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5.1.7 Morphism Spaces in the Homotopy Coherent Nerve

Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$ (Definition 2.4.3.5). Suppose that $\operatorname{\mathcal{C}}$ is locally Kan, so that the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). Our goal in this section is to describe the morphism spaces in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. Our main result (Theorem 5.1.7.5) implies that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there is a canonical homotopy equivalence

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }(X,Y), \]

where $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ denotes the Kan complex of morphisms from $X$ to $Y$ in $\operatorname{\mathcal{C}}$, and $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }(X,Y)$ is given by Construction 5.1.1.1.

Notation 5.1.7.1. Let $K$ be a simplicial set. We define a simplicial category $\operatorname{\mathcal{E}}[K]$ as follows:

  • The category $\operatorname{\mathcal{E}}[K]$ has exactly two objects, which we will denote by $x$ and $y$.

  • The morphism spaces in $\operatorname{\mathcal{E}}[K]$ are given by the formulae

    \[ \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}( x, x)_{\bullet } = \{ \operatorname{id}_{x} \} \quad \quad \operatorname{Hom}_{K}(y,y)_{\bullet } = \{ \operatorname{id}_ y \} \]
    \[ \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}(x,y)_{\bullet } = K \quad \quad \operatorname{Hom}_{K}( y,x)_{\bullet } = \emptyset . \]

Remark 5.1.7.2. The simplicial category $\operatorname{\mathcal{E}}[K]$ is characterized by the following universal property: if $\operatorname{\mathcal{C}}$ is any simplicial category containing a pair of objects $X$ and $Y$, then the natural map

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{E}}[K] \rightarrow \operatorname{\mathcal{C}}$ with $F(x) = X$ and $F(y) = Y$} \} \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ) } \]

is a bijection (see Proposition 2.4.5.9).

Construction 5.1.7.3. Fix an integer $n \geq 0$, let $[n]$ denote the linearly ordered set $\{ 0 < 1 < \cdots < n \} $, and let $\{ x\} \star [n]$ denote the linearly ordered set obtained from $[n]$ by adjoining a new least element $x$. Let $\operatorname{Path}[ \{ x\} \star [n] ]_{\bullet }$ denote the simplicial path category of Notation 2.4.3.1). We define a simplicial functor $\pi : \operatorname{Path}[ \{ x\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{E}}[\Delta ^ n]$ as follows:

  • On objects, the functor $\pi $ is given by the formula

    \[ \pi (i) = \begin{cases} x & \textnormal{ if $i = x$} \\ y & \textnormal{ if $0 \leq i \leq n$.} \end{cases} \]
  • For $0 \leq m \leq n$, the morphism of simplicial sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star [n] ] }(x, m)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}[\Delta ^ n]}( x,y )_{\bullet } = \Delta ^ n \]

    is given by the map of partially ordered sets

    \[ \{ \textnormal{Subsets $S = \{ x < i_0 < \cdots < i_ k = m \} \subseteq \{ x\} \star [n]$} \} ^{\operatorname{op}} \rightarrow [n] \quad \quad S \mapsto i_0. \]

Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$. Then every $n$-simplex $\sigma \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ determines a simplicial functor $F_{\sigma }: \operatorname{\mathcal{E}}[\Delta ^ n] \rightarrow \operatorname{\mathcal{C}}$, given on objects by $F_{\sigma }(x) = X$ and $F_{\sigma }(y) = Y$. The composition $F_{\sigma } \circ \pi $ is a simplicial functor from $\operatorname{Path}[ \{ x\} \star [n]]_{\bullet }$ to $\operatorname{\mathcal{C}}$, which (by Proposition 2.4.4.15) we can view as a map of simplicial sets $f_{\sigma }: \{ x\} \star \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$. By construction, $f_{\sigma }$ carries $x$ to $X$, and the restriction $f_{\sigma }|_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < \dots < n \} ) }$ is the constant map taking the value $Y$. We can therefore identify $f_{\sigma }$ with an $n$-simplex $\theta (\sigma )$ of the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y)$ introduced in Construction 5.1.6.1 (see Remark 5.1.6.2). The construction $\sigma \mapsto \theta (\sigma )$ depends functorially on the object $[n] \in \operatorname{{\bf \Delta }}$, and therefore determines a map of simplicial sets

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y), \]

which we will refer to as the comparison map.

Exercise 5.1.7.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X$ and $Y$, and let $\operatorname{\mathcal{C}}^{\Delta }$ denote the associated simplicial category (Construction 2.5.9.2). Show that the isomorphism $K( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \xrightarrow {\sim } \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ of Example 5.1.6.14 factors as a composition

\[ K( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) = \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet } \xrightarrow {\theta } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } )}^{\mathrm{L}}( X, Y) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X,Y ), \]

where $\theta $ is the comparison map of Construction 5.1.7.3 and $\rho $ is induced by the trivial Kan fibration $\mathfrak {Z}: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ of Proposition 2.5.9.10. Beware that $\theta $ and $\rho $ are generally not isomorphisms.

Our comparison result can now be formulated as follows:

Theorem 5.1.7.5. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Then the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 5.1.7.3 is a homotopy equivalence of Kan complexes.

Remark 5.1.7.6. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Combining Theorem 5.1.7.5 with Proposition 5.1.6.9, we obtain a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$, given by composing the comparison map $\theta $ of Construction 5.1.7.3 with the left-pinch inclusion map of Construction 5.1.6.6.

Before giving the proof of Theorem 5.1.7.5, let us outline some applications.

Definition 5.1.7.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a simplicial functor.

  • We say that $F$ is weakly fully faithful if, for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet }$ is a weak homotopy equivalence of simplicial sets.

  • We say that $F$ is weakly essentially surjective if the induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective (that is, every object of $\operatorname{\mathcal{D}}$ is homotopy equivalent to an object of the form $F(X)$, for some $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$).

  • We say that $F$ is a weak equivalence of simplicial categories if it is weakly fully faithful and weakly essentially surjective.

Corollary 5.1.7.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be locally Kan simplicial categories, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a simplicial functor, and let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F): \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ be the induced functor of $\infty $-categories. Then:

$(1)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is fully faithful (in the sense of Definition 5.1.2.1) if and only if the simplicial functor $F$ is weakly fully faithful (in the sense of Definition 5.1.7.7).

$(2)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is essentially surjective (in the sense of Definition 5.1.2.9) if and only if the simplicial functor $F$ is weakly essentially surjective (in the sense of Definition 5.1.7.7).

$(3)$

The functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10) if and only if $F$ is a weak equivalence of simplicial categories (in the sense of Definition 5.1.7.7).

Proof. For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, we have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r]^-{F} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) )_{\bullet } \ar [d] \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X, Y) \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(F) } & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})}( F(X), F(Y) ), } \]

where the vertical maps are the homotopy equivalences supplied by Remark 5.1.7.6. It follows that the upper horizontal map is a homotopy equivalence if and only if the lower horizontal map is a homotopy equivalence. This proves $(1)$. Assertion $(2)$ follows from Proposition 2.4.6.8. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with the criterion of Theorem 5.1.2.17. $\square$

Theorem 5.1.7.5 is an immediate consequence of the following more general result:

Theorem 5.1.7.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$, and suppose that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$ is also an $\infty $-category, and the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 5.1.7.3 is an equivalence of $\infty $-categories.

Remark 5.1.7.10. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$, and suppose that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Applying Theorem 5.1.7.9 to the opposite simplicial category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (and using Remark 5.1.6.3), we obtain an equivalence of $\infty $-categories

\[ \theta ': \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\operatorname{op}} \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y), \]

which can be described explicitly using a variant of Construction 5.1.7.3.

The remainder of this section is devoted to the proof of Theorem 5.1.7.9.

Construction 5.1.7.11. Let $K$ be a simplicial set. We let $\Sigma (K)$ denote the pushout $(\{ x\} \star K) \coprod _{K} \{ y\} $ (this is a model for the unreduced suspension of $K$). Let $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ denote the simplicial path category of $\Sigma (K)$ (Notation 2.4.4.2). Then $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ has exactly two objects, which we denote by $x$ and $y$. We let $\Phi (K)$ denote the simplicial set $\operatorname{Hom}_{ \operatorname{Path}[ \Sigma (K) ]}(x,y)_{\bullet }$.

Example 5.1.7.12. If $K = \Delta ^0$, then the suspension $\Sigma (K)$ can be identified with $\Delta ^1$. In this case, the simplicial path category $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ can be identified with the ordinary category $[1]$, and the simplicial set $\Phi (K)$ is isomorphic to $\Delta ^{0}$.

Remark 5.1.7.13. Let $K$ be a simplicial set, and let $\operatorname{\mathcal{D}}$ be another simplicial set containing vertices $X$ and $Y$. Unwinding the definitions, we have a canonical bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{ Morphisms $K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{L}}(X,Y)$} \} \ar [d]^{\sim } \\ \{ \textnormal{Morphisms $F: \Sigma (K) \rightarrow \operatorname{\mathcal{D}}$ with $F(x) = X$ and $F(y) = Y$} \} . } \]

Remark 5.1.7.14. Let $K$ be a simplicial set. Note that, for $n > 0$, every nondegenerate simplex $\sigma : \Delta ^ n \rightarrow \Sigma (K)$ satisfies $\sigma (0) = x$ and $\sigma (n) = y$. Using Theorem 2.4.4.10, we see that for each $m \geq 0$, $\operatorname{Path}[ \Sigma (K) ]_{m}$ can be identified with the path category of a directed graph $G_ m$ with vertex set $\operatorname{Vert}(G_ m) = \{ x,y\} $, where each edge of $G_ m$ has source $x$ and target $y$. These path categories are easy to describe: they satisfy

\[ \operatorname{Hom}_{ \operatorname{Path}[G_ m] }( x, x) = \{ \operatorname{id}_{x} \} \quad \quad \operatorname{Hom}_{\operatorname{Path}[G_ m] }( y,y) = \{ \operatorname{id}_{y} \} \]
\[ \operatorname{Hom}_{ \operatorname{Path}[G_ m] }(x,y) = \operatorname{Edge}(G_ m) \quad \quad \operatorname{Hom}_{ \operatorname{Path}[G_ m] }( y,x) = \emptyset . \]

Allowing $m$ to vary, we conclude that the simplicial category $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ satisfies

\[ \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(x,x)_{\bullet } = \{ \operatorname{id}_{x} \} \quad \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(y,x)_{\bullet } = \emptyset \quad \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(y,y)_{\bullet } = \{ \operatorname{id}_{y} \} . \]

That is, $\operatorname{Path}[\Sigma (K)]_{\bullet }$ can be identified with the simplicial category $\operatorname{\mathcal{E}}[ \Phi (X) ]$ of Notation 5.1.7.1. Moreover, we can identify $m$-simplices of $\Phi (X)$ with elements of the set $E( \Sigma (K), m)$ defined in Notation 2.4.4.9.

Remark 5.1.7.15. Let $u: K \rightarrow K'$ be a monomorphism of simplicial sets. Then the induced map $\Phi (u): \Phi (K) \rightarrow \Phi (K')$ is also a monomorphism (this follows immediately from the description given in Remark 5.1.7.14).

Lemma 5.1.7.16. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be a simplicial category containing objects $X$ and $Y$. Then the natural map

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $F: \operatorname{Path}[\Sigma (K)]_{\bullet }$ with $F(x) = X$ and $F(y) = Y$} \} \ar [d] \\ \{ \textnormal{Morphisms $\Phi (K) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$} \} } \]

is a bijection.

Combining Remark 5.1.7.13 with Lemma 5.1.7.16 and invoking the universal property of simplicial path categories, we obtain the following:

Corollary 5.1.7.17. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be a simplicial category containing objects $X$ and $Y$. Then we have a canonical bijection

\[ \{ \textnormal{Morphisms $K \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$} \} \simeq \{ \textnormal{Morphisms $\Phi (K) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$} \} . \]

Corollary 5.1.7.18. Let $A$ and $B$ be simplicial sets, and let $\operatorname{\mathcal{E}}[B]$ be the simplicial category of Notation 5.1.7.1. Then we have a canonical bijection

\[ \{ \textnormal{Morphisms $A \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}[B])}^{\mathrm{L}}(x,y)$} \} \simeq \{ \textnormal{Morphisms $\Phi (A) \rightarrow B$} \} . \]

Proof. Apply Corollary 5.1.7.17 in the special case $\operatorname{\mathcal{C}}= \operatorname{\mathcal{E}}[B]$. $\square$

Corollary 5.1.7.19. The functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad B \mapsto \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}[B] )}(x,y) \]

has a left adjoint, given by the functor $A \mapsto \Phi (A)$ of Construction 5.1.7.11.

Remark 5.1.7.20. The adjunction of Corollary 5.1.7.19 has an interpretation in the framework of Proposition 1.1.8.22. Let $Q^{\bullet }$ denote the cosimplicial object of $\operatorname{Set_{\Delta }}$ given by the construction $[n] \mapsto \Phi ( \Delta ^ n )$. For every simplicial set $B$, Corollary 5.1.7.18 supplies a canonical isomorphism of simplicial sets

\[ \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}[B] )}(x,y) \simeq \operatorname{Sing}_{\bullet }^{Q}( B ), \]

where $\operatorname{Sing}_{\bullet }^{Q}(B)$ is the simplicial set defined in Variant 1.1.7.6. It follows that $\Phi $ can be identified with the generalized geometric realization functor $K \mapsto | K |^{Q}$ of Proposition 1.1.8.22.

Corollary 5.1.7.21. The functor $\Phi : \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 5.1.7.11 preserves colimits.

Construction 5.1.7.22. Let $K$ be a simplicial set, let $\operatorname{\mathcal{E}}[K]$ be the simplicial category of Notation 5.1.7.1, and let

\[ \theta : K = \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}(x,y)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}[K])}( x, y) \]

be the comparison map of Construction 5.1.7.3. We let $\rho _{K}: \Phi (K) \rightarrow K$ denote the image of $\theta $ under the bijection of Corollary 5.1.7.18.

We will deduce Theorem 5.1.7.9 from the following result, which we prove at the end of this section:

Proposition 5.1.7.23. Let $K$ be a simplicial set. Then the morphism $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 5.1.7.22 is a categorical equivalence of simplicial sets.

Corollary 5.1.7.24. Let $u: K \rightarrow K'$ be a categorical equivalence of simplicial sets. Then the induced map $\Phi (u): \Phi (K) \rightarrow \Phi (K')$ is also a categorial equivalence of simplicial sets.

Proof. We have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Phi (K) \ar [r]^-{ \Phi (u) } \ar [d]^{\rho _ K} & \Phi (K') \ar [d]^{ \rho _{K'} } \\ K \ar [r]^-{u} & K' } \]

where $u$ is a categorical equivalence by hypothesis and the vertical maps are categorical equivalences by Proposition 5.1.7.23. Using Remark 4.5.2.6, we conclude that $\Phi (u)$ is a categorical equivalence as well. $\square$

Corollary 5.1.7.25. Let $\operatorname{\mathcal{C}}$ be a simplicial categories containing a pair of objects $X$ and $Y$, and assume that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the simplicial set $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y)$ is also an $\infty $-category.

Proof. Let $i: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. We wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{i} & \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & \Delta ^0 } \]

admits a solution. By virtue of Corollary 5.1.7.17, we can rephrase this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Phi (A) \ar [r] \ar [d]^{\Phi (i)} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d] \\ \Phi (B) \ar [r] \ar@ {-->}[ur] & \Delta ^0. } \]

Note that $\Phi (i)$ is a monomorphism (Remark 5.1.7.15) and a categorical equivalence (Corollary 5.1.7.24), so the desired result follows from Lemma 4.5.6.2. $\square$

Corollary 5.1.7.26. Let $\operatorname{\mathcal{C}}$ be a simplicial categories containing a pair of objects $X$ and $Y$, and assume that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Let $K$ be another simplicial set, and suppose we are given a pair of morphisms $f_0, f_1: K \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }(X,Y)$, which correspond (under the bijection of Corollary 5.1.7.17) to diagrams $f'_0,f'_1: \Phi (K) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. The following conditions are equivalent:

$(1)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}(K, \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }(X,Y) )$.

$(2)$

The diagrams $f'_0$ and $f'_1$ are isomorphic when regarded as objets of the $\infty $-category $\operatorname{Fun}( \Phi (K), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )$.

Proof. Choose a categorical mapping cylinder

\[ K \coprod K \xrightarrow { (s_0, s_1) } \overline{K} \xrightarrow {\pi } K \]

for the simplicial set $K$ (Definition 5.1.4.3). Using Remark 5.1.7.15, Corollary 5.1.7.21, and Corollary 5.1.7.24, we conclude that the induced diagram

\[ \Phi (K) \coprod \Phi (K) \xrightarrow { ( \Phi (s_0), \Phi (s_1) ) } \Phi ( \overline{K} ) \xrightarrow { \Phi (\pi ) } \Phi (K) \]

exhibits $\Phi ( \overline{K} )$ as a categorical mapping cylinder of $K$. Using Corollary 5.1.4.7, we see that $(1)$ and $(2)$ are equivalent to the following:

$(1')$

There exists a diagram $\overline{f}: \overline{K} \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }(X,Y)$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.

$(2')$

There exists a diagram $\overline{f}': \Phi ( \overline{K} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ satisfying $f'_0 = \overline{f}' \circ \Phi (s_0)$ and $f'_1 = \overline{f}' \circ \Phi (s_1)$.

The equivalence of $(1')$ and $(2')$ follows from Corollary 5.1.7.17. $\square$

Proof of Theorem 5.1.7.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$ for which the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Applying Corollary 5.1.7.25, we deduce that the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X, Y)$ is also an $\infty $-category. We wish to show that the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 5.1.7.3 is an equivalence of $\infty $-categories. To prove this, it will suffice to show that for every simplicial set $K$, postcomposition with $\theta $ induces a bijection

\[ \theta _{K}: \pi _0( \operatorname{Fun}( K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(K, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y))^{\simeq } ). \]

By virtue of Corollary 5.1.7.26, we can identify $\pi _0( \operatorname{Fun}(K, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y))^{\simeq } )$ with the set $\pi _0( \operatorname{Fun}( \Phi (K), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } )$. Under this identification, $\theta _{K}$ corresponds to the map

\[ \pi _0( \operatorname{Fun}( K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \Phi (K), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \]

given by precomposition with the map $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 5.1.7.22, which is bijective by virtue of the fact that $\rho _{K}$ is a categorical equivalence of simplicial sets (Proposition 5.1.7.23). $\square$

We now turn to the proof of Proposition 5.1.7.23. Our strategy is to use formal arguments to reduce to the case where the simplicial set $K$ is a standard simplex, which can be analyzed explicitly.

Example 5.1.7.27. Let $m$ and $n$ be nonnegative integers. By virtue of Remark 5.1.7.14, we can identify $m$-simplices of the simplicial set $\Phi ( \Delta ^ n )$ with the set $E( \Sigma (\Delta ^ n), m)$ defined in Notation 2.4.4.9. By definition, the elements of $E( \Sigma (K), m)$ are given by pairs $(\sigma , \overrightarrow {I} )$, where $\sigma : \Delta ^ k \rightarrow \Sigma (\Delta ^ n)$ is a nondegenerate simplex of dimension $k > 0$ and $\overrightarrow {I} = ( I_0 \supseteq I_1 \supseteq \cdots \supseteq I_ m )$ is a chain of subsets of $[k]$ satisfying $I_0 = [k]$ and $I_ m = \{ 0, k\} $.

For each $k > 0$, there is a canonical bijection

\[ \{ \textnormal{Subsets $S \subseteq [n]$ of cardinality $k$} \} \simeq \{ \textnormal{Nondegenerate $k$-Simplices of $\Sigma (\Delta ^ n)$} \} , \]

which carries a subset $S$ to the $k$-simplex $\sigma _{S}$ given by the composite map

\[ \Delta ^ k \simeq \{ x\} \star \operatorname{N}_{\bullet }(S) \hookrightarrow \{ x\} \star \Delta ^ n \twoheadrightarrow \Sigma (\Delta ^ n). \]

For every such subset $S$, let $\iota _{S}: \operatorname{N}_{\bullet }(S) \hookrightarrow \Delta ^ k$ be the inclusion map. Then the construction

\[ ( \sigma _ S, \overrightarrow {I} ) \mapsto ( \sigma _{S}^{-1}(I_0) \supseteq \sigma _{S}^{-1}(I_1) \supseteq \cdots \supseteq \sigma _{S}^{-1}(I_ m) ) \]

induces a bijection from $E( \Sigma ( \Delta ^ n), m)$ to the collection of chains $\overrightarrow {S} = ( S_0 \supseteq S_1 \supseteq \cdots \supseteq S_ m )$ of subsets of $[n]$ which satisfy the following pair of conditions:

$(a)$

The set $S_ m$ contains exactly one element.

$(b)$

For $0 \leq i \leq m$, the unique element of $S_ m$ is the largest element of $S_ i$.

Let us henceforth use this bijection to identify $m$-simplices of $\Phi ( \Delta ^ n )$ with chains $\overrightarrow {S}$ satisfying $(a)$ and $(b)$. In these terms, the face and degeneracy operators for the simplicial set $\Phi ( \Delta ^ n ) = \Phi (\Delta ^ n)_{\bullet }$ can be described explicitly as follows:

  • For $0 \leq i \leq m$, the degeneracy operator $s_ i: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m+1}$ is given by

    \[ s_ i( S_0 \supseteq \cdots \supseteq S_ m ) = (S_0 \supseteq \cdots \supseteq S_{i-1} \subseteq S_{i} \supseteq S_ i \supseteq S_{i+1} \supseteq \cdots \supseteq S_ m) \]
  • For $0 \leq i < m$, the face operator $d_ i: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m-1}$ is given by the construction

    \[ d_ i( S_0 \supseteq \cdots \supseteq S_ m ) = (S_0 \supseteq \cdots \supseteq S_{i-1} \supseteq S_{i+1} \supseteq \cdots \supseteq S_ m). \]
  • For $m > 0$, the face operator $d_ m: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m-1}$ is given by

    \[ d_0( S_0 \supseteq \cdots \supseteq S_ m ) = (S'_0 \supseteq S'_1 \supseteq \cdots \supseteq S'_{m-1} ), \]

    where $S'_{i} = \{ j \in S_ i: j \leq \min (S_{m-1}) \} $.

See Remark 2.4.4.17.

Construction 5.1.7.28. Let $m$ and $n$ be nonnegative integers. Suppose we are given an $m$-simplex of $\Phi (\Delta ^ n)$, which we identify with a chain of subsets $\overrightarrow {S} = (S_0 \supseteq \cdots \supseteq S_ m)$ satisfying conditions $(a)$ and $(b)$ of Example 5.1.7.27. Let $\tau : [m] \rightarrow [1]$ be a nondecreasing function. Let $\overrightarrow {S}' = (S'_0 \supseteq \cdots \supseteq S'_ m)$ be the chain of subsets of $[n+1]$ given by the formula

\[ S'_{i} = \begin{cases} \{ s+1: s \in S_ i \} \text{ if } \tau (i) = 1 \\ \{ 0\} & \text{ if } \tau (m) = 0 \\ \{ 0 \} \cup \{ s+1: s \in S_ i \} & \text{ otherwise. } \end{cases} \]

The construction $(\overrightarrow {S}, \tau ) \mapsto \overrightarrow {S}'$ is compatible with the formation of face and degeneracy operators, and therefore determines a morphism of simplicial sets $\pi : \Phi ( \Delta ^ n) \times \Delta ^1 \rightarrow \Phi ( \Delta ^{n+1} )$.

Lemma 5.1.7.29. Let $n \geq 0$ be an integer, and let $\pi : \Phi ( \Delta ^ n) \times \Delta ^1 \rightarrow \Phi ( \Delta ^{n+1} )$ be the morphism of simplicial sets defined in Construction 5.1.7.28. Then $\pi $ fits into a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \Delta ^ n ) \times \{ 0\} \ar [r] \ar [d] & \Phi ( \Delta ^ n ) \times \Delta ^1 \ar [d]^{\pi } \\ \Delta ^0 \ar [r] & \Phi ( \Delta ^{n+1} ). } \]

Remark 5.1.7.30. It follows from Lemma 5.1.7.29 that the morphism $\pi $ of Construction 5.1.7.28 induces an isomorphism of simplicial sets $\Delta ^{0} \diamond \Phi (\Delta ^ n) \rightarrow \Phi ( \Delta ^{n+1} )$, where the operation $\diamond $ is defined in Notation 4.5.5.3.

Proof of Lemma 5.1.7.29. Fix an integer $m \geq 0$. By construction, the restriction $\pi |_{ \Phi (\Delta ^ n) \times \{ 0\} }$ is the constant map which carries each $m$-simplex of $\Phi ( \Delta ^{n} )$ to the element of $\Phi ( \Delta ^{n+1} )$ given by the constant chain $\overrightarrow {S}_0 = ( \{ 0 \} \subseteq \{ 0 \} \subseteq \cdots \subseteq \{ 0 \} )$. To complete the proof, we must show that for each $m \geq 0$, the map $\pi $ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \Delta ^ n)_ m \times \{ \textnormal{Nondecreasing functions $\tau : [m] \rightarrow [1]$ with $\tau (m) = 1$} \} \ar [d] \\ \Phi ( \Delta ^{n+1})_{m} \setminus \{ \overrightarrow {S}_0 \} }. \]

The inverse bijection can be described explicitly as follows: it carries an $m$-simplex $(S'_0 \supseteq \cdots \supseteq S'_ m) \neq \overrightarrow {S}_0$ of $\Phi ( \Delta ^{n+1} )$ to the pair $( \overrightarrow {S}, \tau )$, where $\overrightarrow {S} = (S_0 \supseteq \cdots \supseteq S_ m)$ is the $m$-simplex of $\Phi ( \Delta ^ n )$ given by

\[ S_{i} = \{ s-1: s \in S'_ i, s > 0 \} \quad \quad \tau (i) = \begin{cases} 0 & \textnormal{ if } 0 \in S'_ i \\ 1 & \textnormal{ if } 0 \notin S'_ i. \end{cases} \]
$\square$

Proof of Proposition 5.1.7.23. Let $K$ be a simplicial set. We wish to show that the map $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 5.1.7.22 is a categorical equivalence of simplicial sets. Using Corollary 5.1.7.21, we can write $\rho _{K}$ as a filtered colimit of morphisms $\rho _{K_{\alpha }}: \Phi ( K_{\alpha } ) \rightarrow K_{\alpha }$, where $K_{\alpha }$ ranges over the collection of all finite simplicial subsets of $K$ (Remark 3.5.1.8). Since the collection of categorical equivalences is closed under the formation of filtered colimits (Corollary 4.5.4.2), it will suffice to show that each $\rho _{ K_{\alpha } }$ is a categorical equivalence. We may therefore replace $K$ by $K_{\alpha }$ and thereby reduce to the case where the simplicial set $K$ is finite.

Since $K$ is a finite simplicial set, it has dimension $\leq n$ for some integer $n \geq -1$. We proceed by induction on $n$. If $n=-1$, then both $K$ and $\Phi (K)$ are empty, and there is nothing to prove. We may therefore assume that $n \geq 0$ and that $\rho _{K'}$ is a categorical equivalence for every simplicial set $K'$ of dimension $< n$. We now proceed by induction on the number $m$ of nondegenerate $n$-simplices of $K$. If $m = 0$, then $K$ has dimension $\leq n-1$ and the desired result holds by virtue of our inductive hypothesis. We may therefore assume that $K$ has at least one nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow K$. Using Proposition 1.1.3.13, we see that there is a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ K' \ar [r] & K, } \]

where $S'$ is a simplicial set of dimension $\leq n$ with exactly $(m-1)$-nondegenerate $m$-simplices. We then have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \operatorname{\partial \Delta }^ n ) \ar [rr] \ar [dd] \ar [dr]^{ \rho _{ \operatorname{\partial \Delta }^ n }} & & \Phi ( \Delta ^ n) \ar [dd] \ar [dr]^{ \rho _{ \Delta ^ n } } & \\ & \operatorname{\partial \Delta }^ n \ar [rr] \ar [dd] & & \Delta ^ n \ar [dd] \\ \Phi (K') \ar [rr] \ar [dr]^{ \rho _{K'} } & & \Phi (K) \ar [dr]^{ \rho _ K} & \\ & K' \ar [rr] & & K } \]

where the front and back faces are pushout squares (Corollary 5.1.7.21) in which the horizontal maps are monomorphisms (Remark 5.1.7.15), and are therefore categorical pushout squares (Example 4.5.3.8). Our inductive hypotheses guarantees that the maps $\rho _{K'}$ and $\rho _{\operatorname{\partial \Delta }^ n}$ are categorical equivalences. Consequently, to show that $\rho _{K}$ is a categorical equivalence, it will suffice to show that $\rho _{\Delta ^ n}$ is a categorical equivalence (Proposition 4.5.3.5). We may therefore replace $K$ by $\Delta ^ n$ and thereby reduce to the case where $K$ is a standard simplex.

If $n = 0$, then the map $\rho _{\Delta ^ n}: \Phi ( \Delta ^ n ) \rightarrow \Delta ^ n$ is an isomorphism (Example 5.1.7.12). We may therefore assume without loss of generality that $n > 0$, so that Lemma 5.1.7.29 supplies an isomorphism of simplicial sets $\Phi ( \Delta ^ n ) \simeq \Delta ^{0} \diamond \Phi ( \Delta ^{n-1} )$. Using this isomorphism, we can identify $\rho _{\Delta ^ n}$ with the composite map

\[ \Delta ^{0} \diamond \Phi ( \Delta ^{n-1} ) \xrightarrow { \operatorname{id}\diamond \rho _{ \Delta ^{n-1} } } \Delta ^{0} \diamond \Delta ^{n-1} \xrightarrow {c} \Delta ^{0} \star \Delta ^{n-1} \simeq \Delta ^ n, \]

where $c$ is the comparison map of Notation 4.5.5.3 (to check this, it suffices to observe that they agree on vertices). Our inductive hypothesis guarantees that $\rho _{ \Delta ^{n-1} }$ is a categorical equivalence of simplicial sets, so that the induced map $\Delta ^{0} \diamond \Phi ( \Delta ^{n-1} ) \xrightarrow { \operatorname{id}\diamond \rho _{ \Delta ^{n-1} }} \Delta ^0 \diamond \Delta ^{n-1}$ is also a categorical equivalence by virtue of Remark 4.5.5.6. We are therefore reduced to showing that $c$ is a categorical equivalence, which is a special case of Proposition 4.5.5.11. $\square$