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 (has finitely many objects and morphisms), 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 (where the elements of the group correspond to the morphisms of the category), 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" (or "R"["C"]) to be the free "R"-module with basis consisting of the maps of "C". In other words, "RC" consists of formal linear combinations (which are finite sums) of the form formula_1, 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 formula_3 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 formula_4 (thought of as functionals on the maps of "C"), then their product is defined as:
The latter sum is finite because the functions are finitely supported, and therefore formula_6.
Incidence algebra-style definition.
The definition used for incidence algebras assumes that the category "C" is locally finite (see below), 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 (not to be confused with the "has finite Hom-sets" meaning). The category algebra (in this sense) 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 formula_8 (they can all be non-zero).
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 (in the group algebra sense of the definition) is the space of all maps from the maps of "C" to "R", denoted "F"("C"), and has a natural coalgebra structure. Thus for a locally finite category, the dual of a category algebra (in the group algebra sense) is the category algebra (in the incidence algebra sense), and has both an algebra and coalgebra structure.