nLab
stabilization hypothesis

Contents

Idea

The Baez-Dolan stabilization hypothesis states that for all k=n+2k = n+2 a k-tuply monoidal n-category is “maximally monoidal”. In other words, for kn+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

in terms of Tamsamani n-categories?.

A proof of the full statement in terms of quasi-categories is sketched in section 43.5 of

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

References

Section 5.1.2 of

Revised on September 2, 2015 13:27:56 by Mike Shulman (192.195.154.58)