New proof and generalization of Licata-Finster theorem #1821
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Why are the equivalences more natural in the other direction? |
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For example, for any type |
Alizter
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Jan 19, 2024
jdchristensen
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Jan 19, 2024
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| destruct n. | ||
| { srapply licata_finster. | ||
| 1: exact _. | ||
| reflexivity. } | ||
| refine ((pequiv_ptr (n:=n.+2))^-1* o*E _). | ||
| symmetry; rapply pequiv_ptr_loop_psusp'. | ||
| 1: srapply (licata_finster (m:=-2)). | ||
| refine (_ o*E pequiv_ptr (n:=n.+2)). | ||
| rapply pequiv_ptr_loop_psusp'. |
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With the new licata_finster lemma, we should be able to avoid the destruct n and make pequiv_ptr_loop_psusp' handle both cases. It's just a matter of some trunc_index juggling. I'll see if I can do that, but probably not today.
jarlg
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Jan 20, 2024
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A few minuscule suggestions/questions; LGTM.
Co-authored-by: Jarl G. Taxerås Flaten <[email protected]>
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With #1814 done, we knew we could give a new proof of the Licata-Finster theorem, while also generalizing it to any truncation level. One way to do this would be to use homotopy groups and the truncated Whitehead theorem, but I noticed that one could use Prop 2.31 from CORS to prove the analogous result for any reflective subuniverse. When I went to formalize it, I found that @mikeshulman had beat me to it almost four years ago with
OO_inverts_conn_map_factor_conn_map! So the additional work to prove Licata-Finster is just a few lines.The first commit contains this, and removes the old encode-decode proof. I'm happy to keep that proof around if people want.
The second commit cleans up EMSpace a bit by reversing some of the equivalences to be in the direction of the natural map, which avoids some inversions in other places.
The third commit contains two results about trunc indices that I didn't end up using but which will likely be useful to someone.