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

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


Relation to algebras with bimodules

Since algebras may be identified with one-object categories internal to vector spaces, it is sometimes useful to regard AlgAlg as a strict 2-category, namely as a full sub-2-category of the 2-category Cat(Vect)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 AlgAlg regarded as a strict 2-category this way there is a canonical 2-functor

AlgBimod Alg \hookrightarrow Bimod

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

category: category

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