Category algebra

Not to be confused with categorical 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. It generalizes the notions of group algebra and incidence algebra, just as category generalizes the notions of group and partially ordered set.

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 such that 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 unit. 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 $\sum a_{i}f_{i}$ , where f_i are maps of C, and a_i are elements of the ring R. Define a multiplication operation on RC as follows, using the composition operation in the category:

\sum a_{i}f_{i}\sum b_{j}g_{j}=\sum a_{i}b_{j}f_{i}g_{j}

where $f_{i}g_{j}=0$ 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 $a,b\in RC$ (thought of as functionals on the maps of C), then their product is defined as:

(a*b)(h):=\sum _{{fg=h}}a(f)b(g).

The latter sum is finite because the functions are finitely supported, and therefore $a*b\in RC$ .

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 only finitely many ways as a product of non-identity maps. 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

\sum _{{f_{i}\in {\mathrm {Hom}}(C)}}a_{i}f_{i},

where there are no restrictions on the $a_{i}$ (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.

Examples

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.
The path algebra of a quiver Q is the category algebra of the free category on Q.

References

Haigh, John. On the Möbius Algebra and the Grothendieck Ring of a Finite Category J. London Math. Soc (2), 21 (1980) 81–92.