stabilization hypothesis in nLab
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Idea
The Baez-Dolan stabilization hypothesis states that for all a k-tuply monoidal n-category is “maximally monoidal”. In other words, for , a -tuply monoidal -category is the same thing as an -tuply monoidal -category. More precisely, the natural inclusion is an equivalence of higher categories.
More generally, we can state a version for (n,k)-categories?: an -tuply monoidal -category is maximally monoidal.
Proof when
An aspect of the proof of this when (i.e. that -tuply monoidal -categories are maximally monoidal) was demonstrated in
- Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak -categories (arXiv:math/9810058)
in terms of Tamsamani n-categories?.
A proof of the full statement in terms of quasi-categories is sketched in section 43.5 of
- André Joyal, Notes on quasi-categories (pdf).
Probably the first full proof in print is given in
where it appears in example 1.2.3 as a direct consequence of a more general statement, corollary 1.1.10.
Proof in general
A proof of the general form for arbitrary , using iterated -categorical enrichment to define -categories, is in
- David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads (arXiv:1312.3178)
See also
-
Jacob Lurie, section 5.1.2 Higher Algebra
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Michael Batanin, An operadic proof of Baez-Dolan stabilization hypothesis (arXiv:1511.09130)
Last revised on November 26, 2017 at 16:00:17. See the history of this page for a list of all contributions to it.