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Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in this category is a linear map such that $\phi_W f = \phi_V f$. Note this category is the same as the category of $\mathbb C [t]$-modules whose underlying space is finite dimensional as a $\mathbb C$-vector space.

I am having some trouble working out how many isomorphism classes there are. The problem is that even if $V \cong W$ as vector spaces, the isomorphism might not respect the structure morphisms in $\mathcal C$. So potentially there are a LOT of isomorphism classes.

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up vote 4 down vote accepted

Jordan normal form tells you what the isomorphism classes look like, but you don't need to know this: it suffices to show that the collection of isomorphism classes with a fixed value of $\dim V$ forms a set, and this is straightforward as specifying the corresponding $\phi_V$ requires at most $(\dim V)^2$ parameters.

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in general, the category of f.g. modules over a ring is essentially small: every such module is isomorphic to a quotient of a f.g. module (and there is a set of these up to isomorphism) by one of its submodules (and each of them has a set of submodules) – Mariano Suárez-Alvarez Oct 1 '12 at 18:31
I'm a bit confused about your last sentence - what do you mean by the corresponding $\phi_V$? – Paul Slevin Oct 1 '12 at 20:06
@Paul: an isomorphism class is specified by a pair $(V, \phi_V)$. I've fixed the dimension of $V$, which is tantamount to fixing $V$, so the only thing I need to do is to fix $\phi_V$. – Qiaochu Yuan Oct 1 '12 at 20:20
where does (dimV)^ 2 come from? – Paul Slevin Oct 1 '12 at 20:36
@Paul: $\phi_V$ can be specified by specifying a matrix. – Qiaochu Yuan Oct 1 '12 at 21:28

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