What is a module over an algebra in the category theoretic sense?

Question

Can someone please confirm this for me. In category theory one defines monoidal categories. One can then consider monoids in a monoidal category $C$. Given such a monoid $M$, one can then consider modules over $M$. We have:

• In the Cartesian monoidal category $Set$:

  monoids are ordinary monoids, and modules over a monoid correspond to monoid actions.

• In the (non-Cartesian) monoidal category $Ab$ of abelian groups, with the tensor product of abelian groups:

  monoids are (unital) rings, and a module over a ring $R$ corresponds to the usual notion of $R$-module in algebra.

• In the monoidal category $R$-$Mod$ of $R$-modules (in the usual algebraic sense), with the tensor product of $R$-modules:

  monoids are (unital associative) $R$-algebras. My question is: what is a module (in the categorical sense) over an associative unital algebra $A$? Is it the usual "module over an algebra" which sometimes crops up in algebra, e.g. in A module over an algebra. Is it a vector space? ?

Thanks

Answer 1

For an $R%$-algebra $A$, an $A$-module is just a module over the underlying ring of $A$. The $R$-module structure on $A$ does not matter.

Answer 2

If you want a really general category theory definition of a module, here you go:

A module over a $k$-linear small category $A$ is a functor from $A$ to $k$ vector spaces $A \to k\mathrm{-Vect}$.

This also naturally extends to modules over other small categories.

Now if you have a $k$-algebra $B$ it is actually a $k$-linear category with one object $$ and endomorphisms of $*$ are $B$, then a module over $B$ as a functor is the same as the classic definition of a module, just as geoffrey said! (which is a nice exercise)

A nice benefit of that is that you can carry over a lot of structures from vector spaces via this interpretation!