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 $e_{a, b, \phi} \cdot e_{c, d, \psi} = 0$ ; otherwise set $e_{a, b, \phi} \cdot e_{b, c, \psi} = e_{a, c, \psi \circ \phi}$ . 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 $P$ .