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.

In case of Ab, this gives us Category of Rings, namely, $Alg_Z$.

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.

Related concepts

• Vect

• Mod, 2Mod, nMod

• dgcAlg

• Eilenberg-Moore category

category: category