Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves

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 ...

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...

Homotopy Limits and Colimits in Nature A Motivation for Derivators

An introduction to the notions of homotopy limit and colimit is given. It is explained how they can be used to neatly describe the “old” distinguished triangles and shift functors of derived categories resp. cofiber and fiber sequences in algebraic topology. One of the goals is to motivate the language of derivators from the perspective of classical homological algebra. Another one is to give e...

Notes on Homotopy Colimits and Homotopy Limits (a Work in Progress)

14. Diagrams of spaces 68 14.1. The geometric realization and total singular complex functors 68 14.2. Filtered diagrams of simplicial sets 70 14.3. Commuting hocolim and holim: simplicial sets 72 14.4. Commuting hocolim and holim: topological spaces 74 References 75

Homotopy colimits in model categories