality of Equation (4a) also follows from the proof above.

By definition fc = ȷdom f c f for any c domf. Since U is an inclusion preserving functor, we have

U(fc) = U(ȷdom f c )U(f) = ȷU(dom f) U(c) U(f) = U(f)U(c).  Since f(c) = im(fc) by definition, we get U f(c) = U im(fc) = imU(fc) = im U(f)U(c) = U(f) U(c).

This proves Equation (4b).

It is clear that there is a (large) category 𝔖𝔠𝔞𝔱 in which objects are 𝒮-categories and morphisms are inclusion-preserving functors that preserve images.

The following is an immediate consequence of the definition of 𝒮-category which will be useful in the sequal.

Corollary 3.2. Let 𝒞 be an 𝒮-category with respect to the forgetful functor U : 𝒞Set. Assume that 𝒟 is a subcategory of 𝒞 having images such that the inclusion ȷ𝒞𝒟 preserve images of morphisms in 𝒟. Then 𝒟 is an 𝒮-category withrespect to U𝒟.

Let 𝒟 be a subcategorty of the 𝒮-category satisfying the requtrements of the corollary above. Ihen 𝒟 will be called an 𝒮-subcategory of 𝒞.

Let 𝒞 be a category with images and U : 𝒞Set be an embedding. Let

imU = {U(c) : c 𝖛𝒞},{U(f) : f 𝒞} (5)

Let f,g 𝒞. Since U is an embedding the composition U(f)U(g) exists in Set if and only if fg exists in 𝒞 and

U(fg) = U(f)U(g).

Consequently imU is a subcategory of Set. Since U is an embedding c d in 𝖛𝒞 if and only if U(c) U(d) in Set. Hence imU is a category with subobjects in which the subobject relation is the set-theoretic inclusion. Notice that imU Set is an embedding of imU into Set. It is routine to verify the following.

Proposition 3.3. Let U : 𝒞Set be an embedding of a category 𝒞 with subobjects to Set. Let imU denote the category with subobjects defined above. Then imU is a subcategory of Set and the inclusion imU Set is an embedding. Moreover U : 𝒞 imU is an isomorphism of categories with subobjects and 𝒞 is a 𝒮-category with respect to U if and only if imU is an 𝒮-category with respect to inclusion.

The result above shows that we may replace a 𝒮-category 𝒞 by imU in which objects are sets, morphisms are mappings of sets and the image-factorizations of morphisms in imU are those factorizations in Set. Consequently it may be possible to replace categorical manipulations in 𝒞 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 Set. In the next proof we use this fact and the properties of 𝒮-categories to extend the validity of Equation (6) to arbitrary 𝒮-categories. Recall that two morphisms f,g 𝒞 are parallel if they belong to the same home-set; that is, domf = domg and codf = codg.

Proposition 3.4. Suppose that 𝒞 is an 𝒮-category and that f,g is a pair of composable morphisms in 𝒞. Then

im(fg) = g imf and(fg) = fgimf. (6)

Moreover, if g and h are parallel and if g = h then (fg) = (fh) for all f for which fg exists and (gk) = (hk) for all k 𝒞for which gk exists.

Proof. The hypothesis implies that f : c d and g : d e for some c,d,e 𝖛𝒞. Then imf d and

U im(fg) = im U(f)U(g)  by Lemma 3.1 = U(g) imU(f)  since (6) hold in Set = U g(imf)  by (4b)

Thus U im(fg) = U g(imf). Since U is an embedding, 𝖛U is injective and so im(fg) = g(imf).

U (fg) = U(fg)  by Lemma 3.1 = U(f)U(g) = U(f)U(g)imU(f)  since (6) holds in Set = U(f)U (gimf)  by (4b) = U f(gimf).

Also by the first equality in (6),

im(fg) = cod(fg) = g(imf) = cod(gimf)

and so (fg),f(gimf)𝒞(domf,im(fg)). Since U is faithful, we have (fg) = f(gimf). This proves Equation (6) for all 𝒮-categories.

Now, to prove the remaining statements, let g = h. Suppose that domg = domh = d. We observe that for all a d, by definition

(ga) = (ȷd agȷcod g im g ) = (ȷd ag) = (ȷd ah) = (ha).

Hence if fg (or fh) exists, then

(fg) = f(gimf) = f(himf) = (fh).

by (6). Since g and h are parallel, gk exists if and only if hk exists. If this holds, we have img = codg = codh = imh and again by Equation (6)

(gk) = g(kimg) = h(kimh) = (hk).

This complets the proof.

Remark 3.1: Since epimorphisms are surjective in the categories Grp, Ab, Vct𝕂, etc., they could regarded as (large) 𝒮-categories. Notice that these categories also have unique factorization property. Also any partial order is a 𝒮-category but don’t have unique factorization.

Example 3.1: Let 𝒞 denote the category with vertex set 𝖛𝒞 = {x,y} and morphism set 𝒞 = {1x,α : x y,1y}. Setting identity relation on 𝖛𝒞 as the subobject relation, 𝒞 becomes a category with factorization in which α is a surjection. Furthermore, 𝒞 has images. Assume that X and Y are disjoint sets and f : X Y is a map such that imf = Y Y (Y Y ). Then the assignments

U : xX,yY  andαf

is an embedding of 𝒞 into Set. However, 𝒞 is not an 𝒮-category with respect to U. On the other hand, if g : X Y is a surjection, then the assignments

V (x) = X,V (y) = Y  andV (α) = g

is an embedding of 𝒞 into Set such that 𝒞 is an 𝒮-category with respect to V . Similarly one can construct examples of distinct functors U and V such that 𝒞 is an 𝒮-category with respect to both U and V .

It is clear from the remarks above that the structure of a 𝒮-category (𝒞,U) depends both on the structure of 𝒞 as well as U. However, the following proposition discuss a class of categories 𝒞 (with images) that are 𝒮-categories with respect to functors U where the functor U is completely determined by the structure of 𝒞 itself. These are useful for discussing regular categories.

Proposition 3.5. Let 𝒞 be a small category with images. Define assignments 𝖛U : 𝖛𝒞 𝖛Set and U : 𝒞Set as follows: for each c 𝖛𝒞

U(c) = {f : f 𝒞,imf c} (7a)

and for each morphism f : c d 𝒞, let

U(f) : g U(c)(gf) U(d). (7b)

Then U : 𝒞Set is a functor satisfying the following:

(a)
The functor U : 𝒞Set is an embedding.
(b)
For a,b 𝖛𝒞 and h = h U(a) U(b) there is c 𝖛𝒞 such that h U(c) U(a) U(b).
(c)
f 𝒞 is a monomorphism if and only if U(f) is injective.
(d)
f 𝒞 is a split-epimorphism if and only if U(f) is surjective.

Consequently (𝒞,U) is an 𝒮-category if and only if every surjection in 𝒞 is a split-epimorphism.

Proof. For each c 𝖛𝒞, U(c) is a set and for each f 𝒞(c,d), U(f) : g U(c)(gf) is a well-defined map by Proposition 3.4. If f : c d and g : d e then for any h U(c),

(h)U(f)U(g) = (hf)U(g) = h(fg) = (h)U(fg).

Hence U : 𝒞Set is a functor. Taking f = ȷdc in the above computation, we see that U(ȷdc) = ȷU(d) U(c). Hence U : 𝒞Set is an inclusion-preserving functor.

(a) If c d and f U(c) then imf c d. Hence f U(d) and so, U(c) U(d). Further, by Proposition 3.4 (see Equation (6))

(f)U(ȷd c) = (fȷd c) = f = (f)ȷU(d) U(c)

Hence U(ȷdc) = ȷU(d) U(c). Conversely if U(c) U(d), then 1c U(c) and so 1c U(d). This gives c d. Consequently, 𝖛U is an order-embedding and so, U : 𝒞Set is an embedding if it can be shown that U is faithful. To this end assume that f,g 𝒞(c,d) and U(f) = U(g). Then by Equation (7b)

1cU(f) = f = 1 cU(g) = g.

Hence f = g. This proves that U is faithful and so (a) is proved.

(b) Let h = h U(c) U(d). Then, clearly, c = codh c and and c d. Hence h U(c) U(c) U(d).

(c) Suppose that f : c d is a monomorphism. If (h)U(f) = (g)U(f) then (hf) = (gf) and so, hf = gf. Since f is a monomorphism, h = g and so, h = g. Therefore U(f) is injective. Conversely if U(f) is injective and if hf = gf then

(h)U(f) = (hf) = (gf) = (g)U(f).

Since U(f) is injective, h = g and so, h = g. Hence f is a monomorphism.

(d) Suppose that f : c d 𝒞 such that U(f) is surjective. Since 1d U(d), there exist h = h : d c c such that (h)U(f) = (hf) = 1d. Since codhf = d, we have hf = 1d and so, f is a split epimorphism. On the other hand, if f : c d is a split epimorphism then hf = 1d for some h : d c. Then U(h)U(f) = 1U(d) and so, U(f) is surjective. This complete the proof of (d).

If 𝒞 has images, statements (a) and (b) shows that (𝒞,U) satisfies axioms (Sb:a) and (Sb:c) of Definition 3.1. Suppose that every surjection in 𝒞 split. Then by (d), U(f) is surjective for all f 𝒞. Hence U(f) = (U(f)) and so by Lemma 3.1, 𝒞 satisfies axiom (Sb:b). Therefore 𝒞 is a 𝒮-category. Conversely, if 𝒞 is an 𝒮-category then by Lemma 3.1, U(f) = U(f) for all f 𝒞. Hence, by (d), every surjection split.

Given a category 𝒞 with images, we shall refer to the embedding U : 𝒞Set constructed above as the natural embedding of 𝒞 into Set. 𝒞 is called a 𝒩𝒮-category if it is a 𝒮-category with respect to its natural embedding.