nLab
linear functor

• This page is about functors between linear categories. For other notions of “linear functor,” see Goodwillie calculus and polynomial functor.

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Definition

A linear category is a category enriched over Vect, and similarly a linear functor is a functor enriched over $\mathrm{Vect}$. Unwrapping this a bit: given objects $x,y$ in a linear category $C$, the homset $\mathrm{hom}(x,y)$ is equipped with the structure of a vector space, and a functor $F:C\to D$ between linear categories is said to be linear if the map

$$F:\mathrm{hom}(x,y)\to \mathrm{hom}(F(x),F(y))$$F: hom(x,y) \to hom(F(x), F(y))

is linear for all $x,y\in C$.

Note that a linear functor between linear additive categories is automatically additive.

Related concepts

• linear function, linear equation