The Baez-Dolan Stabilization Hypothesis was posed in 1995. It involves the relationship between weak $n$-categories as $n$ varies. Specifically, if one has an $n+k$ category $C$ and "forgets" to an $n$-category $D$ then $D$ has extra structure. To make sure we didn't forget anything important, we assume $C$ has exactly one object, one 1-cell, one 2-cell, ..., one $(k-1)$-cell, and we are simply reindexing so that the $k$-cells in $C$ become the objects of $D$. For example, if $k=1$ then the objects in $D$ are the morphisms of $C$, so they have a composition law making them into a monoid. If $k = 2$ then the objects of $D$ have both horizontal and vertical composition rules, so by the Eckmann-Hilton argument, the objects of $D$ have the structure of a commutative monoid. If $k \geq 3$, you still get a commutative monoid. The forgetting process stabilized after $k \geq n+2$. The stabilization hypothesis posits that this always happens, in any reasonable model of a weak $n$-category. It was a hypothesis rather than a conjecture because there was no known model at the time of weak $n$-categories.

The stabilization hypothesis has recently been proven in many different models of weak $n$-categories including:

1. Enriched $\infty$-categories (by a remark in Lurie's book (2009), then a 2013 paper of Gepner and Haugseng)

2. Charles Rezk's $\Theta_n$-space model for weak $n$-categories and $(m,n)$-categories, by a 2015 paper of Michael Batanin.

3. Tamsamani's model, Simpson's higher Segal categories, Ara's $n$-quasicategories, and various models due to Bergner and Rezk, by a 2020 paper by Michael Batanin and me.

The Baez-Dolan Stabilization Hypothesis was posed in 1995. It involves the relationship between weak $n$-categories as $n$ varies. Specifically, if one has an $n+k$ category $C$ and "forgets" to an $n$-category $D$ then $D$ has extra structure. To make sure we didn't forget anything important, we assume $C$ has exactly one object, one 1-cell, one 2-cell, ..., one $(k-1)$-cell, and we are simply reindexing so that the $k$-cells in $C$ become the objects of $D$. For example, if $k=1$ then the objects in $D$ are the morphisms of $C$, so they have a composition law making them into a monoid. If $k = 2$ then the objects of $D$ have both horizontal and vertical composition rules, so by the Eckmann-Hilton argument, the objects of $D$ have the structure of a commutative monoid. If $k \geq 3$, you still get a commutative monoid. The forgetting process stabilized after $k \geq n+2$. The stabilization hypothesis posits that this always happens, in any reasonable model of a weak $n$-category. It was a hypothesis rather than a conjecture because there was no known model at the time of weak $n$-categories.

The stabilization hypothesis has recently been proven in many different models of weak $n$-categories including:

1. Enriched $\infty$-categories (by a remark in Lurie's book (2009), then a 2013 paper of Gepner and Haugseng), published in Advances.

2. Charles Rezk's $\Theta_n$-space model for weak $n$-categories and $(m,n)$-categories, by a 2015 paper of Michael Batanin, published in Proceedings of the AMS.

3. Tamsamani's model, Simpson's higher Segal categories, Ara's $n$-quasicategories, and various models due to Bergner and Rezk, by a 2020 paper by Michael Batanin and me, published in Transactions.