Timeline for Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 1, 2018 at 8:58 | vote | accept | Soham Chowdhury | ||
| Aug 31, 2018 at 23:53 | history | edited | Noah Snyder | CC BY-SA 4.0 |
symmetry was misspelled
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| Aug 31, 2018 at 18:54 | comment | added | Mike Shulman | By the way, the idea of categories as "groupoids with structure" is very explicit in Chapter 9 of the HoTT Book homotopytypetheory.org/book, which in turn owes a lot to Charles Rezk's "complete Segal spaces" as a model for $(\infty,1)$-categories by regarding them as "$\infty$-groupoids with structure". | |
| Aug 31, 2018 at 18:53 | comment | added | Mike Shulman | Maybe worth adding that once you do take that step of regarding groupoids, or more generally $\infty$-groupoids, as a primitive notion to be axiomatized, then it is no longer true that every category can be presented by a set of objects equipped with a set of arrows (unless you assume as an axiom that every groupoid can be so presented, which rules out many naturally-occurring higher topos models). | |
| Aug 31, 2018 at 18:06 | comment | added | David Corfield | The "n-categorical hierarchy" extends in another direction to the (n, r)-hierarchy, or periodic table - ncatlab.org/nlab/show/%28n%2Cr%29-category#the_periodic_table. | |
| Aug 31, 2018 at 17:13 | comment | added | Soham Chowdhury | Thanks for clearing that up! One of the rare cases where something is less subtle than I thought it was. | |
| Aug 31, 2018 at 17:04 | comment | added | Peter LeFanu Lumsdaine | Right, this is exactly my impression. (And I definitely understand Voevodsky there as using breakthrough in the sense of a personal aha moment, not in the sense of a difficult technical triumph — he had much higher standards of what the latter would mean…) | |
| S Aug 31, 2018 at 16:34 | history | suggested | psmears | CC BY-SA 4.0 |
Improve wording and grammar
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| Aug 31, 2018 at 16:28 | review | Suggested edits | |||
| S Aug 31, 2018 at 16:34 | |||||
| Aug 31, 2018 at 14:55 | comment | added | Soham Chowdhury | Seconded. I'd never thought of categories as "groupoids with structure" before, and "To some extent the set of object and of arrows with the appropriate structure is a "presentation" of your category." was a bit of an aha-moment (a breakthrough, if you will!) for me. :) | |
| Aug 31, 2018 at 14:27 | comment | added | Pol van Hoften | This is a great answer! | |
| Aug 31, 2018 at 14:27 | history | edited | Simon Henry | CC BY-SA 4.0 |
added 63 characters in body
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| Aug 31, 2018 at 14:20 | history | answered | Simon Henry | CC BY-SA 4.0 |