Homotopy limits in type theory

Homotopy Limits and Colimits and Enriched Homotopy Theory

Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, s...

Univalence for inverse diagrams and homotopy canonicity

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky...

Homotopy (limits And) Colimits

These notes were written to accompany two talks given in the Algebraic Topology and Category Theory Proseminar at the University of Chicago in Winter 2009. When a category has some notion of limits and colimits associated to it, its ordinary limits and colimits are not necessarily homotopically meaningful. We describe a notion of a “homotopy colimit” for two sorts of categories with a homotopy ...

Topology Proceedings 2 (1977) pp. 461-477: STEENROD HOMOTOPY THEORY, HOMOTOPY INDEMPOTENTS, AND HOMOTOPY LIMITS

The Homology of Homotopy Inverse Limits

The homology of a homotopy inverse limit can be studied by a spectral sequence with has E2 term the derived functors of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonné modules if one has a diagram of connected abelian Hopf algebras. One of the standard problems in homotopy theory is to calculate the homology of a given type of inverse li...