nLab linear functor

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

• There is a notion of map between linearly distributive categories called linearly distributive functor?, which is often shortened to “linear 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 $Vect$.

Unwrapping this a bit: given objects $x, y$ in a linear category $C$, the homset $hom(x,y)$ 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

$$F: hom(x,y) \to hom(F(x), F(y))$$

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

• smooth functor