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I'm learning category theory by self-study. I have a couple of texts, and they both talk about how we ought to try not to think so much about the equality between objects in categories. Rather, the important relation is isomorphism.

Okay, sure.

Then they go on to talk about "Cat", the category of (small) categories. One of them even segues into it by considering "categories themselves as structured objects. The `morphisms' between them that preserve their structure are called functors." But then it becomes clear that equality of functors depends on equality of objects, which isn't part of the structure of categories! (Or at least, a part we're not supposed to think about.)

There are natural transformations, which inherit equality from the underlying equality of morphisms in the codomain category. They induce a natural isomorphism on functors. Treating functors, modulo natural isomorphism, as morphisms in a category of categories, seems to me to be the obvious "Cat".

In fact, not only is this not "Cat", but I couldn't find it among the loads of exotic examples of categories I'm given. Apparently it wasn't even deemed worthy of mention. This is a bit disappointing. Doesn't it at least have a standard name and snappy acronym? Can someone point me somewhere where I can learn about its basic properties somewhere? Or is there something defective about it that I've missed?

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3 Answers 3

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You are correct that in practice one does not care so much whether two functors are equal, but rather whether they are naturally isomorphic. The difficulty is that typically one does not want to completely forget about the natural isomorphism either! (Which is what happens if one quotients out by natural isomorphisms as you suggest.)

A typical example (which I hope will make sense to you) is to consider for each topological space the category of vector bundles on $X$. So we have a category $Vect\_X$. If $f: X \to Y$ is a continuous map of spaces, and $\mathcal V$ is a vector bundle on $Y$, then one can pull-back $\mathcal V$ to form a vector-bundle $f^* \mathcal V$ on $Y$. So we get a functor $f^*: Vect_Y \to Vect_X$. If now $g: Y \to Z$ as well, then one sees that $(g f)^* $ is naturally isomorphic to $f^* g^*$, say by some natural isomorphism $c_{f,g}: (g f)^* \cong f^* g^*.$

Morally, one would like to say that $X \mapsto Vect_X$ and $f \mapsto f^* $ gives a contravariant functor from the category of topological spaces to the category of categories Cat. In practice, because we don't have equality between $(f g)^* $ and $f^* g^* $, we don't get such a functor, although we do get a functor into your suggested category "Cat". The problem is that in practice, one wants to remember the natural isomorphisms $c_{f , g}$, which satisfy some important properties: for example, if $h: Z \to W$ is a third map, then $f^* c_{g,h} \circ c_{f, h g} = c_{f,g}h^* \circ c_{gf, h}.$ (If you write this out, it is a commutative square that relates the various ways to pull back $\mathcal V$ by $h$, $g$, and $f$, taking into account the associative law $(hg)f = h(g f)$.)

So really, one wants to work in a more sophisticated structure then either Cat or "Cat", namely a structure in which the objects are categories, the morphisms are functors, and in which we add an explicit extra layer of structure, so-called 2-morphisms, which are natural isomorphisms between functors. One then develops a theory in which one morally regards functors as equal if they coincide up to natural isomorphism (i.e. up to a diagram involving a 2-morphism), but one also keeps track of the 2-morphisms.

The resulting structure is called a 2-category, and is part of the study of higher category theory. (Everything written above should provide some background for understanding Martin Brandenburg's answer.)

This theory has a lot in common with homotopy theory: in topology, one can pass to a category in which objects are spaces and maps are continuous maps modulo homotopy, but in lots of applications one wants to remember not just that maps are homotopic, but one actually wants to remember the homotopy; often then there are homotopies between homotopies, homotopies between homotopies between homotopies, and so on. Similarly in category theory one can introduce not just the notion of 2-category, but notions of n-categories, in which there are 3-morphisms beweeen the 2-morphisms, etc., up to n-morphisms.

Your structure "Cat" is analogous to passing to the homotopy category in topology; it is interesting, but forgets information one often wants to remember.

If you search for "higher category theory", you will find an enormous amount of material. One good reference is the n-category cafe and the n-Lab. Another place to look is at the various manuscripts on Jacob Lurie's web-page (at Harvard). What you will find is a rather intricate, and rapidbly evolving theory, blending category theory and homotopy theory in a fascinating (although sometimes daunting!) way.

In summary, your idea and your question are far from misguided, but in fact are pointing at one of the most active areas of modern research in category theory and related areas!

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Thanks for the details answer and the pointer to n-Lab. As I say below, I'll need to ponder this. In the meantime, is it correct that: the reason the category of (small) categories is presented this way is that normally we really want to think of it is a 2-category; if we want to think of it as a (1-)category of categories, we would normally use the quotient construction rather than just dropping the 2-morphisms, as this allows things like your vector bundle functor example and Martin's ring module functor example ? –  Forgetful functor Oct 5 '10 at 16:27
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@Forgetful: I think that, without the 2-category formalism, no-one really uses the category of categories in any serious way. The quotient construction you suggest loses too much information to be useful in practice, and asking for exact equalities, on the other hand, greatly restricts its usefulness and applicability. I think that what happened is that in older texts (say Grothendieck's work involving fibred categories) ad hoc ideas that were implicitly 2-categorical were used, and beginning in the late 60s, 2-categorical ideas were explicitly introduced. (E.g. I seemto remember that ... –  Matt E Oct 5 '10 at 17:27
    
... Deligne and Mumford's paper on moduli stacks of algebraic curves use the terminology of 2-categories in explaining stacks. (They are in a situation similar to that of vector bundles, in which one can pull-back families of curves via morphisms, and pulling back by successive morphisms is naturally isomorphic, but not literally identical to, pulling back via the composition of the morphisms.) –  Matt E Oct 5 '10 at 17:34

If $C$ is a $2$-category, every hom-set is a category and isomorphy yields a congruence relation on them. The quotient $\tilde{C}$ is then a $1$-category. It's objects are the objects of $C$, but the morphisms are the $1$-morphisms of $C$ modulo $2$-isomorphy.

When you want to compare $C$ with $2$-categories, it does not make any sense to consider $\tilde{C}$. However, this construction becomes useful when you want to compare with $1$-categories, that is, usual categories. For example, the functor

$Ring \to \tilde{Cat}, R \mapsto Mod(R), \phi \mapsto \phi^*$

Note that $\phi^* \psi^* \cong (\phi \psi)^*$ holds in $Cat$, thus these morphisms are equal in $\tilde{Cat}$ and we get, indeed, a functor.

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This does sound like what I'm talking about. Now, I didn't know what a 2-category was until just now, so I'm going to need to ponder these answers a bit. In the meantime, is $\tilde{C}$ (or $\tilde{Cat}$) standard notation, and/or is there a short name for this construction? –  Forgetful functor Oct 5 '10 at 16:09
    
@Forgetful: one name for this construction is the 1-co-truncation of $\mathrm{cotr}^1(C)$. The “just drop all cells above dimension 1” construction is called “1-truncation”, $\mathrm{tr}^1$. The duality comes via the “locally discrete” 2-category on a category, which has no standard notation but let’s call it $D_2$: Cat $\to$ 2-Cat; then there are adjunctions $\mathrm{cotr}^1 \dashv D_2 \dashv \mathrm{tr}^1$. Actually, there’s another adjoint at least… can you spot it? And these all arise interestingly from functors $\mathrm{ob}$, $D$, $\pi_0$ between Set and Cat. –  Peter LeFanu Lumsdaine Nov 20 '10 at 3:03

I'm not sure to understand what you mean, but maybe the operation for which you are asking for a name is skeleton.

Quoting Mac Lane's "Categories for the working mathematician" (chapter IV, section 4),

In any category ${\cal C}$ [not just $\mathbf{Cat}$], a skeleton of ${\cal C}$ is any full subcategory ${\cal A}$ such that any object of ${\cal C}$ is isomorphic to exactly one object of ${\cal A}$. Then ${\cal A}$ is equivalent to ${\cal C}$ and the inclusion $K: {\cal A} \longrightarrow {\cal C}$ is an equivalence of categories. (...) A category is equivalent to any of its skeletons. For instance, the category of all finite sets has as skeleton the full subcategory with objects all finite ordinal numbers $0, 1, 2, \dots , n , \dots$

Finally, Mac Lane adds

A category is called skeletal when any two isomorphic objects are identical; i.e., when the category is its own skeleton.

For instance, the category with objects all finite ordinal numbers is skeletal: there you can forget about the picky distinction between "equal" and "isomorphic" objects. In real life, there is another circumstance in which isomorphic objects are treated as being "equal" for all purposes, namely, when you have a canonical isomorphism, for instance one which is derived from a universal property such as a the one verified by coproducts.

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The question does not have to do with skeletons. –  Martin Brandenburg Oct 4 '10 at 23:39
    
@Martin. Maybe... –  a.r. Oct 4 '10 at 23:57
    
I can't vote, but Martin has understood what I was getting at. –  Forgetful functor Oct 5 '10 at 16:03

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