Proposition 5.69.
Let \(\varphi\colon T \to S\) be a morphism of logoi. Then \(\lra{\varphi(T)}\) is a logos, and the inclusion \(\lra{\varphi(T)} \hookrightarrow S\) is a morphism of logoi. In particular, \(\varphi\) admits a factorization in \(\Logos\) as
\[T \longrightarrow \lra{\varphi(T)} \lhook\joinrel\longrightarrow S,\]
where the second map is a monomorphism in \(\Logos\).
Proof
Presentability is the substantive point and is proved in [Lurie 2009, Proposition 6.3.6.2]. By construction, the inclusion \(\lra{\varphi(T)}\hookrightarrow S\) preserves all colimits and finite limits. Consequently, colimits in the image are universal, coproducts are disjoint, and groupoid objects are effective because the corresponding statements hold in \(S\) and the inclusion preserves and reflects the diagrams involved. Thus the Giraud conditions of Theorem 2.42 hold in \(\lra{\varphi(T)}\).The inclusion is therefore a morphism of logoi. Since it is fully faithful, it is a monomorphism in \(\Logos\).
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.