Kerodon

Variant 8.1.7.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $R$ be a collection of edges of $\operatorname{\mathcal{C}}$. We let $\operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ denote the simplicial subset $\operatorname{Cospan}^{A,R}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$, where $A$ is the collection of all edges of $\operatorname{\mathcal{C}}$. Similarly, if $L$ is a collection of edges of $\operatorname{\mathcal{C}}$, we let $\operatorname{Cospan}^{L,\mathrm{all}}(\operatorname{\mathcal{C}})$ denote the simplicial subset $\operatorname{Cospan}^{L,A}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{C}})$. Note that the simplicial set $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ of Definition 8.1.6.1 can be recovered as the intersection $\operatorname{Cospan}^{L, \mathrm{all}}(\operatorname{\mathcal{C}}) \cap \operatorname{Cospan}^{\mathrm{all}, R}(\operatorname{\mathcal{C}})$.