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How can we associate a vector space structure to a category with one object ? Is there a canonical way of doing this ?

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    @Qiaochu: Sorry, you're right, I should have said monoid. Still, the same comment applies. It depends what the OP is after.2011-05-22

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You must be able to add vectors, but generally there is no way to "add" morphisms in a general category with one object. Also, what should the base field be?

Let $C$ be your category. What you could do, is to fix a field $k$, and consider the freely generated vector space $k[Hom(C)]$ (funny notation...). That is, the vector space with elements formal $k$-linear combinations of the morphisms in $C$.

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Let's denote $*$ the (unique) object of your category. The (unique) set of maps of this category is the set of endomorphisms $\mathrm{Hom}(*,*)$. By definition of category, this is set is, well, a set. But there are circumstances where it can have more structure.

For instance, if your category is an abelian one, then $\mathrm{Hom}(*,*)$ is an abelian group. This is the case, e.g., for the category of abelian groups, where the set of morphisms between two abelian groups is also an abelian group.

When the sets of morphisms have the structure of $\mathbf{k}$-vector spaces, the category is called $\mathbf{k}$-linear ($\mathbf{k}$ a field). For instance, in the category of $\mathbf{k}$-vector spaces, hom-sets are also $\mathbf{k}$-vector spaces.

Hence, a $\mathbf{k}$-linear category with just one object is essentially the same as a $\mathbf{k}$-vector space. Namely, $\mathrm{Hom}(*,*)$.

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    Wow, a downvote. Any reason for that or just because? :-)2013-05-27