colimit

The direct limit, also called a colimit, of a family of R-modules is the dual notion of an inverse limit and is characterized by the following mapping property. For a directed set I and a family of R-modules {M_i}_(i element I), let (M_i, σ_(i j)) be a direct system. lim_⟶ M_i is some R-module with some homomorphisms σ_i, where for each i element I, i<=j, σ_i :M_i->lim_⟶ M_i with the property σ_i = σ_j °σ_(i j) such that if there exists some R-module N with homomorphisms α_i, where for each i element I, i<=j, α_i :M_i->N with the property α_i = α_j °σ_(i j),

commutative diagram | directed set | direct sum | direct system | inverse limit | module | module homomorphism | quotient module