@EdoardoLanari I might be confused but as far as I understand in section 6 they define Nat(F,G):=lim_{Tw(C)}Fun(F(-),G(-)). My question is if we define Nat(F,G) by considering Fun(C,Cat_{\infty)_{/G} and then taking the fibre (in scaled simplicial sets) over F we obtain the same thing.
You do, but the slice you have to consider is the "lax" slice, which is defined wrt to a suitable join for scaled simplicial sets (you can check Section 2.2 here arxiv.org/pdf/1911.01905.pdf , excuse the self-reference)
Trying to understand the proof that covering dimension bounds the homotopy dimension in HTT, me and a friend found independently counter examples to the validity of construction 7.2.3.5 (i.e. that the sets in the construction satisfy condition (2)), which seem quite important in the proof of hodim<=covdim. Anyone aware of this problem and how to fix it? it seems very technical and minor but I don't know how to...
Take X_1,X_2 and X_3 to be equal to the discrete set {1,...,10}, let k=1 and let all the partitions of the double intersections be trivial except that X_1 V X_2 is given by {1,...,5}V{6,...,10}. Now consider the sets W_{1,(1,3),{1,...,10}} and W_{2,(2,3),{1,...,10}}. By the definitions there the intersection should be {1,...,10} which is not inside any of the sets in the partition associated with (1,2)
the problem is that you are allowed to never choose A_0 in the book's notation to actually be your tuple.
