Homotopy Theories of Coalgebras

Explore the construction and significance of various homotopy theories in the category of simplicial coalgebras over fields of arbitrary characteristic. Delve into a model that fully and faithfully represents the homotopy theory of spaces, considering weak equivalences generated by continuous maps. Examine how this model induces isomorphisms on fundamental groups and, for universal covers, on homology with coefficients in a fixed algebraically closed field. Discover how this work extends existing algebraic models for p-adic homotopy theory of spaces by incorporating the fundamental group comprehensively. Investigate the crucial concept of combining Koszul duality between (dg) coalgebras and algebras to achieve a homological formulation of the fundamental group using chain-level non-linear structure. If time allows, consider G-equivariant analogs of these models in this hour-long lecture presented by Manuel Rivera from Purdue University at the University of Miami.