homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Given a simplicial group , hence a group object internal to SimplicialSets, a group action of on a simplicial set is a -action object internal to SimplicialSets, hence a morphism of simplicial sets of the form
satisfying the action property.
The category of simplicial -action may be understood as the sSet-enriched functor category from the one-object sSet-category to sSet:
(model structure on simplicial group actions)
There is a model category-structure on simplicial group actions (1) whose weak equivalences and fibrations are those in the underlying classical model structure on simplicial sets, hence are the simplicial weak equivalences and Kan fibrations of the underlying simplicial sets.
(DDK 80, Prop. 2.2. (ii), Guillou, Prop. 5.3, Goerss & Jardine 09, V Lem. 2.4).
Emmanuel Dror, William Dwyer, Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (jstor:2043448)
Paul Goerss, J. F. Jardine, Section V.2 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)
Bert Guillou, A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)
Created on July 4, 2021 at 18:18:44. See the history of this page for a list of all contributions to it.