nLab
linear functor


Contents

Definition

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

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

is linear for all x,yCx,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)