4.3.5 Slices of Simplicial Sets
Let $\operatorname{\mathcal{C}}$ be a category. In ยง4.3.1, we associated to every diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ a slice category $\operatorname{\mathcal{C}}_{/F}$ and a coslice category $\operatorname{\mathcal{C}}_{F/}$ (Construction 4.3.1.8). We now introduce a generalization of this construction, where we replace (the nerves of) $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{K}}$ by arbitrary simplicial sets. As our starting point, we recall that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}_{/F}$ can be characterized as the right adjoint of the join functor
\[ \operatorname{Cat}\rightarrow \operatorname{Cat}_{\operatorname{\mathcal{K}}/ } \quad \quad \operatorname{\mathcal{E}}\mapsto \operatorname{\mathcal{E}}\star \operatorname{\mathcal{K}} \]
(see Corollary 4.3.2.17).
Construction 4.3.5.1 (Slice Simplicial Sets). Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a simplicial set $X_{/f}$ as follows:
For each $n \geq 0$, an $n$-simplex of $X_{/f}$ is a map of simplicial sets $\overline{f}: \Delta ^ n \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$.
For every nondecreasing function $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the associated map
\[ \alpha ^{\ast }: \{ \textnormal{$n$-simplices of $X_{/f}$} \} \rightarrow \{ \textnormal{$m$-simplices of $X_{/f}$} \} \]
carries an $n$-simplex $\overline{f}: \Delta ^ n \star K \rightarrow X$ to the composite map
\[ \Delta ^{m} \star K \xrightarrow { \alpha \star \operatorname{id}_ K} \Delta ^{n} \star K \xrightarrow { \overline{f} } X. \]
We will refer to $X_{/f}$ as the slice simplicial set of $X$ over $f$.
Variant 4.3.5.3 (Coslice Simplicial Sets). Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a simplicial set $X_{f/}$ as follows:
For each $n \geq 0$, an $n$-simplex of $X_{f/}$ is a map of simplicial sets $\overline{f}: K \star \Delta ^ n \rightarrow X$ satisfying $\overline{f}|_{K} = f$.
For every nondecreasing function $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the associated map
\[ \alpha ^{\ast }: \{ \textnormal{$n$-simplices of $X_{f/}$} \} \rightarrow \{ \textnormal{$m$-simplices of $X_{f/}$} \} \]
carries an $n$-simplex $\overline{f}: K \star \Delta ^ n \rightarrow X$ to the composite map
\[ K \star \Delta ^{m} \xrightarrow { \operatorname{id}_{K} \star \alpha } K \star \Delta ^{n} \xrightarrow { \overline{f} } X. \]
We will refer to $X_{f/}$ as the coslice simplicial set of $X$ under $f$. As in Remark 4.3.5.2, it is equipped with a projection map $X_{f/} \rightarrow X$.
Notation 4.3.5.6 (Slicing over Vertices). Let $X$ be a simplicial set containing a vertex $x$, and let $f_{x}: \Delta ^{0} \rightarrow X$ be the map carrying the unique vertex of $\Delta ^0$ to $x$. We will generally abuse notation by not distinguishing between the vertex $x$ and the morphism $f_ x$. For example, we will denote the slice simplicial set $X_{/f_ x}$ by $X_{/x}$, and the coslice simplicial set $X_{f_ x/}$ by $X_{x/}$.
Example 4.3.5.7. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f = \operatorname{N}_{\bullet }(F)$ denote the induced morphism of simplicial sets from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For each $n \geq 0$, we have canonical bijections
\begin{eqnarray*} \{ \textnormal{$n$-simplices of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f}$} \} & \simeq & \{ \textnormal{Morphisms $\overline{f}: \Delta ^ n \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Morphisms $\overline{f}: \operatorname{N}_{\bullet }([n]) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Morphisms $\overline{f}: \operatorname{N}_{\bullet }([n] \star \operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Functors $\overline{F}: [n] \star \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ with $\overline{F}|_{\operatorname{\mathcal{K}}} = F$} \} \\ & \simeq & \{ \textnormal{Functors $[n] \rightarrow \operatorname{\mathcal{C}}_{/F}$} \} \\ & \simeq & \{ \textnormal{$n$-simplices of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} )$ } \} . \end{eqnarray*}
Here the third bijection comes from Example 4.3.3.23, the fourth from Proposition 1.3.3.1, and the fifth from Proposition 4.3.2.10. These bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} )$. Similarly, we have a canonical isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{f/} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{F/} )$. For a more general statement, see Corollary 4.3.5.17.
Example 4.3.5.8. Let $\operatorname{\mathcal{C}}$ be a category containing an object $X$, which we also view as a vertex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Specializing Example 4.3.5.7 (and invoking Example 4.3.1.10), we obtain canonical isomorphisms
\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/X} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/X} ) \quad \quad \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{X/} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{X/} ). \]
Example 4.3.5.9. Let $K$ be a simplicial set, let $Y$ be a topological space, and let $f: K \rightarrow \operatorname{Sing}_{\bullet }(Y)$ be a morphism of simplicial sets, which we will identify with a continuous function $F: |K| \rightarrow Y$. For each $n \geq 0$, we have canonical bijections
\begin{eqnarray*} \{ \textnormal{$n$-simplices of $\operatorname{Sing}_{\bullet }(Y)_{/f}$} \} & \simeq & \{ \textnormal{Morphisms $\overline{f}: \Delta ^ n \star K \rightarrow \operatorname{Sing}_{\bullet }(Y)$ with $\overline{f}|_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})} = f$} \} \\ & \simeq & \{ \textnormal{Continuous maps $\overline{F}: | \Delta ^ n \star K | \rightarrow Y$ with $\overline{Y}|_{|K|} = f$} \} \\ & \simeq & \{ \textnormal{Continuous maps $\overline{F}: | \Delta ^ n | \star |K| \rightarrow Y$ with $\overline{F}|_{|K|} = F$} \} \end{eqnarray*}
Here the third bijection is provided by Proposition 4.3.4.11. Using the fact that these bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$ and invoking the universal property $| \Delta ^ n | \star |K|$ (see Remark 4.3.4.3), we obtain an isomorphism of $\operatorname{Sing}_{\bullet }(Y)_{/f}$ with the iterated fiber product
\[ \operatorname{Sing}_{\bullet }(Y) \times _{ \operatorname{Fun}( \{ 0\} \times K, \operatorname{Sing}_{\bullet }(Y) ) } \operatorname{Fun}( \Delta ^1 \times K, \operatorname{Sing}_{\bullet }(Y) ) \times _{ \operatorname{Fun}( \{ 1\} \times K, \operatorname{Sing}_{\bullet }(Y) )} \{ f \} . \]
Example 4.3.5.10. Let $Y$ be a topological space equipped with a base point $y$. Let $P = \{ p: [0,1] \rightarrow Y \} $ denote the collection of all continuous functions from the unit interval $[0,1]$ to $Y$, and let $P_{y} = \{ p \in P: p(1) = y \} $ denote the subset of $P$ consisting of those continuous paths which end at the point $y$. We regard $P$ as a topological space by equipping it with the compact-open topology, so the singular simplicial set $\operatorname{Sing}_{\bullet }(P)$ can be identified with $\operatorname{Fun}( \Delta ^1, \operatorname{Sing}_{\bullet }(Y) )$ (see Warning 2.4.2.18). Identifying $y$ with a vertex of the singular simplicial set $\operatorname{Sing}_{\bullet }(Y)$, Example 4.3.5.9 supplies an isomorphism of simplicial sets
\[ \operatorname{Sing}_{\bullet }(Y)_{/y} \simeq \operatorname{Sing}_{\bullet }(P) \times _{ \operatorname{Sing}_{\bullet }(Y) } \{ y\} = \operatorname{Sing}_{\bullet }(P_ y). \]
In particular, since the topological space $P_ y$ is contractible, the simplicial set $\operatorname{Sing}_{\bullet }(Y)_{/y}$ is a contractible Kan complex (this is a special case of a general phenomenon: see Corollary 4.3.7.14).
Warning 4.3.5.11. Recall that, if $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories, then the slice category $\operatorname{\mathcal{C}}_{/F}$ can be defined as the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}}) } \{ F\} $ (see Remark 4.3.1.11). In the setting of simplicial sets, our definition is somewhat different. Nevertheless, to any morphism of simplicial sets $F: K \rightarrow \operatorname{\mathcal{C}}$, one can associate a comparison map
\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\{ 0\} \times K, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1 \times K, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} \times K, \operatorname{\mathcal{C}}) } \{ F \} \]
which we will refer to as the slice diagonal morphism (see Construction 4.6.4.13). This map has the following features:
When $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, the morphism $\delta _{/F}$ is an isomorphism of simplicial sets.
When $\operatorname{\mathcal{C}}$ is an $\infty $-category, the morphism $\delta _{/F}$ is an equivalence of $\infty $-categories (Theorem 4.6.4.17).
When $\operatorname{\mathcal{C}}= \operatorname{Sing}_{\bullet }(X)$ is the singular simplicial set of a topological space $X$, the morphism $\delta _{/F}$ does not coincide with the isomorphism constructed in Example 4.3.5.9 (however, they are naturally homotopic).
The morphism $\delta _{/F}$ is usually not an isomorphism of simplicial sets (see Warning 4.3.3.33).
The slice simplicial sets of Construction 4.3.5.1 can be characterized by a universal property.
Construction 4.3.5.12. Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a morphism of simplicial sets $c: X_{/f} \star K \rightarrow X$ as follows:
The restriction of $c$ to the simplicial subset $X_{/f} \subseteq X_{/f} \star K$ is equal to the projection map $X_{/f} \rightarrow X$ of Remark 4.3.5.2.
The restriction of $c$ to the simplicial subset $K \subseteq X_{/f} \star K$ is equal to $f$.
Let $\sigma : \Delta ^{n} \rightarrow X_{/f} \star K$ be an $n$-simplex which does not belong to $X_{/f}$ or $K$, so that $\sigma $ factors (uniquely) as a composition
\[ \Delta ^{n} \simeq \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X_{/f} \star K \]
for $p+1+q=n$ (see Remark 4.3.3.15). Using the definition of the simplicial set $X_{/f}$, we can identify $\sigma _{-}$ with a morphism of simplicial sets $\overline{f}: \Delta ^{p} \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$. We then define $c(\sigma )$ to be the $n$-simplex of $X$ given by the composite map
\[ \Delta ^{n} \simeq \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \operatorname{id}\star \sigma _{+} } \Delta ^{p} \star K \xrightarrow { \overline{f} } X. \]
We will refer to $c$ as the slice contraction morphism. Applying a similar construction to the opposite simplicial sets, we obtain a morphism $c': K \star X_{f/} \rightarrow X$ which we will refer to as the coslice contraction morphism.
Proposition 4.3.5.13. Let $f: K \rightarrow X$ be a morphism of simplicial sets, and let $c: X_{/f} \star K \rightarrow X$ be the slice contraction morphism of Construction 4.3.5.12. Then, for any simplicial set $Y$, postcomposition with $c$ induces a bijection
\[ \theta _{Y}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{/f} ) \rightarrow \{ \textnormal{Morphisms $\overline{f}: Y \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$} \} \]
Similarly, postcomposition with the coslice contraction morphism $c': K \star X_{f/} \rightarrow X$ induces a bijection
\[ \theta '_{Y}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{f/} ) \rightarrow \{ \textnormal{Morphisms $\overline{f}: K \star Y \rightarrow X$ satisfying $\overline{f}|_{K} = f$} \} . \]
Proof.
In the case where $Y$ is a standard simplex, both assertions follow immediately from the definition of the simplicial sets $X_{/f}$ and $X_{f/}$. Since every simplicial set can be realized as a colimit of simplices (Remark 1.1.3.13), it will suffice to show that the constructions $Y \mapsto \theta _{Y}$ and $Y \mapsto \theta '_{Y}$ carry colimits of simplicial sets to limits in the arrow category $\operatorname{Fun}([1], \operatorname{Set})$. This follows from the observation that the functors
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto Y \star K, Y \mapsto K \star Y \]
preserve small colimits (see Remark 4.3.3.31).
$\square$
Corollary 4.3.5.14. Let $K$ be a simplicial set. Then the join functor
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto Y \star K \]
admits a right adjoint, given on objects by the slice construction $(f: K \rightarrow X) \mapsto X_{/f}$. Similarly, the join functor
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{K/} \quad \quad Y \mapsto K \star Y \]
admits a right adjoint, given on objects by the coslice construction $(f: K \rightarrow X) \mapsto X_{f/}$.
Example 4.3.5.15. Let $X$ be a simplicial set containing a vertex $x$. Let $Y$ be a simplicial set, and let $v$ and $v'$ denote the cone points of $Y^{\triangleright }$ and $Y^{\triangleleft }$, respectively. Then Proposition 4.3.5.13 supplies bijections
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{/x} ) \simeq \{ \textnormal{Morphisms $f: Y^{\triangleright } \rightarrow X$ with $f(v) = x$} \} \]
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, X_{x/} ) \simeq \{ \textnormal{Morphisms $f: Y^{\triangleleft } \rightarrow X$ with $f(v') = x$} \} . \]
Example 4.3.5.7 can be adapted to describe any slice or coslice of a simplicial set having the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Corollary 4.3.5.17. Let $\operatorname{\mathcal{C}}$ be a category and let $K$ be a simplicial set equipped with a morphism $f: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $u: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{K}}$ as a homotopy category of $K$ (see Definition 1.3.6.1), so that $f$ factors uniquely as a composition $K \xrightarrow {u} \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow {\operatorname{N}_{\bullet }(F)} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ for some functor $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$. Then $u$ induces isomorphisms of simplicial sets
\[ \theta : \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/\operatorname{N}_{\bullet }(F)} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f} \quad \quad \theta ': \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{F/} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{\operatorname{N}_{\bullet }(F)/} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{f/}. \]
Proof.
We will prove that $\theta $ is an isomorphism; the proof for $\theta '$ is similar. Fix an $n$-simplex $\sigma $ of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/f}$, which we identify with a morphism of simplicial sets $\overline{f}: \Delta ^ n \star K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ satisfying $\overline{f}|_{K} = f$. Let $\overline{f}_0 = \overline{f}|_{ \Delta ^ n }$. Using Proposition 4.3.5.13, we can identify $\overline{f}$ with a morphism of simplicial sets $g: K \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{\overline{f}_0 / }$. We wish to show that $\sigma $ can be lifted uniquely to an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ / \operatorname{N}_{\bullet }(F) }$. Equivalently, we wish to show that $g$ admits a unique factorization
\[ K \xrightarrow {u} \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow { \overline{g} } \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{ \overline{f}_0 / } \]
for which the composite map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \xrightarrow { \overline{g} } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ \overline{f}_0 / } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is equal to $\operatorname{N}_{\bullet }(F)$. This follows our assumption that $u$ exhibits $\operatorname{\mathcal{K}}$ as a homotopy category of $K$, since the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ \overline{f}_0 / }$ is isomorphic to the nerve of a category (see Example 4.3.5.7).
$\square$
Corollary 4.3.5.18. Let $A$ and $B$ be simplicial sets, and let $u: A \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}})$ and $v: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{B}})$ be morphisms which exhibit $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{B}}$ as the homotopy categories of $A$ and $B$, respectively. Then the composite map
\[ A \star B \xrightarrow { u \star v } \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{B}}) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{A}}\star \operatorname{\mathcal{B}}) \]
exhibits $\operatorname{\mathcal{A}}\star \operatorname{\mathcal{B}}$ as the homotopy category of $A \star B$.
Proof.
Let $\operatorname{\mathcal{C}}$ be a category, and suppose we are given a map of simplicial sets $f: A \star B \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Applying Corollary 4.3.5.17 to the morphism $f|_{A}$, we deduce that $f$ factors uniquely as a composition
\[ A \star B \xrightarrow {u \star \operatorname{id}} \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \star B \xrightarrow {f'} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). \]
Similarly, $f'$ factors uniquely as a composition
\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \star B \xrightarrow { \operatorname{id}\star v} \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{B}}) \xrightarrow {f''} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). \]
Combining these observations (together with Example 4.3.3.23 and Proposition 1.3.3.1), we conclude that $f$ factors uniquely as a composition
\[ A \star B \xrightarrow {u \star v} \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{B}}) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{A}}\star \operatorname{\mathcal{B}}) \xrightarrow { \operatorname{N}_{\bullet }(F) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]
for some functor $F: \operatorname{\mathcal{A}}\star \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$.
$\square$