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Does someone know offhand the relationship between the stabilization hypothesis “for $(n,1)$-categories” attributed to Joyal and Lurie at stabilization hypothesis and the version that appears in arXiv:1312.3178? It would be nice to add a reference to the latter to the page stabilization hypothesis but I’m not sure how to relate it to what’s already there.
Here is what I know. The strong form of the stabilization hypothesis states the following: (This is what is proven by Rune and David in the arxiv paper you link to).
Thm: There is an equivalence of homotopy theories between pointed k-trivial (infty, n+k)-categories and E_k-algebras in (infty,n)-categories.
Here we can regard the E_k operad as an operad in (infty, 0)-categories = infty-groupoids.
The inclusion of (n+m, n)-categories into all (infty,n)-categories has a left adjoint which is a kind of truncation functor which I will call T. This functor is product preserving. This implies that an (n+m, n)-category is an E_k-algebra if and only if it is a T(E_k)-algebra. What is this operad T(E_k)? it is precisely the operad whose spaces are the (n+m)-types of the spaces of the E_k-operad. Now the map of operads from E_k to E_{k+1} is an equivalence of (n+m)-types if k is sufficiently large compared to (n+m). I believe this happens when k is at least n+m+2, but it is not hard to work out the precise bounds.
Taking m=0, (and using the strong form of the SH to go back to k-trivial (n+k,n+k)-categories) this implies the usual weaker form of the stabilization hypothesis, which is the one stated on the nlab page.
Letting m vary, but taking n=1 recovers the version attributed to Joyal and Lurie.
Thanks! I’ve added this to stabilization hypothesis and also delooping hypothesis.
I have added pointer to the new Batanin 15
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