Category algebras generalize the notions of group algebras and incidence algebras, just as categories generalize the notions of groups and partially ordered sets.
Define RC (or R[C]) to be the free R-module with the set
In other words, RC consists of formal linear combinations (which are finite sums) of the form
, where fi are morphisms 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
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 morphisms of C to R which are finitely supported.
(thought of as functionals on the morphisms of C), then their product is defined as: The latter sum is finite because the functions are finitely supported, and therefore
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.
The category algebra (in this sense) is defined as above, but allowing all coefficients to be non-zero.
The module dual of the category algebra (in the group algebra sense of the definition) is the space of all maps from the morphisms of C to R, denoted F(C), and has a natural coalgebra structure.