I am not an algebraic geometer in the first place, and I am mainly familiar with topology and category theory. Recently I am studying Grothendieck categories and I am struggling with getting acquainted with them.

What is a Grothendieck category?

This question will appear extremely naive, especially to algebraic geometers, thus please give me some time to comment and define a bit of context.

Rem 1. Indeed, following Gabriel-Popescu theorem and the general theory of lex-localizations, it follows that a Grothendieck category is no more than an $\mathbb{Ab}$-topos. By $\mathsf{V}$-topos we intend an enriched lex-reflective subcategory of an (enriched) presheaf category $\mathsf{V}^C$. Even better, following the research line lead by Borceux and later developed by Lowen, even site presentations of these abelian topoi are available.

Rem 2. Yet, in order to study Grothendieck categories with the same intuition of Grothendieck topoi, we should study geometric morphisms between them. This notion does not appear to be the correct one.

What is the correct notion of morphism between Grothendieck categories?

Cocontinuous functors. For example, if we look at the construction of Quasi-coherent sheaves over a scheme $\mathsf{Qcoh}(-)$, it looks natural to study inverse images (that is cocontinuous functors). Yet, quasicoherent sheaves over a scheme are not precisely the abelian analogous of sheaves over a manifold (for several reasons). Maybe a more natural analogous would be to study abelian group objects internal to little Zariski topos. Also in this case, the natural notion of functor seems (to me) to be cocontinuous functors.

Brandenburg's PhD thesis. The global pictures gets messier when we add Brandenburg's Phd thesis to the cocktail. To my superficial understanding, he studies Grothendieck categories equipped with a tensor product (motivated by very natural considerations that look quite convincing to me) and thus considers cocontinuous functors preserving the tensorial structure. This notion seems to me as the most promising notion of geometric morphism where lex-ity is substituted by the preservation of the tensorial structure.

Back to $\mathbb{Ab}$-topoi, flatness and intuiton. Putting all these observations together, the $2$-category of $\mathbb{Ab}$-topoi (namely Grothendieck categories and lex-cocontinuous functors) seems a strange object to look at, and yet it is naturally connected to the study of flat morphisms between schemes.

What should be my geometric intuition (!) on flat morphisms of Grothendieck categories? (And maybe more generally on flat morphisms of schemes? (I did not find satisfying this answer))

Any kind of comment to any statement I have written is extremely welcome, and I encourage everybody to correct my mistakes. As I mentioned above, I am not a specialist in this subject and I am learning.

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    $\begingroup$ Small comment: there are multiple sources of sheaf categories in algebraic geometry. Flatness is automatic when the structure sheaf is constant (e.g. the étale site with sheaves of $\mathbf Z/n$-modules), but in the quasi-coherent setting is related to the change of rings $f^{-1}\mathcal O_Y \to \mathcal O_X$ (see for example Tag 04JA). Which of the two are you trying to capture? $\endgroup$ – R. van Dobben de Bruyn Apr 6 '20 at 17:49
  • $\begingroup$ @R.vanDobbendeBruyn I like your comment and I am not sure about what I should want, I invite you to elaborate it. I know nothing about etàle site. What story do they tell? $\endgroup$ – Ivan Di Liberti Apr 6 '20 at 19:10
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    $\begingroup$ Basically there are two kind of cohomological coefficients continuous and discrete (after Grothendieck). The former are basically quasi-coherent sheaves, the latter local systems in the classical case. As Zariski topology does not capture well the discrete coefficients one has to switch topos and consider the Étale topology and for p-torsion crystalline cohomology. Every theory has an associated Grothendieck category of abelian sheaves (with extra structure). $\endgroup$ – Leo Alonso Apr 6 '20 at 21:15
  • $\begingroup$ Is a "flat morphism" of Grothendieck categories a functor with an exact left adjoint? $\endgroup$ – Tim Campion May 4 '20 at 17:58
  • $\begingroup$ Yes, that's what I mean. $\endgroup$ – Ivan Di Liberti May 4 '20 at 19:11

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