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Construction Let $\operatorname{\mathcal{C}}$ be a simplicial set. For every integer $n \geq 0$, we let $\operatorname{Cospan}_{n}(\operatorname{\mathcal{C}})$ denote the collection of morphisms $\operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{C}}$ in the category of simplicial sets. The construction $[n] \mapsto \operatorname{Cospan}_{n}(\operatorname{\mathcal{C}})$ depends functorially on the set $[n] = \{ 0 < 1 < \cdots < n \} $ as an object of the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$, and can therefore be viewed as a simplicial set. We will denote this simplicial set by $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ and refer to it as the simplicial set of cospans in $\operatorname{\mathcal{C}}$.