Does the Grothendieck construction satisfy Fubini's thorem












12












$begingroup$


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 $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?










share|cite|improve this question











$endgroup$

















    12












    $begingroup$


    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 $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?










    share|cite|improve this question











    $endgroup$















      12












      12








      12


      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 $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?










      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 9 at 15:46









      David White

      13.1k462105




      13.1k462105










      asked Feb 9 at 13:50









      Harry GindiHarry Gindi

      9,720778174




      9,720778174






















          1 Answer
          1






          active

          oldest

          votes


















          9












          $begingroup$

          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
            Feb 9 at 15:27












          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "504"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f322825%2fdoes-the-grothendieck-construction-satisfy-fubinis-thorem%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          9












          $begingroup$

          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
            Feb 9 at 15:27
















          9












          $begingroup$

          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
            Feb 9 at 15:27














          9












          9








          9





          $begingroup$

          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 Feb 9 at 15:45

























          answered Feb 9 at 15:19









          David WhiteDavid White

          13.1k462105




          13.1k462105












          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            Feb 9 at 15:27


















          • $begingroup$
            Great, thanks a bunch!
            $endgroup$
            – Harry Gindi
            Feb 9 at 15:27
















          $begingroup$
          Great, thanks a bunch!
          $endgroup$
          – Harry Gindi
          Feb 9 at 15:27




          $begingroup$
          Great, thanks a bunch!
          $endgroup$
          – Harry Gindi
          Feb 9 at 15:27


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to MathOverflow!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f322825%2fdoes-the-grothendieck-construction-satisfy-fubinis-thorem%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Human spaceflight

          Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

          張江高科駅