Kerodon

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 1.2.1.18 supplies a map of simplicial sets

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 1.2.1.23, 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.

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 2.4.2.8. 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

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

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.