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4.3.3 Joins of Simplicial Sets

Our next goal is to extend the join operation of Definition 4.3.2.1 to the setting of $\infty $-categories (and more general simplicial sets). We begin with a slightly more general discussion. Let $\operatorname{Lin}$ denote the category whose objects are finite linearly ordered sets and whose morphisms are nondecreasing functions. The functor category $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ is equivalent to the category of augmented simplicial sets (see §), and contains a full subcategory which is equivalent to the category of simplicial sets (see Proposition 4.3.3.11 below).

Notation 4.3.3.1. Let $J$ be a linearly ordered set. We say that a subset $I \subseteq J$ is an initial segment of $J$ if it is closed downwards: that is, if, for every pair of elements $i \leq j$ in $J$, we have $(j \in I) \Rightarrow (i \in I)$. We will write $I \sqsubseteq J$ to indicate that $I$ is an initial segment of $J$.

Construction 4.3.3.2 (Joins of Augmented Simplicial Sets). For every pair of functors $X,Y: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we let $(X \star Y): \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote a new functor given objects by the formula

\[ (X \star Y)(J) = \coprod _{I \sqsubseteq J} (X(I) \times Y(J \setminus I)). \]

Here the coproduct indexed by the collection of all initial segments $I \sqsubseteq J$.

More formally, the functor $(X \star Y): \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ can be described as follows:

  • For every finite linearly ordered set $J$, $(X \star Y)(J)$ is the collection of all triples $(I, x, y)$, where $I$ is an initial segment of $J$, $x$ is an element of $X(I)$, and $y$ is an element of $Y(J \setminus I)$.

  • If $\alpha : J' \rightarrow J$ is a nondecreasing function, then the induced map $(X \star Y)(\alpha ): (X \star Y)(J) \rightarrow (X \star Y)(J')$ is given by the constructoin

    \[ (I, x, y) \mapsto ( \alpha ^{-1}(I), X( \alpha |_{ \alpha ^{-1}(I) })(x), Y( \alpha |_{ \alpha ^{-1}(J \setminus I)} )(y) ). \]

We will refer to $X \star Y$ as the join of $X$ and $Y$.

Example 4.3.3.3. Let $E: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor given by

\[ E(I) = \begin{cases} \ast & \textnormal{ if $I = \emptyset $} \\ \emptyset & \textnormal{ otherwise. } \end{cases} \]

For every functor $X: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we have canonical bijections

\[ (X \star E)(J) = \coprod _{I \sqsubseteq J} (X(I) \times E( J \setminus I)) \simeq X(J) \times E(\emptyset ) \simeq X(J) \]
\[ (E \star X)(J) = \coprod _{I \sqsubseteq J} (E(I) \times X(J \setminus I)) \simeq E(\emptyset ) \times X(J) \simeq X(J). \]

These bijections depend functorially on $J$, and therefore determine isomorphisms of functors

\[ X \star E \simeq X \simeq E \star X. \]

Remark 4.3.3.4 (Functoriality). Construction 4.3.3.2 determines a functor

\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \quad \quad (X,Y) \mapsto X \star Y. \]

Note that this functor preserves colimits separately in each variable.

Remark 4.3.3.5 (Associativity). Let $X$, $Y$, and $Z$ be functors from $\operatorname{Lin}^{\operatorname{op}}$ to the category of sets. For every finite linearly ordered set $K$, we have a canonical bijection

\begin{eqnarray*} (X \star (Y \star Z))(K) & = & \coprod _{ I \sqsubseteq K} (X(I) \times (Y \star Z)(K \setminus I) ) \\ & = & \coprod _{I \sqsubseteq K} (X(I) \times \coprod _{J \sqsubseteq K \setminus I} (Y(J) \times Z(K \setminus (I \cup J)) ) ) \\ & \simeq & \coprod _{I \sqsubseteq K} \coprod _{J \sqsubseteq K \setminus I} (X(I) \times Y(J) \times Z(K \setminus (I \cup J) ) ) \\ & \simeq & \coprod _{J' \sqsubseteq K} \coprod _{I \sqsubseteq J'} (X(I) \times Y(J' \setminus I) \times Z(K \setminus J') ) \\ & \simeq & \coprod _{J' \sqsubseteq K} ( \coprod _{I \sqsubseteq J'} (X(I) \times Y(J' \setminus I) \times Z(K \setminus J' ) ) \\ & = & \coprod _{J \sqsubseteq K} ((X \star Y)(J') \star Z(K \setminus J' )) \\ & = & ((X \star Y) \star Z)(K). \end{eqnarray*}

These bijections depend functorially on $K \in \operatorname{Lin}^{\operatorname{op}}$, and therefore supply an isomorphism of functors $\alpha _{X,Y,Z}: X \star (Y \star Z) \simeq (X \star Y) \star Z$.

Remark 4.3.3.6. The join operation of Construction 4.3.3.2 determines a functor

\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}). \]

This functor determines a monoidal structure on the category $\operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$, whose associativity constraints are the isomorphisms $\alpha _{X,Y,Z}$ of Remark 4.3.2.6 and whose unit object is the functor $E$ of Example 4.3.3.3.

Example 4.3.3.7. For every category $\operatorname{\mathcal{C}}$, let $h_{\operatorname{\mathcal{C}}}: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor represented by $\operatorname{\mathcal{C}}$, given by the formula

\[ h_{\operatorname{\mathcal{C}}}(J) = \{ \textnormal{Functors from $J$ to $\operatorname{\mathcal{C}}$} \} . \]

For every pair of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ and every finite linearly ordered set $J$, we have a canonical bijection

\begin{eqnarray*} (h_{\operatorname{\mathcal{C}}} \star h_{\operatorname{\mathcal{D}}})(J) & = & \coprod _{ I \sqsubseteq J} h_{\operatorname{\mathcal{C}}}( I) \times h_{\operatorname{\mathcal{D}}}(J \setminus I) \\ & = & \coprod _{I \sqsubseteq J} \{ \textnormal{Functors $I \rightarrow \operatorname{\mathcal{C}}$} \} \times \{ \textnormal{Functors $(J \setminus I) \rightarrow \operatorname{\mathcal{D}}$} \} \\ & \simeq & \{ \textnormal{Functors $J \rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$} \} \\ & = & h_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(J). \end{eqnarray*}

These bijections depend functorially on $J$, and therefore determine an isomorphism $h_{\operatorname{\mathcal{C}}} \star h_{\operatorname{\mathcal{D}}} \simeq h_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}$ in the category $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$; here $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denotes the join of the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, in the sense of Definition 4.3.2.1.

Remark 4.3.3.8. Let $\operatorname{\mathcal{C}}$ be a small monoidal category. Then the presheaf category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ inherits a monoidal structure given by Day convolution (see §), which is characterized up to equivalence by the following properties:

$(1)$

The Yoneda embedding

\[ h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C) \]

can be promoted to a symmetric monoidal functor.

$(2)$

The tensor product on $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ preserves small colimits separately in each variable.

Let us specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{Lin}$ is the category of finite linearly ordered sets. Note that $\operatorname{Lin}$ can be identified with a full subcategory of $\operatorname{Cat}$ which is closed under the formation of joins (and contains the unit object $\emptyset \in \operatorname{Cat}$), and therefore inherits the structure of a monoidal category (where the tensor product is given by joins). With respect to this monoidal structure, the Yoneda embedding $h: \operatorname{Lin}\rightarrow \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ satisfies condition $(1)$ (Example 4.3.3.7), and the join functor on $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ satisfies $(2)$ by virtue of Remark 4.3.3.4). It follows that the join operation on $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ is given by Day convolution (with respect to the join operation on the category $\operatorname{Lin}$).

We now adapt Construction 4.3.3.2 to the setting of simplicial sets.

Notation 4.3.3.9. Let $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ spanned by those functors $X: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ for which the set $X(\emptyset )$ is a singleton (that is, the full subcategory spanned by those functors which preserve final objects).

Remark 4.3.3.10. For every pair of functors $X,Y: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we have a canonical bijection $(X \star Y)(\emptyset ) = X(\emptyset ) \times Y(\emptyset )$. In particular, if $X$ and $Y$ belong to the subcategory $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \subseteq \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$, then the join $X \star Y$ also belongs to $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$. Moreover, $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ contains the unit object $E$ of Example 4.3.3.3. It follows that $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ inherits the structure of a monoidal category (with respect to the join operation of Construction 4.3.3.2).

Recall that the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2 is the full subcategory of $\operatorname{Lin}$ spanned by objects of the form $[n] = \{ 0 < 1 < \cdots < n \} $ for $n \geq 0$.

Proposition 4.3.3.11. The restriction functor

\[ \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto X|_{ \operatorname{{\bf \Delta }}^{\operatorname{op}} } \]

is an equivalence of categories.

Proof. Let $S$ be a one-element set, and let $\operatorname{Fun}'_{\ast }(\operatorname{Lin}^{\operatorname{op}},\operatorname{Set})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ spanned by those functors $X: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ satisfying $X(\emptyset ) = S$. Since the inclusion functor $\operatorname{Fun}'_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \hookrightarrow \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ is an equivalence of categories, it will suffice to show that the restriction functor

\[ \operatorname{Fun}'_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto X|_{ \operatorname{{\bf \Delta }}^{\operatorname{op}} } \]

is an equivalence of categories. Let $\operatorname{Lin}_{\neq \emptyset }$ denote the full subcategory of $\operatorname{Lin}$ spanned by the nonempty finite linearly ordered sets, so that the category $\operatorname{Lin}$ can be identified with the left cone $\operatorname{Lin}_{\neq \emptyset }^{\triangleleft }$ of Example 4.3.2.5. Using Proposition 4.3.2.13 (and the fact that the forgetful functor $\operatorname{Set}_{/\ast } \rightarrow \operatorname{Set}$ is an isomorphism), we deduce that the restriction functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{Lin}_{\neq \emptyset }^{\operatorname{op}}, \operatorname{Set})$ is an isomorphism of categories. We are therefore reduced to showing that the restriction functor $\operatorname{Fun}( \operatorname{Lin}_{\neq \emptyset }^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set}) = \operatorname{Set_{\Delta }}$ is an equivalence of categories. This is clear, since the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Lin}_{\neq \emptyset }$ is an equivalence (Remark 1.1.1.3). $\square$

Remark 4.3.3.12. The inclusion functor $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Lin}_{\neq \emptyset }$ has a unique left inverse $R: \operatorname{Lin}_{\neq \emptyset } \rightarrow \operatorname{{\bf \Delta }}$, given on objects by the formula $R(I) = [n]$ when $I$ has cardinality $n+1$. It follows that the equivalence $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}$ of Proposition 4.3.3.11 admits an explicit right inverse, which carries a simplicial set $X: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ to the functor $X^{+}: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ given by the formula

\[ X^{+}(I) = \begin{cases} X( R(I) ) & \textnormal{ if $I$ is nonempty } \\ \ast & \textnormal{ otherwise. } \end{cases} \]

Construction 4.3.3.13 (Joins of Simplicial Sets). Let $X$ and $Y$ be simplicial sets. We let $X \star Y$ denote the simplicial set given by the restriction $( X^{+} \star Y^{+} )|_{ \operatorname{{\bf \Delta }}^{\operatorname{op}} }$. Here $X^{+}, Y^{+} \in \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ are given by Remark 4.3.3.12, and $X^{+} \star Y^{+}$ denotes the join of Construction 4.3.3.2. We will refer to $X \star Y$ as the join of $X$ and $Y$. The construction $X,Y \mapsto X \star Y$ determines a functor $\star : \operatorname{Set_{\Delta }}\times \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$, which we will refer to as the join functor. It is characterized (up to isomorphism) by the fact that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \ar [r]^-{\star } \ar [d] & \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \ar [d] \\ \operatorname{Set_{\Delta }}\times \operatorname{Set_{\Delta }}\ar [r]^-{\star } & \operatorname{Set_{\Delta }}} \]

commutes up to isomorphism, where the vertical maps are the equivalences supplied by Proposition 4.3.3.11.

Remark 4.3.3.14. For every pair of simplicial sets $X$ and $Y$, we have canonical monomorphisms

\[ X \simeq X \star \emptyset \hookrightarrow X \star Y \hookleftarrow \emptyset \star Y \simeq Y. \]

We will often abuse notation by identifying $X$ and $Y$ with the simplicial subsets of $X \star Y$ given by the images of these monomorphisms.

Remark 4.3.3.15. Let $X$ and $Y$ be simplicial sets. For each $n$-simplex $\sigma : \Delta ^ n \rightarrow X \star Y$, exactly one of the following conditions holds:

  • The morphism $\sigma $ factors through $X$ (where we identify $X$ with a simplicial subset of $X \star Y$ as in Remark 4.3.3.14).

  • The morphism $\sigma $ factors through $Y$ (where we identify $Y$ with a simplicial subset of $X \star Y$ as in Remark 4.3.3.14).

  • The morphism $\sigma $ factors as a composition

    \[ \Delta ^{n} = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]

    for integers $p,q \geq 0$ satisfying $p+1+q=n$ and simplices $\sigma _{-}: \Delta ^{p} \rightarrow X$ and $\sigma _{+}: \Delta ^{q} \rightarrow Y$ of $X$ and $Y$, respectively. Moreover, in this case, the simplices $\sigma _{-}$ and $\sigma _{+}$ (and the integers $p,q \geq 0$) are uniquely determined.

Remark 4.3.3.16. Let $i: X \hookrightarrow X'$ and $j: Y \hookrightarrow Y'$ be monomorphisms of simplicial sets. From the description of Remark 4.3.3.15, we see that the join $(i \star j): X \star Y \rightarrow X' \star Y'$ is also a monomorphism of simplicial sets.

Remark 4.3.3.17. Let $X_{\bullet }$ and $Y_{\bullet }$ be simplicial sets. By virtue of Remark 4.3.3.15, the join $(X \star Y)_{\bullet }$ can be described explicitly by the formula

\[ (X \star Y)_{n} = X_{n} \amalg ( \coprod _{ p+1+q = n } X_{p} \times Y_ q ) \amalg Y_{n}. \]

In these terms, the face and degeneracy operators $\{ d_ i: (X \star Y)_{n} \rightarrow (X \star Y)_{n-1} \} _{0 \leq i \leq n}$ and $\{ s_ i: (X \star Y)_{n} \rightarrow ( X \star Y)_{n+1} \} $ are given on the first and third summand by the analogous operators for $X_{\bullet }$ and $Y_{\bullet }$, and on elements $(\sigma , \tau ) \in X_{p} \times Y_{q}$ by the forula

\[ d_ i( \sigma , \tau ) = \begin{cases} ( d_ i(\sigma ), \tau ) & \textnormal{ if $i \leq p$ } \\ (\sigma , d_{i-1-p}(\tau )) & \textnormal{ if } i > p. \end{cases} \quad \quad s_ i(\sigma , \tau ) = \begin{cases} (s_ i(\sigma ), \tau ) & \textnormal{ if } i \leq p \\ (\sigma , s_{i-1-p}(\tau ) & \textnormal{ if } i > p. \end{cases} \]

Remark 4.3.3.18. For every pair of simplicial sets $X$ and $Y$, we have a canonical isomorphism $(X \star Y)^{\operatorname{op}} \simeq Y^{\operatorname{op}} \star X^{\operatorname{op}}$.

Example 4.3.3.19. For every simplicial set $X$, we have canonical isomorphisms $X \star \emptyset \simeq X \simeq \emptyset \star X$ (compare with Example 4.3.3.3).

Example 4.3.3.20. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Using Example 4.3.3.7, we obtain a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})$, where $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denotes the join of the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.

In particular, for integers $p, q \geq 0$, there is a unique isomorphism of simplicial sets

\[ \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}, \]

which is given on vertices of $\Delta ^{p}$ by the construction $i \mapsto i$ and on vertices of $\Delta ^{q}$ by $j \mapsto p+1+j$.

Proposition 4.3.3.21. Let $X$ and $Y$ be $\infty $-categories. Then the join $X \star Y$ is an $\infty $-category.

Proof. Fix integers $0 < i < n$ and a map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X \star Y$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex of $X \star Y$. Let $J \subseteq [n]$ denote the collection of vertices $j$ of $\Lambda ^ n_ i \subset \Delta ^ n$ for which $\sigma _0(j)$ belongs to $X$. We first claim that $J$ is an initial segment of $[n]$. If not, then there exists an integer $0 < j \leq n$ such that $j \in V$ and $j-1 \notin V$. Let $e$ be the edge of $\Lambda ^ n_ i$ having source $j-1$ and target $j$. Then $\sigma _0(e)$ is an edge of $X \star Y$ having source $\sigma _0(j-1) \in Y$ and target $\sigma _0(j) \in X$, which is impossible.

We now consider three cases:

  • Suppose that $J = [n]$: that is, the morphism $\sigma _0$ carries each vertex of $\Lambda ^ n_{i}$ to a vertex of the simplicial subset $X \subseteq X \star Y$. Then $\sigma _0$ factors through $X$, so we can extend $\sigma _0$ to a morphism of simplicial sets $\sigma : \Delta ^ n \rightarrow X$ by virtue of our assumption that $X$ is an $\infty $-category.

  • Suppose that $J = \emptyset $: that is, the morphism $\sigma _0$ carries each vertex of $\Lambda ^ n_{i}$ to a vertex of the simplicial subset $Y \subseteq X \star Y$. Then $\sigma _0$ factors through $Y$, so we can extend $\sigma _0$ to a morphism of simplicial sets $\sigma : \Delta ^ n \rightarrow Y$ by virtue of our assumption that $Y$ is an $\infty $-category.

  • Suppose that $\emptyset \subsetneq J \subsetneq [n]$. Let us identify the nerve $\operatorname{N}_{\bullet }(J)$ with its image in $\operatorname{N}_{\bullet }([n]) \simeq \Delta ^ n$, so that $\operatorname{N}_{\bullet }(J)$ is contained in the horn $\Lambda ^ n_{i}$. Then the composite map

    \[ \operatorname{N}_{\bullet }(J) \hookrightarrow \Lambda ^ n_{i} \xrightarrow { \sigma _0} X \star Y \]

    factors through $X$, and therefore determines a map of simplicial sets $\sigma _{-}: \operatorname{N}_{\bullet }(J) \rightarrow X$. Similarly, the composite map

    \[ \operatorname{N}_{\bullet }( [n] \setminus J) \hookrightarrow \Lambda ^ n_{i} \xrightarrow {\sigma _0} X \star Y \]

    determines a map of simplicial sets $\sigma _{+}: \operatorname{N}_{\bullet }( [n] \setminus J) \rightarrow Y$. The map $\sigma _0$ then admits a unique extension $\sigma : \Delta ^ n \rightarrow X \star Y$, given concretely by the composition

    \[ \Delta ^ n = \operatorname{N}_{\bullet }( [n] ) \simeq \operatorname{N}_{\bullet }( J ) \star \operatorname{N}_{\bullet }( [n] \setminus J) \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y. \]
$\square$

Construction 4.3.3.22. Let $X$ be a simplicial set. We will denote the join $\Delta ^{0} \star X$ by $X^{\triangleleft }$ and refer to it as the left cone of $X$. Similarly, we denote the join $X \star \Delta ^0$ by $X^{\triangleright }$ and refer to it as the right cone of $X$. We will often abuse notation by using Remark 4.3.3.14 to identify $X$ with its image in the cones $X^{\triangleleft }$ and $X^{\triangleright }$. Moreover, Remark 4.3.3.14 also supplies morphisms of simplicial sets $X^{\triangleleft } \hookleftarrow \Delta ^0 \hookrightarrow X^{\triangleright }$, which we can identify with vertices which we refer to as the cone points of $X^{\triangleleft }$ and $X^{\triangleright }$, respectively.

Example 4.3.3.23. Let $\operatorname{\mathcal{C}}$ be a category. Then Example 4.3.3.20 supplies canonical isomorphisms

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

where $\operatorname{\mathcal{C}}^{\triangleleft }$ and $\operatorname{\mathcal{C}}^{\triangleright }$ denote the left and right cones of $\operatorname{\mathcal{C}}$ (see Example 4.3.2.5).

Example 4.3.3.24. Let $n \geq 0$, and let $\Delta ^{n}$ denote the standard $n$-simplex. Using Example 4.3.3.23, we see that there is a unique isomorphism of simplicial sets $(\Delta ^{n})^{\triangleright } \simeq \Delta ^{n+1}$, which carries each vertex $i \in \{ 0, 1, \ldots , n \} $ to itself and the cone point of $( \Delta ^{n} )^{\triangleright }$ to the final vertex $n+1$. This isomorphism carries the simplicial subset $(\operatorname{\partial \Delta }^{n})^{\triangleright } \subseteq ( \Delta ^{n} )^{\triangleright }$ to the horn $\Lambda ^{n+1}_{n+1} \subseteq \Delta ^{n+1}$. Similarly, the left cone $( \operatorname{\partial \Delta }^{n} )^{\triangleleft }$ is isomorphic to the horn $\Lambda ^{n+1}_{0}$.

Remark 4.3.3.25. For every simplicial set $X$, Remark 4.3.3.18 supplies a canonical isomorphism $(X^{\triangleleft })^{\operatorname{op}} \simeq (X^{\operatorname{op}})^{\triangleright }$, carrying the cone point of $X^{\triangleleft }$ to the cone point of $(X^{\operatorname{op}})^{\triangleright }$.

Remark 4.3.3.26. Let $X$ be a simplicial set. Then the construction $Y \mapsto X \star Y$ determines a functor

\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \]

which preserves small colimits. It follows that the composite functor

\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto X \star Y \]

preserves filtered colimits and pushouts. Beware that it does not preserve colimits in general (for example, it carries the initial object $\emptyset \in \operatorname{Set_{\Delta }}$ to the simplicial set $X$, which need not be initial).

Remark 4.3.3.27 (Associativity). Let $X$, $Y$, and $Z$ be simplicial sets. Then Remark 4.3.3.5 supplies a canonical isomorphism of simplicial sets $\alpha _{X,Y,Z}: X \star (Y \star Z) \simeq (X \star Y) \star Z$. These isomorphisms are associativity constraints for a monoidal structure on the category of simplicial sets, which is characterized (up to isomorphism) by the requirement that the equivalence $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}$ of Proposition 4.3.3.11 can be promoted to a monoidal functor.

Warning 4.3.3.28. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Then the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ of Definition 4.3.2.1 is characterized (up to isomorphism) by the existence of a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ (\{ 0\} \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}) \coprod ( \{ 1\} \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}) \ar [r] \ar [d] & [1] \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [d] \\ ( \{ 0\} \times \operatorname{\mathcal{C}}) \coprod ( \{ 1\} \times \operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}} \]

in the category $\operatorname{Cat}$ (see Remark 4.3.2.14). Beware that, in the setting of simplicial sets, the analogous statement is not quite true. To every pair of simplicial sets $X$ and $Y$, one can associate a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ (\{ 0\} \times X \times Y) \coprod ( \{ 1\} \times X \times Y) \ar [r] \ar [d] & \Delta ^1 \times X \times Y \ar [d] \\ ( \{ 0\} \times X ) \coprod ( \{ 1\} \times Y) \ar [r] & X \star Y } \]

(see Construction 4.5.5.1). This diagram is almost never a pushout square (Example ). Nevertheless, the pushout can be regarded as a good approximation to the join $X \star Y$: see Proposition 4.5.5.2 and Theorem 4.5.5.7.