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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  • 5




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    3 hours ago










  • $begingroup$
    In that case, you still necessarily have compositions, or am I misunderstanding something?
    $endgroup$
    – APR
    3 hours ago






  • 1




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    2 hours ago
















3












$begingroup$


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











$endgroup$








  • 5




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    3 hours ago










  • $begingroup$
    In that case, you still necessarily have compositions, or am I misunderstanding something?
    $endgroup$
    – APR
    3 hours ago






  • 1




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    2 hours ago














3












3








3





$begingroup$


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 3 hours ago







APR

















asked 4 hours ago









APRAPR

724




724








  • 5




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    3 hours ago










  • $begingroup$
    In that case, you still necessarily have compositions, or am I misunderstanding something?
    $endgroup$
    – APR
    3 hours ago






  • 1




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    2 hours ago














  • 5




    $begingroup$
    Shouldn't this be equivalent to a category enriched over pointed sets?
    $endgroup$
    – Qiaochu Yuan
    3 hours ago










  • $begingroup$
    In that case, you still necessarily have compositions, or am I misunderstanding something?
    $endgroup$
    – APR
    3 hours ago






  • 1




    $begingroup$
    There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
    $endgroup$
    – Tim Campion
    2 hours ago








5




5




$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago




$begingroup$
Shouldn't this be equivalent to a category enriched over pointed sets?
$endgroup$
– Qiaochu Yuan
3 hours ago












$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago




$begingroup$
In that case, you still necessarily have compositions, or am I misunderstanding something?
$endgroup$
– APR
3 hours ago




1




1




$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago




$begingroup$
There's an $infty$-categorical version: 2-Segal spaces in the sense of Dyckerhoff-Kapranov
$endgroup$
– Tim Campion
2 hours ago










2 Answers
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$begingroup$

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









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    0












    $begingroup$

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






    share|cite|improve this answer











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      2 Answers
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      2 Answers
      2






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      active

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      5












      $begingroup$

      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









      $endgroup$


















        5












        $begingroup$

        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









        $endgroup$
















          5












          5








          5





          $begingroup$

          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 3 hours ago









          Mike ShulmanMike Shulman

          37.8k485235




          37.8k485235























              0












              $begingroup$

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






              share|cite|improve this answer











              $endgroup$


















                0












                $begingroup$

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






                share|cite|improve this answer











                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  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 2 hours ago









                  Dan Petersen

                  26.1k277142




                  26.1k277142










                  answered 3 hours ago









                  Theo Johnson-FreydTheo Johnson-Freyd

                  29.8k881252




                  29.8k881252






























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