Does the Grothendieck construction satisfy Fubini's thorem












10












$begingroup$


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











$endgroup$

















    10












    $begingroup$


    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











    $endgroup$















      10












      10








      10


      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 8 hours ago









      David White

      11.7k460102




      11.7k460102










      asked 10 hours ago









      Harry GindiHarry Gindi

      9,271778171




      9,271778171






















          1 Answer
          1






          active

          oldest

          votes


















          6












          $begingroup$

          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
            8 hours ago











          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









          6












          $begingroup$

          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
            8 hours ago
















          6












          $begingroup$

          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
            8 hours ago














          6












          6








          6





          $begingroup$

          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 8 hours ago

























          answered 9 hours ago









          David WhiteDavid White

          11.7k460102




          11.7k460102












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


















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
















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




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


















          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

          How to label and detect the document text images

          Image
          1 $begingroup$ This is what I mean as document text image: I want to label the texts in image as separate blocks and my model should detect these labels as classes. NOTE: This is how the end result should be like: The labels like Block 1, Block 2, Block 3,.. should be Logo, Title, Date,.. Others, etc. Work done: First approach : I tried to implement this method via Object Detection, it didn't work. It didn't even detect any text. Second approach : Then I tried it using PixelLink. As this model is build for scene text detection, it detected each and every text in the image. But this method can detect multiple lines of text if the threshold values are increased. But I have no idea how do I add labels to the text blocks. PIXEL_CLS_WEIGHT_all_ones = 'PIXEL_CLS_WEIGHT_all_ones' PIXEL_C...
          Read more

          Vallis Paradisi

          Image
          Coordinata: .mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap} 33°03′S 71°37′W  /  33.05°S 71.617°W  / -33.05; -71.617 Vallis Paradisi Res apud Vicidata repertae: Civitas : Chilia Numerus incolarum : 296 655 Zona horaria : UTC-4 Situs interretialis Nomen officiale : Valparaíso Gubernium Consilium : municipal council of Valparaíso Geographia Superficies : 401 600 000 Metrum quadratum Coniunctiones urbium Urbes gemellae : Ovetum , Badalona , Santa Fe , Ora longa , Sciamhaevum , Quantunia , Callao , Bat Yam , Melaka , Novorossiysk , Busan , Corduba , Odessa , Rosario , Malacca , Barcino , Soteropolis , Franciscopolis (California) , Guanajuato, Silao Tabula aut despectus Despectus in portum sinumque et urbem Vallis Paradisi Vallis Para...
          Read more

          Tabula Rosettana

          Image
          Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a...
          Read more