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?
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
The equivariant version of rational homotopy theory, dealing with rationalization in (proper) equivariant homotopy theory, detected by Bredon equivariant rational cohomology.
equivariant chain complex, model structure on equivariant chain complexes
Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras
fundamental theorem of dg-algebraic equivariant rational homotopy theory
The original reference for finite groups is
Georgia Triantafillou, Equivariant rational homotopy theory, chapter III of Peter May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf, ISBN: 978-0-8218-0319-6)
Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)
but beware that Scull 01 claims that the statement about minimal models there is not correct. Corrrected statements for finite groups as well as generalization to compact Lie groups, at least to the circle group, is due to
Further discussion in:
Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, pdf)
Marek Golasiński, Injective models of $G$-disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)
The model structure on equivariant dgc-algebras, generalizing the projective model structure on dgc-algebras, in which equivariant minimal Sullivan models are cofibrant objects:
See also
Peter J. Kahn, Rational Moore G-Spaces, Transactions of the American Mathematical Society Vol. 298, No. 1 (1986), pp. 245-271 (jstor:2000619)
C. Allday, V. Puppe, sections 3.3 and 3.4 of Cohomological methods in transformation groups, Cambridge 1993 (doi:10.1017/CBO9780511526275)
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