Proposition 6.66. ({cf. [Lurie 2018, Proposition A.7.1.5 and Corollary A.7.2.8]})
The functor \(L_*\colon \Topos \to \lim_n \Topos_n\) admits fully faithful adjoints on both sides.
A topos \(T\) lies in the image of the left adjoint if and only if it is Postnikov complete.
A topos \(T\) lies in the image of the right adjoint if and only if it is bounded.
Proof
The right adjoint sends a compatible system \((T_n)_n\) to \(\lim_nT_n\). By Lemma 6.65,
\[L_m\bigl(\lim_nT_n\bigr)\iso\lim_nL_mT_n\iso T_m.\]
Thus the right adjoint is fully faithful, and its essential image consists precisely of the bounded topoi. In particular, the bounded reflection of an arbitrary topos is \[T^b:=\lim_nL_nT.\]
We next identify the left adjoint. For arbitrary topoi \(T\) and \(S\), the universal properties of truncation and localic reflection give \begin{align*} \Geom(\widehat T,S) &\iso \lim_n\Fun^{\lex,\colim}(S_{\leq n},T_{\leq n}), \\ \Geom(T,S^b) &\iso \lim_n\Geom(T,L_nS) \\ &\iso \lim_n\Fun^{\lex,\colim}(S_{\leq n-1},T_{\leq n-1}). \end{align*} Reindexing gives a natural equivalence \(\Geom(\widehat T,S)\iso\Geom(T,S^b)\). Thus Postnikov-completion is left adjoint to bounded reflection.The Postnikov-completion \(\widehat T\) is Postnikov-complete and has the same truncated objects as \(T\). Conversely, if \(T\) is Postnikov-complete, then \(T\iso\widehat T\iso\widehat{T^b}\). Hence the essential image of the fully faithful left adjoint is exactly \(\Topos^{\post}\), while the essential image of the right adjoint is \(\Topos^b\).References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.