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 $J$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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).
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 on objects by the formula Here the coproduct is 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 construction
\[ (X \star Y)(J) = \coprod _{I \sqsubseteq J} (X(I) \times Y(J \setminus I)). \]
\[ (I, x, y) \mapsto ( \alpha ^{-1}(I), X( \alpha |_{ \alpha ^{-1}(I) })(x), Y( \alpha |_{ \alpha ^{-1}(J \setminus I)} )(y) ). \]
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 For every functor $X: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we have canonical bijections These bijections depend functorially on $J$, and therefore determine isomorphisms of functors
\[ E(I) = \begin{cases} \ast & \textnormal{ if $I = \emptyset $} \\ \emptyset & \textnormal{ otherwise. } \end{cases} \]
\[ (X \star E)(J) = \coprod _{I \sqsubseteq J} (X(I) \times E( J \setminus I)) \simeq X(J) \times E(\emptyset ) \simeq X(J) \]
\[ (E \star X)(J) = \coprod _{I \sqsubseteq J} (E(I) \times X(J \setminus I)) \simeq E(\emptyset ) \times X(J) \simeq X(J). \]
\[ X \star E \simeq X \simeq E \star X. \]
Remark 4.3.3.4 (Functoriality). Construction 4.3.3.2 determines a functor Note that this functor preserves colimits separately in each variable.
\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \quad \quad (X,Y) \mapsto X \star Y. \]
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 These bijections depend functorially on $K \in \operatorname{Lin}^{\operatorname{op}}$, and therefore supply an isomorphism of functors $\alpha _{X,Y,Z}: X \star (Y \star Z) \simeq (X \star Y) \star Z$.
\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*}
Remark 4.3.3.6. The join operation of Construction 4.3.3.2 determines a functor 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 4.3.3.5 and whose unit object is the functor $E$ of Example 4.3.3.3.
\[ \star : \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Fun}(\operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}). \]
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 For every pair of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ and every finite linearly ordered set $J$, we have a canonical bijection 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.
\[ h_{\operatorname{\mathcal{C}}}(J) = \{ \textnormal{Functors from $J$ to $\operatorname{\mathcal{C}}$} \} . \]
\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*}
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 §
The Yoneda embedding
can be promoted to a monoidal functor.
The tensor product on $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ preserves small colimits separately in each variable.
), which is characterized up to equivalence by the following properties:
\[ h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C) \]
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 1.1.0.2 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 is an equivalence of categories.
\[ \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto X|_{ \operatorname{{\bf \Delta }}^{\operatorname{op}} } \]
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 1.1.0.3). $\square$
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 commutes up to isomorphism, where the vertical maps are the equivalences supplied by Proposition 4.3.3.11.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \times \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \ar [r]^-{\star } \ar [d] & \operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \ar [d] \\ \operatorname{Set_{\Delta }}\times \operatorname{Set_{\Delta }}\ar [r]^-{\star } & \operatorname{Set_{\Delta }}} \]
Example 4.3.3.14. 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 which is given on vertices of $\Delta ^{p}$ by the construction $i \mapsto i$ and on vertices of $\Delta ^{q}$ by $j \mapsto p+1+j$.
\[ \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}, \]
Example 4.3.3.15. 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).
Remark 4.3.3.16. For every pair of simplicial sets $X$ and $Y$, we have canonical monomorphisms 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.
\[ X \simeq X \star \emptyset \hookrightarrow X \star Y \hookleftarrow \emptyset \star Y \simeq Y. \]
Remark 4.3.3.17. 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 4.3.3.16).
The morphism $\sigma $ factors through $Y$ (where we identify $Y$ with a simplicial subset of $X \star Y$ as in Remark 4.3.3.16).
The morphism $\sigma $ factors as a composition
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.
\[ \Delta ^{n} = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]
Remark 4.3.3.18. Let $i: X \hookrightarrow X'$ and $j: Y \hookrightarrow Y'$ be monomorphisms of simplicial sets. From the description of Remark 4.3.3.17, we see that the join $(i \star j): X \star Y \rightarrow X' \star Y'$ is also a monomorphism of simplicial sets.
Remark 4.3.3.19. Let $X_{\bullet }$ and $Y_{\bullet }$ be simplicial sets. By virtue of Remark 4.3.3.17, the join $(X \star Y)_{\bullet }$ can be described explicitly by the formula In these terms, the face and degeneracy operators $\{ d^{n}_ i: (X \star Y)_{n} \rightarrow (X \star Y)_{n-1} \} _{0 \leq i \leq n}$ and $\{ s^{n}_ 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 formula
\[ (X \star Y)_{n} = X_{n} \amalg ( \coprod _{ p+1+q = n } X_{p} \times Y_ q ) \amalg Y_{n}. \]
\[ d^{n}_ i( \sigma , \tau ) = \begin{cases} ( d^{p}_ i(\sigma ), \tau ) & \textnormal{ if $i \leq p$ } \\ (\sigma , d^{q}_{i-1-p}(\tau )) & \textnormal{ if } i > p. \end{cases} \quad \quad s^{n}_ i(\sigma , \tau ) = \begin{cases} (s^{p}_ i(\sigma ), \tau ) & \textnormal{ if } i \leq p \\ (\sigma , s^{q}_{i-1-p}(\tau )) & \textnormal{ if } i > p. \end{cases} \]
Remark 4.3.3.20. Let $X$ and $Y$ be simplicial sets, and let $\sigma : \Delta ^{m} \rightarrow X \star Y$ be an $m$-simplex which factors as a composition for some integers $m_{-}, m_{+} \geq 0$ satisfying $m_{-} + 1 + m_{+} = m$. Then $\sigma $ is nondegenerate if and only if both $\sigma _{-}$ and $\sigma _{+}$ are nondegenerate. It follows that, for every integer $n$, the $n$-skeleton of $X \star Y$ is given by the union In particular, we have an equality provided that we adopt the convention that an empty simplicial set has dimension $-1$.
\[ \Delta ^{m} \simeq \Delta ^{m_{-}} \star \Delta ^{m_{+}} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y \]
\[ \operatorname{sk}_{n}(X) \cup (\bigcup _{p+1+q =n} \operatorname{sk}_{p}(X) \star \operatorname{sk}_{q}(Y) ) \cup \operatorname{sk}_{n}(Y). \]
\[ \mathrm{dim}(X \star Y) = \mathrm{dim}(X) + 1 + \mathrm{dim}(Y), \]
Remark 4.3.3.21. Let $X$ and $Y$ be finite simplicial sets. Then the join $X \star Y$ is also finite.
Remark 4.3.3.22. For every pair of simplicial sets $X$ and $Y$, we have a canonical isomorphism $(X \star Y)^{\operatorname{op}} \simeq Y^{\operatorname{op}} \star X^{\operatorname{op}}$.
Remark 4.3.3.23. Let $X$, $Y$, and $K$ be simplicial sets. Unwinding the definitions, we see that morphisms from $K$ to $X \star Y$ can be identified with triples $(\pi , f_{-}, f_{+} )$, where are morphisms of simplicial sets (note that, when $K$ is a simplex, this recovers the description of Remark 4.3.3.19).
\[ \pi : K \rightarrow \Delta ^1 \quad \quad f_{-}: \{ 0\} \times _{\Delta ^1} K \rightarrow X \quad \quad f_{+}: \{ 1\} \times _{\Delta ^1} K \rightarrow Y \]
Proposition 4.3.3.24. Let $u: X \rightarrow X'$ and $v: Y \rightarrow Y'$ be inner fibrations of simplicial sets. Then the join $(u \star v): X \star Y \rightarrow X' \star Y'$ is also an inner fibration of simplicial sets.
Corollary 4.3.3.25. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category.
Proof. Since $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, the projection maps $u: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ and $v: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ are inner fibrations (Example 4.1.1.2). Applying Proposition 4.3.3.24, we deduce that the join
\[ (u \star v): \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \Delta ^0 \star \Delta ^0 \simeq \Delta ^1 \]
is also an inner fibration. Since $\Delta ^1$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category (Remark 4.1.1.9). $\square$
Proof of Proposition 4.3.3.24. Let $u: X \rightarrow X'$ and $v: Y \rightarrow Y'$ be inner fibrations of simplicial sets and let $0 < i < n$ be integers; we wish to show that every lifting problem
4.14
\begin{equation} \begin{gathered}\label{equation:join-of-inner-fibration} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & X \star Y \ar [d]^{u \star v } \\ \Delta ^{n} \ar [r]^-{ \sigma ' } \ar@ {-->}[ur]^{ \sigma } & X' \star Y' } \end{gathered} \end{equation}
admits a solution. If $\sigma '$ factors through either $X'$ or $Y'$, this follows immediately from our assumption that $u$ and $v$ are inner fibrations. We may therefore assume without loss of generality that $\sigma '$ factors as a composition
\[ \Delta ^{n} = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma '_{-} \star \sigma '_{+} } X' \star Y' \]
for some pair of integers $p,q \geq 0$ satisfying $p+1+q=n$ and simplices $\sigma '_{-}: \Delta ^{p} \rightarrow X'$ and $\sigma '_{+}: \Delta ^{q} \rightarrow Y'$. Let $\iota _{-}$ denote the inclusion map
\[ \Delta ^{p} \hookrightarrow \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q} = \Delta ^{n}, \]
and define $\iota _{+}: \Delta ^{q} \hookrightarrow \Delta ^ n$ similarly. Note that both $\iota _{-}$ and $\iota _{+}$ factor through the inner horn $\Lambda ^{n}_{i} \subseteq \Delta ^ n$. Set $\sigma _{-} = \sigma _0 \circ \iota _{-}$ and $\sigma _{+} = \sigma _0 \circ \iota _{+}$. Unwinding the definitions, we see that the composite map
\[ \Delta ^{n} = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y \]
determines an $n$-simplex $\sigma $ of $X \star Y$ which is a solution to the lifting problem (4.14). $\square$
Construction 4.3.3.26. 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 4.3.3.16 to identify $X$ with its image in the cones $X^{\triangleleft }$ and $X^{\triangleright }$. Moreover, Remark 4.3.3.16 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 4.3.3.27. Let $\operatorname{\mathcal{C}}$ be a category. Then Example 4.3.3.14 supplies canonical isomorphisms 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).
\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\triangleleft } \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\triangleleft } ) \quad \quad \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\triangleright } \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\triangleright } ), \]
Example 4.3.3.28. Let $n \geq 0$, and let $\Delta ^{n}$ denote the standard $n$-simplex. Using Example 4.3.3.27, 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 4.3.3.29. For every simplicial set $X$, Remark 4.3.3.22 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 4.3.3.30. Let $X$ be a simplicial set. Then, for every nonnegative integer $n$, the $n$-skeleton of the cone $X^{\triangleright }$ fits into a pushout diagram see Remark 4.3.3.20. In particular, $X^{\triangleright }$ has dimension $\leq n$ if and only if $X$ has dimension $\leq n-1$.
\[ \xymatrix { \operatorname{sk}_{n-1}(X) \ar [r] \ar [d] & \operatorname{sk}_{n-1}(X)^{\triangleright } \ar [d] \\ \operatorname{sk}_{n}(X) \ar [r] & \operatorname{sk}_{n}(X)^{\triangleright }; } \]
Remark 4.3.3.31. Let $X$ be a simplicial set. Then the construction $Y \mapsto X \star Y$ determines a functor which preserves small colimits. It follows that the composite functor 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).
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \]
\[ \operatorname{Set_{\Delta }}\rightarrow (\operatorname{Set_{\Delta }})_{X/} \rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto X \star Y \]
Remark 4.3.3.32 (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 4.3.3.33. 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 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 (see Construction 4.5.8.1), which is almost never a pushout square. Nevertheless, the pushout can be regarded as a good approximation to the join $X \star Y$: see Proposition 4.5.8.2 and Theorem 4.5.8.8.
\[ \xymatrix@R =50pt@C=50pt{ (\{ 0\} \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}) \coprod ( \{ 1\} \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}) \ar [r] \ar [d] & [1] \times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\ar [d] \\ ( \{ 0\} \times \operatorname{\mathcal{C}}) \coprod ( \{ 1\} \times \operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}} \]
\[ \xymatrix@R =50pt@C=50pt{ (\{ 0\} \times X \times Y) \coprod ( \{ 1\} \times X \times Y) \ar [r] \ar [d] & \Delta ^1 \times X \times Y \ar [d] \\ ( \{ 0\} \times X ) \coprod ( \{ 1\} \times Y) \ar [r] & X \star Y } \]