Section 1. (0MLL) · ScalingStacks

1. Introduction[0MLL]

Any model for the theory of $(\infty,n)$ -categories must, at a minimum, form an $\infty$ -category $\mathcal{C}$ .¹¹ 1 By this we mean a quasicategory in the sense of Michael Boardman and Rainer Vogt, André Joyal, and Jacob Lurie; we freely use the language and technology of Lurie’s books [28, 26]. Such an $\infty$ -category must contain the gaunt $n$ -categories (Definition 3.1) as a full subcategory – these are strict $n$ -categories with no nontrivial isomorphisms at any level. In particular, $\mathcal{C}$ contains the $k$ -cells $C_{k}$ for $0\leq k\leq n$ . These are the strict $n$ -categories with the universal property that the set of $k$ -morphisms of a strict $n$ -category $D$ is the set of functors $C_{k}\to D$ .

In order for the objects of $\mathcal{C}$ to be considered as $(\infty,n)$ -categories, they must be built from cells together with composition operations. These composition operations are governed by pasting diagrams, possibly quite general ones. The largest conceivable collection of these pasting diagrams is the class of gaunt $n$ -categories itself, so we encode these properties via a pair of axioms:

C.1

Strong generation. Every object of $\mathcal{C}$ is the canonical colimit (in the $\infty$ -categorical sense) of the diagram of all the gaunt $n$ -categories that map to it.
C.2

Weak generation. Every object of $\mathcal{C}$ admits a ‘cell decomposition’ – i.e., it is some colimit of cells (again in the $\infty$ -categorical sense).

Correspondences are configurations of $(\infty,n)$ -categories parametrized by cells. These must be well-behaved:

C.3

Internal Homs for correspondences. For any $0\leq k\leq n$ , the $\infty$ -category of objects of $\mathcal{C}$ over the $k$ -cell $C_{k}$ has internal Homs.

Our pasting diagrams are constructed by repeatedly applying certain gluing operations. These operations must remain colimits when viewed in $\mathcal{C}$ . To ensure this, we identify (6.5) a finite list $S_{00}$ of pushouts of gaunt $n$ -categories, and we introduce another axiom:

C.4

Fundamental pushouts. The image of the diagrams $S_{00}$ are pushouts in $\mathcal{C}$ .

Finally, the $\infty$ -category $\mathcal{C}$ must be minimal (in a rather weak sense) with these features:

C.5

Versality. If $\mathcal{D}$ is an $\infty$ -category that contains the gaunt $n$ -categories as a full subcategory and satisfies the axioms above, then there is a left adjoint $K\colon\mathcal{C}\to\mathcal{D}$ and a natural transformation $\eta$ between the restriction of $K$ to the gaunt $n$ -categories and the inclusion of the gaunt $n$ -categories into $\mathcal{D}$ such that $\eta$ is an equivalence on cells.

An $\infty$ -category $\mathcal{C}$ that contains a copy of $\gaunt_{n}$ and satisfies these axioms is called a theory of $(\infty,n)$ -categories. In this paper

•

we prove that there is a unique theory of $(\infty,n)$ -categories, up to equivalence;
•

we prove that, up to the formation of opposites, there is a contractible space of equivalences of the theory of $(\infty,n)$ -categories; and
•

we prove that all the best-known purported models of $(\infty,n)$ -categories satisfy these axioms and are therefore equivalent in an essentially unique manner.

In more detail, the main theorem is:

[0MH7]

Theorem 1.1 (Unicity). The moduli space $\thy_{(\infty,n)}$ of theories of $(\infty,n)$ -categories is a $B(\mathbb{Z}/2)^{n}$ .

Any theory of $(\infty,n)$ -categories has internal Homs, and is thus canonically enriched in itself. Results of David Gepner and Rune Haugseng [20, 21] show that categories enriched in $(\infty,n)$ -categories are a model of $(\infty,n+1)$ -categories. Thus the unicity theorem for the $\infty$ -category of $(\infty,n)$ -categories implies the unicity of the $(\infty,n+1)$ -category of $(\infty,n)$ -categories.

Other axiomatizations of higher categories[0MLM]

Carlos Simpson [36, Conjectures 2 and 3] conjectured a similar unicity result for the theory of $n$ -categories. Simpson suggests ten axioms (Properties 1–10), which are extremely different from those here. Nevertheless, the kind of unicity that Simpson proposed (and even the idea that one could axiomatize the homotopy theory of higher categories itself) was of course a direct inspiration for our work here.

Bertrand Toën [38] later proved a Unicity Theorem of the kind above for the theory of $(\infty,1)$ -categories. His framework provides seven axioms. The basic data is that of a homotopy theory $\mathcal{M}$ containing a cosimiplicial interval object. A subset of his axioms (A2, A6, A7) imply that this interval object can be used to define a right adjoint $N$ from $\mathcal{M}$ to the homotopy theory $\CSS$ of complete Segal spaces. The axioms (A6) is that $N$ is conservative, and the rest of the axioms are used to show that the left adjoint of $N$ is fully faithful, which shows this is an equivalence. Since $\CSS$ satisfies our axioms for $n=1$ (see 14.6), one knows a posteriori that Toën’s axioms and ours specify the same homotopy theory.

One may ask whether this is clear a priori. It seems not: there doesn’t seem to be any simple mechanism by which one could translate Toën’s axioms into ours or vice versa. While our axioms do have the one point in common that we each require the existence of a well-behaved (presentable) homotopy theory and internal Homs, the similarities end here. Even the basic data we are axiomatizing is not the same: while Toën’s axioms are precisely adapted to the comparison with $\CSS$ , our axioms remain agnostic about the “shapes” of the basic objects that are used to generate models of higher categories.

Plan[0MLN]

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:

(a)

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

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

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

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

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

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

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

Acknowledgements[0MLP]

We are grateful to Charles Rezk, who noticed an error in a very early version of this paper. We also thank the referee, who made a number of helpful suggestions to make our arguments more legible.

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

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Original source · 1112.0040v6