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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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1