What is the topology associated with the algebras for the ultrafilter monad? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a way to make tangent bundle a monad?Coproducts and pushouts of Boolean algebras and Heyting algebrasProof of theorems in the field of banach-and $c^*$-algebras in a categorial languageEpimorphisms of locally compact spacesThis is just the Eilenberg-Moore category, right?Which spaces can be used as “test spaces” for the Stone-Čech compactification?What is the opposite category of $operatornameTop$?Finitely presentable objects and the Kleisli categoryultrafilter convergence versus non-standard topologyMorita theory for algebras for a monad $T$

Project Euler #1 in C++

Sum letters are not two different

Effects on objects due to a brief relocation of massive amounts of mass

How does the math work when buying airline miles?

If windows 7 doesn't support WSL, then what does Linux subsystem option mean?

Are all finite dimensional hilbert spaces isomorphic to spaces with Euclidean norms?

How do I use the new nonlinear finite element in Mathematica 12 for this equation?

What is the appropriate index architecture when forced to implement IsDeleted (soft deletes)?

Source for Esri sample data from 911 Hot Spot Analysis

Using audio cues to encourage good posture

The code below, is it ill-formed NDR or is it well formed?

A term for a woman complaining about things/begging in a cute/childish way

How come Sam didn't become Lord of Horn Hill?

How do I find out the mythology and history of my Fortress?

Why wasn't DOSKEY integrated with COMMAND.COM?

Has negative voting ever been officially implemented in elections, or seriously proposed, or even studied?

Selecting user stories during sprint planning

Maximum summed subsequences with non-adjacent items

How to install press fit bottom bracket into new frame

Crossing US/Canada Border for less than 24 hours

Did Deadpool rescue all of the X-Force?

Drawing without replacement: why the order of draw is irrelevant?

Take 2! Is this homebrew Lady of Pain warlock patron balanced?

What is a fractional matching?



What is the topology associated with the algebras for the ultrafilter monad?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a way to make tangent bundle a monad?Coproducts and pushouts of Boolean algebras and Heyting algebrasProof of theorems in the field of banach-and $c^*$-algebras in a categorial languageEpimorphisms of locally compact spacesThis is just the Eilenberg-Moore category, right?Which spaces can be used as “test spaces” for the Stone-Čech compactification?What is the opposite category of $operatornameTop$?Finitely presentable objects and the Kleisli categoryultrafilter convergence versus non-standard topologyMorita theory for algebras for a monad $T$










7












$begingroup$


It is easy to find references stating that the category of compact Hausdorff spaces $mathbfCompHaus$ is equivalent to the category of algebras for the ultrafilter monad, $mathbfbeta Alg$. After doing some digging, the $mathbfCompHausto mathbfbeta Alg$ half of the equivalence is simple enough, but I haven't been able to find a description of the $mathbfbeta Algto mathbfCompHaus$ half of this equivalence. I have tried to work it out, but I have little experience with topological spaces and am not sure what the associated topology ought to look like.



I'm wondering if anyone has a good reference that describes the $mathbfbeta Algto mathbfCompHaus$ half of the equivalence, or can describe it here.










share|cite|improve this question









$endgroup$
















    7












    $begingroup$


    It is easy to find references stating that the category of compact Hausdorff spaces $mathbfCompHaus$ is equivalent to the category of algebras for the ultrafilter monad, $mathbfbeta Alg$. After doing some digging, the $mathbfCompHausto mathbfbeta Alg$ half of the equivalence is simple enough, but I haven't been able to find a description of the $mathbfbeta Algto mathbfCompHaus$ half of this equivalence. I have tried to work it out, but I have little experience with topological spaces and am not sure what the associated topology ought to look like.



    I'm wondering if anyone has a good reference that describes the $mathbfbeta Algto mathbfCompHaus$ half of the equivalence, or can describe it here.










    share|cite|improve this question









    $endgroup$














      7












      7








      7





      $begingroup$


      It is easy to find references stating that the category of compact Hausdorff spaces $mathbfCompHaus$ is equivalent to the category of algebras for the ultrafilter monad, $mathbfbeta Alg$. After doing some digging, the $mathbfCompHausto mathbfbeta Alg$ half of the equivalence is simple enough, but I haven't been able to find a description of the $mathbfbeta Algto mathbfCompHaus$ half of this equivalence. I have tried to work it out, but I have little experience with topological spaces and am not sure what the associated topology ought to look like.



      I'm wondering if anyone has a good reference that describes the $mathbfbeta Algto mathbfCompHaus$ half of the equivalence, or can describe it here.










      share|cite|improve this question









      $endgroup$




      It is easy to find references stating that the category of compact Hausdorff spaces $mathbfCompHaus$ is equivalent to the category of algebras for the ultrafilter monad, $mathbfbeta Alg$. After doing some digging, the $mathbfCompHausto mathbfbeta Alg$ half of the equivalence is simple enough, but I haven't been able to find a description of the $mathbfbeta Algto mathbfCompHaus$ half of this equivalence. I have tried to work it out, but I have little experience with topological spaces and am not sure what the associated topology ought to look like.



      I'm wondering if anyone has a good reference that describes the $mathbfbeta Algto mathbfCompHaus$ half of the equivalence, or can describe it here.







      general-topology category-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 hours ago









      Malice VidrineMalice Vidrine

      6,34121123




      6,34121123




















          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          The other half of the equivalence is described on the nLab page on ultrafilters.



          Given an algebra structure $xicolon beta X to X$, we define the topology on $X$ by declaring that a subset $Usubseteq X$ is open if and only if for every point $xin U$ and every ultrafilter $Fin beta X$ such that $xi(F) = x$, we have $Uin F$.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
            $endgroup$
            – Malice Vidrine
            1 hour ago











          Your Answer








          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          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%2fmath.stackexchange.com%2fquestions%2f3192728%2fwhat-is-the-topology-associated-with-the-algebras-for-the-ultrafilter-monad%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









          5












          $begingroup$

          The other half of the equivalence is described on the nLab page on ultrafilters.



          Given an algebra structure $xicolon beta X to X$, we define the topology on $X$ by declaring that a subset $Usubseteq X$ is open if and only if for every point $xin U$ and every ultrafilter $Fin beta X$ such that $xi(F) = x$, we have $Uin F$.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
            $endgroup$
            – Malice Vidrine
            1 hour ago















          5












          $begingroup$

          The other half of the equivalence is described on the nLab page on ultrafilters.



          Given an algebra structure $xicolon beta X to X$, we define the topology on $X$ by declaring that a subset $Usubseteq X$ is open if and only if for every point $xin U$ and every ultrafilter $Fin beta X$ such that $xi(F) = x$, we have $Uin F$.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
            $endgroup$
            – Malice Vidrine
            1 hour ago













          5












          5








          5





          $begingroup$

          The other half of the equivalence is described on the nLab page on ultrafilters.



          Given an algebra structure $xicolon beta X to X$, we define the topology on $X$ by declaring that a subset $Usubseteq X$ is open if and only if for every point $xin U$ and every ultrafilter $Fin beta X$ such that $xi(F) = x$, we have $Uin F$.






          share|cite|improve this answer









          $endgroup$



          The other half of the equivalence is described on the nLab page on ultrafilters.



          Given an algebra structure $xicolon beta X to X$, we define the topology on $X$ by declaring that a subset $Usubseteq X$ is open if and only if for every point $xin U$ and every ultrafilter $Fin beta X$ such that $xi(F) = x$, we have $Uin F$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 hours ago









          Alex KruckmanAlex Kruckman

          28.8k32758




          28.8k32758











          • $begingroup$
            Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
            $endgroup$
            – Malice Vidrine
            1 hour ago
















          • $begingroup$
            Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
            $endgroup$
            – Malice Vidrine
            1 hour ago















          $begingroup$
          Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
          $endgroup$
          – Malice Vidrine
          1 hour ago




          $begingroup$
          Of course I looked at precisely the wrong nLab pages to answer my question :P Thanks!
          $endgroup$
          – Malice Vidrine
          1 hour ago

















          draft saved

          draft discarded
















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • 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%2fmath.stackexchange.com%2fquestions%2f3192728%2fwhat-is-the-topology-associated-with-the-algebras-for-the-ultrafilter-monad%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

          Oświęcim Innehåll Historia | Källor | Externa länkar | Navigeringsmeny50°2′18″N 19°13′17″Ö / 50.03833°N 19.22139°Ö / 50.03833; 19.2213950°2′18″N 19°13′17″Ö / 50.03833°N 19.22139°Ö / 50.03833; 19.221393089658Nordisk familjebok, AuschwitzInsidan tro och existensJewish Community i OświęcimAuschwitz Jewish Center: MuseumAuschwitz Jewish Center

          Valle di Casies Indice Geografia fisica | Origini del nome | Storia | Società | Amministrazione | Sport | Note | Bibliografia | Voci correlate | Altri progetti | Collegamenti esterni | Menu di navigazione46°46′N 12°11′E / 46.766667°N 12.183333°E46.766667; 12.183333 (Valle di Casies)46°46′N 12°11′E / 46.766667°N 12.183333°E46.766667; 12.183333 (Valle di Casies)Sito istituzionaleAstat Censimento della popolazione 2011 - Determinazione della consistenza dei tre gruppi linguistici della Provincia Autonoma di Bolzano-Alto Adige - giugno 2012Numeri e fattiValle di CasiesDato IstatTabella dei gradi/giorno dei Comuni italiani raggruppati per Regione e Provincia26 agosto 1993, n. 412Heraldry of the World: GsiesStatistiche I.StatValCasies.comWikimedia CommonsWikimedia CommonsValle di CasiesSito ufficialeValle di CasiesMM14870458910042978-6

          Typsetting diagram chases (with TikZ?) Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)How to define the default vertical distance between nodes?Draw edge on arcNumerical conditional within tikz keys?TikZ: Drawing an arc from an intersection to an intersectionDrawing rectilinear curves in Tikz, aka an Etch-a-Sketch drawingLine up nested tikz enviroments or how to get rid of themHow to place nodes in an absolute coordinate system in tikzCommutative diagram with curve connecting between nodesTikz with standalone: pinning tikz coordinates to page cmDrawing a Decision Diagram with Tikz and layout manager