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[parent] 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. [ full author list (2) ]
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Cross-references: partially ordered set, group algebra, group, composition, associativity, multiplication, scalar, addition, basis, morphism, objects, coefficients, linear combinations, finite, algebra, ring, category
There are 2 references to this entry.

This is version 3 of algebra formed from a category, born on 2006-12-25, modified 2006-12-25.
Object id is 8686, canonical name is AlgebraFormedFromACategory.
Accessed 933 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
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