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2.4.6 The Homotopy Category of a Simplicial Category

For every simplicial set $S_{\bullet }$, we let $\pi _{0}( S_{\bullet } )$ denote the set of connected components of $S_{\bullet }$ (Definition 1.1.6.8). Recall that the functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ preserves finite products (Corollary 1.1.6.26). Applying Remark 2.1.7.4, we obtain the following:

Construction 2.4.6.1 (The Homotopy Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define an ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, we have

    \[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ). \]
  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition map

    \[ \circ : \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Z ) \]

    is given by the composition

    \begin{eqnarray*} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) & = & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } ) \times \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \\ & \xleftarrow {\sim } & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } ) \\ & \rightarrow & \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } ) \\ & = & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z). \end{eqnarray*}

We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of $\operatorname{\mathcal{C}}$.

Remark 2.4.6.2 (The Component Functor). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category (Construction 2.4.6.1). For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, Construction 1.1.6.18 supplies a map of simplicial sets

\[ u_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \underline{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) }_{\bullet }. \]

Here $\underline{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) }_{\bullet }$ denotes the constant simplicial set associated to the set $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$, and $u_{X,Y}$ carries each $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to the connected component which contains it. Allowing $X$ and $Y$ to vary, we obtain a simplicial functor $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ which is the identity on objects; we will refer to $u$ as the component functor.

Remark 2.4.6.3. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category with underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$. Then the simplicial functor $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ induces a functor of ordinary categories $u_0: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which can be described as follows:

  • On objects, the functor $u_0$ is the identity map from $\operatorname{Ob}( \operatorname{\mathcal{C}}) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ to itself.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}) = \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is a surjection, which we will denote by $f \mapsto [f]$.

  • Given a pair of morphisms $f,g; X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ having the same source and target, we have $[f] = [g]$ if and only if $f$ and $g$ belong to the same connected component of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

Remark 2.4.6.4. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category with underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$, and let $f,g: X \rightarrow Y$ be a pair of morphisms of $\operatorname{\mathcal{C}}$ having the same source and target. Using Remark 1.1.6.23, we see that the following conditions are equivalent:

$(a)$

The morphisms $f$ and $g$ represent the same morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$: that is, we have $[f] = [g]$.

$(b)$

There exists a sequence of morphisms $f = f_0, f_1, f_2, \ldots , f_ n = g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ such that, for $1 \leq i \leq n$, either there exists a homotopy from $f_{i-1}$ to $f_{i}$ or a homotopy from $f_{i}$ to $f_{i-1}$ (in the sense of Definition 2.4.1.6).

If $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan, then we can replace $(b)$ by the following simpler condition:

$(c)$

There exists a homotopy from $f$ to $g$ (in the sense of Definition 2.4.1.6).

See Remark 2.4.1.9.

Example 2.4.6.5. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category (Definition 2.2.0.1) and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.7). Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}_{\bullet }}$ of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Construction 2.4.6.1) can be identified with the coarse homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of $\operatorname{\mathcal{C}}$ (in the sense of Construction 2.2.8.2).

Example 2.4.6.6 (The Homotopy Category of $\operatorname{Top}$). Let $\operatorname{Top}$ denote the category of topological spaces and continuous functions, endowed with the simplicial enrichment $\operatorname{Top}_{\bullet }$ described in Example 2.4.1.5. Then the homotopy category $\mathrm{h} \mathit{\operatorname{Top}}$ is the homotopy category of all topological spaces: the objects of $\mathrm{h} \mathit{\operatorname{Top}}$ are topological spaces, and the morphisms of $\mathrm{h} \mathit{\operatorname{Top}}$ are homotopy classes of continuous maps.

The homotopy category of a simplicial category can be characterized by a universal mapping property:

Proposition 2.4.6.7. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ be the simplicial functor described in Remark 2.4.6.2. Then, for any category $\operatorname{\mathcal{D}}$, composition with $u$ induces a bijection

\[ \{ \textnormal{Ordinary Functors $f: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Simplicial Functors $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{\operatorname{\mathcal{D}}}_{\bullet }$} \} . \]

Proof. Use Proposition 1.1.6.19. $\square$

Corollary 2.4.6.8. The fully faithful embedding

\[ \operatorname{Cat}\hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{D}}\mapsto \underline{\operatorname{\mathcal{D}}}_{\bullet } \]

of Example 2.4.2.3 admits a left adjoint, given on objects by the formation of homotopy categories $\operatorname{\mathcal{C}}_{\bullet } \mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

We have now introduced two different notions of homotopy category:

  • The homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, given by Construction 2.4.6.1.

  • The homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ of a simplicial set $S_{\bullet }$, defined in Definition 1.2.5.1 (and described more explicitly in §1.3.5 when $S_{\bullet }$ is an $\infty $-category).

These constructions are related. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Applying the homotopy coherent nerve to the component functor $u$ of Remark 2.4.6.2, we obtain a map of simplicial sets

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{N}_{\bullet }^{\operatorname{hc}}(u)} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } ) \simeq \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), \]

which we can identify with a functor of ordinary categories $U: \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Proposition 2.4.6.9. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category. Then the construction above induces an isomorphism of categories $U: \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

To prove Proposition 2.4.6.9, we need to analyze the $2$-simplices of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$. Recall that the vertices and edges of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ can be identified with objects and morphisms in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ (Example 2.4.3.9). In particular, a map of simplicial sets $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with a (possibly noncommutative) diagram

\[ \xymatrix { & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]_{ f_{20} }& & X_2 } \]

in the category $\operatorname{\mathcal{C}}$. We will need the following:

Proposition 2.4.6.10. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we identify with a diagram

\[ \xymatrix { & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]_{ f_{20} } & & X_2 } \]

as above. Then the construction of Example 2.4.3.10 induces a bijection

\[ \xymatrix { \{ \textnormal{Maps $\sigma : \Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ with $\sigma |_{ \operatorname{\partial }\Delta ^2} = \sigma _0$} \} \ar [d]^{\sim } \\ \{ \textnormal{Homotopies from $f_{21} \circ f_{10}$ to $f_{20}$} \} . } \]

It is not difficult to deduce Proposition 2.4.6.10 directly from the definition of the homotopy coherent nerve. We will instead deduce it from a more general result (Corollary 2.4.6.13), which supplies an analogous description of the $n$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ for all $n > 0$. First, let us note some consequences of Proposition 2.4.6.10.

Example 2.4.6.11. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category, so that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). Suppose we are given a pair of morphisms $f,g: X \rightarrow Y$ in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ having the same source and target. Let $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be the map corresponding to the (possibly noncommutative) diagram

\[ \xymatrix { & Y \ar [dr]^{ \operatorname{id}_ Y } & \\ X \ar [ur]^{f} \ar [rr]^{g} & & Y. } \]

Applying Proposition 2.4.6.10 we obtain a bijection from the set of homotopies from $f$ to $g$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (in the sense of Definition 1.3.3.1) to the set of homotopies from $f$ to $g$ in the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Definition 2.4.1.6). In particular, we see that $f$ and $g$ are homotopic in $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if they are homotopic in $\operatorname{\mathcal{C}}_{\bullet }$.

Proof of Proposition 2.4.6.9. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category; we wish to show that the comparison map $U: \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is an isomorphism of categories. By construction, $U$ is bijective on objects. It will therefore suffice to show that for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the induced map

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

is a bijection. This is precisely the content of Example 2.4.6.11. $\square$

We will deduce Proposition 2.4.6.10 from the following variant of Proposition 2.4.5.8:

Proposition 2.4.6.12. Let $n$ be a positive integer and let $F: \operatorname{Path}[ \partial \Delta ^ n ]_{\bullet } \rightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet }$ be the simplicial functor induced by the boundary inclusion $\partial \Delta ^ n \hookrightarrow \Delta ^ n$. Then:

$(a)$

The functor $F$ is bijective on objects; in particular, we can identify objects of $\operatorname{Path}[ \operatorname{\partial \Delta }^ n ]_{\bullet }$ with elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.

$(b)$

For $(j,k) \neq (0,n)$, the functor $F$ induces an isomorphism of simplicial sets

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }(j, k)_{\bullet } \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(j, k)_{\bullet }. \]
$(c)$

The functor $F$ induces a monomorphism of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }(0,n)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n)_{\bullet }$, whose image can be identified with the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n)_{\bullet }$ introduced in Notation 2.4.5.5.

Proof. Assertion $(a)$ is immediate from Theorem 2.4.4.10. To prove $(b)$ and $(c)$, fix an integer $m \geq 0$ and let us identify $\operatorname{Path}[ \Delta ^ n ]_{m}$ with the path category $\operatorname{Path}[G]$ of the directed graph $G$ appearing in the proof of Proposition 2.4.5.8. Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \partial \Delta ^ n ]_{m}$ can be identified with the path category of the directed subgraph $G' \subseteq G$ having the same vertices, where an edge $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ of $G$ belongs to $G'$ unless $I_0 = [n]$. In particular, we see that for $(j,k) \neq (0,n)$, every edge of $G$ with source $j$ and target $k$ is contained in $G'$. It follows that the simplicial functor $F$ induces a bijection

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( j, k )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(j, k)_{m} \]

for $(j, k) \neq (0,n)$, which proves $(b)$. Moreover, the map

\[ \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( 0, n )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{m} \]

is a monomorphism, whose image consists of those chains

\[ \overrightarrow {I} = (I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m}) \]

where either $I_0 \neq [n]$ or $I_{m} \neq \{ 0, n\} $. Under the identification of $\operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{\bullet }$ with the cube $\operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{N}_{\bullet }( P( \{ 1, \ldots , n-1\} ))$ described in Remark 2.4.5.4, this is exactly the set of $m$-simplices which belong to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$. $\square$

Combining Propositions 2.4.6.12 and 2.4.5.9, we obtain the following:

Corollary 2.4.6.13. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $n > 0$, and let $\sigma _0: \partial \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we identify with a simplicial functor $F: \operatorname{Path}[ \partial \Delta ^ n ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ inducing a map of simplicial sets

\[ \lambda _0: \operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( 0, n)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }. \]

Then we have a canonical bijection

\[ \xymatrix { \{ \textnormal{Maps $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma |_{ \partial \Delta ^ n }$} \} \ar [d] \\ \{ \textnormal{Maps $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda |_{ \operatorname{\partial \raise {0.1ex}{\square }}^{n-1}}$ } \} }. \]

Example 2.4.6.14 ($1$-Simplices of the Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. By definition, giving a map of simplicial sets $\sigma _0: \operatorname{\partial }\Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to giving a pair of objects $X_0 = \sigma _0(0)$ and $X_1=\sigma _0(1)$ of the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$. Applying Corollary 2.4.6.13, we deduce that extending $\sigma _0$ to a $1$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to supplying a morphism $f: X_0 \rightarrow X_1$ in the category $\operatorname{\mathcal{C}}$ (see Example 2.4.3.9).

Proof of Proposition 2.4.6.10. Apply Corollary 2.4.6.13 in the case $n = 2$. $\square$

Example 2.4.6.15 ($3$-Simplices of the Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Using Proposition 2.4.6.10, we see that a map of simplicial sets $\sigma _0: \operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with the following data:

  • A collection of four objects $\{ X_ i \in \operatorname{\mathcal{C}}\} _{0 \leq i \leq 3}$.

  • A collection of six morphisms $\{ f_{ji} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_ i, X_ j) \} _{0 \leq i < j \leq 3}$.

  • A collection of four $1$-simplices $\{ h_{kji} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_ i, X_ k)_{1} \} _{ 0 \leq i < j < k \leq 3}$, where each $h_{kji}$ is a homotopy from $f_{kj} \circ f_{ji}$ to $f_{ki}$.

From this data, we can assemble a map of simplicial sets $\lambda _0: \operatorname{\partial \raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_3)_{\bullet }$, which is represented by the diagram

\[ \xymatrix@C =50pt@R=50pt{ f_{32} \circ f_{21} \circ f_{10} \ar [r]^-{ h_{321} \circ \operatorname{id}_{f_{10}} } \ar [d]^-{\operatorname{id}_{f_{32}} \circ h_{210} } & f_{31} \circ f_{10} \ar [d]^-{h_{310}} \\ f_{32} \circ f_{20} \ar [r]^-{ h_{320} } & f_{30}. } \]

Corollary 2.4.6.13 then asserts that extending $\sigma _0$ to a $3$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to extending $\lambda _0$ to a map of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^2 \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_3)_{\bullet }$. Stated more informally, the map $\sigma _0$ supplies two potentially different paths from the composition $f_{32} \circ f_{21} \circ f_{10}$ to $f_{30}$ in the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_3)_{\bullet }$. To extend $\sigma _0$ to a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, one must supply additional data which “witnesses” that these paths are homotopic.

We close this section with a refinement of Construction 2.4.6.1:

Construction 2.4.6.16 (The Homotopy $2$-Category of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ are the objects of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the category $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is the homotopy category of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition map

    \[ \circ : \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X, Z ) \]

    is given by the composition

    \begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}( X,Y) & = & (\mathrm{h} \mathit{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }) \times (\mathrm{h} \mathit{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet }) \\ & \xleftarrow {\sim } & \mathrm{h} \mathit{( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )} \\ & \rightarrow & \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }} \\ & = & \operatorname{Hom}_{\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}}(X,Z), \end{eqnarray*}

    where the isomorphism is supplied by Corollary 1.4.3.6.

We will refer to $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy $2$-category of $\operatorname{\mathcal{C}}$.

Remark 2.4.6.17. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy $2$-category of $\operatorname{\mathcal{C}}$. Then the underlying category $\operatorname{\mathcal{C}}_0$ of $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Example 2.4.1.4) coincides with the underlying category of the strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Remark 2.2.0.3).

Remark 2.4.6.18. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Then the homotopy category of $\operatorname{\mathcal{C}}_{\bullet }$ can be identified with the coarse homotopy category of the homotopy $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ of Construction 2.4.6.16, in the sense of Construction 2.2.8.2. That is, we have a canonical isomorphism $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \simeq \mathrm{h} \mathit{( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )}$.

Remark 2.4.6.19. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ be the homotopy $2$-category of $\operatorname{\mathcal{C}}$, and let $(\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}})_{\bullet }$ denote the simplicial category obtained from $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}}$ by applying the construction of Example 2.4.2.7. Then there is a simplicial functor $U: \operatorname{\mathcal{C}}_{\bullet } \rightarrow ( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )_{\bullet }$, given on objects by the identity map and on morphism spaces by the tautological maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } }). \]

Passing to the homotopy coherent nerve (and invoking Example 2.4.3.11), we obtain a map of simplicial sets $V: \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$, which restricts to the identity on the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (which we can regard as a simplicial subset of both $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$).

Remark 2.4.6.20. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. The comparison map $V: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$ of Remark 2.4.6.19 is always bijective at the level of vertices (which can be identified with the objects of the category $\operatorname{\mathcal{C}}_0$ underlying $\operatorname{\mathcal{C}}_{\bullet }$) and edges (which can be identified with morphisms of $\operatorname{\mathcal{C}}_0$). Suppose that, for every pair of objects $C,D \in \operatorname{\mathcal{C}}_0$, the simplicial set $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$ is an $\infty $-category. In this case, the map $V$ is also surjective (but not necessarily injective) at the level of $2$-simplices. By virtue of Example 2.3.1.15, we can identify $2$-simplices $\overline{\sigma }$ of $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$ with diagrams

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{[\mu ]} & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z, } \]

where $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ are morphisms in $\operatorname{\mathcal{C}}_0$, and $[\mu ]: g \circ f \rightarrow h$ is a morphism in the homotopy category of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. To lift $\overline{\sigma }$ to a $2$-simplex $\sigma $ of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, one must choose a morphism $\mu : g \circ f \rightarrow h$ in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ which represents the homotopy class $[\mu ]$ (see Example 2.4.3.10). Such a representative always exists, but is not necessarily unique.

Using the universal property of the homotopy category, we immediately obtain the following variant of Proposition 2.4.6.7:

Proposition 2.4.6.21. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $U: \operatorname{\mathcal{C}}_{\bullet } \rightarrow ( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )_{\bullet }$ be the simplicial functor described in Remark 2.4.6.19. Then, for any strict $2$-category $\operatorname{\mathcal{D}}$, composition with $U$ induces a bijection category $\operatorname{\mathcal{D}}$, composition with $U$ induces a bijection

\[ \{ \textnormal{Strict functors $f: \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Simplicial Functors $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$} \} ; \]

here $\operatorname{\mathcal{D}}_{\bullet }$ denote the simplicial category associated to $\operatorname{\mathcal{D}}$ by Example 2.4.2.7.