Fix a field $k$ and suppose $\mathcal{C}$ and $\mathcal{D}$ are $k$-linear additive categories and are enriched over the category $\mathcal{V}$ of finite-dimensional $k$-vector spaces. So we have copower functors $$ \mathcal{C} \times \mathcal{V} \to \mathcal{C} \quad \text{and} \quad \mathcal{D} \times \mathcal{V} \to \mathcal{D}. $$ Now suppose $F \colon \mathcal{C} \to \mathcal{D}$ is a $k$-linear functor. Does it follow that the diagram $\require{AMScd}$ \begin{CD} \mathcal{C} \times \mathcal{V} @>>> \mathcal{C} \\ @V F \times \text{id}_{\mathcal{V}} VV & @VV F V \\ \mathcal{D} \times \mathcal{V} @>>> \mathcal{D} \end{CD} commutes (where the horizontal arrows are the copower functors)? If not, are there some natural assumptions on $F$ that make this diagram commute? I have a particular example in mind, where $\mathcal{C}$ and $\mathcal{D}$ are categories of modules over finite-dimensional $k$-algebras and $F$ is a functor given by tensoring with a bimodule. So I'd be happy with assumptions that are natural in that setting.
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4$\begingroup$ Tensoring with a bimodule even preserves all weighted colimits, and in particular preserves copowers with infinite-dimensional vector spaces. $\endgroup$– Qiaochu YuanCommented Aug 6, 2016 at 17:09
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1$\begingroup$ "Then they are enriched over ... finite-dimensional vector spaces." - This should be an extra assumption, since in general the dimensions may be infinite. $\endgroup$– HeinrichDCommented Sep 12, 2016 at 8:40
1 Answer
Yes, this is true (up to natural isomorphism). The simplest way to see this is just to write down what the copowers are explicitly. Let us take a skeleton of $\mathcal{V}$ consisting of all vector spaces of the form $k^n$, and write $\otimes$ for copowers. Then on objects, $A\otimes k^n$ is just a direct sum of $n$ copies of $A$. And on morphisms, given $f:A\to B$ and $T:k^n\to k^m$, the induced map $f\otimes T:A\otimes k^n\to B\otimes k^m$ can be described as follows. The domain is $A^n$ and the codomain is $B^m$, so we can describe $f\otimes T$ as an $n\times m$ matrix of morphisms $A\to B$. The $ij$ entry of this matrix is just $T_{ij}f$, where $T_{ij}\in k$ is the $ij$ matrix entry of $T$.
All of this structure depends only on the $k$-linear structure of $\mathcal{C}$, so it is preserved by any $k$-linear functor. Explicitly, morphisms $A^n\to B^m$ are in bijection with matrices of morphisms $A\to B$ by composing with the inclusion and projection maps $A\to A^n$ and $B^m\to B$, and this correspondence is preserved by any additive functor. ($k$-linearity is also needed so that you know the scalar products $T_{ij}f$ are preserved.)
