A category-like structure without composition?












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Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!






reference-request ct.category-theory







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$endgroup$








  • 9




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    Apr 3 at 23:10






  • 3




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    Apr 4 at 0:05






  • 1




    $begingroup$
    Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
    $endgroup$
    – Tim Campion
    2 days ago
















8












$begingroup$


Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!






reference-request ct.category-theory







share|cite|improve this question











$endgroup$








  • 9




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    Apr 3 at 23:10






  • 3




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    Apr 4 at 0:05






  • 1




    $begingroup$
    Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
    $endgroup$
    – Tim Campion
    2 days ago














8












8








8


2



$begingroup$


Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!






reference-request ct.category-theory







share|cite|improve this question











$endgroup$




Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!






reference-request ct.category-theory




reference-request ct.category-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 3 at 23:36







APR

















asked Apr 3 at 22:28









APRAPR

995




995








  • 9




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    Apr 3 at 23:10






  • 3




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    Apr 4 at 0:05






  • 1




    $begingroup$
    Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
    $endgroup$
    – Tim Campion
    2 days ago














  • 9




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    Apr 3 at 23:10






  • 3




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    Apr 4 at 0:05






  • 1




    $begingroup$
    Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
    $endgroup$
    – Tim Campion
    2 days ago








9




9




$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10




$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
Apr 3 at 23:10




3




3




$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05




$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
Apr 4 at 0:05




1




1




$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
2 days ago




$begingroup$
Er -- I should clarify that 2-Segal spaces are a bit more general -- in addition to allowing composition to be undefined, they also allow composition to be multiply-defined. So they're like a category enriched in spans rather than pointed sets.
$endgroup$
– Tim Campion
2 days ago










2 Answers
2






active

oldest

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12












$begingroup$

As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.



A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.






share|cite|improve this answer









$endgroup$





















    4












    $begingroup$

    Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.






    share|cite|improve this answer











    $endgroup$









    • 3




      $begingroup$
      I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
      $endgroup$
      – Kevin Walker
      2 days ago






    • 1




      $begingroup$
      I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
      $endgroup$
      – Kevin Walker
      2 days ago










    • $begingroup$
      @Dan Petersen Thanks for the correction.
      $endgroup$
      – Theo Johnson-Freyd
      2 days ago










    • $begingroup$
      @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
      $endgroup$
      – Theo Johnson-Freyd
      2 days ago










    • $begingroup$
      @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
      $endgroup$
      – Hurkyl
      yesterday














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    2 Answers
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    2 Answers
    2






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    active

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    12












    $begingroup$

    As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.



    A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.






    share|cite|improve this answer









    $endgroup$


















      12












      $begingroup$

      As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.



      A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.






      share|cite|improve this answer









      $endgroup$
















        12












        12








        12





        $begingroup$

        As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.



        A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.






        share|cite|improve this answer









        $endgroup$



        As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.



        A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Apr 3 at 23:45









        Mike ShulmanMike Shulman

        37.9k487236




        37.9k487236























            4












            $begingroup$

            Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.






            share|cite|improve this answer











            $endgroup$









            • 3




              $begingroup$
              I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
              $endgroup$
              – Kevin Walker
              2 days ago






            • 1




              $begingroup$
              I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
              $endgroup$
              – Kevin Walker
              2 days ago










            • $begingroup$
              @Dan Petersen Thanks for the correction.
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
              $endgroup$
              – Hurkyl
              yesterday


















            4












            $begingroup$

            Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.






            share|cite|improve this answer











            $endgroup$









            • 3




              $begingroup$
              I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
              $endgroup$
              – Kevin Walker
              2 days ago






            • 1




              $begingroup$
              I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
              $endgroup$
              – Kevin Walker
              2 days ago










            • $begingroup$
              @Dan Petersen Thanks for the correction.
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
              $endgroup$
              – Hurkyl
              yesterday
















            4












            4








            4





            $begingroup$

            Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.






            share|cite|improve this answer











            $endgroup$



            Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Apr 4 at 0:22









            Dan Petersen

            26.1k277142




            26.1k277142










            answered Apr 3 at 23:46









            Theo Johnson-FreydTheo Johnson-Freyd

            29.8k881253




            29.8k881253








            • 3




              $begingroup$
              I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
              $endgroup$
              – Kevin Walker
              2 days ago






            • 1




              $begingroup$
              I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
              $endgroup$
              – Kevin Walker
              2 days ago










            • $begingroup$
              @Dan Petersen Thanks for the correction.
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
              $endgroup$
              – Hurkyl
              yesterday
















            • 3




              $begingroup$
              I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
              $endgroup$
              – Kevin Walker
              2 days ago






            • 1




              $begingroup$
              I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
              $endgroup$
              – Kevin Walker
              2 days ago










            • $begingroup$
              @Dan Petersen Thanks for the correction.
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
              $endgroup$
              – Theo Johnson-Freyd
              2 days ago










            • $begingroup$
              @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
              $endgroup$
              – Hurkyl
              yesterday










            3




            3




            $begingroup$
            I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
            $endgroup$
            – Kevin Walker
            2 days ago




            $begingroup$
            I've long been in favor of replacing "category" with "monoidoid" (in analogy to "group" $to$ "groupoid"). If we combine this with Jørgen's idea then the subject of the above question would be called a "monoidoidoid", which makes this nomenclature reform proposal all the more attractive. The only downside I see is that there are relatively few opportunities to refer to monoidoidoids. Perhaps we could start calling monoidoids (i.e. categories) "special monoidoidoids", in which composition just happens to be always defined.
            $endgroup$
            – Kevin Walker
            2 days ago




            1




            1




            $begingroup$
            I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
            $endgroup$
            – Kevin Walker
            2 days ago




            $begingroup$
            I also think we should give our fingers a rest and just write "mplete" instead of "cocomplete".
            $endgroup$
            – Kevin Walker
            2 days ago












            $begingroup$
            @Dan Petersen Thanks for the correction.
            $endgroup$
            – Theo Johnson-Freyd
            2 days ago




            $begingroup$
            @Dan Petersen Thanks for the correction.
            $endgroup$
            – Theo Johnson-Freyd
            2 days ago












            $begingroup$
            @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
            $endgroup$
            – Theo Johnson-Freyd
            2 days ago




            $begingroup$
            @KevinWalker I could certainly get behind "mplete". Mpleteness is more basic, IMHO, than completeness. I'm reminded of a seminar talk years ago in which Noah Snyder and I were sitting next to each other in the back row. The speaker said that the next result as a "co-corollary", and explained that that meant that the Theorem followed immediately from it. Noah and I turned to each other and simultaneously said "rollary".
            $endgroup$
            – Theo Johnson-Freyd
            2 days ago












            $begingroup$
            @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
            $endgroup$
            – Hurkyl
            yesterday






            $begingroup$
            @TheoJohnson-Freyd: It's an interesting topic. I think finite completeness is far more basic than finite mpleteness, due to its extremely tight connection with logic. I would agree with you for the case of small limits/colimits, but the more I reflect on it, the more I think that's an expression of the habit of approaching the infinite via "synthezising" things as a filtered colimit of understandable pieces rather than "analyzing" them as a cofiltered limit of partial information.
            $endgroup$
            – Hurkyl
            yesterday




















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