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algebra formed from a category (Definition)

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 $P$.



"algebra formed from a category" is owned by rspuzio.

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Cross-references: group, composition, scalar, morphism, objects, category
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This is version 1 of algebra formed from a category, born on 2009-03-11.
Object id is 589, canonical name is AlgebraFormedFromACategory.
Accessed 297 times total.

Classification:
Physics Classification: 02.10.-v (Logic, set theory, and algebra)
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