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Todd Trimble (Jul 10 2024 at 17:36):In cases typically arising in practice, a [[convenient category of spaces]] is coreflective in the larger category of all topological spaces. A category that is coreflective in a [[cototal category]] (such as the category of topological spaces) is also cototal, so on these grounds a convenient category of spaces is typically going to be cototal. But is it total? I'm guessing probably not, but can someone say for sure?
Morgan Rogers (he/him) (Jul 11 2024 at 07:26):I presume your question is whether there exists a coreflective category of Top which is not total, rather than whether failing to be total would make the category inconvenient :stuck_out_tongue_wink:
Benedikt Peterseim (Jul 11 2024 at 07:55):If you believe the nlab, the answer is in fact that every coreflective subcategory of Top (Edit: for which the coreflectors are bijective, which is typically the case for convenient categories of spaces) is total. Every topological concrete category is both total and cototal (according to https://ncatlab.org/nlab/show/total+category) and both reflective and coreflective subcategories of topological concrete categories (Edit: for which the (co)reflectors are bijections) are again topological concrete categories (according to https://ncatlab.org/nlab/show/topological+concrete+category#further_properties).
Todd Trimble (Jul 11 2024 at 11:19):Ah, thank you Benedikt! I knew that topological categories are both total and cototal, but I'd missed that "further property". (Probably a lot of this is in The Joy of Cats, so I'll look there next.)