nLab
algebroid

Algebroids (linear categories)

Idea

A linear category, or algebroid, is a category whose hom-sets are all vector spaces (or modules) and whose composition operation is bilinear. This concept is a horizontal categorification of the concept of (unital associative) algebra.

Definitions

Fix a commutative ring KK. (Often we want KK to be a field, such as the field \mathbb{C} of complex numbers.)

A KK-linear category, or KK-algebroid, is a category enriched over KK\,Mod, the monoidal category of KK-modules with the usual tensor product. (Note that we usually speak of KK\,Vect instead of KModK\,Mod when KK is a field.)

Just as a \mathbb{Z}-algebra is the same thing as a ring, so a \mathbb{Z}-algebroid is the same thing as a ringoid.

Remarks

Generalizations