In algebraic geometry, what kind of theory can only be described by topos but not a site?












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A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books about Etale cohomology don't use the notion of topos. It seems study of sites is enough for Etale cohomology. But topos was invented in algebraic geometry. I wonder in what area of algebraic geometry, study of topoi can not be replaced by study of sites?






algebraic-geometry topos-theory big-picture







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    4












    $begingroup$


    A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books about Etale cohomology don't use the notion of topos. It seems study of sites is enough for Etale cohomology. But topos was invented in algebraic geometry. I wonder in what area of algebraic geometry, study of topoi can not be replaced by study of sites?






    algebraic-geometry topos-theory big-picture







    share|cite|improve this question











    $endgroup$















      4












      4








      4





      $begingroup$


      A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books about Etale cohomology don't use the notion of topos. It seems study of sites is enough for Etale cohomology. But topos was invented in algebraic geometry. I wonder in what area of algebraic geometry, study of topoi can not be replaced by study of sites?






      algebraic-geometry topos-theory big-picture







      share|cite|improve this question











      $endgroup$




      A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books about Etale cohomology don't use the notion of topos. It seems study of sites is enough for Etale cohomology. But topos was invented in algebraic geometry. I wonder in what area of algebraic geometry, study of topoi can not be replaced by study of sites?






      algebraic-geometry topos-theory big-picture




      algebraic-geometry topos-theory big-picture






      share|cite|improve this question















      share|cite|improve this question













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      share|cite|improve this question








      edited Dec 26 '18 at 14:34







      Nicky

















      asked Dec 26 '18 at 12:51









      NickyNicky

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          2 Answers
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          $begingroup$

          Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.



          However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.



          There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
            $endgroup$
            – Nicky
            Dec 26 '18 at 19:28



















          0












          $begingroup$

          The theorem "topos= sheaf in some site" fails for $infty$-topoi. So I guess spectral algebraic geometry might be a natural answer.



          Edit: Also, even in the ordinary theory, often it's easier to think of cohomology in the topos than in the site. The silliest way to explain this would be: for cohomology in a site which does not have a final object, cohomology of an abelian sheaf $F$ is naturally $text{Ext}^i (*, F).$ The topos has an initial object(*) even though the site doesn't have a final object. The way to describe this by staying in the site would involve derived inverse limits.






          share|cite|improve this answer











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            2 Answers
            2






            active

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            2 Answers
            2






            active

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            active

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            active

            oldest

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            1












            $begingroup$

            Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.



            However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.



            There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
              $endgroup$
              – Nicky
              Dec 26 '18 at 19:28
















            1












            $begingroup$

            Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.



            However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.



            There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
              $endgroup$
              – Nicky
              Dec 26 '18 at 19:28














            1












            1








            1





            $begingroup$

            Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.



            However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.



            There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).






            share|cite|improve this answer









            $endgroup$



            Indeed, all the topoi I personally know to appear in algebraic geometry are Grothendieck toposes, hence toposes of sheaves over some site.



            However, there is NO one-to-one correspondence between morphisms of sites and morphisms of the toposes over them. In particular the morphisms between crystalline toposes can not be realized by morphisms between the underlying sites. This is one reason why we're very much interested in toposes.



            There are also other reasons, for instance that they provide a "base-independent" (site-independent) notion; exploiting that the same topos can be presented by vastly different sites is the heart of Olivia Caramello's program of toposes as bridges. We can also exploit the internal language of a topos allows to reduce some notions and theorems of algebraic geometry to notions and theorems of linear algebra and to develop a synthetic account of scheme theory (see these notes).







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 26 '18 at 17:03









            Ingo BlechschmidtIngo Blechschmidt

            1,395815




            1,395815












            • $begingroup$
              I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
              $endgroup$
              – Nicky
              Dec 26 '18 at 19:28


















            • $begingroup$
              I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
              $endgroup$
              – Nicky
              Dec 26 '18 at 19:28
















            $begingroup$
            I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
            $endgroup$
            – Nicky
            Dec 26 '18 at 19:28




            $begingroup$
            I studied a bit topos theory from logic point of view and read that topos arose in algebraic geometry to define cohomology to solve Weil conjecture. But books on Etale cohomology based on SGA4 don't seem to use the notion of topos. Could you say about the ideas of topos in SGA4?
            $endgroup$
            – Nicky
            Dec 26 '18 at 19:28











            0












            $begingroup$

            The theorem "topos= sheaf in some site" fails for $infty$-topoi. So I guess spectral algebraic geometry might be a natural answer.



            Edit: Also, even in the ordinary theory, often it's easier to think of cohomology in the topos than in the site. The silliest way to explain this would be: for cohomology in a site which does not have a final object, cohomology of an abelian sheaf $F$ is naturally $text{Ext}^i (*, F).$ The topos has an initial object(*) even though the site doesn't have a final object. The way to describe this by staying in the site would involve derived inverse limits.






            share|cite|improve this answer











            $endgroup$


















              0












              $begingroup$

              The theorem "topos= sheaf in some site" fails for $infty$-topoi. So I guess spectral algebraic geometry might be a natural answer.



              Edit: Also, even in the ordinary theory, often it's easier to think of cohomology in the topos than in the site. The silliest way to explain this would be: for cohomology in a site which does not have a final object, cohomology of an abelian sheaf $F$ is naturally $text{Ext}^i (*, F).$ The topos has an initial object(*) even though the site doesn't have a final object. The way to describe this by staying in the site would involve derived inverse limits.






              share|cite|improve this answer











              $endgroup$
















                0












                0








                0





                $begingroup$

                The theorem "topos= sheaf in some site" fails for $infty$-topoi. So I guess spectral algebraic geometry might be a natural answer.



                Edit: Also, even in the ordinary theory, often it's easier to think of cohomology in the topos than in the site. The silliest way to explain this would be: for cohomology in a site which does not have a final object, cohomology of an abelian sheaf $F$ is naturally $text{Ext}^i (*, F).$ The topos has an initial object(*) even though the site doesn't have a final object. The way to describe this by staying in the site would involve derived inverse limits.






                share|cite|improve this answer











                $endgroup$



                The theorem "topos= sheaf in some site" fails for $infty$-topoi. So I guess spectral algebraic geometry might be a natural answer.



                Edit: Also, even in the ordinary theory, often it's easier to think of cohomology in the topos than in the site. The silliest way to explain this would be: for cohomology in a site which does not have a final object, cohomology of an abelian sheaf $F$ is naturally $text{Ext}^i (*, F).$ The topos has an initial object(*) even though the site doesn't have a final object. The way to describe this by staying in the site would involve derived inverse limits.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 29 '18 at 7:25

























                answered Dec 29 '18 at 7:06









                Shubhodip MondalShubhodip Mondal

                2,186916




                2,186916






























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