The join operation on simplicial sets admits a topological interpretation.
Construction 4.3.4.1. Let $X$ and $Y$ be topological spaces, and let $[0,1] = | \Delta ^1 |$ denote the unit interval. We let $X \star Y$ denote the topological space given by the iterated pushout We will refer to $X \star Y$ as the join of $X$ and $Y$.
\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times [0,1] \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y. \]
Remark 4.3.4.2. Let $X$ and $Y$ be topological spaces. Then the join $X \star Y$ of Construction 4.3.4.1 is equipped with a pair of maps $\iota _{X}: X \hookrightarrow X \star Y$ and $\iota _{Y}: Y \hookrightarrow X \star Y$. It is not difficult to see that these maps are closed embeddings: that is, they induce homeomorphisms from $X$ and $Y$ onto closed subsets of $X \star Y$. We will generally abuse notation by identifying $X$ and $Y$ with their images under $\iota _{X}$ and $\iota _{Y}$, respectively.
Remark 4.3.4.3. Let $X$, $Y$, and $Z$ be topological spaces. Then the datum of a continuous function $X \star Y \rightarrow Z$ is equivalent to the datum of a triple $(f_ X, f_ Y, h)$, where $f_ X: X \rightarrow Z$ and $f_ Y: Y \rightarrow Z$ are continuous functions and $h: X \times [0,1] \times Y \rightarrow Z$ is a homotopy from $f_{X} \circ \pi _{X}$ to $f_{Y} \circ \pi _{Y}$; here $\pi _{X}: X \times Y \rightarrow X$ and $\pi _{Y}: X \times Y \rightarrow Y$ denote the projection maps.
Remark 4.3.4.4 (Symmetry). Let $X$ and $Y$ be topological spaces. Then there is a canonical homeomorphism $X \star Y \simeq Y \star X$, which is induced by the homeomorphism
\[ X \times [0,1] \times Y \rightarrow Y \times [0,1] \times X \quad \quad (x,t,y) \mapsto (y, 1-t, x). \]
Example 4.3.4.5 (Cones). Let $\ast $ denote the topological space consisting of a single point. For any topological space $X$, we write $X^{\triangleleft }$ for the join $\ast \star X$, and $X^{\triangleright }$ for the join $X \star \ast $, given more concretely by the formulae We will refer to both $X^{\triangleleft }$ and $X^{\triangleright }$ as the cone on $X$ (note that they are canonically homeomorphic, by virtue of Remark 4.3.4.4).
\[ X^{\triangleleft } = \ast \coprod _{ (\{ 0\} \times X) }( [0,1] \times X) \quad \quad X^{\triangleright } = (X \times [0,1]) \coprod _{ (X \times \{ 1\} ) } \ast . \]
Remark 4.3.4.6. Let $X$ be a locally compact Hausdorff space. Then the functor preserves colimits. This follows from the fact that the functors $Y \mapsto X \times Y$ and $Y \mapsto X \times [0,1] \times Y$ preserve colimits.
\[ \operatorname{Top}\rightarrow \operatorname{Top}_{X/} \quad \quad Y \mapsto X \star Y \]
Example 4.3.4.7. For each integer $n \geq 0$, let $| \Delta ^{n} | = \{ (u_0, \ldots , u_ n) \in \operatorname{\mathbf{R}}_{\geq 0}: u_0 + \cdots + u_ n = 1 \} $ denote the topological $n$-simplex. For $p,q \geq 0$, we have maps $| \Delta ^{p} | \xrightarrow {\iota } | \Delta ^{p+1+q} | \xleftarrow {\iota '} | \Delta ^{q} |$ given by the formulae There is a “straight-line” homotopy $h: | \Delta ^{p} | \times [0,1] \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ from $\iota \circ \pi _{| \Delta ^{p} | }$ to $\iota ' \circ \pi _{ | \Delta ^{q} |}$, given concretely by the formula By virtue of Remark 4.3.4.3, the triple $(\iota , \iota ', h)$ can be identified with a continuous function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$.
\[ \iota ( u_0, \ldots , u_ p) = (u_0, \ldots , u_ p, 0, \ldots , 0) \quad \quad \iota '( v_0, \ldots , v_ q) = (0, \ldots , 0, v_0, \ldots , v_ q). \]
\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]
Proposition 4.3.4.8. Let $p$ and $q$ be nonnegative integers. Then the function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ of Example 4.3.4.7 is a homeomorphism of topological spaces.
Proof. Since $| \Delta ^ p | \star | \Delta ^ q |$ is compact and $| \Delta ^{p+1+q} |$ is Hausdorff, the continuous function $H_{p,q}$ is automatically closed. To complete the proof, it will suffice to show that $H_{p,q}$ is bijective. Fix a point $x$ of $| \Delta ^{p+1+q} |$, given by a sequence of nonnegative real numbers $( \overline{u}_0, \ldots , \overline{u}_ m, \overline{v}_0, \overline{v}_1, \ldots , \overline{v_ n} )$ satisfying
Set $t = \overline{v}_0 + \cdots + \overline{v_ n}$. If $t = 0$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{u}_0, \ldots , \overline{u}_ m)$. If $t = 1$, the set $H_{p,q}^{-1} \{ x \} $ consists of a single point of $| \Delta ^ q |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{v}_0, \ldots , \overline{v}_ m)$. In the case $0 < t < 1$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} | \star | \Delta ^{q} |$, given as the image of the triple
We now compare the join operation on topological spaces (given by Construction 4.3.4.1) to the join operation on simplicial sets (given by Construction 4.3.3.13).
Construction 4.3.4.9. Let $X$ and $Y$ be simplicial sets, and let $\sigma : \Delta ^ n \rightarrow X \star Y$ be a morphism. We define a continuous function $f(\sigma ): | \Delta ^ n | \rightarrow |X| \star |Y|$ as follows (see Remark 4.3.3.17):
If $\sigma $ factors through $X$, we let $f(\sigma )$ denote the composition
where the second map is the inclusion of Remark 4.3.4.2.
If $\sigma $ factors through $Y$, we let $f(\sigma )$ denote the composition
where the second map is the inclusion of Remark 4.3.4.2.
If $\sigma $ factors as a composition
then we let $f(\sigma )$ denote the composite map
where $H_{p,q}$ denotes the homeomorphism of Proposition 4.3.4.8.
\[ | \Delta ^ n | \xrightarrow { | \sigma | } |X| \xrightarrow {\iota _{|X|}} |X| \star |Y|, \]
\[ | \Delta ^ n | \xrightarrow { | \sigma | } |Y| \xrightarrow { \iota _{|Y|}} |X| \star |Y|, \]
\[ \Delta ^ n = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]
\[ | \Delta ^ n | = | \Delta ^{p+1+q} | \xrightarrow { H_{p,q}^{-1} } | \Delta ^{p} | \star | \Delta ^{q} | \xrightarrow { | \sigma _{-} | \star | \sigma _{+} | } |X| \star |Y|, \]
The construction $\sigma \mapsto f(\sigma )$ is compatible with face and degeneracy operators, and therefore determines a morphism of simplicial sets $f: X \star Y \rightarrow \operatorname{Sing}_{\bullet }( |X| \star |Y| )$. We will identify $f$ with a continuous function $T_{X,Y}: | X \star Y | \rightarrow |X| \star |Y|$, which we will refer to as the join comparison map.
Example 4.3.4.10. Let $X = \Delta ^{p}$ and $Y = \Delta ^{q}$ be standard simplices. Then the join comparison map $T_{X,Y}: | \Delta ^{p} \star \Delta ^{q} | \rightarrow |\Delta ^{p}| \star | \Delta ^{q} |$ fits into a commutative diagram where $\rho : \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ denotes the isomorphism of simplicial sets appearing in Example 4.3.3.14 and $H_{p,q}$ is the homeomorphism of Proposition 4.3.4.8. In particular, $T_{X,Y}$ is a homeomorphism.
\[ \xymatrix@R =50pt@C=50pt{ | \Delta ^{p} \star \Delta ^{q} | \ar [dr]^{ | \rho | } \ar [rr]^{ T_{X,Y} } & & | \Delta ^{p} | \star | \Delta ^{q} | \ar [dl]_{ H_{p,q} } \\ & | \Delta ^{p+1+q} |, & } \]
Proposition 4.3.4.11. Let $X$ and $Y$ be simplicial sets. If either $X$ or $Y$ is finite, then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ of Construction 4.3.4.9 is a homeomorphism.
Proof. Without loss of generality, we may assume that $X$ is finite. Then the geometric realization $|X|$ is a compact Hausdorff space (Corollary 3.6.1.12). Using Remarks 4.3.4.6 and 4.3.3.31, we see that the functors
preserve colimits. Consequently, if we regard $X$ as fixed, then the collection of simplicial sets $Y$ for which $T_{X,Y}$ is a homeomorphism is closed under colimits. Since every simplicial set can be realized as a colimit of standard simplices (Remark 1.1.3.13), it will suffice to prove Proposition 4.3.4.11 in the special case where $Y = \Delta ^{q}$ is a standard simplex. In this case, $Y$ is also finite. Repeating the preceding argument (with the roles of $X$ and $Y$ reversed), we are reduced to proving that $T_{X,Y}$ is a homeomorphism in the case where $X = \Delta ^{p}$ is also a standard simplex. In this case, the desired result follows from Example 4.3.4.10. $\square$
Corollary 4.3.4.12. Let $X$ be a simplicial set. Then the join comparison maps $T_{\Delta ^0,X}$ and $T_{X, \Delta ^0}$ supply homeomorphisms of topological spaces Here $X^{\triangleleft }$ and $X^{\triangleright }$ denote the left and right cones on $X$ in the category of simplicial sets (Construction 4.3.3.26), while $|X|^{\triangleleft }$ and $|X|^{\triangleright }$ denote the cone $|X|$ in the category of topological spaces (Example 4.3.4.5).
\[ | X^{\triangleleft } | \simeq |X|^{\triangleleft } \quad \quad | X^{\triangleright } | \simeq | X |^{\triangleright }. \]
The join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ need not be a homeomorphism in general. However, we do have the following:
Corollary 4.3.4.13. Let $X$ and $Y$ be simplicial sets. Then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ is a bijection.
Proof. As a map of sets, we can realize $T_{X,Y}$ as a filtered colimit of join comparison maps $T_{X', Y}$, where $X'$ ranges over the finite simplicial subsets of $X$ (Remark 3.6.1.8). Each of these maps is bijective (even a homeomorphism), by virtue of Proposition 4.3.4.11. $\square$
Warning 4.3.4.14. Let $X$ and $Y$ be simplicial sets, and let $X \diamond Y$ denote the simplicial set given by the iterated coproduct (see Notation 4.5.8.3). Since the formation of geometric realization commutes with the formation of colimits, we have an evident comparison map of topological spaces This map is always bijective, and is a homeomorphism if either $X$ or $Y$ is finite (see Corollary 3.6.2.2). In this case, Corollary 4.3.4.13 supplies a homeomorphism of geometric realizations $| X \diamond Y | \simeq | X \star Y |$. Beware that this homeomorphism does not arise from a morphism of simplicial sets. In the case $X = \Delta ^ p$ and $Y = \Delta ^{q}$, it arises from the homotopy appearing in Example 4.3.4.7, which is not piecewise-linear with respect to the natural triangulation of the polysimplex $| \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^ q |$.
\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y \]
\[ |X \diamond Y | \rightarrow |X| \star |Y|. \]
\[ h: | \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} | \]
\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]