nLab stabilization hypothesis

Idea

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

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

Proof when $n=1$

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

• Carlos Simpson, On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak $n$-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 $m,n,k$, using iterated $(\infty,1)$-categorical enrichment to define $(\infty,n)$-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)