Contents

Idea

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

Proof for $(n,1)$-categories

An aspect of the proof of this for (n,1)-categories 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.

References

Section 5.1.2 of

• Jacob Lurie, Higher Algebra