ScalingStacks

This paper is divided into three parts. The first part concerns various aspects of strict $n$-category theory, most particularly including the theory of gaunt $n$-categories. It does not make use of any $\infty$-category theory.

The second part concerns the axiomatization. We first introduce our axioms. Then we show that $\thy_{(\infty,n)}$ is nonempty by explicitly constructing a theory of $(\infty,n)$-categories that satisfies our axioms. We show that any other theory of $(\infty,n)$-categories is equivalent to this given theory – $\thy_{(\infty,n)}$ is connected. We then compute the based loopspace at the point we constructed – that is, the space of autoequivalences of the model of $(\infty,n)$-categories. There are $n$ obvious involutions, which are given by forming the opposite at each categorical level; it turns out that up to a contractible space of identifications, these are all of the autoequivalences.

In the third and final part of this paper we prove that most of the purported models of $(\infty,n)$-categories in the literature satisfy our axioms. These include:

1. (a)

   Charles Rezk’s complete Segal $\Theta_{n}$-spaces,

2. (b)

   the $n$-fold complete Segal spaces of the first-named author,

3. (c)

   André Hirschowitz and Simpson’s Segal $n$-categories,

4. (d)

   the $n$-relative categories of the first-named author and Dan Kan,

5. (e)

   categories enriched in any internal model category whose underlying homotopy theory is a homotopy theory of $(\infty,n)$-categories,

6. (f)

   when $n=1$, Boardman and Vogt’s quasicategories,

7. (g)

   when $n=1$, Lurie’s marked simplicial sets, and

8. (h)

   when $n=2$, Lurie’s scaled simplicial sets,

Consequently they are all equivalent to our model, in a manner that is unique up to the formation of the opposites at the various levels. This also confirms that any model categories that these $\infty$-categories underlie are Quillen equivalent. In fact, Quillen equivalences between model categories of $(\infty,n)$-categories are easily recognized (Proposition 15.10): a Quillen adjunction between two model categories of $(\infty,n)$-categories is a Quillen equivalence if and only if it preserves the cells up to weak equivalence. This implies that many of the known Quillen functors relating various models are in fact Quillen equivalences.