Localic reflection

The category \(T_{\leq n-1}\) is an \(n\)-topos. The presentation theorem for \(n\)-topoi provides a small \((n,1)\)-category \(C\) with finite limits and a Grothendieck topology \(\tau\) for which One may obtain such a site by choosing a sufficiently large regular cardinal \(\kappa\), taking a small finite-limit-closed subcategory of \(\kappa\)-compact objects which generates \(T_{\leq n-1}\) as an \(n\)-topos, and equipping it with the effective epimorphism topology. This proves (1); see [Lurie 2009, Theorem 6.4.1.5 and Proposition 6.4.5.9] for the presentation theorem and its localic refinement.Set \(L_nT:=\Shv_{\tau}(C)\). By [Lurie 2009, Proposition 6.4.5.9], this topos is \(n\)-localic. Its \((n-1)\)-truncated objects are identified with those of \(T\), independently of the chosen presentation. If \(S\) is any \(n\)-localic topos, then the defining property of \(n\)-localic topoi gives natural equivalences \begin{align*} \Geom(L_nT,S) &\iso \Fun^{\lex,\colim}(S_{\leq n-1},(L_nT)_{\leq n-1}) \\ &\iso \Fun^{\lex,\colim}(S_{\leq n-1},T_{\leq n-1}) \\ &\iso \Geom(T,S). \end{align*} Thus \(L_n\) is left adjoint to the inclusion \(\Topos_n\hookrightarrow\Topos\), proving (2).