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.
Tom Hirschowitz (Oct 27 2023 at 09:32):@Yoann Barszezak and I are having a hard time understanding Street and Walters's proof of comprehensive factorisation, Theorem 3 in this paper.
We can follow the whole proof, except the last step. They obtain a pasting of two triangles, the pasting and the right-hand triangle (λ and γ in the attached diagram) are known to be left extensions, and they conclude that the left-hand triangle also is a left extension.
Screenshot_2023-10-27_11-21-27.png
Why is this proof step valid?
Tom Hirschowitz (Oct 27 2023 at 10:26):(We know how to proceed by hand, it's on the nlab.)
Nathanael Arkor (Oct 27 2023 at 19:17):It's a mistake. This is mentioned in Kelly's Basic Concepts of Enriched Category Theory on page 74:
This seems to be best done by a direct calculation using Theorem 4.67(v). (The purported proof in [74] is erroneous; it supposes that, in the situation of our Theorem 4.47, if X and θ are left Kan extensions, so is ϕ; that this is false is easily seen by taking D = 1 and using (4.34).)