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4.5.5 Application: Universal Property of the Join

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, and let $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denote their join (Definition 4.3.2.1). Proposition 4.3.2.13 (and Remark 4.3.2.14) supplies a pushout diagram of categories.

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

Passing to nerves, we obtain a commutative diagram of simplicial sets

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

Beware that this diagram is generally not a pushout square. However, we will show in this section that it is nevertheless a categorical pushout square, in the sense of Definition 4.5.3.1. Moreover, an analogous statement holds if we replace $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ by arbitrary simplicial sets $X$ and $Y$.

Construction 4.5.5.1. Let $X$ and $Y$ be simplicial sets, let $\pi _{X}: X \times Y \rightarrow X$ and $\pi _{Y}: X \times Y \rightarrow Y$ denote the projection maps, and let $\iota _{X}: X \hookrightarrow X \star Y$ and $\iota _{Y}: Y \hookrightarrow X \star Y$ denote the inclusion maps. Then there is a unique map of simplicial sets $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ with the property that $c|_{ X \times \{ 0\} \times Y} = \iota _{X} \circ \pi _{X}$ and $c|_{ X \times \{ 1\} \times Y} = \iota _{Y} \circ \pi _{Y}$. Concretely, if $\sigma = ( \sigma _{X}, \sigma _{ \Delta ^1}, \sigma _{Y} )$ is an $n$-simplex of the product $X \times \Delta ^1 \times Y$, then $c(\sigma )$ is the $n$-simplex of $X \star Y$ given by the composition

\[ \Delta ^ n \simeq ( \sigma _{\Delta ^1}^{-1}\{ 0\} ) \star ( \sigma _{\Delta ^1}^{-1} \{ 1\} ) \xrightarrow {\sigma _{X} \star \sigma _{Y} } X \star Y. \]

We will refer to $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ as the collapse map.

Proposition 4.5.5.2. Let $X$ and $Y$ be simplicial sets, and let $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ denote the collapse map of Construction 4.5.5.1. Then the commutative diagram of simplicial sets

4.23
\begin{equation} \begin{gathered}\label{equation:diagram-of-collapse} \xymatrix@R =50pt@C=50pt{ (X \times \{ 0 \} \times Y) \coprod (X \times \{ 1\} \times Y) \ar [r] \ar [d]^{\pi _{X} \coprod \pi _{Y}} & X \times \Delta ^1 \times Y \ar [d]^{c} \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [r]^-{( \iota _{X}, \iota _{Y} )} & X \star Y} \end{gathered} \end{equation}

is a categorical pushout square.

It will be convenient to state Proposition 4.5.5.2 in a slightly different form.

Notation 4.5.5.3. Let $X$ and $Y$ be simplicial sets. We let $X \diamond Y$ denote the simplicial set given by the iterated pushout

\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ ( X \times \{ 1\} \times Y) } Y, \]

so that we have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \times \operatorname{\partial \Delta }^1 \times Y \ar [r] \ar [d]^{\pi _{X} \coprod \pi _{Y}} & X \times \Delta ^1 \times Y \ar [d] \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [r] & X \diamond Y. } \]

It follows that the commutative diagram (4.23) determines a morphism of simplicial sets $c_{X,Y}: X \diamond Y \rightarrow X \star Y$, which we will refer to as the comparison map.

Example 4.5.5.4. Let $X$ and $Y$ be simplicial sets. If $X$ is empty, then the simplicial set $X \diamond Y$ can be identified with $Y$. If $Y$ is empty, then the simplicial set $X \diamond Y$ can be identified with $X$. In either case, the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ is an isomorphism of simplicial sets.

Remark 4.5.5.5 (Functoriality). The construction $(X,Y) \mapsto X \diamond Y$ determines a functor $\diamond : \operatorname{Set_{\Delta }}\times \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$. Moreover:

  • For fixed $X$, the functor

    \[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto X \star Y \]

    preserves monomorphisms, filtered colimits and pushout diagrams.

  • For fixed $Y$, the functor

    \[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto X \star Y \]

    preserves monomorphisms, filtered colimits, and pushout diagrams.

Remark 4.5.5.6. Let $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$ be categorical equivalences of simplicial sets. Then the induced map $(f \diamond g): X \diamond Y \rightarrow X' \diamond Y'$ is also a categorical equivalence. This follows by applying Proposition 4.5.3.5 to the diagram

\[ \xymatrix@C =20pt@R=50pt{ X \times \operatorname{\partial \Delta }^1 \times Y \ar [rr] \ar [dd] \ar [dr] & & X \times \Delta ^1 \times Y \ar [dd] \ar [dr] & \\ & X' \times \operatorname{\partial \Delta }^1 \times Y' \ar [rr] \ar [dd] & & X' \times \Delta ^1 \times Y' \ar [dd] \\ ( X \times \{ 0\} ) \coprod ( \{ 1\} \times Y ) \ar [rr] \ar [dr] & & X \diamond Y \ar [dr] & \\ & (X' \times \{ 0\} ) \coprod ( \{ 1\} \times Y' ) \ar [rr] & & X' \diamond Y'. } \]

By virtue of Proposition 4.5.3.7, Proposition 4.5.5.2 can be restated as follows:

Theorem 4.5.5.7. Let $X$ and $Y$ be simplicial sets. Then the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ of Notation 4.5.5.3 is a categorical equivalence of simplicial sets.

Corollary 4.5.5.8. Let $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$ be categorical equivalences of simplicial sets. Then the induced map $(f \star g): X \star Y \rightarrow X' \star Y'$ is also a categorical equivalence of simplicial sets.

Proof. We have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [r]^-{f \diamond g} \ar [d]^{ c_{X,Y} } & X' \diamond Y' \ar [d]^{ c_{X',Y'} } \\ X \star Y \ar [r]^-{ f \star g} & X' \star Y',} \]

where $f \diamond g$, $c_{X,Y}$, and $c_{X',Y'}$ are categorical equivalences (Remark 4.5.5.6 and Theorem 4.5.5.7). Invoking the two-out-of-three property (Remark 4.5.2.6), we conclude that $f \star g$ is also a categorical equivalence. $\square$

The proof of Theorem 4.5.5.7 will require some preliminaries. We begin by reducing to the special case where $X = \Delta ^1$.

Lemma 4.5.5.9. Let $Y$ be a simplicial set, and suppose that the comparison map $c_{\Delta ^1, Y}: \Delta ^1 \diamond Y \rightarrow \Delta ^1 \star Y$ is a categorical equivalence. Then, for every simplicial set $X$, the comparison map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ is a categorical equivalence.

Proof. Throughout the proof, we regard the simplicial set $Y$ as fixed. Let us say that a simplicial set $X$ is good if $c_{X,Y}$ is a categorical equivalence. We begin with some elementary observations:

$(a)$

The collection of good simplicial sets is closed under the formation of filtered colimits (since the collection of categorical equivalences is closed under filtered colimits, by virtue of Corollary 4.5.4.2).

$(b)$

Suppose we are given a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d] & X(0) \ar [d] \\ X(1) \ar [r] & X(01), } \]

where $f$ is a monomorphism. If $X$, $X(0)$, and $X(1)$ are good, then $X(01)$ is good. This follows by applying Proposition 4.5.3.5 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [dr]^{ c_{X,Y} } \ar [rr] \ar [dd] & & X(0) \diamond Y \ar [dr]^{ c_{X(0),Y} } \ar [dd] & \\ & X \star Y \ar [rr] \ar [dd] & & X(0) \star Y \ar [dd] \\ X(1) \diamond Y \ar [dr]^{ c_{X(1),Y}} \ar [rr] & & X(01) \diamond Y \ar [dr]^{ c_{X(01),Y} } & \\ & X(1) \star Y \ar [rr] & & X(01) \star Y, } \]

noting that the upper and lower squares are categorical pushouts by virtue of Example 4.5.3.8).

$(c)$

Let $f: X \rightarrow X'$ be an inner anodyne morphism of simplicial sets. Then $X$ is good if and only if $X'$ is good. To prove this, we observe that there is a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [r]^-{ c_{X,Y} } \ar [d]^{f \diamond \operatorname{id}_ Y} & X \star Y \ar [d]^{f \star \operatorname{id}_{Y}} \\ X' \diamond Y \ar [r]^-{c_{Y} } & X' \star Y. } \]

By the two-out-of-three property (Remark 4.5.2.6), it will suffice to show that the morphisms $f \diamond \operatorname{id}_{Y}$ and $f \star \operatorname{id}_{Y}$ are categorical equivalences. In the first case, this follows from Remark 4.5.5.6. For the second, we observe that $f \star \operatorname{id}_{Y}$ is actually inner anodyne, since it factors as a composition

\[ X \star Y \xrightarrow {u} X' \coprod _{X} (X \star Y ) \xrightarrow {v} X' \star Y, \]

where $u$ is a pushout of $f$ (hence inner anodyne because $f$ is inner anodyne) and $v$ is left anodyne by virtue of Proposition 4.3.7.1 (hence also inner anodyne).

We wish to show that if the $1$-simplex $\Delta ^1$ is good, then every simplicial set $X$ is good. Writing $X$ as the filtered colimit its finite simplicial subsets (Remark 3.5.1.8), we can use $(a)$ to reduce to the case where $X$ is finite. We now proceed by induction on the dimension of $X$. If $X = \emptyset $, then $c_{X}$ is an isomorphism (Example 4.5.5.4). Otherwise, the simplicial set $X$ has dimension $n \geq 0$. We now proceed by induction on the number of nondegenerate $n$-simplices of $X$. Using Proposition 1.1.3.13, we can choose a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \Delta ^{n} \ar [d] \\ X' \ar [r] & X } \]

where $X' \subseteq X$ is a simplicial subset having one fewer nondegenerate $n$-simplex. It then follows from our inductive hypothesis that $\operatorname{\partial \Delta }^ n$ and $X'$ are good. By virtue of $(b)$, it will suffice to show that $\Delta ^ n$ is good. This holds for $n=1$ by assumption, and also for $n=0$ because $\Delta ^{0}$ is a retract of $\Delta ^{1}$. We may therefore assume that $n \geq 2$, so that the horn inclusion $\Lambda ^{n}_{1} \hookrightarrow \Delta ^ n$ is inner anodyne. Our inductive hypothesis guarantees that $\Lambda ^{n}_{1}$ is good, so that $\Delta ^ n$ is good by virtue of $(c)$. $\square$

Lemma 4.5.5.10. The comparison map $c_{ \Delta ^1, \Delta ^0}: \Delta ^{1} \diamond \Delta ^0 \rightarrow \Delta ^{1} \star \Delta ^0$ is a categorical equivalence.

Proof. Unwinding the definitions, we can identify $\Delta ^1 \diamond \Delta ^0$ with the simplicial set $(\Delta ^1 \times \Delta ^1) \coprod _{ \Delta ^{1} \times \{ 1\} } \Delta ^0$, which we represent informally by the diagram

\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [dr] \ar [d] & \bullet \ar@ {=}[d] \\ \bullet \ar [r] & \bullet . } \]

Let $X$ denote the simplicial set $\Delta ^{2} \coprod _{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) } \Delta ^0$ obtained from the standard $2$-simplex by collapsing the final edge to a point. We then have an inclusion map $\iota : X \hookrightarrow \Delta ^1 \diamond \Delta ^0$ (corresponding to the triangle in the upper right of the preceding diagram), which fits into a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{r} \ar [d] & \Delta ^1 \ar [d]^{u} \\ \Delta ^1 \diamond \Delta ^0 \ar [r]^-{c_{ \Delta ^1, \Delta ^0}} & \Delta ^1 \star \Delta ^0; } \]

here $u$ classifies to the “long edge” of the $2$-simplex $\Delta ^1 \star \Delta ^0 \simeq \Delta ^2$. Since the vertical maps are monomorphisms and $r$ is a categorical equivalence (see Example 4.5.2.13), it follows that $c_{\Delta ^1, \Delta ^0}$ is also a categorical equivalence (Remark 4.5.3.9). $\square$

Proposition 4.5.5.11. Let $X$ be a simplicial set. Then the comparison map $c_{X, \Delta ^0}: X \diamond \Delta ^{0} \rightarrow X \star \Delta ^{0} = X^{\triangleright }$ is a categorical equivalence of simplicial sets.

Corollary 4.5.5.12. Let $f: A \hookrightarrow B$ be a right anodyne morphism of simplicial sets. Then the induced map

\[ \theta : B \coprod _{A} (A \diamond \Delta ^0) \hookrightarrow B \diamond \Delta ^0 \]

is a categorical equivalence of simplicial sets.

Proof. Proposition 4.3.6.4 guarantees that the natural map $B \coprod _{A} A^{\triangleright } \hookrightarrow B^{\triangleright }$ is inner anodyne, and therefore a categorical equivalence (Corollary 4.5.2.11). Using Proposition 4.5.3.7, we conclude that the diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A^{\triangleright } \ar [d]^{f^{\triangleright }} \\ B \ar [r] & B^{\triangleright } } \]

is categorical pushout square. It then follows from Theorem 4.5.5.7 and Proposition 4.5.3.5 that the equivalent diagram

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{f} & A \diamond \Delta ^0 \ar [d]^{f \diamond \operatorname{id}_{\Delta ^{0}}} \\ B \ar [r] & B \diamond \Delta ^{0} } \]

is also categorical pushout square, so that $\theta $ is a categorical equivalence by virtue of Proposition 4.5.3.7. $\square$

Proof of Theorem 4.5.5.7. Let $X$ and $Y$ be arbitrary simplicial sets; we wish to show that the collapse map $c_{X,Y}: X \diamond Y \rightarrow X \star Y$ is a categorical equivalence. By virtue of Lemma 4.5.5.9, we may assume without loss of generality that $X = \Delta ^1$. Note that the map the $c_{X,Y}$ fits into a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \times Y \ar [r] \ar [d] & (X \diamond \Delta ^0) \times Y \ar [d] \ar [r]^-{c_{X,\Delta ^0} \times \operatorname{id}_ Y} & X^{\triangleright } \times Y \ar [d] \\ X \ar [r] & X \diamond Y \ar [r]^-{ c_{X,Y} } & X \star Y. } \]

Note that the morphism $c_{X,\Delta ^{0}} \times \operatorname{id}_{Y}$ is a categorical equivalence by virtue of Proposition 4.5.5.11 and Remark 4.5.2.7. Consequently, to show that $c_{X,Y}$ is a categorical equivalence, it will suffice to show that the square on the right is a categorical pushout diagram (Proposition 4.5.3.6). Note that the square on the left is a pushout diagram in which the horizontal maps are monomorphisms, hence also a categorical pushout diagram (Proposition 4.5.3.7). We are therefore reduced to showing that the outer rectangle is a categorical pushout square (Proposition 4.5.3.4).

Specializing now to the case $X = \Delta ^1$, we wish to show that the lower square in the commutative diagram

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

is a categorical pushout diagram. We first claim that the upper square is a categorical pushout: by virtue of Proposition 4.5.3.7, this is equivalent to the assertion that the induced map

\[ \theta : (\Delta ^1 \times Y) \coprod _{ \{ 1\} \times Y} (\{ 1\} ^{\triangleright } \times Y) \rightarrow (\Delta ^1)^{\triangleright } \times Y \]

is a categorical equivalence. This follows from Remark 4.5.2.7, since $\theta $ factors as a product of the identity map $\operatorname{id}_{Y}$ with the inner horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. To complete the proof, it will suffice to show that the outer rectangle is a categorical pushout diagram. Using the criterion of Proposition 4.5.3.7, we are reduced to showing that the map

\[ \rho : \Delta ^1 \coprod _{ \{ 1\} \times Y} ( \{ 1\} ^{\triangleright } \times Y) \rightarrow \Delta ^1 \star Y \]

is a categorical equivalence. Unwinding the definitions, we can identify $\rho $ with the composition

\[ \Delta ^{1} \coprod _{ \{ 1\} } ( \{ 1\} \diamond Y ) \xrightarrow {\rho '} \Delta ^1 \coprod _{ \{ 1\} } ( \{ 1\} \star Y) \xrightarrow {\rho ''} \Delta ^1 \star Y. \]

Here the map $\rho '$ is a categorical equivalence by virtue of Proposition 4.5.5.11 (together with Remark 4.5.3.9), and the map $\rho ''$ is inner anodyne by virtue of Proposition 4.3.6.4. $\square$