I have a category with two classes of weak equivalences (neither class is contained in the other, and I believe they actually both form homotopical categories). I'm wondering if this notion exists in the literature. I'd like to combine them into one class to obtain a category with weak equivalences (or a homotopical category). I believe this can be done by taking the smallest class of morphisms which contains both classes and which forms such a subcategory. Is there anywhere I can read about something like this?
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1$\begingroup$ This isn't entirely unlike the model-category-theoretic version of Bousfield localization. If you have a model category $C$, a Bousfield localization of $C$ is a model structure on the underlying category of $C$ with the same cofibrations as the original model structure, but with more weak equivalences. Hirschhorn's book "Model categories and their localizations" is a good reference for those ideas. But in your case, neither class of weak equivalences is contained in the other. I doubt that the mixing operation you describe always extends to an operation on the model structures, unfortunately. $\endgroup$– user164898May 7, 2022 at 19:38
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$\begingroup$ @A.S. Oh that's fine, I don't have a model structure in mind anyway. Thanks for the reference. I'm wondering if there is any subtle issue that I'm not thinking of that would prevent me from constructing what I want. $\endgroup$– Josh LackmanMay 7, 2022 at 20:18
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4$\begingroup$ Any collection of arrows in a category will form a (directed) graph, and you can always use the image of this graph under the left adjoint to the forgetful functor ${\bf Cat}\to{\bf Graph}$ to 'complete' it into the smallest subcategory of the original category containing those arrows (up to isomorphism), as discussed e.g. here on the nlab. That being said, you may want to try to form a double category where horizontal arrows are one type of equivalence and vertical arrows are the other type. $\endgroup$– Alec RheaMay 7, 2022 at 20:37
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$\begingroup$ Perhaps a 2-relative category is the sort of thing you have in mind. Dan Kan and I introduced these here: arxiv.org/pdf/1102.0186.pdf $\endgroup$– Clark BarwickMay 7, 2022 at 21:34
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$\begingroup$ This happens quite often. Bousfield localizations have already been mentioned. At a more elementary level, on topological spaces, you have homotopy equivalencies and weak equivalences. Same with chain complexes. $\endgroup$– Fernando MuroMay 8, 2022 at 10:55
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