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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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A category with one object is just a group. So it's just the same as asking if you can associate a vector space to a group. And there's several ways of doing that, it depends what you're after. –  George Lowther May 22 '11 at 15:28
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@George: a locally small category with one object is just a monoid. @Christiaan: I don't really understand the question as stated. What do you want this vector space to do? –  Qiaochu Yuan May 22 '11 at 15:30
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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. –  George Lowther May 22 '11 at 15:37
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2 Answers 2

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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Hom(*,*) is better than a set, it's a monoid under composition! You forgot the same structure for k-linear categories: a k-linear category with one object is essentially the same as a k-algebra, not a mere k-vector space. –  Omar Antolín-Camarena May 24 '11 at 0:33
    
Wow, a downvote. Any reason for that or just because? :-) –  a.r. May 27 '13 at 17:56
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