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Is there such a thing as a category of isomorphism classes of, say, modules?

First step in definining morphisms in such a category would be to identify two morphisms $f:M\rightarrow N$ and $f':M'\rightarrow N'$ if there are isomorphisms $i:M\simeq M'$ and $j:N\simeq N'$ such that $j\circ f=f'\circ i$. But this equivalence realtion doesn't satisfy properties like $$f\sim g\implies f\circ h\sim g\circ h,$$ so I don't know how to compose two morphisms in such a category. This question arose because Ext and Tor (or derived functors in general) depend on the choice of resolutions, though not so modulo isomorphism classes. They don't seem to be functors in the standard sense, but a ``functor from isomorphism classes to isomorphism classes.''

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Are you looking for the notion of a skeleton? Alternately, I think using derived categories ( fixes the problem you describe, but I don't really know anything about this. – Qiaochu Yuan Jul 1 '11 at 17:28
I don't think the first sentence matches the rest of the question. In the rest of the question I think you are identifying too many morphisms. The construction you propose doesn't seem reasonable even in, for example, the category of finite-dimensional vector spaces. – Qiaochu Yuan Jul 1 '11 at 17:30
The problem with this definition is that there are often automorphisms, which, under this definition, which makes this stronger than the skeleton category. – Thomas Andrews Jul 1 '11 at 17:49
Yes, the idea is that you want to fix isomorphisms $M \cong M'$ and $N \cong N'$ and only identify morphisms that correspond under those particular isomorphisms, not under all isomorphisms. Of course it's easier to just talk about taking a subcategory containing one element from each isomorphism class. – Qiaochu Yuan Jul 1 '11 at 20:47
up vote 6 down vote accepted

Perhaps part of the issue is (to my perception) a confounding of three different, but related, issues. The second question asked about the dependence of Ext, Tor (and, presumably, other derived functors) on resolutions: the usual arguments not only show that the isomorphism classes of the Ext groups do not depend on the resolution, but that the (natural) maps among Ext groups of different objects are independent of the resolutions. One point is that this "naturality" is often confused with a more colloquial (and still useful) sense of "not making irrelevant choices", etc.

This is related to one of the earlier comments, that too many things were being identified by isomorphisms, and that various categories of finite-dimensional vector spaces already illustrate relevant issues. Namely, we can certainly consider the category with objects $k^0$, $k^1$, $k^2$, ... with a field $k$, that is, the elementary-linear-algebra finite-dimensional vector spaces over $k$. And we can consider a great variety of maps among these. As in the earlier comment, this is a "skeleton" of the category of all finite-dimensional $k$-vectorspaces. However, notice that we do not declare every isomorphism of $k^n$ with itself to be the identity.

The third point, mentioned in a comment, but which I suspect was not a primary part of the question(s) as asked, might indeed be about derived categories: how to make (e.g.) projective resolutions _modulo_homotopy_ the objects in a category, since we know that homotopic resolutions produce the same outcomes. My reaction is that the "natural" dependence of computing derived functors "up to isomorphism" should seem reasonable before worrying about derived categories.

The standard useful exercise in appreciating genuine naturality is to show that finite-dimensional vector spaces are definitely not naturally isomorphic to their duals... by showing that no list (over f.d. $k$-vectorspaces $V$) of isomorphisms $\phi_V:V\rightarrow V^*$ is compatible with all homs among vectorspaces. (And/but, equally standard is writing down the natural isomorphisms of these to their second duals.)

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