PlanetPhysics/Algebra Formed From a Category

Given a [[../Cod/|category]] $𝒞$ and a ring $R$, one can construct an algebra $𝒜$ as follows. Let $𝒜$ 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 [[../TrivialGroupoid/|objects]] $a$ and $b$ of $𝒞$ and every [[../TrivialGroupoid/|morphism]] $\mu$ from $a$ to $b$, there corresponds a basis element $e_{a,b,\mu }$. Addition and [[../Vectors/|scalar]] multiplication are defined in the usual way. Multiplication of elements of $𝒜$ may be defined by specifying how to multiply basis elements. If $b=c$, then set $e_{a,b,\varphi }\cdot e_{c,d,\psi }=0<math>;otherwiseset$e_{a, b, \phi} \cdot e_{b, c, \psi} = e_{a, c, \psi \circ \phi}</math>. Because of the associativity of [[../Cod/|composition]] of morphisms, $𝒜$ will be an associative algebra over $R$.

Two instances of this construction are worth noting. If $G$ is a [[../TrivialGroupoid/|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 $P$.