Construction 13.1.10. Let \((C,C_L,C_R)\) be an adequate triple. We define the simplicial anima \(\NSpan _{L,R}(C) \in \sAn \) as the following composite: \[ \simp \catop \xrightarrow {\Tw ^r} \AdTrip \catop \xrightarrow {\Hom _{\AdTrip }(-,(C,C_L,C_R))} \An . \] In other words, for \(n \geq 0\) we consider the subanima \[ \NSpan _{L,R}(C)_n \subseteq \Map (\Tw ^r([n]),C) \] consisting of the morphisms of adequate triples. This construction is natural in the adequate triple, hence defines a functor \[ \NSpan \colon \AdTrip \to \sAn . \]
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