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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 3.1.5.12). 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.

Remark 3.1.6.3. In the situation of Corollary 3.1.6.2, the Kan complex $Q_{}$ (and the anodyne morphism $f$) can be chosen to depend functorially on $X_{}$. This follows from the proof of Proposition 3.1.6.1 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 3.1.6.4. 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 3.1.5.15). 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 3.1.3.6). 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). $\square$

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 3.1.5.12) and therefore homotopy equivalences (Proposition 3.1.5.11), 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 $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{Kan}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$.

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

Remark 3.1.6.9. Corollary 3.1.6.7 and Variant 3.1.6.8 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 3.1.6.7 even characterizes $\mathrm{h} \mathit{\operatorname{Kan}}$ up to isomorphism).

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 3.1.5.12). It follows that if $f$ is a weak homotopy equivalence, then $Q(f)$ is also a weak homotopy equivalence (Remark 3.1.5.15) and therefore a homotopy equivalence (Proposition 3.1.5.11). 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.