Timeline for Is there a shape-independent definition of (∞,1)-categories?
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| Oct 10, 2023 at 16:11 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
Removed "et al." and added a link to the paper.
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| Oct 10, 2023 at 16:07 | answer | added | Dmitri Pavlov | timeline score: 2 | |
| Oct 10, 2023 at 3:34 | comment | added | lemmanade | Looking around this seems to be similar to what nlab calls Trimble ∞-categories, but I am not sure how these relate to the simplicial definitions. I think my question can be rephrased as, is there a definition in terms of terminal coalgebra ala Trimble n-categories, such that its theory coincides with quasi-categories, or perhaps such a definition is known to be impossible. I have to give this a bit of thought. | |
| Oct 10, 2023 at 3:30 | comment | added | lemmanade | @TimCampion I think the point about Δ being the smallest dense full subcategory is what I was looking for. When I said shape-independent what I think I meant is some process only working with some definition of n-categories and taking a limit for n going to infinity, such that it coincides with the topological (shape-based) definitions. In other words the categories involved would be the (n+1)-categories of n-categories, instead of mentioning test categories. | |
| Oct 10, 2023 at 2:53 | comment | added | Tim Campion | Over here I argue that $\Delta \subseteq Cat_{(\infty,1)}$ can be characterized as the smallest dense full subcategory of $Cat_{(\infty,1)}$ which is closed under retracts. So somehow $\Delta$ is pretty inevitable. | |
| Oct 10, 2023 at 2:37 | comment | added | Tim Campion | I'm not sure what "shape-independent" means in general. For example, are topologically-enriched categories a "shape-independent" model? Or are we restricting attention to presheaf models? | |
| Oct 10, 2023 at 1:28 | comment | added | Simon Henry | All known definition involves some kind of concrete combinatorics and shape of the cells. I feel like there is more to say on this question, like try to argue that some shapes will automatically be present, but I'm not sure what to say exactly. Definitely, any kind of model categories will come with some notion of "cells". | |
| Oct 10, 2023 at 0:46 | history | edited | lemmanade | CC BY-SA 4.0 |
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| Oct 10, 2023 at 0:44 | history | edited | lemmanade | CC BY-SA 4.0 |
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| Oct 10, 2023 at 0:31 | history | edited | lemmanade | CC BY-SA 4.0 |
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| S Oct 10, 2023 at 0:29 | review | First questions | |||
| Oct 10, 2023 at 4:35 | |||||
| S Oct 10, 2023 at 0:29 | history | asked | lemmanade | CC BY-SA 4.0 |