Morita Cohomology

This talk introduces two categorifications of the cohomology of a topological space $X$ obtained by taking coefficients in the model category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent. Moreover this categorified cohomology is quasi-equivalent to homotopy locally constant sheaves on $X$ and to representations in perfect complexes of chains on the based loop space of $X$.