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In algebraic topology, a continuous function between topological spaces is called $n$-connected if it induces isomorphisms on all homotopy groups in degree $\lt n$ and epimorphisms in degree $n$.
In older literature this is often called an $n$-equivalence, since an $\infty$-equivalence in this sense is a weak homotopy equivalence.
In terms of the homotopy theory presented by the classical model structure on topological spaces, an $n$-connected function represents an n-connected morphism in the (∞,1)-topos ∞Gpd.
A map of topological spaces $f \colon X \to Y$ is called $n$-connected (e.g. tomDieck 08, p. 144) or an $n$-equivalence (older literature) if the following equivalent definitions hold:
The induced morphism on homotopy groups $\pi_\bullet(X,x)\to \pi_\bullet(Y,f(x))$ is, for all $x\in X$
an isomorphism in degree $\lt n$;
an epimorphism in degree $n$.
for all $i \le n$ and all commutative squares
there exists a map $w \colon D^i \to X$ such that $w | S^{i-1} = u$ and $f w$ is homotopic to $v$ relative to $S^{i-1}$.
Hence an $\infty$-connected map is a weak homotopy equivalence.
For a map $f \colon X \to Y$ and an integer $n \ge -1$ the following conditions are equivalent.
$f$ is $n$-connected.
All homotopy fibers of $f$ are $(n-1)$-connected.
Let $f \colon X \to Y$ and $g \colon Y \to Z$ be maps of spaces.
If $f$ and $g$ are $n$-connected, then so is $g f$.
If $f$ is $(n-1)$-connected and $g f$ is $n$-connected, then $g$ is $n$-connected.
If $g$ is $(n+1)$-connected and $g f$ is $n$-connected, then $f$ is $n$-connected.
Let
be a commutative diagram of maps of spaces. If $f$ is $(n-1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between homotopy pushouts $B \sqcup_A^h C \to B' \sqcup_{A'}^h C'$ is $n$-connected.
This is (tom Dieck, Theorem 6.7.9).
Let
be a commutative diagram of maps of spaces. If $f$ is $(n+1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between homotopy pullbacks $Y \times_X^h Z \to Y' \times_{X'}^h Z'$ is $n$-connected.
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