Does the Grothendieck construction satisfy Fubini's thorem












11












$begingroup$


Suppose we are given a functor



$F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.



It's well-known that the Grothendieck construction in this case evaluates as



$int_{Atimes B}F = (Atimes B)/F$.



We could also apply this construction pointwise to obtain a functor



$int_A F:B^{op}to operatorname{Cat}$



sending $bmapsto A/F(b)$



and similarly



$int_B F:A^{op}to operatorname{Cat}$



We can apply the Grothendieck construction again to each of these functors to obtain categories



$int_Aint_B F$



and



$int_Bint_A F$



Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?






at.algebraic-topology ct.category-theory grothendieck-construction







share|cite|improve this question











$endgroup$

















    11












    $begingroup$


    Suppose we are given a functor



    $F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.



    It's well-known that the Grothendieck construction in this case evaluates as



    $int_{Atimes B}F = (Atimes B)/F$.



    We could also apply this construction pointwise to obtain a functor



    $int_A F:B^{op}to operatorname{Cat}$



    sending $bmapsto A/F(b)$



    and similarly



    $int_B F:A^{op}to operatorname{Cat}$



    We can apply the Grothendieck construction again to each of these functors to obtain categories



    $int_Aint_B F$



    and



    $int_Bint_A F$



    Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?






    at.algebraic-topology ct.category-theory grothendieck-construction







    share|cite|improve this question











    $endgroup$















      11












      11








      11


      1



      $begingroup$


      Suppose we are given a functor



      $F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.



      It's well-known that the Grothendieck construction in this case evaluates as



      $int_{Atimes B}F = (Atimes B)/F$.



      We could also apply this construction pointwise to obtain a functor



      $int_A F:B^{op}to operatorname{Cat}$



      sending $bmapsto A/F(b)$



      and similarly



      $int_B F:A^{op}to operatorname{Cat}$



      We can apply the Grothendieck construction again to each of these functors to obtain categories



      $int_Aint_B F$



      and



      $int_Bint_A F$



      Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?






      at.algebraic-topology ct.category-theory grothendieck-construction







      share|cite|improve this question











      $endgroup$




      Suppose we are given a functor



      $F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.



      It's well-known that the Grothendieck construction in this case evaluates as



      $int_{Atimes B}F = (Atimes B)/F$.



      We could also apply this construction pointwise to obtain a functor



      $int_A F:B^{op}to operatorname{Cat}$



      sending $bmapsto A/F(b)$



      and similarly



      $int_B F:A^{op}to operatorname{Cat}$



      We can apply the Grothendieck construction again to each of these functors to obtain categories



      $int_Aint_B F$



      and



      $int_Bint_A F$



      Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?






      at.algebraic-topology ct.category-theory grothendieck-construction




      at.algebraic-topology ct.category-theory grothendieck-construction






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 13 hours ago









      David White

      11.7k460102




      11.7k460102










      asked 15 hours ago









      Harry GindiHarry Gindi

      9,276778171




      9,276778171






















          1 Answer
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          active

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          7












          $begingroup$

          Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.



          Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            13 hours ago











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          1 Answer
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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          7












          $begingroup$

          Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.



          Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            13 hours ago
















          7












          $begingroup$

          Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.



          Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            13 hours ago














          7












          7








          7





          $begingroup$

          Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.



          Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.






          share|cite|improve this answer











          $endgroup$



          Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.



          Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 13 hours ago

























          answered 13 hours ago









          David WhiteDavid White

          11.7k460102




          11.7k460102












          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            13 hours ago


















          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            13 hours ago
















          $begingroup$
          Great, thanks a bunch!
          $endgroup$
          – Harry Gindi
          13 hours ago




          $begingroup$
          Great, thanks a bunch!
          $endgroup$
          – Harry Gindi
          13 hours ago


















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