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Stream: theory: category theory

Topic: isocofibrations lifting property

view this post on Zulip Daniel Teixeira (Aug 25 2023 at 18:32):

The cofibrations Cof\text{Cof} of the canonical model structure of Cat\mathsf{Cat} are the functors injective on objects. The trivial fibrations Fib∩W\text{Fib}\cap \mathcal W are the equivalences surjective on objects.

With these definitions, I'm trying to show that indeed Cof=LLP(Fib∩W)\text{Cof} = LLP(\text{Fib}\cap \mathcal W), but I'm struggling in both directions. Any ideas?

view this post on Zulip Kevin Arlin (Aug 25 2023 at 20:12):

For ⊆\subseteq and ff your cofibration and gg your trivial fibration, you have no choice where to send objects in the image of f,f, while all you know for objects not in the image is that they have to lift into a certain fiber of g.g. Since there's no way to prefer any particular elements of this fiber, perhaps any choice will work?