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Stream: learning: questions

Topic: weak equivalences without model structures

Daniel Teixeira (Nov 22 2021 at 22:39):

Are there interesting examples of categories with weak equivalences $(C,W)$ that do not admit any model structure? What tools are available then, besides Gabriel-Zisman or simplicial localization?

Zhen Lin Low (Nov 22 2021 at 23:10):

Well, the category of diagrams of a given shape in a model category always has the structure of a category with weak equivalences, but it is not always known whether it has a model structure per se. (Are there any explicit counterexamples? I forget.)

Tim Hosgood (Nov 22 2021 at 23:12):

as a sort of equivalent suggestion, some functor category between homotopical categories might give a good example, but I also don't know of one off the top of my head (which disappoints me!)

Tim Hosgood (Nov 22 2021 at 23:12):

note that, in order to not admit any model structure at all, it must be the case that the category is not bicomplete (otherwise you have the maximal/minimal model structures)

Daniel Teixeira (Nov 22 2021 at 23:15):

Tim Hosgood said:

note that, in order to not admit any model structure at all, it must be the case that the category is not bicomplete (otherwise you have the maximal/minimal model structures)

this should work, iirc biequivalences provide weak equivalences for bicategories, but $\text{Bicat}$ is never complete or cocomplete unless you only consider strict pseudofunctors

Daniel Teixeira (Nov 22 2021 at 23:16):

that is why Lack's model structure only exists for $\text{Bicat}_{strict}$ in the first place

Daniel Teixeira (Nov 22 2021 at 23:17):

the obvious follow up is a counterexample in the case of bicompleteness, and I bet it would be of the kind that Zhen Lin suggested as presheaves with values in C are bicomplete if C is, e.g. diagrams in Set, Top, Ab etc

Daniel Teixeira (Nov 22 2021 at 23:53):

Tim Hosgood (Nov 23 2021 at 00:05):

Daniel Teixeira said:

related https://mathoverflow.net/questions/133051/a-category-with-weak-equivalences-which-is-not-a-model-category

oh, so semi-simplicial sets are a perfect example! I can't believe I didn't know this one!