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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?

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"I don't want groups to be a subcategory of sets." Why not? (I do.) –  Pete L. Clark May 12 '10 at 19:58
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Haha, it's even crazier than that: <Groups> is a subcategory of <Sets>, but also, <Sets> is a subcategory of <Groups>, via the "free group generated by X"--functor. –  Xandi Tuni May 12 '10 at 20:08
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The question "should Grp really be a subcategory of Set?" is a problem under either definition: with a bit of hacking, we can find a category equivalent to Grp as a Mac Lane subcat of Set. Define, say, an "ersatz group" to be a set X containing a unique element x (call it the "code" of X) such that x is a pair (X\{x},\mu), where \mu is a multiplication making X\{x} a group; and a map or these is a function preserving the code and giving a group homomorphism on the rest. I like your question, but I don't think the "Grp" example really motivates it as well as one might initially think. –  Peter LeFanu Lumsdaine May 12 '10 at 21:34
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@Xandi: That's surely no crazier than the fact that $\mathbb{Q}$ "is" a subset of $\mathbb{N}$ in the categorical sense; it's just another illustration of how "$A$ is a sub-foo of $B$" is almost always a statement not just about $A$ and $B$ but also about an implicit inclusion map. :-P –  Peter LeFanu Lumsdaine May 12 '10 at 22:41
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@Ben: I was confused. I don't want groups to be a subcategory of sets either. See my response below for clarification. –  Pete L. Clark May 12 '10 at 23:35
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6 Answers 6

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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).

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Absolutely agreed on the second paragraph! An example here of an interesting subcategory inclusion that is not full on isomorphisms: "differential manifolds; smooth maps" inside "differential manifolds; continuous maps". On the other hand, "pseudomonic" isn't the only notion of subcategory that's invariant under equivalence --- "faithful" is fine for that too, as are the other candidates discussed on the nlab at Andrew S's link. –  Peter LeFanu Lumsdaine May 12 '10 at 22:35
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Absolutely disagreed on the second paragraph! Requiring a categorical notion to be invariant under equivalence is really much more like requiring the notion of inclusion of topological spaces to be invariant under homeomorphism. Homotopy equivalence is a weaker notion of equivalence imposed on topological spaces when we use them as models for "homotopy types;" the "natural" notion of sameness for topological spaces is homeomorphism, just as the natural notion of sameness for categories is equivalence. –  Mike Shulman May 13 '10 at 6:20
    
@Mike: we can probably argue endlessly on this one, but if you take the geometric realization of a category, equivalences go to homotopy equivalences and not to homeomorphisms. So it's not just a vague analogy I'm referring to here, it's an honest-to-god functor -- although it's of course true that geometric realization is by no means a subcategory inclusion... –  Tilman May 13 '10 at 6:41
    
It's interesting that you're the only person to defend the official definition (maybe implicitly TJF). The second paragraph helped me: homotopy theorists find subspaces helpful, even if they are "evil." There are two useful notions of subcategory, so it's bad that people use the same word. I think you're right that pseudomonic is what people should mean if they want a categorical notion, but I'm not convinced it is what they mean. –  Ben Wieland May 13 '10 at 16:34
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If I want a "full subcategory", I say simply say so.

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(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".

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There is some very nice discussion there, but I don't think in actual mathematical practice "subcategory" is used for any of those definitions besides the ones mentioned in this question. –  Peter LeFanu Lumsdaine May 12 '10 at 21:40
    
Actually, I was probably overstating things in that last comment... most subcategories used in practice do fall under some of the other definitions discussed there, and it's very rare to see quantification over arbitrary subcategories, so I guess this point is often moot! –  Peter LeFanu Lumsdaine May 12 '10 at 22:48
    
That's a great resource and I should have thought of looking there first, but, in the interest of closure, I'm choosing Tilman's definite answer. –  Ben Wieland May 13 '10 at 16:24
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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.)


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

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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.

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I was thinking of modifying the ambient category, replacing NonEmptySets with the category whose objects are groups, but whose morphisms are maps of sets, ignoring the groups structure. It should be possible to define groups as special sets, but if you change the cardinality, as you and Peter LeFanu Lumsdaine (in the comments at top) do, this changes the functor from groups to sets. But I think you can stuff the multiplication table into the identity element. You just have to change the name of the identity in the table. But, mainly, I was just thinking thinking that Grps -> Sets is faithful. –  Ben Wieland May 13 '10 at 0:14
    
yes, you're right --- my suggested coding did unintentionally change the functor. but as you say, this can be fixed --- you have to hack a little harder (to get around the fact that the multiplication table wants to talk about itself, which in the standard ZFC implementation it can't "literally" do) but yep, it does still work. –  Peter LeFanu Lumsdaine May 13 '10 at 3:32
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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.)

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