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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeApr 14th 2018

    There seem to be a lot of questions on the Internet recently about disjointness of homsets of categories, so I thought this definition would be worth recording.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2018

    Sorry for asking a stupid question, but is a protocategory essentially the same thing as a category enriched in SetSet (which doesn’t require disjointness of hom-sets)? If not, can you give a simple example to illustrate the distinction?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 14th 2018

    I think that’s about right as long as by SetSet you mean the category of sets in a material set theory. But the definition of protocategory makes sense even in a structural set theory.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2018

    Thanks; I think I sorted out my confusion, especially with the help of the example of one category being structured over another, as GrpGrp is structured over SetSet. In that type of example, some protomorphisms do not name any morphism of the generated category. Whereas in the examples I had in mind of SetSet-enriched categories, we get a protocategory whose protomorphisms are elements of the union of the hom-sets – but in that type of example every protomorphism names some morphism. (Besides the fact that the operation “taking the union” works a little differently when working over a structural set theory from how it does in a material set theory, as you say – about all we have available in a structural set theory is taking a disjoint union.)

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeApr 15th 2018

    Typo fix (g=gf written when h=gf meant)

    diff, v2, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeApr 16th 2018

    Actually, thinking about it some more, I think it’s not necessarily always true that we can make a protocategory by taking unions of hom-sets, because the composition predicate of a protocategory isn’t parametrized by objects. So if we take unions of homsets, then the only way available to define gf=hg\circ f = h is that there exists some A,B,CA,B,C such that f:ABf:A\to B, g:BCg:B\to C, h:ACh:A\to C, and g A,B,Cf=hg\circ_{A,B,C} f = h. But consider an example like this:

    • objects 0,1,2,0,1,20,1,2,0',1',2'
    • hom(0,1)=hom(0,1)={f}\hom(0,1) = \hom(0',1') = \{f\}, hom(1,2)=hom(1,2)={g}\hom(1,2)=\hom(1',2')=\{g\}, hom(0,2)=hom(0,2)={h,h}\hom(0,2)=\hom(0',2')=\{h,h'\}, no other nonidentity morphisms, all identity morphisms distinct
    • g 0,1,2f=hg\circ_{0,1,2} f = h and g 0,1,2f=hg\circ_{0',1',2'} f = h'.

    Then if we take unions of homsets we have gf=hg\circ f = h and also gf=hg\circ f = h', but then the protocategory composition axiom fails: we have f:01f:0\to 1 and g:12g:1\to 2, but there does not exist a unique kk such that k:02k:0\to 2 and gf=kg \circ f = k.

    I’ve added this to the entry.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 16th 2018

    Added a further remark hoping to guide the reader to a proper perspective on protocategories.

    diff, v3, current