Blakers–Massey for Goodwillie equivalences

Corollary 5.98.

Consider a pushout square in $\Fun(C,T)$ of the form

where $$f$$ is a $$P_n$$ -equivalence and $$g$$ is a $$P_m$$ -equivalence. Then the gap map $F \to G \times_K H$ is a $P_{n+m+1}$ -equivalence.

Proof

The two maps belong to

K^{n+1}

and

K^{m+1}

, respectively. The generalized Blakers–Massey theorem places the gap map in their acyclic product, which is

K^{n+m+2}

. This is the class of

P_{n+m+1}

-equivalences. Compare [Anel et al. 2018, Theorem 3.4.1].

References

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Goodwillie's calculus of functors and higher topos theory. J. Topol., 11 (4), 1100–1132. 2018.