On the unicity of the theory of higher categories

Abstract

We axiomatise the theory of ( ∞ , n ) (\infty ,n) -categories. We prove that the space of theories of ( ∞ , n ) (\infty ,n) -categories is a B ( Z / 2 ) n B(\mathbb {Z}/2)^n . We prove that Rezk’s complete Segal Θ n \Theta _n spaces, Simpson and Tamsamani’s Segal n n -categories, the first author’s n n -fold complete Segal spaces, Kan and the first author’s n n -relative categories, and complete Segal space objects in any model of ( ∞ , n − 1 ) (\infty , n-1) -categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of ( Z / 2 ) n (\mathbb {Z}/2)^n .

Stacks

“…The descent conditions generalize directly to the case of n-truncated 5 presheaves, saying that the space of sections over an object is equal to the homotopy limit of the space of sections over the nerve of a covering. An n-stack in n-groupoids is an n-truncated presheaf of spaces satisfying this descent condition.…”

“…We restrict here to the case of stacks of groupoids. There are now many different definitions of n-category, which have been shown to be equivalent by the axiomatic approach of Barwick and Schommer-Pries [5], but for groupoids this can be understood easily in that an n-groupoid may be viewed as the same thing, up to homotopy, as an n-truncated space. Definition 5.1.…”

“…This has strict pullbacks corresponding to restriction of sections, so it is a presheaf of groupoids in the sense of 5.2. In the other direction, given a presheaf of groupoids F we can let F := C F be the Grothendieck integral, also known as the 5 Recall that a space Y is said to be n-truncated…”

On straightening for Segal spaces

“…Recall that the theory of d-categories is currently much less developed than that of 1-categories; it is already not straightforward to compare the various different homotopy-theoretic models for d-categories (see e.g. [40] for a textbook account and [4] for a more recent state of affairs). Nonetheless, d-categories have been of interest in various areas of mathematics, also outside of the realm of category theory itself.…”

“…Theorem A (Corollary 4.31 and Theorem 6.1). For each (d + 1)-category C, there exist equivalences of 1-categories (here c 1 takes the 1-category underlying a (d + 1)-category) To prove Theorem A, we make use of the model for (d + 1)-categories given by iterated complete Segal spaces [4]. Recall that these are certain types of (d + 1)-fold simplicial spaces…”