Span pattern

Convention 17.3.1. Throughout this section, we fix an adequate triple \(\Ff = (F,F_L,F_R)\) which satisfies the following three conditions:

(1)

The \(\infty \)-category \(F\) is extensive in the sense of Definition 13.3.1, and the triple \((F,F_L,F_R)\) is weakly coextensive in the sense of Definition 13.3.3;

(2)

Morphisms in \(F_L\) satisfy left cancellation: given \(l\colon I \to J\) and \(l'\colon J \to K\), if \(l' \in F_L\) and \(l'l \in F_L\) then also \(l \in F_L\);

(3)

For every \(I\in F\), the morphism \(\emptyset \to I\) is contained in \(F_L\).

Note that this in particular implies that \(F_L\) contains all inclusions \(I_i \hookrightarrow \bigsqcup _{j=1}^n I_j\) for \(I_1, \dots , I_n \in F\): by weak coextensivity, \(F_L\) is closed under finite coproducts, so the inclusion is the coproduct of \(\id _{I_i}\) with the morphisms \(\emptyset \to I_j\) for \(j \neq i\).