Timeline for Does the simplex map to the cube?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 7, 2023 at 23:28 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
added note of Street’s error, Forest’s correction
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| Dec 7, 2023 at 21:54 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
added links to cited papers
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| Dec 7, 2023 at 21:51 | comment | added | Peter LeFanu Lumsdaine | @willie: The full answer to your questions is too long for comments, so I’ve edited to add it to the answer. The short answer: as you say, this isn’t a map of parity complexes in the sense of sending atomic cells to atomic cells — this is a map of the $\omega$-categories they generate, and the fact that this data specifies such a map is exactly by the universal property of $\mathcal{O}_n$ as freely generated by its parity complex. The details are slightly gnarly, and can be found in the papers of Street. | |
| Dec 7, 2023 at 21:47 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
added clarification in response to OP’s comment
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| Dec 5, 2023 at 22:29 | comment | added | Peter LeFanu Lumsdaine | @willie: Exactly — cells of the generated $n$-category are represented by suitable sets of atomic cells, i.e. cells of the parity complex itself. And the $n$-category is sufficiently “freely” presented that to map out of it into any $n$-category, it suffices to specify the map on atomic cells. | |
| Dec 5, 2023 at 19:55 | comment | added | willie | @ Peter LeFanu Lumsdaine: Thanks a lot for your answer. This looks very helpful. I agree that the term "oriented simplex" is the better terminology. | |
| Dec 4, 2023 at 11:56 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
added composability condition overlooked
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| Dec 4, 2023 at 11:26 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |