k-linear category - Knowlpedia

A $k$ -linear category, for a field $k$ , is a category $\mathcal C$ with a $k$ -vector space structure on each $\operatorname{Hom}_{\mathcal C}(X,Y)$ , such that composition is $k$ -bilinear. Thus, whenever the composites are defined,

g\circ(af+bf')=a(g\circ f)+b(g\circ f'),\qquad (ag+bg')\circ f=a(g\circ f)+b(g'\circ f)

for $a,b\in k$ . The vector space structures are part of the data.

What linearity does and does not supply

One can add parallel arrows and multiply them by scalars. In particular the category is preadditive. This does not by itself provide direct sums of objects, kernels, or finite-dimensional Hom spaces. Some authors include additivity in their convention for “linear category”; here it is a separate condition.

For example, a unital $k$ -algebra gives a one-object $k$ -linear category with composition given by multiplication. It need not have a zero object. The category of finite-dimensional vector spaces has all these additional additive structures.

Compatibility with a tensor product

When a $k$ -linear category is also monoidal, compatibility requires the map on arrows $(f,g)\mapsto f\otimes g$ to be $k$ -bilinear. Linearity of composition alone does not impose that compatibility. It is included in the definition of a fusion category.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §1.2; author-hosted text.