$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.4.1.12 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@C =50pt{ X_{01} \ar [rr] \ar [dd] \ar [dr]^-{w_{01}} & & X_{0} \ar [dd] \ar [dr]^-{ w_{0} } & \\ & Y_{01} \ar [rr] \ar [dd] & & Y_0 \ar [dd] \\ X_1 \ar [rr] \ar [dr]^-{ w_1} & & X \ar [dr]^-{w} & \\ & Y_1 \ar [rr] & & Y, } \]
where the morphisms $w_0$, $w_1$, and $w$ are weak homotopy equivalences. Then any two of the following conditions imply the third:
- $(1)$
The back face
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]
is a homotopy pullback square.
- $(2)$
The front face
\[ \xymatrix@R =50pt@C=50pt{ Y_{01} \ar [r] \ar [d] & Y_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square.
- $(3)$
The morphism $w_{01}: X_{01} \rightarrow Y_{01}$ is a weak homotopy equivalence of simplicial sets.
Proof.
Using Corollary 3.4.1.5, we see that the bottom square in the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] \ar [d]^{w_1} & X \ar [d]^{w} \\ Y_1 \ar [r] & Y, } \]
is a homotopy pullback square. Applying Propositions 3.4.1.11 and 3.4.1.9, we see that $(1)$ is equivalent to the following:
- $(1')$
The diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square.
If condition $(3)$ is satisfied, then the equivalence $(1') \Leftrightarrow (2)$ is a special case of Remark 3.4.1.10. Conversely, if $(1')$ and $(2)$ are satisfied, then Propositions 3.4.1.11 and 3.4.1.9 guarantee that the upper half of the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d]^{w_{01}} & X_0 \ar [d]^{w_0} \\ Y_{01} \ar [r] \ar [d] & Y_0 \ar [d] \\ Y_1 \ar [r] & Y } \]
is a homotopy pullback square, so that $w_{01}$ is a weak homotopy equivalence by virtue of Corollary 3.4.1.5.
$\square$