linear functor

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

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.

Revised on September 19, 2012 20:23:37
by Urs Schreiber
(131.174.188.151)