Notation 6.41.
For each natural number \(n \geq 0\), let \(\simp^{\leq n}\) denote the full subcategory of \(\simp\) spanned by the objects \(\{[0], [1], \dots, [n]\}\), and set \(\simp^{\leq -1}:=\emptyset\). If \(T\) is a topos, the restriction functor
admits a right adjoint given by right Kan extension along the inclusion \((\simp^{\leq n})\catop \hookrightarrow \simp\catop\). We let
denote the composition of the restriction functor with its right adjoint and refer to \(\cosk_n\) as the \(n\)-coskeleton functor. In particular, \(\cosk_{-1}\) is the constant simplicial object at the terminal object of \(T\). A simplicial object \(U_\bullet\) is called \(n\)-coskeletal if the unit map \(U_\bullet \to \cosk_n(U_\bullet)\) is an isomorphism, or equivalently if \(U_\bullet\) is a right Kan extension of its restriction to \((\simp^{\leq n})\catop\).