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

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)

• Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (i-hate-the-pi-star-kan-condition.pdf, pdf)