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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?
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.
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.)
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:
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.
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