# 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 3.1.6.14). 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

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

commutes with filtered colimits.

Before giving the proof of Proposition 3.1.7.1, let us note some of its consequences. Applying Proposition 3.1.7.1 in the special case $Y_{} = \Delta ^0$, 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 given below, but there are other (arguably more elegant) ways to achieve the same result. For example, we can take $Q$ to be the simplicial set $\operatorname{Ex}^{\infty }(X)$ of Construction 3.3.6.1 (see Propositions 3.3.6.9 and 3.3.6.7), or the singular simplicial set $\operatorname{Sing}_{\bullet }(|X|)$ (see Proposition 1.1.9.8 and Theorem 3.5.4.1). These constructions also have non-aesthetic advantages: for example, the functors $X \mapsto \operatorname{Ex}^{\infty }(X)$ and $X \mapsto \operatorname{Sing}_{\bullet }(|X|)$ both preserve finite limits.

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

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & Z \ar [d]^{g} \\ Y \ar [r] \ar@ {-->}[ur] & S }$

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 3.1.2.7. 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

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r]^-{f'} & Q \ar [d]^{f''} \\ Y \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & Y }$

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

Recall that the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.5.10 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 (see §6.3).

Proposition 3.1.7.5. 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 3.1.7.1, 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 3.1.6.14) and therefore homotopy equivalences (Proposition 3.1.6.13), 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 3.1.7.6. 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 $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$.

Variant 3.1.7.7. 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 $\operatorname{\mathcal{E}}' \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}})$.

Remark 3.1.7.8. Corollary 3.1.7.6 and Variant 3.1.7.7 can be stated more informally as follows:

• The homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be obtained from the category $\operatorname{Kan}$ of Kan complexes by formally adjoining inverses to all homotopy equivalences.

• The homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ can be obtained from the category $\operatorname{Set_{\Delta }}$ of simplicial sets by formally adjoining inverses to all weak homotopy equivalences.

Either of these assertions characterizes the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ up to equivalence (in fact, Corollary 3.1.7.6 even characterizes $\mathrm{h} \mathit{\operatorname{Kan}}$ up to isomorphism).

Proof of Variant 3.1.7.7. 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 3.1.7.6, 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 3.1.7.1, 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 simplicial set $X$, 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

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{u_ X} & Y \ar [d]^{ u_ Y} \\ Q(X) \ar [r]^-{ Q(f) } & Q(Y), }$

where the vertical maps are weak homotopy equivalences (Proposition 3.1.6.14). It follows that if $f$ is a weak homotopy equivalence, then $Q(f)$ is also a weak homotopy equivalence (Remark 3.1.6.16) and therefore a homotopy equivalence (Proposition 3.1.6.13). 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 3.1.7.1. We will use an easy version of Quillen's “small object argument” (which we will revisit in greater generality in § ).

Proof of Proposition 3.1.7.1. 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. 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 3.1.7.9 (Path Fibrations). If $f: X \rightarrow Y$ is a morphism of Kan complexes, then we can give a much more explicit proof of Proposition 3.1.7.1. Let $P(f)$ denote the fiber product $X \times _{ \operatorname{Fun}( \{ 0\} , Y)} \operatorname{Fun}( \Delta ^1, Y)$. Then $f$ factors as a composition $X \xrightarrow {f'} P(f) \xrightarrow {f''} Y$, 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 $P(f) \rightarrow X$, 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

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

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.

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

Exercise 3.1.7.10. 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. Moreover, this factorization can be chosen to depend functorially on $f$ (as an object of the arrow category $\operatorname{Fun}( , \operatorname{Set_{\Delta }})$).