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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.

• 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 $Set$ (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 $Set$ 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 $Grp$ is structured over $Set$. 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 $Set$-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)

• 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 $g\circ f = h$ is that there exists some $A,B,C$ such that $f:A\to B$, $g:B\to C$, $h:A\to C$, and $g\circ_{A,B,C} f = h$. But consider an example like this:

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

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

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.