nLab Alg

Idea

By Alg is usually meant a category with associative algebras as objects and algebra homomorphisms as morphisms.

(This depends on a choice of ground field or even ground ring, which is often left implicit and determined by the context. Over the ground ring $\mathbb{Z}$ of integers, $Alg_{\mathbb{Z}} \simeq$ Ring is the category of rings.)

Beware that many other types of algebras exist besides associative algebras (e.g. Lie algebras or generally algebras over an operad or algebras over a monad) and all of them form categories which may in corresponding contexts be denoted “$Alg$” or similar.

Properties

Relation with Bimod

Since associative algebras may be identified with one-object categories enriched in modules over the ground ring), it is sometimes useful to regard $Alg$ as the strict full sub-2-category of the 2-category $Mod Cat$ of Mod-enriched categories. In this case the 2-morphisms between morphisms of algebras come from “intertwiners”: inner endomorphisms of the codomain algebra.

(Analogous statements hold for the category Grp of groups when the latter are regarded as their delooping groupoids.)

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

$$Alg \hookrightarrow Bimod$$

to the category Bimod of bimodules, 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.

Related concepts

• Vect

• Mod, 2Mod, nMod

• dgcAlg

• Eilenberg-Moore category

category: category