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I got annoyed with the fact that these links did not exist, and so I created now stubs for them:
To the latter entry I moved the references on (∞,1)-topos theory that had been lsited at higher topos theory.
Mike has in his pages much about his thinking about 2-topoi…
Yeah, and I’ve been planning to move it to the main nLab for a while. But for now, a link will have to do.
The ordered sum is the collage of the terminal profunctor, and also the cocomma object of the product projections A←A×B→B (a.k.a. the join).
The best I can do right now for the ordered product is that it’s the (generalized) Grothendieck construction of the lax functor B→Prof that sends all objects to A and all nonidentity morphisms to the terminal profunctor. Note that well-definedness of the latter depends on antisymmetry of B; it’s not clear to me that it has a generalization to non-posets, or even to other (1,2)-toposes (since the definition as I gave it uses excluded middle).
Ok, thanks! So is there anything to learn from it being a (1,2)-topos, a topic we have absolutely nothing about on nLab.
Talking with John Baez elsewhere, seeing these poset composition operations in terms of operads might shed some light.
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