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Warning 4.3.3.31. 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 4.5.8.1), 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.