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

## Mathematics Subject Classification

18A05*no label found*

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