ScalingStacks

We now introduce our axioms for the theory of $(\infty,n)$-categories.

The first axiom states that every object of $\mathcal{C}$ can be written as a colimit of gaunt $n$-categories in a canonical manner; this is called strong generation.

Equivalently, any object $X\in\mathcal{C}$ is the canonical colimit of the gaunt $n$-categories mapping to it:

This is equivalent to the condition that the functor $f$ induces an localization $\pre(\gaunt_{n})\to\mathcal{C}$; that is, $\mathcal{C}$ can be written as $W^{-1}\pre(\gaunt_{n})$ for some class of maps $W$ of small generation ([27, Remark 20.4.1.5], [28, Proposition 5.5.4.16]).

The next axiom states that every object of $\mathcal{C}$ can be written as a colimit of objects of $\mathbb{G}_{n}$, but not necessarily in this canonical manner.

We also demand that $\mathcal{C}$ admit internal Homs for correspondences.

Equivalently, Axiom C.3 states that every morphism to $C_{k}$ is, in the language of Ayala–Francis [3], an exponentiable fibration. (We are grateful to the referee for suggesting this observation.)

We introduce a special collection of maps of $\mathcal{C}$. Denote $S_{00}$ consist of the union $A\cup B\cup C\cup D$ of the following four finite sets of maps of $\mathcal{C}$:

(when $i=0$, we interpret this as the initial object of $\mathcal{C}$ mapping to the image under $f$ of the empty $n$-category),

and, lastly,

Finally, we will require that $\mathcal{C}$ be versal with Axioms C.1–4, so that any other presentable $\infty$-category satisfying Axioms C.1–4 admits some left adjoint from $\mathcal{C}$:

The $\infty$-category $\thy_{(\infty,n)}$ is an $\infty$-groupoid and thus a homotopy type; it is the moduli space of theories of $(\infty,n)$-categories. Our main theorem is a computation of this homotopy type.

The proof will occupy the next few sections, and is organized as follows.

1. (a)

   In Section 8 we introduce a particular $\infty$-category and verify that it satisfies Axioms C.1-5. This shows that $\thy_{(\infty,n)}$ is non-empty, and we denote our chosen model by $\cat_{(\infty,n)}$. For Axioms C.1-4 this means simply varifying the corresponding properties of our model. For Axiom C.5, Versality, we suppose another $\infty$-category $\mathcal{D}$ satisfying Axioms C.1-4 and must then show how those axioms guerentee the existence of the desired functor and transformation out of $\cat_{(\infty,n)}$. In fact a stronger versality property holds: $\mathcal{D}$ need only satisfy C.1, C.3, and C.4.

2. (b)

   In Section 9 we show that the space $\thy_{(\infty,n)}$ is connected, i.e., any two $\infty$-categories satisfying C.1-5 are equivalent. The proof relies essentially on our $\infty$-categories satisfying Axioms C.1, C.2, and C.5. We note, however, that Axiom C.5 is quantified over the other four axioms, and so this step in fact relies on all five axioms.

3. (c)

   Having shown $\thy_{(\infty,n)}$ is non-empty and connected, it remains to compute its loopspace. This is done in Section 10 by directly computing the automorphism space of our prefered model.