You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
James Deikun (Jan 12 2024 at 17:30):... by their underlying category? My instinct says yes, my attempts to prove it say maybe not.
Mike Shulman (Jan 12 2024 at 18:57):My instinct says no...
Sridhar Ramesh (Jan 12 2024 at 20:04):Is the question whether the "underlying category" functor from the category of multicategories to the category of categories is a Grothendieck fibration?
Mike Shulman (Jan 12 2024 at 20:11):That's what I took it to be.
James Deikun (Jan 12 2024 at 20:28):Yes, that was the question. The answer is "no", btw. Take the multicategory X generated by two objects A and B, a morphism and a "constant" g on B. The underlying category is the walking arrow. There is no Cartesian lift of the injection of the discrete category on two points into the walking arrow because there is no common factorization of the arrow from the multicategory generated by only f and the arrow from the multicategory generated by only g.
Mike Shulman (Jan 12 2024 at 20:31):It might be an opfibration, though.