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algebra formed from a category∗
rspuzio†
2013-03-21 21:41:14
Given a 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
X
ci eai ,bi ,µi ,
i
where the coefficients ci lie in R and, to every pair of objects a and b of C
and every morphism µ from a to b, there corresponds a basis element ea,b,µ .
Addition and 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 6= c, then set ea,b,φ ·ec,d,ψ = 0; otherwise set ea,b,φ ·eb,c,ψ = ea,c,ψ◦φ . Because of
the associativity of composition of morphisms, 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 .
∗ hAlgebraFormedFromACategoryi created: h2013-03-21i by: hrspuzioi version: h38686i
Privacy setting: h1i hDefinitioni h18A05i
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