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I’ve just edited topological concrete category to correct the claim that topological functors create limits, which is not quite true: for instance, the forgetful fails to reflect limits because choosing a finer topology on the limit vertex yields a non-limiting cone with the same image in . This is correctly reported on wikipedia and in Joy of Cats, p. 227.
It is true that topological functors allow you to calculate limits using the image of the diagram under the functor, which is quite powerful. In Joy of Cats, a topological functor is said to “uniquely lift limits” (definition p. 227, proven p. 363). There doesn’t seem to be an nlab page for this property – I suppose it’s not much used by most category theorists.
Thanks, it looks like I conflated creating limits with uniquely lifting them.
I’m afraid I’m not very familiar with this material yet, but comparing the definition in the nLab (existence of initial lifts) with that of Abstract and Concrete Categories, it looks like we forgot to specify unique initial lifts. I’ll put in “unique”, but let me know if I’ve missed something.
Hmm, one might argue that from a univalent point of view, uniqueness ought to be left out, since initial lifts are already unique up to isomorphism.
Well, I’m not saying I’m committed to putting in uniqueness, only that it’s different from what I see in ACC.
Yeah, if we do leave out uniqueness, we should comment on the difference. I’m not sure what the right thing to do is; leaving it out is arguably more in the nSpirit, but is it worth differing from the literature?
Okay, I took out uniqueness and added a remark which I hope is accurate.
I'm pretty sure that I left that out for that reason. I like your remark; I added one sentence.
Looks good, thanks!
I would like to modify the “default” definition of topological concrete category to include the “evil” condition that , corresponding to the “usual” notion of Grothendieck fibration rather than the weaker one of Street fibration. I think this is usually included in definitions in the literature, and satisfied by most examples, and I don’t think there is a good reason to change it; the weaker notion can be called “weakly topological” or something. The proper way to formulate this sort of condition without equality of objects is probably using displayed categories rather than weakening the equality.
But if anyone has an objection, please raise it!
Ok, nobody objected, so I did this.
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