# nLab Alg

## Idea

Alg is the category with algebras as objects and algebra homomorphisms as morphisms.

More abstractly, we can think of $Alg$ as the full subcategory of $Cat(Vect)$, internal categories in Vect, with algebras as objects.

## Properties

### Relation to algebras with bimodules

Since algebras may be identified with one-object categories internal to vector spaces, it is sometimes useful to regard $Alg$ as a strict 2-category, namely as a full sub-2-category of the 2-category $Cat(Vect)$. In this case the 2-morphisms between morphisms of algebras come from “intertwiners”: inner endomorphisms of the target algebra.

Precisely analogous statements hold for the category Grp of groups.

With $Alg$ regarded as a strict 2-category this way there is a canonical 2-functor

$Alg \hookrightarrow Bimod$

to the category Bimod, which sends algebra homomorphisms $f : A \to B$ to the $A$-$B$ bimodule ${}_f B$. This exhibits $Bimod$ as a framed bicategory in the sense of Shulman.

category: category

Revised on February 3, 2013 22:08:20 by Urs Schreiber (89.204.153.115)