What It Means for a Coalgebra To Be Simply Connected

Abstract

This chapter discusses what it means for a coalgebra to be simply connected. If n ∈ ℤ, n > 0, and C is an n-connected coalgebra, then there is a morphism of coalgebras g: C → D such that Dq = 0 for 0 < q ≤ n and Ω(g): Ω(C)→ Ω(D) is a chain equivalence. A coalgebra C is reduced if the differential module P(C) of primitive chains of C is isomorphic with a coproduct of elementary complexes, no one of which is acyclic. If R is a local ring, then the reduction of an elementary complex to a complex over the residue field is an elementary complex, and the reduction of a reduced coalgebra to a coalgebra over the residue field is again a reduced coalgebra.