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1:20 PM
Suppose we have a commutative square of simplicial sets such that: The left-most vertical map is a cofibration and the right-most vertical map is a weak equivalence (Joyal) between infty-categories. It obvious that one can find a solution to this lifting problem in the homotopy category of Set_{\Delta} with respect to the Joyal model structure.

Can I lift this solution to the original commutative square?
 
@F.Abellan Don't you basically need that the rightmost vertical map is a trivial fibration? (In general, ofc in special cases there might be lifts regardless)
 
@DenisNardin I was thinking about an improvement of Proposition A.2.3.1 in HTT but maybe this is not possible.
 
@F.Abellan I think you can, but you need to replace the condition that $X$ is fibrant with the condition that the right vertical map is a fibration
But I haven't checked all the details, this just feels what the natural generalization should be
 
2:08 PM
@F.Abellan Take a map between quasicategories which is a cofibration and a Joyal weak equivalence. (For example, the inclusion of a full subcategory which contains a representative of every isomorphism class of objects.) If you can solve the lifting problem of this map against itself, it has to be an isomorphism.
What you can say under these hypotheses is that you can find a lift in which the top triangle commutes, and the bottom triangle commutes up to homotopy. See Proposition 3.5 of tac.mta.ca/tac/volumes/33/3/33-03.pdf
 
2:25 PM
I guess there is a second, implied question: suppose given a square with a cofibration on the left and fibrant objects (quasicategories) on the right, but neither map is assumed to be a weak equivalence; furthermore suppose given a solution to the lifting problem in the homotopy category. Then to what extent can it be strictified to a solution to the lifting problem in the original category?
I think the answer is the same (upper triangle commutes while lower triangle does so up to homotopy) but I don't know a reference for this.
 
3:07 PM
I think you can even characterize weak equivalences (between fibrant objects?) as maps with the following lifting property against cofibration: the upper triangle strictly commutes and the lower one commutes up to relative homotopy.
 

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