nLab Alg



Category theory




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.

In case of Ab, this gives us Category of Rings, namely, Alg ZAlg_Z.


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

Last revised on February 16, 2021 at 09:56:24. See the history of this page for a list of all contributions to it.