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Construction (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, and $X^{+} \star Y^{+}$ denotes the join of Construction 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