What is the universal property of associated graded?
Given a filtered vector space (or module over a ring) 0=V0⊆V1⊆...⊆V, you can construct the associated graded vector space gr(V)=⊕iVi+1/Vi. Does gr(V) satisfy a universal property? What is it?
Before anybody hastily says, "it's the universal graded vector space with a filtered map from V," let me point out that it's not so simple. A map of filtered vector spaces is a map of vector spaces which respects the filtration. It's clear what the map Vi+1→Vi+1/Vi should be, but what would the map ∪Vi→⊕iVi+1/Vi be?