Span category

Notation 13.1.12. By Proposition 13.1.11 and Theorem 24.1.10, the functor \(\NSpan \colon \AdTrip \to \sAn \) factors uniquely as

We refer to the resulting \(\infty \)-category \(\Span _{L,R}(C)\) as the span category of the adequate triple.

If \(C\) admits pullbacks, we write \[ \Span (C) := \Span _{\all ,\all }(C) \] for the span category associated with the adequate triple \((C,C,C)\). In particular, \(\Span (\Fin )\) is the rigorous construction of the span category of finite sets promised in Construction 5.3.1.