Homotopy colimits in stable representation 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...

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 colimits in model categories

DERIVED REPRESENTATION THEORY AND STABLE HOMOTOPY CATEGORIFICATION OF slk

We set up foundations of representation theory over S, the stable sphere, which is the “initial ring” of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, HarishChandra pairs and Zuckermann functors. As an application, we construct a Khovanov slk-stable homotopy type with a large prime hypothesis, which is...

Stable A1-homotopy Theory

Definition 1. A T -prespectrum E is a sequence En of pointed spaces together with structure morphisms σn : En ∧ T → En+1 for all n. A morphism λ : E → F of T -prespectra is a sequence of maps λn : En → Fn commuting with the structure morphisms. We will denote the category of T -prespectra by PSpT (k). Example 1. For any based space X , we have the suspension prespectrum Σ∞(X ) of X given by Σ∞(...