first (non-art) cohost post! (lets hope i get this right)

for a decent while now ive been trying to research cocategories and cocategory objects, since nobody else really does much. i dont really think its cause theyre genuinely useless or anything, but more because theyre unfamiliar. im currently writing a paper on the subject, but had a question regarding naming convention.

since this hasnt really been something studied much before there arent any existing conventions regarding cocategorical structures, so i wanted to take the opportunity to get some kind of consensus beforehand. we basically have a choice of naming things either by resemblance or reflection (at least if we intend to avoid prefix soup)

for example, the cocategory homomorphism that tends to be most useful more closely resemble cofunctors in category theory, but then of course they reflect functors as a dual in the same way that cofunctors normally do. if we go by resemblence it allows us to, for example, just call thing which look most like products as products. on the other hand, if we go by reflection the word "functor" better translates that they have the same usage in category theory as they do in cocategory theory.

in any case cocategories are at least conceptually very interesting. a small cocategory, for example has, boolean algebras representing its objects and morphisms, and each coclosed category gives rise to one in a similar way to how closed categories produce new categories. hell, even Hask is a cocategory if you define negations.

heres the intuition i currently have so far (though id be delighted to hear if you have any others!):
cocategories "contract" all endomorphisms. exactly what that means depends on the ambient category, but it may mean the empty set for one example
they also provide a way to take one morphism and split it into two along an arbitrary point
if that point is chosen to be one of the existing points on either side of the original morphism, the endomorphic segment contracts, and the other segment is equal to the original (figure 1)
if you split a morphism into three seperate segments, it does not matter whether you split from the left or right on the second subdivision, because each pair of possible segments must be the same for all three segments (figure 2)

excited to see what people think about this, and cocategory theory in general!