ScalingStacks

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.