nForum - Discussion Feed (properties of flat functors) 2023-05-29T09:05:25+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher DavidCarchedi comments on "properties of flat functors" (17639) https://nforum.ncatlab.org/discussion/2045/?Focus=17639#Comment_17639 2010-10-31T15:02:37+00:00 2023-05-29T09:05:25+00:00 DavidCarchedi https://nforum.ncatlab.org/account/186/ Damn, we're actually going in circles then, because Urs and I are trying to PROVE a certain reflective subcategory is a topos by showing that its reflector (which may be written as a Kan-extension) ... Damn, we're actually going in circles then, because Urs and I are trying to PROVE a certain reflective subcategory is a topos by showing that its reflector (which may be written as a Kan-extension) is left-exact. ]]> Mike Shulman comments on "properties of flat functors" (17632) https://nforum.ncatlab.org/discussion/2045/?Focus=17632#Comment_17632 2010-10-29T22:54:16+00:00 2023-05-29T09:05:25+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ It ought to be true when E is a (Grothendieck) topos (prove it for presheaf categories and then for left-exact-reflective subcategories). Conversely, if E is cocomplete and finitely complete, has ...

It ought to be true when E is a (Grothendieck) topos (prove it for presheaf categories and then for left-exact-reflective subcategories). Conversely, if E is cocomplete and finitely complete, has the stated property (i.e.the Yoneda extension of any left exact functor from a finitely complete category into E is left exact), and moreover has a small dense subcategory C, then if C’ denotes the closure of C in E under finite limits, the Yoneda extension of the inclusion C’→E must be a left exact reflection of the embedding of E into Psh(C’), so that E is a topos.

In fact, even if instead of having a small dense subcategory we assume that E is totally cocomplete, meaning that its own Yoneda embedding E→Psh(E) has a left adjoint, then the stated property implies that that left adjoint is left exact, i.e. E is “lex total.” And as long as ob(E) is no bigger than ob(Set), then lex-totality in fact also implies that E is a topos. So morally, at least, I think the extra condition is just “being a topos.”

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Urs comments on "properties of flat functors" (17626) https://nforum.ncatlab.org/discussion/2045/?Focus=17626#Comment_17626 2010-10-29T14:49:44+00:00 2023-05-29T09:05:25+00:00 Urs https://nforum.ncatlab.org/account/4/ At flat functor there is a statement that a functor F:C&rightarrow;&Escr;F : C \to \mathcal{E} on a complete category to a cocomplete category is left exact precisely if its Yoneda extension ...

At flat functor there is a statement that a functor $F : C \to \mathcal{E}$ on a complete category to a cocomplete category is left exact precisely if its Yoneda extension is.

I know this for $\mathcal{E} = Set$. There must be some extra conditions on $\mathcal{E}$ that have to be mentioned here.

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