### 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.

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.