The Stacks project

Definition 7.29.2. Let $\mathcal{C}$, $\mathcal{D}$ be sites. A special cocontinuous functor $u$ from $\mathcal{C}$ to $\mathcal{D}$ is a cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$ satisfying the assumptions and conclusions of Lemma 7.29.1.

Comments (4)

Comment #2041 by Ruian Chen on

Just out of curiosity, why do we emphasize cocontinuous in the name, as the functor is both cocontinuous and continuous by the assumptions of Lemma 7.28.1 anyway?

Comment #2079 by on

Just a choice and not a very good one. The idea is that these types of functors are easy to study and easy to construct as the material in this section shows. Then we'll use them later to massage any morphism of topoi into a sequence of morphisms of topoi given by these ones or their inverses and morphisms of topoi coming from morphisms of sites. Anyway, can anybody suggest a better terminology?

Comment #5362 by Mike Shulman on

"Dense morphism"? (cf. e.g. section 11 of my paper http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html, which may not be exactly the same but is very closely related)

Comment #5599 by on

Sorry, I don't like dense for this particular notion. It does seem your definition is rather close.

There are also:

  • 7 comment(s) on Section 7.29: Morphisms of topoi

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View Definition 7.29.2 as pdf