PlanetPhysics/Algebra Formed From a Category

Given a category $$\mathcal{C}$$ and a ring $$R$$, one can construct an algebra $$\mathcal{A}$$ as follows. Let $$\mathcal{A}$$ be the set of all formal finite linear combinations of the form "$\sum_i c_i e_{a_i, b_i, \mu_i},$" where the coefficients $$c_i$$ lie in $$R$$ and, to every pair of objects $$a$$ and $$b$$ of $$\mathcal{C}$$ and every morphism $$\mu$$ from $$a$$ to $$b$$, there corresponds a basis element $$e_{a,b,\mu}$$. Addition and scalar multiplication are defined in the usual way. Multiplication of elements of $$\mathcal{A}$$ may be defined by specifying how to multiply basis elements. If $$b \not= c$$, then set $$e_{a, b, \phi} \cdot e_{c, d, \psi} = 0e_{a, b, \phi} \cdot e_{b, c, \psi} = e_{a, c, \psi \circ \phi}$$. Because of the associativity of composition of morphisms, $$\mathcal{A}$$ will be an associative algebra over $$R$$.

Two instances of this construction are worth noting. If $$G$$ is a group, we may regard $$G$$ as a category with one object. Then this construction gives us the group algebra of $$G$$. If $$P$$ is a partially ordered set, we may view $$P$$ as a category with at most one morphism between any two objects. Then this construction provides us with the incidence algebra of $$P$$.