Timeline for Useful ideas in category theory which violate the principle of equivalence
Current License: CC BY-SA 4.0
21 events
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| Sep 26, 2023 at 23:03 | comment | added | LSpice | You referred to the "barr construction". I know that there's a Michael Barr whose work is not disjoint from this topic, but I think you meant the bar construction, and edited accordingly. I hope that was correct. | |
| Sep 26, 2023 at 23:02 | history | edited | LSpice | CC BY-SA 4.0 |
barr -> bar; consistency in n-category vs. $n$-category
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| Sep 26, 2023 at 18:31 | vote | accept | Brendan Murphy | ||
| Sep 26, 2023 at 14:16 | history | edited | Simon Henry | CC BY-SA 4.0 |
added 258 characters in body
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| Sep 26, 2023 at 14:14 | comment | added | Simon Henry | @Mike maybe you are right. I was thinking about it in the sense of giving category theoretic meaning to the question of whether "a given weak n-category can be represented by a strict n-category", which at first sight looks like a non-CT problem but can be phrased in terms of existence of a certain filtration on the object (or maybe as well in terms of vanishing of certain higher order operation?). I edited to try to clarify that point. | |
| Sep 26, 2023 at 6:51 | history | edited | Mike Shulman | CC BY-SA 4.0 |
Added mention of dssi's to (B).
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| Sep 26, 2023 at 3:22 | comment | added | Mike Shulman | But then in what sense is your description of strict $n$-categories "about categories"? | |
| Sep 26, 2023 at 2:55 | comment | added | Simon Henry | @MikeShulman, I Agree, but that's somehow what I mean when I say "it is not a result about categories anymore". For e.g. when you phrase Arlin's work in these terms then what he is considering is a pair of a set valued functor and a cat valued functor with a level-wise essentially surjectif functor, and most of what he is doing - or at least the key steps - revolves around the set valued functor, and not the cat valued functor. | |
| Sep 26, 2023 at 2:06 | comment | added | Mike Shulman | On the other hand, if you're willing to admit your description of strict $n$-categories as valid, then for $n=1$ it gives you a notion of strict category, and basically any non-equivalence-invariant notion you like can be formulated in terms of those. For instance, an Arlin prederivator could be defined as a 2-functor taking values in strict categories. | |
| Sep 26, 2023 at 2:02 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Sep 26, 2023 at 2:01 | comment | added | LSpice | I got a bit confused about numbering, and so didn't want to edit for fear of disrupting meaning, but did you really mean to refer to "the second second example"? | |
| Sep 26, 2023 at 2:00 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Sep 26, 2023 at 1:53 | history | edited | Simon Henry | CC BY-SA 4.0 |
added 739 characters in body
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| Sep 26, 2023 at 0:56 | comment | added | Mike Shulman | For another example of (B), how about lax functors between bicategories, which are not invariant under equivalence of bicategories (or even 2-equivalence of 2-categories)? (Of course one may claim they are "really" about double categories instead...) | |
| Sep 26, 2023 at 0:55 | comment | added | Mike Shulman | Interesting point about Arlin's work; that's a good explanation of why it always made me uncomfortable, more so than your examples of (A). | |
| Sep 26, 2023 at 0:52 | comment | added | Mike Shulman | However, the matching up of the source and target in composition is not an issue with the principle of equivalence, because when the definition is stated correctly using a family of hom-sets indexed by the objects, there is no equality of objects mentioned. | |
| Sep 25, 2023 at 19:50 | comment | added | Simon Henry | In a sense yes. When we say "a category is a set of object together with..." we break the equivalence principle, as the set of objects is not an invariant under equivalence. See here mathoverflow.net/q/309524 and once you take the step of saying that a category is "a groupoid of objects together with..." then you have to contend with the issue you mention seriously | |
| Sep 25, 2023 at 19:50 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading and names of links
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| Sep 25, 2023 at 19:44 | comment | added | Will Sawin | Do the strictness problems you mention for higher categories also show up to some extent for 1-categories? Like the notion of composition as a function on pairs of morphisms with the same source and target is strict because we demand the source and target are equal on the nose and not just isomorphic. | |
| Sep 25, 2023 at 19:22 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Sep 24, 2023 at 3:22 | history | answered | Simon Henry | CC BY-SA 4.0 |