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Let $C$ be a finite semisimple $k$-linear category, where $k$ is a field. Let $V$ be a (finite dimensional) vector space over $k$.

Let $X$ be an objet of $C$. I would like to know what $V\otimes X$ means, or would like to know natural way to define it.

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    $\begingroup$ The easiest way of saying it is that $V \otimes X$ is the coproduct of $(\dim V)$-many copies of $X$. Of course, this is really abusing the fact that you have a vector space over a field... $\endgroup$
    – Zhen Lin
    Commented Feb 23, 2016 at 7:44

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