### 3.1.6 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 3.1.6.2). Our goal in this section is to prove a “fiberwise” version of this result, which can be stated as follows:

Proposition 3.1.6.1. 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 3.1.5.13). 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.

Before giving the proof of Proposition 3.1.6.1, let us note some of its consequences. Applying Proposition 3.1.6.1 in the special case $Y_{} = \Delta ^0$, we obtain the following:

Corollary 3.1.6.2. Let $X_{}$ be a simplicial set. Then there exists an anodyne morphism $f: X_{} \rightarrow Q_{}$, where $Q_{}$ is a Kan complex.

Corollary 3.1.6.4. 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 3.1.2.6. To deduce the converse, we first apply Proposition 3.1.6.1 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 3.1.6.5. 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 3.1.6.2, we can choose an anodyne morphism $g: Y \hookrightarrow Y'$, where $Y'$ is a Kan complex. Using Proposition 3.1.6.1, 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 3.1.5.16). It follows that $f'$ is a homotopy equivalence (Proposition 3.1.5.12). 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 3.1.3.6). In particular, the vertical maps are homotopy equivalences (Proposition 3.1.5.10), so the two-out-of-three property guarantees that the upper horizontal map is also a homotopy equivalence (Remark 3.1.5.7).
$\square$

Recall that the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.4.10 is defined as a quotient of the category of Kan complexes $\operatorname{Kan}$ (by identifying morphisms which are homotopic). However, it can also be described as a *localization* of $\operatorname{Kan}$, obtained by inverting the class of homotopy equivalences.

Proposition 3.1.6.6. Let $\operatorname{\mathcal{C}}$ be a category and let $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ be a functor. The following conditions are equivalent:

- $(\ast )$
If $X$ and $Y$ are Kan complexes and $u_0, u_1: X \rightarrow Y$ are homotopic morphisms, then $F(u_0) = F(u_1)$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(X), F(Y) )$.

- $(\ast ')$
For every homotopy equivalence of Kan complexes $u: X \rightarrow Y$, the induced map $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in the category $\operatorname{\mathcal{C}}$.

**Proof.**
The implication $(\ast ) \Rightarrow (\ast ')$ is immediate (note that a morphism of Kan complexes $u: X \rightarrow Y$ is a homotopy equivalence if and only if its homotopy class $[u]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). For the converse, assume that $(\ast ')$ is satisfied, let $X$ and $Y$ be Kan complexes, and let $u_0, u_1: X \rightarrow Y$ be a pair of homotopic morphisms. Let us regard $u_0$ and $u_1$ as vertices of the Kan complex $\operatorname{Fun}(X,Y)$. Since $u_0$ and $u_1$ are homotopic, there exists an edge $e: \Delta ^1 \rightarrow \operatorname{Fun}(X,Y)$ satisfying $e(0) = u_0$ and $e(1) = u_1$. By virtue of Proposition 3.1.6.1, this morphism factors as a composition $\Delta ^1 \xrightarrow {e'} Q \xrightarrow {e''} \operatorname{Fun}(X,Y)$, where $e'$ is anodyne and $e''$ is a Kan fibration. Since $\operatorname{Fun}(X,Y)$ is a Kan complex (Corollary 3.1.3.4), it follows that $Q$ is also a Kan complex. Let us identify $e''$ with a morphism of Kan complexes $h: Q \times X \rightarrow Y$. Let $i_0: X \hookrightarrow Q \times X$ be the product of the identity map $\operatorname{id}_{X}$ with the inclusion $\{ e'(0) \} \hookrightarrow Q$, and define $i_1: X \hookrightarrow Q \times X$ similarly. Since $e'$ is anodyne, the restrictions $e'|_{\{ 0\} }$ and $e'|_{\{ 1\} }$ are anodyne. In particular, they are weak homotopy equivalences (Proposition 3.1.5.13) and therefore homotopy equivalences (Proposition 3.1.5.12), since $Q$ is a Kan complex. It follows that $i_0$ and $i_1$ are also homotopy equivalences, so that $F(i_0)$ and $F(i_1)$ are isomorphisms (by virtue of assumption $(\ast ')$). Using the fact that $i_0$ and $i_1$ are left inverse to the projection map $\pi : Q \times X \rightarrow X$, we see that $F(\pi )$ is an isomorphism in $\operatorname{\mathcal{C}}$ and that we have

\[ F(u_0) = F(h) \circ F(i_0) = F(h) \circ F(\pi )^{-1} = F(h) \circ F(i_1) = F(u_1), \]

as desired.
$\square$

Corollary 3.1.6.7. Let $\operatorname{\mathcal{C}}$ be a category, let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ which carry homotopy equivalences of Kan complexes to isomorphisms in the category $\operatorname{\mathcal{C}}$. Then precomposition with the quotient map $\operatorname{Kan}\rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ induces an isomorphism of categories $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$.

**Proof.**
Combine Remark 3.1.4.11 with Proposition 3.1.6.6.
$\square$

Variant 3.1.6.8. Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{Fun}( \operatorname{Set_{\Delta }}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$. which carry weak homotopy equivalences of simplicial sets to isomorphisms in the category $\operatorname{\mathcal{C}}$. Then:

- $(a)$
For every functor $F \in \operatorname{\mathcal{E}}'$, the restriction $F|_{\operatorname{Kan}}$ factors (uniquely) as a composition $\operatorname{Kan}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{Kan}} \xrightarrow { \overline{F} } \operatorname{\mathcal{C}}$.

- $(b)$
The construction $F \mapsto \overline{F}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$.

**Proof of Variant 3.1.6.8.**
Let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{Kan}, \operatorname{\mathcal{C}})$ be the full subcategory spanned by those functors $F: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$ which carry homotopy equivalences of Kan complexes to isomorphisms in $\operatorname{\mathcal{C}}$. By virtue of Corollary 3.1.6.7, it will suffice to show that the restriction functor $F \mapsto F|_{\operatorname{Kan}}$ induces an equivalence of categories $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. Using Proposition 3.1.6.1, we can choose a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Kan}$ and a natural transformation $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the property that, for every Kan complex $X$, the induced map $u_{X}: X \rightarrow Q(X)$ is anodyne. For every morphism of simplicial sets $f: X \rightarrow Y$, we have a commutative diagram

\[ \xymatrix { X \ar [r]^-{f} \ar [d]^{u_ X} & Y \ar [d]^{ u_ Y} \\ Q(X) \ar [r]^{ Q(f) } & Q(Y), } \]

where the vertical maps are weak homotopy equivalences (Proposition 3.1.5.13). It follows that if $f$ is a weak homotopy equivalence, then $Q(f)$ is also a weak homotopy equivalence (Remark 3.1.5.16) and therefore a homotopy equivalence (Proposition 3.1.5.12). In other words, the functor $Q$ carries weak homotopy equivalences of simplicial sets to homotopy equivalences of Kan complexes. It follows that precomposition with $Q$ induces a functor $\theta : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$. We claim that $\theta $ is homotopy inverse to the restriction functor $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$. This follows from the following pair of observations:

For every functor $F: \operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$, $u$ induces a natural transformation $F \rightarrow F|_{\operatorname{Kan}} \circ Q$, which depends functorially on $F$ and is an isomorphism for $F \in \operatorname{\mathcal{E}}'$.

For every functor $F_0: \operatorname{Kan}\rightarrow \operatorname{\mathcal{C}}$, $u$ induces a natural transformation $F_0 \rightarrow (F_0 \circ Q)|_{\operatorname{Kan}}$, which depends functorially on $F_0$ and is an isomorphism for $F_0 \in \operatorname{\mathcal{E}}$.

$\square$
We now turn to the proof of Proposition 3.1.6.1. We will use an easy version of Quillen's “small object argument” (which we will revisit in greater generality in §).

**Proof of Proposition 3.1.6.1.**
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. 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.
$\square$

Example 3.1.6.10 (Path Fibrations). If $f: X \rightarrow Y$ is a morphism of Kan complexes, then we can give a much more explicit proof of Proposition 3.1.6.1. Let $Q(f)$ denote the fiber product $X \times _{ \operatorname{Fun}( \{ 0\} , Y)} \operatorname{Fun}( \Delta ^1, Y)$. Then $f$ factors as a composition $X \xrightarrow {f'} Q(f) \xrightarrow {f''} Y$, where $f''$ is given by evaluation at the vertex $\{ 1\} \subseteq \Delta ^1$ and $f'$ is obtained by amalgamating the identity morphism $\operatorname{id}_{X}$ with the composition $X \xrightarrow {f} Y \xrightarrow {\delta } \operatorname{Fun}( \Delta ^1, Y)$. Moreover:

The morphism $f'$ is a section of the projection map $Q(f) \rightarrow X$, which is a pullback of the evaluation map $\operatorname{Fun}( \Delta ^1, Y) \rightarrow \operatorname{Fun}( \{ 0\} , Y)$ and therefore a trivial Kan fibration (Corollary 3.1.3.6). It follows that $f'$ is a weak homotopy equivalence. Since it is also a monomorphism, it is anodyne (see Corollary 3.3.7.5).

The morphism $f''$ factors as a composition

\[ Q(f) = X \times _{ \operatorname{Fun}( \{ 0\} , Y)} \operatorname{Fun}(\Delta ^1, Y) \xrightarrow {u} X \times \operatorname{Fun}(\{ 1\} , Y) \xrightarrow {v} Y, \]

where $u$ is a pullback of the restriction map $\operatorname{Fun}(\Delta ^1, Y) \rightarrow \operatorname{Fun}(\operatorname{\partial \Delta }^{1}, Y)$ (and therefore a Kan fibration by virtue of Corollary 3.1.3.3) and $v$ is a pullback of the projection map $X \rightarrow \Delta ^0$ (and therefore a Kan fibration by virtue of our assumption that $X$ is a Kan complex). It follows that $f''$ is also a Kan fibration.

Then $q$ admits a section $f': X \rightarrow Q(f)$ (whose second component is obtained by composing $f$ with the diagonal embedding $Y \hookrightarrow \operatorname{Fun}( \Delta ^1, Y)$).

The proof of Proposition 3.1.6.1 can be repurposed to obtain many analogous results.

Exercise 3.1.6.11. 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.