stabilization hypothesis in nLab

Contents

Contents

Idea

The Baez-Dolan stabilization hypothesis states that for all k≥n+2k \geq n+2 a k-tuply monoidal n-category is “maximally monoidal”. In other words, for k≥n+2k \geq n + 2, a kk-tuply monoidal nn-category is the same thing as an (n+2)(n+2)-tuply monoidal nn-category. More precisely, the natural inclusion kMonnCat↪(n+2)MonnCatk Mon n Cat \hookrightarrow (n+2) Mon n Cat is an equivalence of higher categories.

More generally, we can state a version for (n,k)-categories?: an (m+2)(m+2)-tuply monoidal (m,n)(m,n)-category is maximally monoidal.

Proof when n=1n=1

An aspect of the proof of this when n=1n=1 (i.e. that m+2m+2-tuply monoidal (m,1)(m,1)-categories are maximally monoidal) was demonstrated in

• Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak nn-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

• Jacob Lurie, Ek-Algebras ,

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 m,n,km,n,k, using iterated (∞,1)(\infty,1)-categorical enrichment to define (∞,n)(\infty,n)-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

• Michael Batanin, An operadic proof of Baez-Dolan stabilization hypothesis (arXiv:1511.09130)