Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 3.4.1.10 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ Y \ar [rr] \ar [dd] \ar [dr]^-{w_ Y} & & X \ar [dd] \ar [dr]^-{ w_{X} } & \\ & Y' \ar [rr] \ar [dd] & & X' \ar [dd] \\ T \ar [rr] \ar [dr]^-{ w_ T} & & S \ar [dr]^-{w_ S} & \\ & T' \ar [rr] & & S', } \]

where the morphisms $w_{X}$, $w_{T}$, and $w_{S}$ are weak homotopy equivalences. Then any two of the following conditions imply the third:

$(1)$

The commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T \ar [r] & S } \]

is homotopy Cartesian.

$(2)$

The commutative diagram

\[ \xymatrix { Y' \ar [r] \ar [d] & X' \ar [d] \\ T' \ar [r] & S' } \]

is homotopy Cartesian.

$(3)$

The morphism $w_{Y}$ is a weak homotopy equivalence.

Proof. By virtue of Corollary 3.4.1.3, the bottom square in the commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T \ar [r] \ar [d]^{w_ T} & S \ar [d]^{w_ S} \\ T' \ar [r] & S', } \]

is homotopy Cartesian. Applying Propositions 3.4.1.9 and 3.4.1.7, we see that $(1)$ is equivalent to the following:

  • The diagram

    \[ \xymatrix { Y \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & S' } \]

    is homotopy Cartesian.

If condition $(3)$ is satisfied, then the equivalence $(1') \Leftrightarrow (2)$ is a special case of Remark 3.4.1.8. Conversely, if $(1')$ and $(2)$ are satisfied, then Propositions 3.4.1.9 and 3.4.1.7 guarantee that the upper square in the commutative diagram

\[ \xymatrix { Y \ar [r] \ar [d]^{w_ Y} & X \ar [d]^{w_ X} \\ Y' \ar [r] \ar [d] & X \ar [d] \\ T' \ar [r] & S' } \]

is homotopy Cartesian, so that $w_{Y}$ is a weak homotopy equivalence by virtue of Corollary 3.4.1.3. $\square$