Kerodon

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3.1.7 Fibrant Replacement

The formalism of Kan complexes is extremely useful as a combinatorial foundation for homotopy theory. However, when studying the homotopy theory of Kan complexes, it is often necessary to contemplate more general simplicial sets. For example, if $f_0, f_1: S_{} \rightarrow T_{}$ are morphisms of Kan complexes, then a homotopy from $f_0$ to $f_1$ is defined as a morphism of simplicial sets $h: \Delta ^1 \times S_{} \rightarrow T_{}$; here neither $\Delta ^1$ nor the product $\Delta ^1 \times S_{}$ is a Kan complex (except in the trivial case $S_{} = \emptyset $; see Exercise 1.1.9.2). When working with a simplicial set $X_{}$ which is not a Kan complex, it is often convenient to replace $X_{}$ by a Kan complex having the same weak homotopy type. This can always be achieved: more precisely, one can always find a weak homotopy equivalence $X_{} \rightarrow Q_{}$, where $Q_{}$ is a Kan complex (Corollary 3.1.7.2). Our goal in this section is to prove a “fiberwise” version of this result, which can be stated as follows:

Proposition 3.1.7.1. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Then $f$ can be factored as a composition $X_{} \xrightarrow {f'} Q_{}(f) \xrightarrow {f''} Y_{}$, where $f''$ is a Kan fibration and $f'$ is anodyne (hence a weak homotopy equivalence, by virtue of Proposition 3.1.6.7). Moreover, the simplicial set $Q_{}(f)$ (and the morphisms $f'$ and $f''$) can be chosen to depend functorially on $f$, in such a way that the functor

\[ \operatorname{Fun}( [1], \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad (f: X_{} \rightarrow Y_{} ) \rightarrow Q_{}(f) \]

commutes with filtered colimits.

Our proof of Proposition 3.1.7.1 will use an easy version of Quillen's “small object argument” (which we will revisit in greater generality in §).

Proof of Proposition 3.1.7.1. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We construct a sequence of simplicial sets $\{ X(m)_{} \} _{m \geq 0}$ and morphisms $f(m): X(m)_{} \rightarrow Y_{}$ by recursion. Set $X(0)_{} = X_{}$ and $f(0) = f$. Assuming that $f(m): X(m)_{} \rightarrow Y_{}$ has been defined, let $S(m)$ denote the set of all commutative diagrams $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^ n_ i \ar [r] \ar [d] & X(m)_{} \ar [d]^{f(m)} \\ \Delta ^ n \ar [r]^-{u_{\sigma }} & Y_{}, } \]

where $0 \leq i \leq n$, $n > 0$, and the left vertical map is the inclusion map. For every such commutative diagram $\sigma $, let $C_{\sigma } = \Lambda ^ n_{i}$ denote the upper left hand corner of the diagram $\sigma $, and $D_{\sigma } = \Delta ^ n$ the lower left hand corner. Form a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in S(m)} C_{\sigma } \ar [r] \ar [d] & X(m)_{} \ar [d] \\ \coprod _{\sigma \in S(m)} D_{\sigma } \ar [r] & X(m+1)_{} } \]

and let $f(m+1): X(m+1)_{} \rightarrow Y_{}$ be the unique map whose restriction to $X(m)_{}$ is equal to $f(m)$ and whose restriction to each $D_{\sigma }$ is equal to $u_{\sigma }$. By construction, we have a direct system of anodyne morphisms

\[ X_{} = X(0)_{} \hookrightarrow X(1)_{} \hookrightarrow X(2)_{} \hookrightarrow \cdots \]

Set $Q_{}(f) = \varinjlim _{m} X(m)_{}$. Then the natural map $f': X_{} \rightarrow Q_{}(f)$ is anodyne (since the collection of anodyne maps is closed under transfinite composition), and the system of morphisms $\{ f(m) \} _{m \geq 0}$ can be amalgamated to a single map $f'': Q_{}(f) \rightarrow Y_{}$ satisfying $f = f'' \circ f'$. It is clear from the definition that the construction $f \mapsto Q_{}(f)$ is functorial and commutes with filtered colimits. To complete the proof, it will suffice to show that $f''$ is a Kan fibration: that is, that every lifting problem $\sigma :$

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{v} \ar [d] & Q_{}(f) \ar [d]^{f''} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & Y_{} } \]

admits a solution (provided that $n > 0$). Let us abuse notation by identifying each $X(m)_{}$ with its image in $Q_{}(f)$. Since $\Lambda ^{n}_{i}$ is a finite simplicial set, its image under $v$ is contained in $X(m)_{}$ for some $m \gg 0$. In this case, we can identify $\sigma $ with an element of the set $S(m)$, so that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{v} \ar [d] & X(m+1)_{} \ar [d]^{f(m+1)} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & Y_{} } \]

admits a solution by construction. $\square$

Taking $Y_{} = \Delta ^0$, we obtain the following:

Corollary 3.1.7.2. Let $X_{}$ be a simplicial set. Then there exists an anodyne morphism $f: X_{} \rightarrow Q_{}$, where $Q_{}$ is a Kan complex.

Remark 3.1.7.3. In the situation of Corollary 3.1.7.2, the Kan complex $Q_{}$ (and the anodyne morphism $f$) can be chosen to depend functorially on $X_{}$. This follows from the proof of Proposition 3.1.7.1, but there are other (arguably more elegant) ways to achieve the same result. We will return to this point in §3.3.6 (see Propositions 3.3.6.7 and 3.3.6.9) and in §.

Corollary 3.1.7.4. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is anodyne.

$(2)$

The morphism $f$ has the left lifting property with respect to Kan fibrations. That is, if $g: Z \rightarrow S$ is a Kan fibration of simplicial sets, then every lifting problem

\[ \xymatrix { X \ar [d]^{f} \ar [r] & Z \ar [d]^{g} \\ Y \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 3.1.6.6. To deduce the converse, we first apply Proposition 3.1.7.1 to write $f$ as a composition $X \xrightarrow {f'} Q \xrightarrow {f''} Y$, where $f'$ is anodyne and $f''$ is a Kan fibration. If $f$ satisfies condition $(2)$, then the lifting problem

\[ \xymatrix { X \ar [d]^{f} \ar [r]^-{f'} & Q \ar [d]^{f''} \\ Y \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & Y } \]

admits a solution. It follows that $f$ is a retract of $f'$ (in the arrow category $\operatorname{Fun}([1], \operatorname{Set_{\Delta }})$). Since the collection of anodyne morphisms is closed under retracts, it follows that $f$ is anodyne. $\square$

Corollary 3.1.7.5. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $Z$ be a Kan complex. If $f$ is a weak homotopy equivalence, then composition with $f$ induces a homotopy equivalence $\operatorname{Fun}( Y, Z) \rightarrow \operatorname{Fun}(X,Z)$.

Proof. Using Corollary 3.1.7.2, we can choose an anodyne morphism $g: Y \hookrightarrow Y'$, where $Y'$ is a Kan complex. Using Proposition 3.1.7.1, we can factor $g \circ f$ as a composition $X \xrightarrow {g'} X' \xrightarrow {f'} Y'$, where $g'$ is anodyne and $f'$ is a Kan fibration. We then have a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ X \ar [r]^-{f} \ar [d]^{g'} & Y \ar [d]^{g} \\ X' \ar [r]^-{ f' } & Y', } \]

where $f'$ is a Kan fibration between Kan complexes and the vertical maps are anodyne, and therefore weak homotopy equivalences. Using the two-out-of-three property, we deduce that $f'$ is also a weak homotopy equivalence (Remark 3.1.5.13). It follows that $f'$ is a homotopy equivalence (Proposition 3.1.5.11). We then obtain a commutative diagram of Kan complexes

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}(X,Z) & \ar [l] \operatorname{Fun}(Y,Z) \\ \operatorname{Fun}(X', Z) \ar [u] & \operatorname{Fun}(Y', Z) \ar [l] \ar [u] } \]

where the lower horizontal map is a homotopy equivalence, and the vertical maps are trivial Kan fibrations (Corollary 3.1.6.12). In particular, the vertical maps are homotopy equivalences (Proposition 3.1.5.9), so the two-out-of-three property guarantees that the upper horizontal map is also a homotopy equivalence (Remark 3.1.5.7). $\square$

The proof of Proposition 3.1.7.1 can be repurposed to obtain many analogous results.

Exercise 3.1.7.6. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Show that $f$ can be factored as a composition $X_{} \xrightarrow {f'} P_{}(f) \xrightarrow {f''} Y_{}$, where $f'$ is a monomorphism and $f''$ is a trivial Kan fibration.