FlorianAdler
commented
3 years ago Hi Bastiaan,
Thanks for your feedback!
I believe we defined the fat slice E_{f//} to be ({f} --> Fun(C,E))/(const: E --> Fun(C,E)), where in general F/G for functors F: C --> E and G: D --> E is the pullback of Ar(E) --> E x E along C x D --> E x E. In particular, the slice f/E in question is actually abuse of notation for f/idE. I'm quite sure that f/E and E{f//} are different in general, unless f is the inclusion {e} --> E of a 0-simplex, in which case we obtain the usual slice e/E. Let's ask Fabian in the lecture today!
For the second point, you're totally right that Omega is not a functor on An, but on /An. What I meant to say was that once a basepoint x --> X is chosen, it doesn't matter if we form the pullback defining Omega_xX in /An or in An (and then equip it with its canonical basepoint). I should definitely reformulate that bit to make it less confusing. (The whole point of Remark II.11a was that in order to prove the suspension-loop adjunction one would like to take the pushouts/pullbacks in /An.)
BastiaanCnossen
Hi Ferdinand,
Thanks for your amazing notes!
A few points I noticed when reading:
In the "Proof of Theorem I.52*", you say that the slice f/E you define is not the same as the fat slice E_{f//}. This confuses me a little, because I thought we defined the fat slice as the pullback of Ar(E) -> E x E along C x E -> E x E, and since C x E is the pullback of C -> E <- E x E this would give the same as your definition right? (Also a couple of lines later there is a typo: the "t^{−1}{c} \simeq f / e" should have been t^{-1}(e).)
In "II.11a. Remark" I fully agree with your analysis of Sigma already being defined without the basepoint, but I disagree with your remark that the dualized statement for Omega also holds. The situation isn't self-dual: is a final object but not an initial object in An. So although we can always form a pushout along the canonical map X -> , we can only form a pullback along a map -> X if we have chosen a basepoint in X.