3.3 The $\operatorname{Ex}^{\infty }$ Functor
Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. If $S$ is a Kan complex, then $X$ is also a Kan complex. Moreover, for every vertex $x \in X$ having image $s = f(x) \in S$, Theorem 3.2.6.1 supplies an exact sequence of homotopy groups
\[ \cdots \rightarrow \pi _{2}(S,s) \xrightarrow {\partial } \pi _{1}(X_ s, x) \rightarrow \pi _{1}( X, x) \rightarrow \pi _1(S,s) \xrightarrow {\partial } \pi _{0}(X_ s, x) \rightarrow \pi _0( X,x) \rightarrow \pi _0(S,s). \]
If $S$ is not a Kan complex, then the results of §3.2.6 do not apply directly. However, one can obtain similar information by replacing $f$ by a Kan fibration $f': X' \rightarrow S'$ between Kan complexes, using the following result:
Theorem 3.3.0.1. Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. Then there exists a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [d]^{f} & X' \ar [d]^{f'} \\ S \ar [r]^-{g} & S' } \]
with the following properties:
- $(a)$
The simplicial sets $S'$ and $X'$ are Kan complexes.
- $(b)$
The morphisms $g$ and $g'$ are weak homotopy equivalences.
- $(c)$
The morphism $f'$ is a Kan fibration.
- $(d)$
For every vertex $s \in S$, the induced map $g'_{s}: X_{s} \rightarrow X'_{ g(s)}$ is a homotopy equivalence of Kan complexes.
Note that we can almost deduce Theorem 3.3.0.1 formally from the results of §3.1.7. Given a Kan fibration $f: X \rightarrow S$, we can always choose an anodyne map $g: S \rightarrow S'$, where $S'$ is a Kan complex (Corollary 3.1.7.2). Applying Proposition 3.1.7.1, we deduce that $g \circ f$ factors as a composition $X \xrightarrow {g'} X' \xrightarrow {f'} S'$, where $f'$ is a Kan fibration and $g'$ is anodyne. The resulting commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{g'} \ar [d]^{f} & X' \ar [d]^{f'} \\ S \ar [r]^-{g} & S' } \]
then satisfies conditions $(a)$, $(b)$, and $(c)$ of Theorem 3.3.0.1. However, it is not so obvious that this diagram also satisfies condition $(d)$. To guarantee this, it is convenient to adopt a different approach to the results of §3.1.7. Following Kan ([MR90047]), we will introduce a functor $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ and a natural transformation of functors $\rho ^{\infty }: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow \operatorname{Ex}^{\infty }$ with the following properties:
- $(a')$
For every simplicial set $S$, the simplicial set $\operatorname{Ex}^{\infty }(S)$ is a Kan complex (Proposition 3.3.6.9).
- $(b')$
For every simplicial set $S$, the morphism $\rho _{S}^{\infty }: S \rightarrow \operatorname{Ex}^{\infty }(S)$ is a weak homotopy equivalence (Proposition 3.3.6.7).
- $(c')$
For every Kan fibration of simplicial sets $f: X \rightarrow S$, the induced map $\operatorname{Ex}^{\infty }(f): \operatorname{Ex}^{\infty }(X) \rightarrow \operatorname{Ex}^{\infty }(S)$ is a Kan fibration (Proposition 3.3.6.6).
- $(d')$
The functor $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ commutes with finite limits (Proposition 3.3.6.4). In particular, for every morphism of simplicial sets $f: X \rightarrow S$ and every vertex $s \in S$, the canonical map $\operatorname{Ex}^{\infty }( X_{s} ) \rightarrow \{ s\} \times _{ \operatorname{Ex}^{\infty }(S) } \operatorname{Ex}^{\infty }(X)$ is an isomorphism (Corollary 3.3.6.5).
It follows from these assertions that for any Kan fibration $f: X \rightarrow S$, the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r]^-{ \rho ^{\infty }_{X} } & \operatorname{Ex}^{\infty }(X) \ar [d]^{ \operatorname{Ex}^{\infty }(f) } \\ S \ar [r]^-{ \rho ^{\infty }_{S} } & \operatorname{Ex}^{\infty }(S) } \]
satisfies the requirements of Theorem 3.3.0.1.
Most of this section is devoted to the definition of the functor $\operatorname{Ex}^{\infty }$ (and the natural transformation $\rho ^{\infty }$) and the verification of assertions $(a')$ through $(d')$. The construction is rooted in classical geometric ideas. Let $n$ be a nonnegative integer, let
\[ | \Delta ^{n} | = \{ (t_0, t_1, \ldots , t_ n) \in [0,1]^{n+1} : t_0 + t_1 + \cdots + t_ n = 1 \} \]
denote the topological simplex of dimension $n$. This topological space admits a triangulation whose vertices are the barycenters of its faces. More precisely, there is a canonical homeomorphism of topological spaces $| \operatorname{Sd}( \Delta ^{n} ) | \xrightarrow {\sim } | \Delta ^{n} |$, where $\operatorname{Sd}( \Delta ^{n} )$ denotes the nerve of the partially ordered set of faces of $\Delta ^{n}$ (Proposition 3.3.2.3). For every topological space $Y$, composition with this homeomorphism induces a bijection
\[ \varphi _{n}: \operatorname{Sing}_{n}(Y) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}( \Delta ^{n} ), \operatorname{Sing}_{\bullet }(Y) ). \]
Motivated by this observation, we define a functor $X \mapsto \operatorname{Ex}(X) = \operatorname{Ex}_{\bullet }(X)$ from the category of simplicial sets to itself by the formula $\operatorname{Ex}_ n(X) = \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}( \Delta ^{n} ), X)$ (Construction 3.3.2.5). The preceding discussion can then be summarized by noting that, when $X = \operatorname{Sing}_{\bullet }(Y)$ is the singular simplicial set of a topological space $Y$, the bijections $\{ \varphi _ n \} _{n \geq 0}$ determine an isomorphism of semisimplicial sets $\varphi : X \rightarrow \operatorname{Ex}( X )$ (Example 3.3.2.9). Beware that $\varphi $ is generally not an isomorphism of simplicial sets: that is, it need not be compatible with degeneracy operators.
In §3.3.3, we show that the functor $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint (Corollary 3.3.3.4). We denote the value of this left adjoint on a simplicial set $X$ by $\operatorname{Sd}(X)$, and refer to it as the subdivision of $X$. It is essentially immediate from the definition that, in the special case where $X = \Delta ^{n}$ is a standard simplex, we recover the simplicial set $\operatorname{Sd}( \Delta ^{n})$ defined above. More generally, we will say that a simplicial set $X$ is braced if the collection of nondegenerate simplices of $X$ is closed under face operators (Definition 3.3.1.1). If this condition is satisfied, then the subdivision $\operatorname{Sd}(X)$ can be identified with the nerve of the category $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ of nondegenerate simplices of $X$ (Proposition 3.3.3.17). Moreover, we also have a canonical homeomorphism of topological spaces $| \operatorname{Sd}(X) | \rightarrow |X|$, which carries each vertex of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} )$ to the barycenter of the corresponding simplex of $|X|$ (Proposition 3.3.3.7).
In §3.3.4, we associate to every simplicial set $X$ a pair of comparison maps
\[ \lambda _{X}: \operatorname{Sd}(X) \rightarrow X \quad \quad \rho _{X}: X \rightarrow \operatorname{Ex}(X); \]
we refer to $\lambda _{X}$ as the last vertex map of $X$ (Construction 3.3.4.3). In the special case $X = \Delta ^ n$, the source and target of $\lambda _{X}$ are both weakly contractible, so $\lambda _{X}$ is automatically a weak homotopy equivalence. From this observation, it follows from a simple formal argument that $\lambda _{X}$ is a weak homotopy equivalence for every simplicial set $X$ (Proposition 3.3.4.8). In §3.3.5, we exploit this to show that the functor $\operatorname{Ex}$ carries Kan fibrations to Kan fibrations (Corollary 3.3.5.4), and that the comparison map $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ is a weak homotopy equivalence for every simplicial set $X$ (Theorem 3.3.5.1). Consequently, the functor $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ satisfies analogues of properties $(b')$, $(c')$, and $(d')$ above.
Unfortunately, the functor $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ does not satisfy the analogue of condition $(a')$: in general, a simplicial set of the form $\operatorname{Ex}(X)$ need not satisfy the Kan extension condition. However, one can show that it satisfies a slightly weaker condition: for any morphism of simplicial sets $f_0: \Lambda ^{n}_{i} \rightarrow \operatorname{Ex}(X)$, the composite map $\Lambda ^{n}_{i} \xrightarrow {f_0} \operatorname{Ex}(X) \xrightarrow { \rho _{ \operatorname{Ex}(X)} } \operatorname{Ex}^2(X)$ can be extended to an $n$-simplex of the simplicial set $\operatorname{Ex}^2(X) = \operatorname{Ex}( \operatorname{Ex}(X) )$. We apply this observation in §3.3.6 to deduce that the direct limit
\[ \operatorname{Ex}^{\infty }(X) = \varinjlim ( X \xrightarrow { \rho _{X} } \operatorname{Ex}(X) \xrightarrow { \rho _{ \operatorname{Ex}(X) } } \operatorname{Ex}^2(X) \xrightarrow { \rho _{ \operatorname{Ex}^2(X)} } \operatorname{Ex}^3(X) \rightarrow \cdots ) \]
is a Kan complex (Proposition 3.3.6.9). Moreover, properties $(b')$, $(c')$, and $(d')$ for the functor $X \mapsto \operatorname{Ex}^{\infty }(X)$ are immediate consequences of the analogous properties of the functor $X \mapsto \operatorname{Ex}(X)$.
We close this section by outlining some applications of the functor $\operatorname{Ex}^{\infty }$. In §3.3.7 we prove that, in the situation of Theorem 3.3.0.1, assertion $(d)$ is a formal consequence of $(b)$ and $(c)$ (Proposition 3.3.7.1). Using this, we show that a Kan fibration of simplicial sets $f: X \rightarrow S$ is a weak homotopy equivalence if and only if it is a trivial Kan fibration (Proposition 3.3.7.6), and that a monomorphism of simplicial sets $g: X \hookrightarrow Y$ is a weak homotopy equivalence if and only if it is anodyne (Corollary 3.3.7.7). In §3.3.8 we prove a refinement of Theorem 3.3.0.1, which guarantees that every Kan fibration $f: X \rightarrow S$ is actually isomorphic to the pullback of a Kan fibration $f': X' \rightarrow S'$ between Kan complexes (Theorem 3.3.8.1).
Structure
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Subsection 3.3.1: Digression: Braced Simplicial Sets
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Subsection 3.3.2: The Subdivision of a Simplex
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Subsection 3.3.3: The Subdivision of a Simplicial Set
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Subsection 3.3.4: The Last Vertex Map
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Subsection 3.3.5: Comparison of $X$ with $\operatorname{Ex}(X)$
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Subsection 3.3.6: The $\operatorname{Ex}^{\infty }$ Functor
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Subsection 3.3.7: Application: Characterizations of Weak Homotopy Equivalences
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Subsection 3.3.8: Application: Extending Kan Fibrations