Category algebra


In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the notions of group algebras and incidence algebras, just as categories generalize the notions of groups and partially ordered sets.

Definition

If the given category is finite, then the following two definitions of the category algebra agree.

Group algebra-style definition

Given a group G and a commutative ring R, one can construct RG, known as the group algebra; it is an R-module equipped with a multiplication. A group is the same as a category with a single object in which all morphisms are isomorphisms, so the following construction generalizes the definition of the group algebra from groups to arbitrary categories.
Let C be a category and R be a commutative ring with unity. Define RC to be the free R-module with basis consisting of the maps of C. In other words, RC consists of formal linear combinations of the form, where fi are maps of C, and ai are elements of the ring R. Define a multiplication operation on RC as follows, using the composition operation in the category:
where if their composition is not defined. This defines a binary operation on RC, and moreover makes RC into an associative algebra over the ring R. This algebra is called the category algebra of C.
From a different perspective, elements of the free module RC could also be considered as functions from the maps of C to R which are finitely supported. Then the multiplication is described by a convolution: if , then their product is defined as:
The latter sum is finite because the functions are finitely supported, and therefore.

Incidence algebra-style definition

The definition used for incidence algebras assumes that the category C is locally finite, is dual to the above definition, and defines a different object. This isn't a useful assumption for groups, as a group that is locally finite as a category is finite.
A locally finite category is one where every map can be written in only finitely many ways as the composition of two non-identity maps. The category algebra is defined as above, but allowing all coefficients to be non-zero.
In terms of formal sums, the elements are all formal sums
where there are no restrictions on the .
In terms of functions, the elements are any functions from the maps of C to R, and multiplication is defined as convolution. The sum in the convolution is always finite because of the local finiteness assumption.

Dual

The module dual of the category algebra is the space of all maps from the maps of C to R, denoted F, and has a natural coalgebra structure. Thus for a locally finite category, the dual of a category algebra is the category algebra, and has both an algebra and coalgebra structure.

Examples