All geometric morphisms between Grothendieck topoi admit an essentially unique (terminally connected, pro-etale) factorization, extending the classical (connected, etale) factorization.
Sketches of an Elephant: A Topos Theory Compendium
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On a (terminally connected, pro-etale) factorization of geometric morphisms
All geometric morphisms between Grothendieck topoi admit an essentially unique (terminally connected, pro-etale) factorization, extending the classical (connected, etale) factorization.