nLab π-Kan condition

Idea

The $\pi_\ast$-Kan condition (Bousfield & Friedlander 1978, B.3) on a bisimplicial set guarantees that its geometric realization behaves well with respect to homotopy pullback (Bousfield & Friedlander 1978, Thm. B.4).

See this Prop. at geometric realization of simplicial topological spaces.

Related entries

• commutativity of homotopy limits and colimits

References

• Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf, pdf)

• Donald Werner Anderson, Fibrations and geometric realization, Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (euclid:1183541139)

• Jacob Lurie, Simplicial spaces, Lecture 7 of: Algebraic L-theory and Surgery, 2011 (pdf)

• Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (pdf, pdf)

• Edoardo Lanari, Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves (pdf, pdf)

  (expanded version of Rezk 14)

• Aaron Mazel-Gee, Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces (arXiv:1412.8411)

• Jacob Lurie, around Lem. 5.5.6.17 in: Higher Algebra (~2017)

• Jacob Lurie, Thms. A.5.4.1, Spectral Algebraic Geometry (2018) [pdf]

• Daniel Berwick-Evans, Pedro Boavida, Dmitri Pavlov, §3 of: Classifying spaces of infinity-sheaves, Algebraic & Geometric Topology [arXiv:1912.10544]