Why are epimorphisms in the category of Monoids not necessarily surjections?












3












$begingroup$


The book "basic category theory" states that in the category Mon, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?






category-theory







share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    The book "basic category theory" states that in the category Mon, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?






    category-theory







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      The book "basic category theory" states that in the category Mon, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?






      category-theory







      share|cite|improve this question









      $endgroup$




      The book "basic category theory" states that in the category Mon, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?






      category-theory




      category-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Feb 1 at 17:40









      user56834user56834

      3,10521252




      3,10521252






















          3 Answers
          3






          active

          oldest

          votes


















          4












          $begingroup$

          Let $Bbb Z$ be the additive group of integers, and $Bbb N_0={0,1,2,ldots}$.
          Then the inclusion $Bbb N_0toBbb Z$ is an epimorphism in Mon.






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            Quoting Wikipedia, we see:



            In the category of monoids, Mon, the inclusion map $mathbb{N}tomathbb{Z}$ is a non-surjective epimorphism. To see this, suppose that $g_1$ and $g_2$ are two distinct maps from $mathbb{Z}$ to some monoid $M$. Then for some $ninmathbb{Z}$, $g_1(n)neq g_2(n)$, so $g_1(-n)neq g_2(-n)$. Either $n$ or $-n$ is in $mathbb{N}$,, so the restrictions of $g_1$ and $g_2$ to $mathbb{N}$ are unequal.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
              $endgroup$
              – user56834
              Feb 1 at 18:17








            • 1




              $begingroup$
              $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
              $endgroup$
              – vadim123
              Feb 1 at 20:53





















            1












            $begingroup$

            Consider the inclusion $i: mathbb{N}hookrightarrowmathbb{Z}$. This is clearly not surjective. However, for any monoid $M$ and any two maps $f,g:mathbb{Z} to M$ such that $f circ i = g circ i$, we must have $f = g$, by a simple argument.






            share|cite|improve this answer









            $endgroup$













              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: "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%2f3096526%2fwhy-are-epimorphisms-in-the-category-of-monoids-not-necessarily-surjections%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              4












              $begingroup$

              Let $Bbb Z$ be the additive group of integers, and $Bbb N_0={0,1,2,ldots}$.
              Then the inclusion $Bbb N_0toBbb Z$ is an epimorphism in Mon.






              share|cite|improve this answer









              $endgroup$


















                4












                $begingroup$

                Let $Bbb Z$ be the additive group of integers, and $Bbb N_0={0,1,2,ldots}$.
                Then the inclusion $Bbb N_0toBbb Z$ is an epimorphism in Mon.






                share|cite|improve this answer









                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  Let $Bbb Z$ be the additive group of integers, and $Bbb N_0={0,1,2,ldots}$.
                  Then the inclusion $Bbb N_0toBbb Z$ is an epimorphism in Mon.






                  share|cite|improve this answer









                  $endgroup$



                  Let $Bbb Z$ be the additive group of integers, and $Bbb N_0={0,1,2,ldots}$.
                  Then the inclusion $Bbb N_0toBbb Z$ is an epimorphism in Mon.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 1 at 17:45









                  Lord Shark the UnknownLord Shark the Unknown

                  104k1160132




                  104k1160132























                      3












                      $begingroup$

                      Quoting Wikipedia, we see:



                      In the category of monoids, Mon, the inclusion map $mathbb{N}tomathbb{Z}$ is a non-surjective epimorphism. To see this, suppose that $g_1$ and $g_2$ are two distinct maps from $mathbb{Z}$ to some monoid $M$. Then for some $ninmathbb{Z}$, $g_1(n)neq g_2(n)$, so $g_1(-n)neq g_2(-n)$. Either $n$ or $-n$ is in $mathbb{N}$,, so the restrictions of $g_1$ and $g_2$ to $mathbb{N}$ are unequal.






                      share|cite|improve this answer









                      $endgroup$













                      • $begingroup$
                        I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                        $endgroup$
                        – user56834
                        Feb 1 at 18:17








                      • 1




                        $begingroup$
                        $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                        $endgroup$
                        – vadim123
                        Feb 1 at 20:53


















                      3












                      $begingroup$

                      Quoting Wikipedia, we see:



                      In the category of monoids, Mon, the inclusion map $mathbb{N}tomathbb{Z}$ is a non-surjective epimorphism. To see this, suppose that $g_1$ and $g_2$ are two distinct maps from $mathbb{Z}$ to some monoid $M$. Then for some $ninmathbb{Z}$, $g_1(n)neq g_2(n)$, so $g_1(-n)neq g_2(-n)$. Either $n$ or $-n$ is in $mathbb{N}$,, so the restrictions of $g_1$ and $g_2$ to $mathbb{N}$ are unequal.






                      share|cite|improve this answer









                      $endgroup$













                      • $begingroup$
                        I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                        $endgroup$
                        – user56834
                        Feb 1 at 18:17








                      • 1




                        $begingroup$
                        $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                        $endgroup$
                        – vadim123
                        Feb 1 at 20:53
















                      3












                      3








                      3





                      $begingroup$

                      Quoting Wikipedia, we see:



                      In the category of monoids, Mon, the inclusion map $mathbb{N}tomathbb{Z}$ is a non-surjective epimorphism. To see this, suppose that $g_1$ and $g_2$ are two distinct maps from $mathbb{Z}$ to some monoid $M$. Then for some $ninmathbb{Z}$, $g_1(n)neq g_2(n)$, so $g_1(-n)neq g_2(-n)$. Either $n$ or $-n$ is in $mathbb{N}$,, so the restrictions of $g_1$ and $g_2$ to $mathbb{N}$ are unequal.






                      share|cite|improve this answer









                      $endgroup$



                      Quoting Wikipedia, we see:



                      In the category of monoids, Mon, the inclusion map $mathbb{N}tomathbb{Z}$ is a non-surjective epimorphism. To see this, suppose that $g_1$ and $g_2$ are two distinct maps from $mathbb{Z}$ to some monoid $M$. Then for some $ninmathbb{Z}$, $g_1(n)neq g_2(n)$, so $g_1(-n)neq g_2(-n)$. Either $n$ or $-n$ is in $mathbb{N}$,, so the restrictions of $g_1$ and $g_2$ to $mathbb{N}$ are unequal.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Feb 1 at 17:46









                      vadim123vadim123

                      76k897189




                      76k897189












                      • $begingroup$
                        I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                        $endgroup$
                        – user56834
                        Feb 1 at 18:17








                      • 1




                        $begingroup$
                        $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                        $endgroup$
                        – vadim123
                        Feb 1 at 20:53




















                      • $begingroup$
                        I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                        $endgroup$
                        – user56834
                        Feb 1 at 18:17








                      • 1




                        $begingroup$
                        $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                        $endgroup$
                        – vadim123
                        Feb 1 at 20:53


















                      $begingroup$
                      I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                      $endgroup$
                      – user56834
                      Feb 1 at 18:17






                      $begingroup$
                      I don't understand the argument? specifically, why does $g_1(-n)neq g_2(-n)$ need to hold? That would hold if we were in the category of groups, I see that. But I don't see why it would hold in the category of monoids. also, I don't get the last sentence.
                      $endgroup$
                      – user56834
                      Feb 1 at 18:17






                      1




                      1




                      $begingroup$
                      $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                      $endgroup$
                      – vadim123
                      Feb 1 at 20:53






                      $begingroup$
                      $0=g_1(0)=g_1(n+(-n))=g_1(n)+g_1(-n)$
                      $endgroup$
                      – vadim123
                      Feb 1 at 20:53













                      1












                      $begingroup$

                      Consider the inclusion $i: mathbb{N}hookrightarrowmathbb{Z}$. This is clearly not surjective. However, for any monoid $M$ and any two maps $f,g:mathbb{Z} to M$ such that $f circ i = g circ i$, we must have $f = g$, by a simple argument.






                      share|cite|improve this answer









                      $endgroup$


















                        1












                        $begingroup$

                        Consider the inclusion $i: mathbb{N}hookrightarrowmathbb{Z}$. This is clearly not surjective. However, for any monoid $M$ and any two maps $f,g:mathbb{Z} to M$ such that $f circ i = g circ i$, we must have $f = g$, by a simple argument.






                        share|cite|improve this answer









                        $endgroup$
















                          1












                          1








                          1





                          $begingroup$

                          Consider the inclusion $i: mathbb{N}hookrightarrowmathbb{Z}$. This is clearly not surjective. However, for any monoid $M$ and any two maps $f,g:mathbb{Z} to M$ such that $f circ i = g circ i$, we must have $f = g$, by a simple argument.






                          share|cite|improve this answer









                          $endgroup$



                          Consider the inclusion $i: mathbb{N}hookrightarrowmathbb{Z}$. This is clearly not surjective. However, for any monoid $M$ and any two maps $f,g:mathbb{Z} to M$ such that $f circ i = g circ i$, we must have $f = g$, by a simple argument.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Feb 1 at 17:47









                          user3482749user3482749

                          4,266919




                          4,266919






























                              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%2f3096526%2fwhy-are-epimorphisms-in-the-category-of-monoids-not-necessarily-surjections%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 change which sound is reproduced for terminal bell?

                              Image
                              3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
                              Read more

                              Can I use Tabulator js library in my java Spring + Thymeleaf project?

                              Image
                              0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
                              Read more

                              Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                              Image
                              up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx...
                              Read more