Lemma 6.65. ([Lurie 2018, Lemma A.7.1.4])
The functor \(L_n \colon \Topos \to \Topos_n\) preserves colimits and small limits.
Proof
Preservation of colimits follows because \(L_n\) is a left adjoint. The limit-preservation statement is [Lurie 2018, Lemma A.7.1.4]; it is the additional input needed to commute localic reflection with the inverse tower of localic approximations.
References
- Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.