In category theory, a field of mathematics, a categorical algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. It generalizes the notions of group algebra and incidence algebra, just as category generalizes the notions of group and partially ordered set.
Infinite categories are conventionally treated differently for group algebras and incidence algebras; the definitions agree for finite categories. We first present the definition that generalizes the group algebra.
Group algebra-style definition
Let C be a category and R be a commutative ring with unit. Then as a set and as a module, the categorical algebra RC (or R[C]) is the free module on the maps of C.
The multiplication on RC can be understood in several ways, depending on how one presents a free module.
Thinking of the free module as formal linear combinations (which are finite sums), the multiplication is the multiplication (composition) of the category, where defined:
where if their composition is not defined. This is defined for any finite sum.
Thinking of the free module as finitely supported functions, the multiplication is defined as a convolution: if (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.
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 only finitely many ways as a product of non-identity maps. The categorical 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 (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.
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 categorical algebra (in the group algebra sense) is the categorical algebra (in the incidence algebra sense), and has both an algebra and coalgebra structure.
- If C is a group (thought of as a groupoid with a single object), then RC is the group algebra.
- If C is a monoid (thought of as a category with a single object), then RC is the monoid ring.
- If C is a partially ordered set, then (using the appropriate definition), RC is the incidence algebra.
- Haigh, John. On the Möbius Algebra and the Grothendieck Ring of a Finite Category J. London Math. Soc (2), 21 (1980) 81-92.
- Categorical Algebra at PlanetMath.
- Locally Finite Category at PlanetMath.