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

$\displaystyle \sum_i c_i e_{a_i, b_i, \mu_i},$

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$ .



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