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 $e_{a,b,\phi }\cdot e_{c,d,\psi }=0<math>;otherwiseset$e_{a, b, \phi} \cdot e_{b, c, \psi} = e_{a, c, \psi \circ \phi}</math>. 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 $P$.