10:35 AM
Is there a (relatively formal) way to prove that if $X$ and $Y$ are $n$-connected (for $n \ge 1$) then a map $f : X \to Y$ is an $m$-equivalence iff $\Sigma f : \Sigma X \to \Sigma Y$ is an $(m+1)$ equivalence? By formal I mean using as little as possible (no Hurewicz/Freudnethal etc). In particular i'm interested in whether this property is true in every infinity topos.
Equivalently you can prove that a map between simply connected spaces is an $n$-equivalence iff the cofiber is $n$-connected.
(I know that freudenthal works in infinity topoi but preferably this result xould be proved without using it...)