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Owen Lynch (Nov 30 2020 at 12:57):Hello, I don't know if this should be in algebraic topology instead of this, but I recently read https://erischel.com/the-homotopy-theory-of-groups/, and although it's supposed to be a pedagogical joke, I'm wondering if their approach could be generalized to model categories of any Generalized Algebraic Theory/Lawvere theory/insert your favorite way of talking about generators and relations. Does anyone have any thoughts about this?
Maxime Lucas (Dec 01 2020 at 13:44):Well I don't know if you noticed but in this model structure any map is both a cofibration and a fibration, and weak equivalences are the isomorphisms (that's the joke behind remark 2.4). This model structure exists on any category...
Reid Barton (Dec 01 2020 at 14:02):I don't think that's right, otherwise the situation in Example 4.1 could not happen. The real point is that Grp is not the category of groups.
Maxime Lucas (Dec 03 2020 at 07:45):Ah I hadn't noticed that Gpd wasn't the usual category of groups, so for me Example 4.1 was just plain false.