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

where the coefficients c_i 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 e_a,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≠c, then set e_a,b,ϕ ⋅ e_c,d,ψ = 0; otherwise set e_a,b,ϕ ⋅ e_b,c,ψ = e_a,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.