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I have several questions about geometric morphisms of topoi. It was recommended that I move my question here from Math.Stackexchange, since it would be good to get an expert on topos theory to answer.

1) A geometric morphism $g \dashv f : C \rightarrow D$, with $f : C \rightarrow D$ is called strongly connected if $g$ has a further left adjoint $h : C \rightarrow D$ which preserves finite products. This is equivalently an essential geometric morphism in which the left-most adjoint preserves finite products. Do people ever consider geometric morphisms $g \dashv f : C \rightarrow D$, with $f : C \rightarrow D$, where $f : C \rightarrow D$ has a right adjoint $h : C \rightarrow D$ which preserves finite coproducts? This seems like the natural dual of strongly connected, but does it have any significance in topos theory?

2) Is there the opposite notion of a local geometric morphism? That would seem to be a geometric morphism $g \dashv f : C \rightarrow D$ which has a further left adjoint $h \dashv g$, $h : C \rightarrow D$ such that $h$ is fully faithful.

3) I am curious about the dual to essential morphisms as well.

I expect that local geometric morphisms are more significant than their opposite cousins, and the same for strongly connected morphisms, since topoi are not self dual. Still, an explanation of the "asymmetrical significance" would help me to understand why we consider one and not the other.

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  1. I don't know of any significance to this notion. I would explain this by the fact that the notion of topos is not self-dual. Finite limits (including finite products) are an integral part of the notion of topos (and geometric morphism), but finite colimits are not; so we can't expect to always get a meaningful notion by dualizing a definition and applying it to toposes.

  2. There is a dual notion to that of local geometric morphism, but to get it we have to choose a different characterization of local geometric morphisms to dualize. A geometric morphism $f_* : C \to D : f^*$ is local just when it has a left adjoint in the slice category ${\rm Top}/D$ of the 2-category of toposes and geometric morphisms. Dually, if $f$ has a right adjoint in that 2-category, it is called totally connected. You can read a bit more about totally connected morphisms in C3.6.16-19 of Sketches of an Elephant.

  3. Essential-ness is not a very topos-theoretic notion, so I don't expect its dual to be important either. A more topos-theoretic variant of essential geometric morphism is to require the left adjoint $f_!$ of $f^*$ to be $D$-indexed, which is equivalent to saying that $f$ is a locally connected morphism. The "dual" of this would be to ask that $f_*$ has a right adjoint that is $D$-indexed, which turns out to be another equivalent characterization of when $f$ is local!

    This "duality" is not really a topos-theoretic "duality" though, although it is formally dual in a 2-categorical sense. Topos theoretic duality tends to interchange locally-connected-type conditions and proper-type conditions, whereas both local and locally-connected geometric morphisms are of the first type.

I highly recommend chapter C3 of Sketches of an Elephant if you haven't read it yet. You may also be interested in some remarks here and here about its categorifications.

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  • $\begingroup$ Thanks @Mike, this really helped. Do you know of any other formal dualities in the 2-category Topos like these two? $\endgroup$ – Kind Bubble Jan 22 '20 at 21:52
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    $\begingroup$ @DeanYoung, diliberti.github.io/Talk/Scott.pdf $\endgroup$ – Ivan Di Liberti Jan 22 '20 at 21:58
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    $\begingroup$ @DeanYoung Nothing else offhand. $\endgroup$ – Mike Shulman Jan 24 '20 at 3:35

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