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5.5.2 Digression: Slicing and the Homotopy Coherent Nerve

Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote its nerve. For every object $X \in \operatorname{\mathcal{C}}$, Example 4.3.5.8 supplies canonical isomorphisms

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/X} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/X} \quad \quad \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{X/} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}. \]

Our goal in this section is to establish a counterpart of this result in the case where $\operatorname{\mathcal{C}}$ is a (locally Kan) simplicial category. In this case, the slice and coslice categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ inherit simplicial enrichments (Construction 5.5.2.1), and there are natural comparison maps

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}. \]

Beware that these maps are generally not isomorphisms at the level of simplicial sets (Warning 5.5.2.19). However, we will show that, under some mild assumptions, they are equivalences of $\infty $-categories (Theorem 5.5.2.21).

Construction 5.5.2.1 (Slices of Simplicial-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We define a simpicial category $\operatorname{\mathcal{C}}_{/X}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{/X}$ are pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f: C \rightarrow X$ is a vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet }$.

  • Let $(C,f)$ and $(D,g)$ be objects of $\operatorname{\mathcal{C}}_{/X}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial set given by the fiber product

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } } \{ f \} , \]

    which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$. More precisely, we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$ consisting of those $n$-simplices $\sigma $ for which the composite map is equal to the constant map $\Delta ^{n} \twoheadrightarrow \{ f \} $.

  • Let $(C,f)$, $(D,g)$, and $(E,h)$ be objects of $\operatorname{\mathcal{C}}_{/X}$. Then the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (D,g) , (E,h) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f) , (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f) , (E,h) )_{\bullet } \]

    for the simplicial category $\operatorname{\mathcal{C}}_{/X}$ is given by the restriction of the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, E)_{\bullet } \]

    for the simplicial category $\operatorname{\mathcal{C}}$.

Exercise 5.5.2.2. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing an object $X$. Show that the simplicial categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ of Construction 5.5.2.1 are well-defined (that is, the composition law of Construction 5.5.2.1 is unital and associative).

Variant 5.5.2.3 (Coslices of Simplicial-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We define a simpicial category $\operatorname{\mathcal{C}}_{X/}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{X/}$ are pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f$ is a vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)_{\bullet }$.

  • Let $(C,f)$ and $(D,g)$ be objects of $\operatorname{\mathcal{C}}_{X/}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial set given by the fiber product

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,D)_{\bullet } } \{ g \} , \]

    which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$.

  • Let $(C,f)$, $(D,g)$, and $(E,h)$ be objects of $\operatorname{\mathcal{C}}_{X/}$. Then the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (D,g) , (E,h) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f) , (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f) , (E,h) )_{\bullet } \]

    for the simplicial category $\operatorname{\mathcal{C}}_{X/}$ is given by the restriction of the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, E)_{\bullet } \]

    for the simplicial category $\operatorname{\mathcal{C}}$.

Remark 5.5.2.4. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing an object $X$, which we also regard as an object of the opposite simplicial category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then there is a canonical isomorphism of simplicial categories $(\operatorname{\mathcal{C}})_{X/})^{\operatorname{op}} \simeq (\operatorname{\mathcal{C}}^{\operatorname{op}})_{/X}$.

Remark 5.5.2.5. For every simplicial category $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}^{\circ }$ denote the underlying ordinary category of $\operatorname{\mathcal{C}}$ (Example 2.4.1.4). If $X$ is an object of $\operatorname{\mathcal{C}}$, then we have canonical isomorphisms

\[ (\operatorname{\mathcal{C}}_{/X})^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })_{/X} \quad \quad ( \operatorname{\mathcal{C}}_{X/})^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })_{X/}, \]

where the left hand sides are defined using the slice and coslice operations on simplicial categories (Construction 5.5.2.1 and Variant 5.5.2.3) and the right hand sides are defined using the slice and coslice operations on ordinary categories (Construction 4.3.1.1 and Variant 4.3.1.4). In other words, the slice and coslice constructions are compatible with the forgetful functor from simplicial categories to ordinary categories. We can summarize the situation more informally as follows: if $\operatorname{\mathcal{C}}$ is a category and $X$ is an object of $\operatorname{\mathcal{C}}$, then any simplicial enrichment of $\operatorname{\mathcal{C}}$ determines a simplicial enrichment on the slice and coslice categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$.

Remark 5.5.2.6. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\underline{\operatorname{\mathcal{C}}}$ denote the associated constant simplicial category (Example 2.4.2.3). Then the simplicial categories $\underline{\operatorname{\mathcal{C}}}_{/X}$ and $\underline{\operatorname{\mathcal{C}}}_{X/}$ of Construction 5.5.2.1 and Variant 5.5.2.3 are also constant, associated to the ordinary categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{X/}$ of Construction 4.3.1.1 and Variant 4.3.1.4, respectively. In other words, the slice and coslice constructions are compatible with the operation of regarding an ordinary category as a (constant) simplicial category.

Warning 5.5.2.7. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$ (Construction 2.4.6.1). For every object $X \in \operatorname{\mathcal{C}}$, there is a natural comparison map $\mathrm{h} \mathit{(\operatorname{\mathcal{C}}_{/X})} \rightarrow (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$, which carries an object $(C,f)$ of the slice simplicial category $\operatorname{\mathcal{C}}_{/X}$ to the object $(C, [f] )$ of the slice category $(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})_{/X}$, where $[f] \in \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } )$ denotes the homotopy class of $f$. Beware that this functor is generally not an equivalence of categories (see Warning 3.2.1.9).

We now characterize the simplicial category $\operatorname{\mathcal{C}}_{/X}$ of Construction 5.5.2.1 by a universal property.

Notation 5.5.2.8. Let $\operatorname{\mathcal{C}}$ be a simplicial category. We define a simplicial category $\operatorname{\mathcal{C}}^{\triangleleft }$ as follows:

  • The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}}^{\triangleleft })$ is given by the (disjoint) union $\operatorname{Ob}(\operatorname{\mathcal{C}}) \cup \{ Y_0 \} $, where $X_0$ is an auxiliary symbol.

  • The simplicial morphism sets in $\operatorname{\mathcal{C}}^{\triangleleft }$ are given by

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleleft }}( C,D )_{\bullet } = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } & \text{ if } C,D \in \operatorname{Ob}(\operatorname{\mathcal{C}}) \\ \Delta ^{0} & \text{ if } C = X_0 \\ \emptyset & \text{ otherwise. } \end{cases} \]
  • For objects $C,D,E \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\triangleright } )$, the composition law

    \[ \circ : \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleleft } }( D, E)_{\bullet } \times \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleleft } }( C, D)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleleft } }( C, E)_{\bullet } \]

    is given by the composition law on $\operatorname{\mathcal{C}}$ in the case where $C,D,E \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and is otherwise uniquely determined (since either the left hand side is empty or the right hand side is $\Delta ^0$).

More informally, the simplicial category $\operatorname{\mathcal{C}}^{\triangleleft }$ is obtained from $\operatorname{\mathcal{C}}$ by adjoining a (new) initial object $X_0$. We will refer to $\operatorname{\mathcal{C}}^{\triangleleft }$ as the left cone on $\operatorname{\mathcal{C}}$, and to the object $X_0 \in \operatorname{\mathcal{C}}^{\triangleleft }$ as the cone point.

Variant 5.5.2.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category. We define a simplicial category $\operatorname{\mathcal{C}}^{\triangleright }$ as follows:

  • The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}}^{\triangleright })$ is given by the (disjoint) union $\operatorname{Ob}(\operatorname{\mathcal{C}}) \cup \{ Y_0 \} $, where $Y_0$ is an auxiliary symbol.

  • The simplicial morphism sets in $\operatorname{\mathcal{C}}^{\triangleright }$ are given by

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleright }}( C,D )_{\bullet } = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } & \text{ if } C,D \in \operatorname{Ob}(\operatorname{\mathcal{C}}) \\ \Delta ^{0} & \text{ if } D = Y_0 \\ \emptyset & \text{ otherwise. } \end{cases} \]
  • For objects $C,D,E \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\triangleright } )$, the composition law

    \[ \circ : \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( D, E)_{\bullet } \times \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( C, D)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( C, E)_{\bullet } \]

    is given by the composition law on $\operatorname{\mathcal{C}}$ in the case where $C,D,E \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and is otherwise uniquely determined.

More informally, the simplicial category $\operatorname{\mathcal{C}}^{\triangleright }$ is obtained from $\operatorname{\mathcal{C}}$ by adjoining a (new) final object $Y_0$. We will refer to $\operatorname{\mathcal{C}}^{\triangleright }$ as the right cone on $\operatorname{\mathcal{C}}$, and to the object $Y_0 \in \operatorname{\mathcal{C}}^{\triangleright }$ as the cone point.

Remark 5.5.2.10. Let $\operatorname{\mathcal{C}}$ be a simplicial category. Then there is a canonical isomorphism of simpicial categories $(\operatorname{\mathcal{C}}^{\triangleleft })^{\operatorname{op}} \simeq (\operatorname{\mathcal{C}}^{\operatorname{op}})^{\triangleright }$.

Remark 5.5.2.11. For every simplicial category $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}^{\circ }$ denote the underlying ordinary category of $\operatorname{\mathcal{C}}$ (Example 2.4.1.4). Then we have canonical isomorphisms

\[ ( \operatorname{\mathcal{C}}^{\triangleleft } )^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })^{\triangleleft } \quad \quad ( \operatorname{\mathcal{C}}^{\triangleright } )^{\circ } \simeq (\operatorname{\mathcal{C}}^{\circ })^{\triangleright }, \]

where the left hand sides are defined using Notation 5.5.2.8 and Variant 5.5.2.9, and the right hand sides are defined in Example 4.3.2.5. In other words, the formation of cones is compatible with the forgetful functor from simplicial categories to ordinary categories.

Remark 5.5.2.12. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\underline{\operatorname{\mathcal{C}}}$ denote the associated constant simplicial category (Example 2.4.2.3). Then the simplicial categories $\underline{\operatorname{\mathcal{C}}}^{\triangleleft }$ and $\underline{\operatorname{\mathcal{C}}}^{\triangleright }$ of Notation 5.5.2.8 and Variant 5.5.2.9 are also constant, associated to the ordinary categories $\operatorname{\mathcal{C}}^{\triangleleft }$ and $\operatorname{\mathcal{C}}^{\triangleright }$ of Example 4.3.2.5. In other words, the formation of cones is compatible with the operation of regarding an ordinary category as a (constant) simplicial category.

Remark 5.5.2.13. For every simplicial category $\operatorname{\mathcal{C}}$, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. Then there are canonical isomorphism of categories

\[ \mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\triangleleft } )} \simeq (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\triangleleft } \quad \quad \mathrm{h} \mathit{( \operatorname{\mathcal{C}}^{\triangleright } )} \simeq (\mathrm{h} \mathit{\operatorname{\mathcal{C}}})^{\triangleright }. \]

In other words, the formation of cones is compatible with the passage from a simplicial category to its homotopy category.

Remark 5.5.2.14. For every simplicial category $\operatorname{\mathcal{C}}$, let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$. Then there are canonical isomorphisms of simplicial sets

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\triangleleft } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\triangleleft } \quad \quad \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\triangleright } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\triangleright }, \]

which are uniquely determined by the requirements that they restrict to the identity on $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and preserve the cone points. In other words, the formation of cones is compatible with the homotopy coherent nerve.

Construction 5.5.2.15. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. We define a simplicial functor $V: ( \operatorname{\mathcal{C}}_{/Y })^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ as follows:

  • The functor $V$ carries each object $(C,f) \in \operatorname{\mathcal{C}}_{/Y}$ to the object $C \in \operatorname{\mathcal{C}}$, and carries the cone point $Y_0 \in (\operatorname{\mathcal{C}}_{/Y})^{\triangleright }$ to the object $Y \in \operatorname{\mathcal{C}}$.

  • If $(C,f)$ and $(D,g)$ are objects of $\operatorname{\mathcal{C}}_{/Y}$, then the induced map of simplicial sets

    \[ \operatorname{Hom}_{(\operatorname{\mathcal{C}}_{/Y})^{\triangleright } }( (C,f), (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( V(C,f), V(D,g) )_{\bullet } \]

    is equal to the inclusion map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Y} }( (C,f), (D,g) )_{\bullet } \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, D)_{\bullet }$.

  • If $(C,f)$ is an object of $\operatorname{\mathcal{C}}_{/Y}$, then the induced map

    \[ \Delta ^{0} = \operatorname{Hom}_{(\operatorname{\mathcal{C}}_{/Y})^{\triangleright } }( (C,f), Y_0 )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( V(C,f), V(Y_0) )_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)_{\bullet } \]

    is equal to the vertex $f$.

We will refer to $V$ as the right cone contraction functor. Similarly, to every object $X \in \operatorname{\mathcal{C}}$ we can associate a simplicial functor $V': ( \operatorname{\mathcal{C}}_{X/} )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point of $( \operatorname{\mathcal{C}}_{X/})^{\triangleleft }$ to the object $X$, which we will refer to as the left cone contraction functor.

Proposition 5.5.2.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial categories. Let $X_0$ and $Y_0$ denote the cone points of $\operatorname{\mathcal{D}}^{\triangleleft }$ and $\operatorname{\mathcal{D}}^{\triangleright }$, respectively. Then:

  • For every object $Y \in \operatorname{\mathcal{C}}$, postcomposition with the right cone contraction functor $V: (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ of Construction 5.5.2.15 induces a bijection

    \[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$} \ar [d]^{\sim } \\ \{ \textnormal{Simplicial functors $G: \operatorname{\mathcal{D}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ with $G(Y_0)=Y$} \} } \]
  • For every object $X \in \operatorname{\mathcal{C}}$, postcomposition with the left cone contraction functor $V': (\operatorname{\mathcal{C}}_{X/})^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of Construction 5.5.2.15 induces a bijection

    \[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{X/}$} \ar [d]^{\sim } \\ \{ \textnormal{Simplicial functors $G: \operatorname{\mathcal{D}}^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $G(X_0)=X$} \} } \]

Proof. We will prove the first assertion; the proof of the second is similar. Fix a simplicial functor $G: \operatorname{\mathcal{D}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ and set $Y = G(Y_0)$. We wish to show that there is a unique simplicial functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$ for which the composition

\[ \operatorname{\mathcal{D}}^{\triangleright } \xrightarrow { F^{\triangleright } } ( \operatorname{\mathcal{C}}_{/Y} )^{\triangleright } \xrightarrow {V} \operatorname{\mathcal{C}} \]

is equal to $G$. For each object $D \in \operatorname{\mathcal{D}}$, the simplicial functor $G$ induces a morphism of simplicial sets

\[ \Delta ^{0} = \operatorname{Hom}_{ \operatorname{\mathcal{D}}^{\triangleright } }( D, Y_0)_{\bullet } \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), G(Y_0) )_{\bullet }, \]

which we can identify with a vertex $f$ of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), Y)_{\bullet }$. The simplicial functor $F$ is then given on objects by the formula $F(D) = (G(D), f)$, and is determined on morphisms by the requirement that the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, E)_{\bullet } \xrightarrow {F} \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Y} }( F(D), F(E) )_{\bullet } \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), G(E) )_{\bullet } \]

coincides with the map of simplicial sets determined by the simplicial functor $G$. $\square$

Construction 5.5.2.17. Let $\operatorname{\mathcal{C}}$ be a simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $V: (\operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the right cone contraction functor of Construction 5.5.2.15. Passing to homotopy coherent nerves (and invoking Remark 5.5.2.14, we obtain a map

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \]

carrying the cone point to the vertex $X$, which we can further identify with a morphism of simplicial sets $c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X}$. We will refer to $c$ as the slice comparison morphism. Similarly, the left cone contraction functor $V': : (\operatorname{\mathcal{C}}_{X/} )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ induces a morphism of simplicial sets $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/}$, which we will refer to as the coslice comparison morphism.

Example 5.5.2.18. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we identify with the associated constant simplicial category $\underline{\operatorname{\mathcal{C}}}$ of Example 2.4.2.3. For every object $X \in \operatorname{\mathcal{C}}$, the slice and coslice comparison morphisms

\[ c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

of Construction 5.5.2.17 can be identified with the isomorphisms $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{/X}) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/X}$ and $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{X'}) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}$ described in Example 4.3.5.8.

Warning 5.5.2.19. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing an object $X$. Then the slice and coslice comparison morphisms

\[ c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

of Construction 5.5.2.17 are always bijective at the level of vertices (on the left side, vertices of either of the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} )$ and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X}$ can be identified with pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f$ is a morphism from $C$ to $X$). Beware that $c$ and $c'$ are generally not bijective on simplices of dimension $\geq 1$. Unwinding the definitions, we see that edges of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{/X})$ can be identified with diagrams

\[ \xymatrix@R =50pt@C=50pt{ C \ar [dr]^{f} \ar [rr]^{h} & & D \ar [dl]_{g} \\ & X & } \]

in the category $\operatorname{\mathcal{C}}$ which are strictly commutative, while edges of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{/X}$ can be identified with diagrams which commute up to a specified homotopy $\mu : g \circ h \rightarrow f$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet }$.

Exercise 5.5.2.20. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Show that the slice and coslice comparison morphisms

\[ c: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{/X} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{/X} \quad \quad c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/} \]

are monomorphisms of simplicial sets.

We are now ready to state the main result of this section. For the sake of brevity, we will formulate the statement only for coslice categories (one can deduce a dual statement for slice categories by replacing $\operatorname{\mathcal{C}}$ by its opposite).

Theorem 5.5.2.21. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$ with the following property:

$(\ast )$

For every morphism $f: X \rightarrow Y$ and every object $Z \in \operatorname{\mathcal{C}}$, the morphism of simplicial sets $ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ is a Kan fibration.

Then the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ of Construction 5.5.2.17 is an equivalence of $\infty $-categories.

For many applications, hypothesis $(\ast )$ of Theorem 5.5.2.21 is too strong: it is often satisfied only for morphisms $f: X \rightarrow Y$ which are sufficiently well-behaved. We therefore consider a somewhat more general situation:

Proposition 5.5.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ be a full simplicial subcategory of $\operatorname{\mathcal{C}}_{X/}$ with the following property:

$(\ast )$

For every pair of objects $(Y,f)$ and $(Z,g)$ of the simplicial category $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}_{X/}$, the morphism of simplicial sets $ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ is a Kan fibration.

Then the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $\infty $-category, and the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ of Construction 5.5.2.17 restricts to a fully faithful functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$.

Proof of Theorem 5.5.2.21 from Proposition 5.5.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which satisfies hypothesis $(\ast )$ of Theorem 5.5.2.21. Applying Proposition 5.5.2.22 in the case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{X/}$, we conclude that the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ is fully faithful. Since $c'$ is bijective on vertices, it is also essentially surjective, and is therefore an equivalence of $\infty $-categories by virtue of Theorem 4.6.2.17. $\square$

Proof of Proposition 5.5.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing an object $X$, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}_{X/}$ be a full simplicial subcategory satisfying hypothesis $(\ast )$ of Proposition 5.5.2.22. For every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet }$ is the fiber of the Kan fibration

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \]

over the vertex $g$, and is therefore a Kan complex (Remark 3.1.1.9). Applying Theorem 2.4.5.1, we conclude that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $\infty $-category. We wish to show that, for every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$ as above, the coslice comparison morphism $c'$ induces a homotopy equivalence of morphism spaces

\[ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/ } }( (Y,f), (Z,g) ). \]

By virtue of Proposition 4.6.6.9, this is equivalent to the requirement that $c'$ induces a homotopy equivalence $\rho : \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})}( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} }( (Y,f), (Z,g) )$ of left-pinched morphism spaces.

Construction 4.6.7.3 supplies comparison maps

\[ \overline{\theta }: \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }^{\mathrm{L}}( (Y,f), (Z,g) ) \]

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

which are homotopy equivalences of Kan complexes by virtue of Theorem 4.6.7.5. Let us regard $f: X \rightarrow Y$ as an edge of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, and let $Q$ denote the fiber $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z\} $. Since the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ is right anodyne, the restriction map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/}$ is a trivial Kan fibration (Proposition 4.3.6.9), and therefore restricts to a trivial Kan fibration

\[ \pi : Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (Y, Z). \]

In particular, $Q$ is a Kan complex and $\pi $ is a homotopy equivalence. Let $\pi '$ denote the restriction map

\[ Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (X, Z). \]

Note that $\pi '$ is a pullback of the left fibration $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/}$ (Corollary 4.3.6.8), and is therefore also a left fibration (Remark 4.2.1.9). Since the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z)$ is a Kan complex (Proposition 4.6.6.4), the morphism $\pi '$ is a Kan fibration (Corollary 4.4.3.7). We will construct an auxiliary map of Kan complexes $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ with the following properties:

$(a)$

The composition $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \lambda } Q \xrightarrow {\pi } \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ is equal to $\theta _{Y,Z}$.

$(b)$

The cubical diagram of Kan complexes

5.28
\begin{equation} \begin{gathered}\label{equation:diagram-Kan-slice-compatibility} \xymatrix@C =-10pt@R=40pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \ar [rr] \ar [dr]^{\rho \circ \overline{\theta } } \ar [dd] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [dr]^{\lambda } \ar [dd]^(.6){\circ f} & \\ & \operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}}^{\mathrm{L}}( (Y,f), (Z,g) ) \ar [rr] \ar [dd] & & Q \ar [dd]^{\pi '} \\ \{ g\} \ar@ {=}[dr] \ar [rr] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [dr]_-{\theta _{X,Z}} & \\ & \{ g\} \ar [rr] & & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) } \end{gathered} \end{equation}

is commutative.

Suppose that such a map has been constructed. It follows from $(a)$ that $\lambda $ is a homotopy equivalence. Moreover, the front and back faces of the diagram (5.28) are pullback squares of simplicial sets. Since the vertical maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \quad \quad \pi ': Q \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) \]

are Kan fibrations, these faces are also homotopy pullback squares (Example 3.4.1.5). Since $\lambda $, $\theta _{X,Z}$, and the identity map $\operatorname{id}: \{ g\} \rightarrow \{ g\} $ are homotopy equivalences of Kan complexes, it follows from Corollary 3.4.1.10 that the map $\rho \circ \overline{\theta }$ is also a homotopy equivalence of Kan complexes. Since $\overline{\theta }$ is a homotopy equivalence, we conclude that $\rho $ is a homotopy equivalence as desired.

We now complete the proof by constructing the morphism $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$, so that $\theta _{Y,Z}( \sigma )$ is an $n$-simplex of the left-pinched morphism space $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ which we can identify with a simplicial functor $F_{\sigma }: \operatorname{Path}[ \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ such that $F_{\sigma }(y) = Y$ and $F_{\sigma } |_{ \operatorname{Path}[n]_{\bullet } }$ is the constant functor taking the value $Z$ (see Construction 4.6.7.3). We extend $F_{\sigma }$ to a simplicial functor $F_{\sigma }^{+}: \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as follows:

  • The functor $F_{\sigma }^{+}$ carries $x$ to the object $X \in \operatorname{\mathcal{C}}$.

  • For every element $i \in \{ y\} \star [n]$, the induced map of simplicial sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet } \]

    is given by the composition

    \begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } & \xrightarrow {u} & \operatorname{Hom}_{ \operatorname{Path}[ \{ y\} \star [n] ] }( y, i)_{\bullet } \\ & \xrightarrow { F_{\sigma } } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, F_{\sigma }(i) )_{\bullet } \\ & \xrightarrow { \circ f} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet }, \end{eqnarray*}

    where $u$ is induced by the map of partially ordered sets $\{ x\} \star \{ y\} \star [n] \rightarrow \{ y\} \star [n]$ which is the identity on $\{ y\} \star [n]$ and carries $x$ to $y$.

Then $F_{\sigma }^{+}$ determines a morphism of simplicial sets $\{ x \} \star \{ y\} \star \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carrying $\{ x\} \star \{ y\} $ to the edge $f$ and $\Delta ^{n}$ to the vertex $Z$, which we can identify with an $n$-simplex $\lambda (\sigma )$ of the Kan complex $Q$. The construction $\sigma \mapsto \lambda (\sigma )$ depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore induces a morphism of simplicial sets $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ which is easily verified to satisfy conditions $(a)$ and $(b)$. $\square$