# Kerodon

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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. $\square$