What do people mean by "subcategory"?

Question

Mac Lane defines a subcategory as a subset of objects and a subset of morphisms that form a category. But the first rule of category theory is that you do not talk about equality of objects. Up to equivalence, the definition becomes a faithful functor. This is a useful concept, but I don't think it fits the name. I don't want groups to be a subcategory of sets!

This is not a question about aesthetics, but about usage. I don't think people tend to use Mac Lane's definition. Maybe they're just wrong, but I'd like to know if there is another definition which fits the usage better. All the time I see people say things like "we may assume that our subcategory contains every object isomorphic to an object of the subcategory." I guess we can expand to an equivalent subcategory to achieve this, but we probably have to choose how the objects are isomorphic (though it may be easier if to change the ambient category). This is a much more natural thing to do (and safer) if the subcategory is full, or at least contains all the automorphisms of its objects. This leads me to suspect that people are assuming or thinking of some stronger definition than faithful.

Do people tend to mean the official definition? or do they also require full? containing all the automorphisms? Are there other useful intermediate notions?

Answer 1

Do you want the notion of "subcategory" to be invariant under categorical equivalence? If so, then "pseudomonic" functors are the right thing: faithful, and full on isomorphisms.

But I don't think one would want this any more than the notion of "inclusion" of topological spaces to be invariant under homotopy equivalence (which would make it meaningless).

Answer 2

If I want a "full subcategory", I say simply say so.

Answer 3

(This is not my answer (as I didn't have anything to do with this page), hence the community-wiki tag.)

http://ncatlab.org/nlab/show/subcategory

In particular, you'll find some sympathy for the viewpoint that "groups" should not be a subcategory of "sets".

Answer 4

Do people tend to mean the official definition?

I think "official" belongs in scare-quotes... I tend to think that "subcategory" is an evil notion. I'm not published anywhere, but in my notes I use "subcategory" to mean "a subset of $\operatorname{Hom}$, closed under $1_{-}$ and composition." Then you can suitably mimic Mac Lane's definition by specifying $$\operatorname{SubC}(A,X)=\operatorname{SubC}(X,A) = \left\{\begin{array}{c} \{1_A\} & A=X \\\\ \emptyset & A\neq X \end{array} \right. $$ for any objects $A$ you might want to ignore; they become isolated and trivial. That does feel a bit kludgy, though.

or do they also require full? containing all the automorphisms?

Probably not, in either case. That just looks weird to me... but what do I know?

Are there other useful intermediate notions?

Definitely.

Mac Lane's definition of subcategory given in the question corresponds to a functor which is strongly faithful in the sense that $F(g)=F(h)\implies g=h$ without hypotheses on the sources and targets of $g,h$; this strong property comes from the fact that functors don't generally have sensible image categories: what do you do if all objects are mapped to the same object!?

The ordinary sense of faithful is a slightly less strict condition: if $f,g:S\to T$ and $F(f)=F(g)$ then $f=g$.

Consider functors of groupoids $F:\mathcal{A}\to\mathcal{B}$. Every such functor factors in an essentially unique way as $$ F = G_3 \circ G_2 \circ G_1 $$ where $G_i$ omits only property $i$ among

1. Faithful
2. Full
3. Essentially Surjective

The same construction${}^1$ that gives this factorization makes good sense for general categories as well, although it's then complicated by other functor properties you might want to consider (reflects isomorphisms, reflects isomorphy, etc.). $G_3$ might be called "full objectwise-subcategory" (the reference calls it "forgets only property") and $G_2$ ("only structure"), I think, generalizes the notion I describe at the top of this answer.

(To be clear, "$G:\mathcal{X}\to \mathcal{Y}$ is essentially surjective" means that every morphism of $\mathcal{Y}$ factors as $i\circ G(\varphi) \circ j$ for isomorphisms $i,j$ in $\mathcal{Y}$. This implies a property of $G$ relative to objects which isn't worth spelling out here.)

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${}^1$take a day to read this page

Answer 5

Upon request, I will clarify -- and partially take back! -- my earlier comment.

What I find completely unobjectionable is Mac Lane's definition of a subcategory $\mathcal{D}$ of a category $\mathcal{C}$ as being given by subclasses of objects and morphisms which forms a category under the induced composition. I don't see what else you would want a subcategory to be. I do agree that the notion of "subcategory" is not one of the more useful categorical concepts I know, and it even has some potential to be evil in the sense that modern categorists use the word. (Surely it would be more in the spirit of things to talk about a functor from $\mathcal{C}$ to $\mathcal{D}$ which satisfies certain "injectivity" properties.)

Now let's return to the statement "I don't want Groups to be a subcategory of Sets". In my comment I said that I did want this, but I don't now know why I said that: I think I must simply have been confused. Indeed, it is not obvious to me that this definition makes Groups a subcategory of Sets, at least not in any unique or benign way.

If you asked me to spell out the most evident categorical relationship between sets and groups, I would first of all point to the category NonEmptySets -- now that's a subcategory of Sets! -- and then the "forgetful" functor from Groups to NonEmptySets. This functor is (at least assuming the Axiom of Choice) surjective: every nonempty set is the underlying set of some group. But most sets can be endowed with a group law in multiple (usually nonisomorphic) ways, so this is not an "inclusion functor".

(Even the other way around, namely the free group functor from Sets to Groups seems not to quite make Sets into a subcategory of groups, because the class of sets is not a subclass of the class of groups.)

Maybe you are thinking of doing something tricky: defining a group to be an ordered pair [identifying ordered pairs with sets in one of the usual -- silly! -- ways] $(S,\circ)$ where $S$ is a set and $\circ$ is a subset of $S \times S \times S$ satisfying certain axioms. (Note that this is definitely incompatible with the above way of thinking about groups as having -- but not being -- an "underlying set".) But isn't this especially evil?

Comments more than welcome.

Answer 6

I think in practice, most people would (if pressed) describe the definition they're using as something like the Mac Lane definition, plus being allowed to replace either category with another equivalent category [where "equivalent" means "with a chosen equivalence in mind"].

This is clearly not equivalent to the unmodified Mac Lane definition. (E.g. the "walking isomorphism" $I$ has a ML-subobject that is discrete on two objects, but $I$ is equivalent to to the terminal category $1$, which has no subcategory equivalent to a discrete cat. on two objects.)

This is equivalent to the definition "a subobject is a faithful functor". Precisely, given a category $\mathcal{C}$, we can define two 2-categories, $\mathrm{SubCat}(\mathcal{C})$ and $(\mathrm{Cat}/C)_\mathrm{faithful}$, with objects respectively

• an object of $(\mathrm{Cat}/C)_\mathrm{faithful}$ is a faithful functor into $\mathcal{C}$;
• an object of $\mathrm{SubCat}(\mathcal{C})$ is a chain $\mathcal{D} \simeq \mathcal{D}' \subseteq \mathcal{C}' \simeq \mathcal{C}$, where $\subseteq$ denotes the inclusion of a literal Mac Lane subcategory;

and an arrow is a triangle/"ladder-triangle", commutative up to specified natural isomorphism(s); and 2-cells in each case are natural isomorphisms commuting with everything in sight.

Then (if I'm not mistaken) these two 2-categories are 2-equivalent.

I think this usually what "subcategory" is used to mean in practice. "Repletification" --- making your subcategory "closed under isomorphisms" --- is then unproblematic: it just involves composing in an extra equivalence $D'' \simeq D'$ on the front, where an object of $D''$ is an object of $C$ together with a chosen isomorphism to some object of your original subcategory.

(As you point out, for literal Mac Lane subcategories this is problematic except when the subcategory is full on isomorphisms.)