Kerodon

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

9.9.1 Towers of Kan Fibrations

We now study the behavior of Kan complexes with respect to inverse limits.

Proposition 9.9.1.1. Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

If each of the morphisms $q_ n$ is a Kan fibration, then the canonical map $q: X \rightarrow X(0)$ is a Kan fibration.

Proof. This is a special case of Proposition 1.4.4.14 (which guarantees that the collection of all Kan fibrations is closed under transfinite composition in the opposite of the category of simplicial sets). $\square$

Corollary 9.9.1.2. Suppose we are given a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

If $X(0)$ is a Kan complex and each $q_ n$ is a Kan fibration, then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is a Kan complex.

Variant 9.9.1.3. Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

If each of the morphisms $q_ n$ is a trivial Kan fibration, then the canonical map $q: X \rightarrow X(0)$ is a trivial Kan fibration.

Proof. By virtue of Proposition 1.4.4.14, the collection of trivial Kan fibrations is closed under transfinite composition in the opposite of the category of simplicial sets. $\square$

Corollary 9.9.1.4. Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

Assume that each of the simplicial sets $X(n)$ is weakly contractible and that each of the morphisms $q_ n$ is a Kan fibration. Then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is weakly contractible.

Proof. If each of these simplicial sets $X(n)$ is weakly contractible, then each of the morphisms $q_ n: X(n+1) \rightarrow X(n)$ is a weak homotopy equivalence. Applying Proposition 3.3.7.4, we conclude that each $q_ n$ is a trivial Kan fibration. Variant 9.9.1.3 then guarantees that the projection map $X \rightarrow X(0)$ is also a trivial Kan fibration. Since $X(0)$ is weakly contractible, it follows that $X$ is also weakly contractible. $\square$

Corollary 9.9.1.5. Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

Assume that each of the simplicial sets $X(n)$ is a contractible Kan complex and that each of the morphisms $q_ n$ is a Kan fibration. Then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is a contractible Kan complex.

Proposition 9.9.1.6. Suppose we are given a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r]^-{ p_3} & X(3) \ar [d]^{ f_3} \ar [r]^-{ p_2} & X(2) \ar [r]^-{p_1} \ar [d]^{f_2} & X(1) \ar [r]^-{ p_0} \ar [d]^{f_1} & X(0) \ar [d]^{f_0} \\ \cdots \ar [r]^-{ q_3 } & Y(3) \ar [r]^-{ q_2} & Y(2) \ar [r]^-{q_1} & Y(1) \ar [r]^-{ q_0 } & Y(0). } \]

Assume that for each $n \geq 0$, the morphisms $p_ n$ and $q_ n$ are Kan fibrations, and the morphism $f_ n$ is a homotopy equivalence. Then the induced map $f: \varprojlim \{ X(n) \} _{n \geq 0} \rightarrow \varprojlim \{ Y(n) \} _{n \geq 0}$ is a homotopy equivalence of Kan complexes.

Proof. Set $X = \varprojlim \{ X(n) \} _{n \geq 0}$ and $Y = \varprojlim \{ Y(n) \} _{n \geq 0}$, so that $X$ and $Y$ are Kan complexes (Corollary 9.9.1.2). By virtue of Remark 3.4.0.6, it will suffice to show that for each vertex $y \in Y$, the homotopy fiber $F = X \times ^{\mathrm{h}}_{Y} \{ y\} $ is a contractible Kan complex. For each $n \geq 0$, set $F(n) = X(n) \times ^{\mathrm{h}}_{Y(n)} \{ y\} $, so that $F$ can be identified with the limit $\varprojlim \{ F(n) \} _{n \geq 0}$. Since $f_{n}$ is a homotopy equivalence, $F(n)$ is a contractible Kan complex (Remark 3.4.0.6). By virtue of Corollary None, it will suffice to show that each of the transition maps $\theta _{n}: F(n+1) \rightarrow F(n)$ is a Kan fibration. Unwinding the definitions, we see that $\theta _{n}$ factors as a composition

\[ X(n+1) \times _{Y(n+1)}^{\mathrm{h}} \{ y\} \xrightarrow {\theta '} X(n+1) \times _{Y(n)}^{\mathrm{h}} \{ y\} \xrightarrow {\theta ''} X(n) \times _{ Y(n) }^{\mathrm{h}} \{ y\} . \]

Here $\theta '$ is a pullback of the map $p_{n}$ (and therefore a Kan fibration by assumption), while $\theta ''$ is a pullback of the map

\[ \operatorname{Fun}( \Delta ^1, Y(n+1) ) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^1, Y(n+1) ) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, Y(n) ) } \operatorname{Fun}( \Delta ^1, Y(n) ), \]

and is therefore a Kan fibration by virtue of Theorem 3.1.3.1. $\square$