In what categories does the “classical” notion of function make sense?












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$begingroup$


I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?






functions category-theory







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    5












    $begingroup$


    I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?






    functions category-theory







    share|cite|improve this question









    $endgroup$















      5












      5








      5


      1



      $begingroup$


      I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?






      functions category-theory







      share|cite|improve this question









      $endgroup$




      I'm new to category theory, and I often struggle to choose the right level of abstraction when working with categories. I also found that many textbooks are rather inconsistent in their conventions with regards to the terminology (eg. they often interchangeably use terms like epimorphism and surjection, etc). So I wondered what's a minimal set of requirements on a category so that it makes sense to say that morphism are functions? How about Abelian categories?






      functions category-theory




      functions category-theory






      share|cite|improve this question













      share|cite|improve this question











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      share|cite|improve this question










      asked Dec 28 '18 at 21:58









      gengen

      4772521




      4772521






















          2 Answers
          2






          active

          oldest

          votes


















          12












          $begingroup$

          The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
            $endgroup$
            – Asaf Karagila♦
            Dec 29 '18 at 7:51






          • 1




            $begingroup$
            @AsafKaragila: $Ab$ certainly satisfies it.
            $endgroup$
            – Berci
            Dec 30 '18 at 23:48










          • $begingroup$
            @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
            $endgroup$
            – Asaf Karagila♦
            Dec 30 '18 at 23:49






          • 1




            $begingroup$
            Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
            $endgroup$
            – Berci
            Dec 30 '18 at 23:53





















          4












          $begingroup$

          Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.



          For example, consider




          • the category Mon where objects are monoids and morphisms their algebraic homomorphisms

          • and the forgetful functor making it concrete being the obvious $U: mathrm{Mon} to mathrm{Set}$.


          Now the embedding morphism $m: (mathbb{Z}, cdot, 1) to (mathbb{Q}, cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): mathbb{Z} to mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            (Not an answer, but too long for a comment.)
            $endgroup$
            – ComFreek
            Dec 29 '18 at 7:38












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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          12












          $begingroup$

          The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
            $endgroup$
            – Asaf Karagila♦
            Dec 29 '18 at 7:51






          • 1




            $begingroup$
            @AsafKaragila: $Ab$ certainly satisfies it.
            $endgroup$
            – Berci
            Dec 30 '18 at 23:48










          • $begingroup$
            @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
            $endgroup$
            – Asaf Karagila♦
            Dec 30 '18 at 23:49






          • 1




            $begingroup$
            Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
            $endgroup$
            – Berci
            Dec 30 '18 at 23:53


















          12












          $begingroup$

          The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
            $endgroup$
            – Asaf Karagila♦
            Dec 29 '18 at 7:51






          • 1




            $begingroup$
            @AsafKaragila: $Ab$ certainly satisfies it.
            $endgroup$
            – Berci
            Dec 30 '18 at 23:48










          • $begingroup$
            @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
            $endgroup$
            – Asaf Karagila♦
            Dec 30 '18 at 23:49






          • 1




            $begingroup$
            Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
            $endgroup$
            – Berci
            Dec 30 '18 at 23:53
















          12












          12








          12





          $begingroup$

          The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.






          share|cite|improve this answer









          $endgroup$



          The notion you are looking for is probably that of a concrete category. A concrete category is a category that is embedded in the category of sets; thus its objects are associated with actual sets, and its morphisms are associated with actual functions.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 28 '18 at 22:10









          Pierre-Guy PlamondonPierre-Guy Plamondon

          8,91511739




          8,91511739








          • 2




            $begingroup$
            Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
            $endgroup$
            – Asaf Karagila♦
            Dec 29 '18 at 7:51






          • 1




            $begingroup$
            @AsafKaragila: $Ab$ certainly satisfies it.
            $endgroup$
            – Berci
            Dec 30 '18 at 23:48










          • $begingroup$
            @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
            $endgroup$
            – Asaf Karagila♦
            Dec 30 '18 at 23:49






          • 1




            $begingroup$
            Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
            $endgroup$
            – Berci
            Dec 30 '18 at 23:53
















          • 2




            $begingroup$
            Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
            $endgroup$
            – Asaf Karagila♦
            Dec 29 '18 at 7:51






          • 1




            $begingroup$
            @AsafKaragila: $Ab$ certainly satisfies it.
            $endgroup$
            – Berci
            Dec 30 '18 at 23:48










          • $begingroup$
            @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
            $endgroup$
            – Asaf Karagila♦
            Dec 30 '18 at 23:49






          • 1




            $begingroup$
            Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
            $endgroup$
            – Berci
            Dec 30 '18 at 23:53










          2




          2




          $begingroup$
          Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
          $endgroup$
          – Asaf Karagila♦
          Dec 29 '18 at 7:51




          $begingroup$
          Top and Ab are perhaps two of the most concrete categories out there (after Set, anyway). But neither satisfies that epi = surjection.
          $endgroup$
          – Asaf Karagila♦
          Dec 29 '18 at 7:51




          1




          1




          $begingroup$
          @AsafKaragila: $Ab$ certainly satisfies it.
          $endgroup$
          – Berci
          Dec 30 '18 at 23:48




          $begingroup$
          @AsafKaragila: $Ab$ certainly satisfies it.
          $endgroup$
          – Berci
          Dec 30 '18 at 23:48












          $begingroup$
          @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
          $endgroup$
          – Asaf Karagila♦
          Dec 30 '18 at 23:49




          $begingroup$
          @Berci: I thought that the identity from $Bbb Z$ to $Bbb Q$ is an epimorphism. No?
          $endgroup$
          – Asaf Karagila♦
          Dec 30 '18 at 23:49




          1




          1




          $begingroup$
          Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
          $endgroup$
          – Berci
          Dec 30 '18 at 23:53






          $begingroup$
          Not in the category of Abelian groups. For rings or monoids (according to the other answer), it is indeed an epimorphism. In an Abelian category, a mono and epi morphism must be isomorphism. (Specifically we can take e.g. the quotient map and the zero map $Bbb Qto Bbb Q/Bbb Z$ to obtain the same composition $Bbb ZtoBbb Q/Bbb Z$.)
          $endgroup$
          – Berci
          Dec 30 '18 at 23:53













          4












          $begingroup$

          Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.



          For example, consider




          • the category Mon where objects are monoids and morphisms their algebraic homomorphisms

          • and the forgetful functor making it concrete being the obvious $U: mathrm{Mon} to mathrm{Set}$.


          Now the embedding morphism $m: (mathbb{Z}, cdot, 1) to (mathbb{Q}, cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): mathbb{Z} to mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            (Not an answer, but too long for a comment.)
            $endgroup$
            – ComFreek
            Dec 29 '18 at 7:38
















          4












          $begingroup$

          Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.



          For example, consider




          • the category Mon where objects are monoids and morphisms their algebraic homomorphisms

          • and the forgetful functor making it concrete being the obvious $U: mathrm{Mon} to mathrm{Set}$.


          Now the embedding morphism $m: (mathbb{Z}, cdot, 1) to (mathbb{Q}, cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): mathbb{Z} to mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            (Not an answer, but too long for a comment.)
            $endgroup$
            – ComFreek
            Dec 29 '18 at 7:38














          4












          4








          4





          $begingroup$

          Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.



          For example, consider




          • the category Mon where objects are monoids and morphisms their algebraic homomorphisms

          • and the forgetful functor making it concrete being the obvious $U: mathrm{Mon} to mathrm{Set}$.


          Now the embedding morphism $m: (mathbb{Z}, cdot, 1) to (mathbb{Q}, cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): mathbb{Z} to mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.






          share|cite|improve this answer









          $endgroup$



          Beware that even in conrete categories the terms epimorphism and surjection are not interchangeable.



          For example, consider




          • the category Mon where objects are monoids and morphisms their algebraic homomorphisms

          • and the forgetful functor making it concrete being the obvious $U: mathrm{Mon} to mathrm{Set}$.


          Now the embedding morphism $m: (mathbb{Z}, cdot, 1) to (mathbb{Q}, cdot, 1)$ is a monomorphism and epimorphism as can be seen by some small, but tedious calculations. However, the associated actual function given by $U(m): mathbb{Z} to mathbb{Q}$ is obviously not an epimorphism in Set. Notably, in Set the terms epimorphism and surjection really coincide — at least with AoC.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 29 '18 at 7:37









          ComFreekComFreek

          5521411




          5521411












          • $begingroup$
            (Not an answer, but too long for a comment.)
            $endgroup$
            – ComFreek
            Dec 29 '18 at 7:38


















          • $begingroup$
            (Not an answer, but too long for a comment.)
            $endgroup$
            – ComFreek
            Dec 29 '18 at 7:38
















          $begingroup$
          (Not an answer, but too long for a comment.)
          $endgroup$
          – ComFreek
          Dec 29 '18 at 7:38




          $begingroup$
          (Not an answer, but too long for a comment.)
          $endgroup$
          – ComFreek
          Dec 29 '18 at 7:38


















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