On the Unicity of the Homotopy Theory of Higher Categories

By Clark Barwick, Christopher J. Schommer-Pries. 2011

108 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/c16c6d76e2d5f8cd93477bf8bad1e0d8f0cb5324

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Citations in the corpus

Fibrations and Yoneda's lemma in an ∞-cosmos (2017, background): “model categories [1] or having equivalent quasi-categories [2].”
Univalent categories and the Rezk completion (2013, background): “A category of the latter sort is sometimes called gaunt (this term was introduced by Barwick and Schommer-Pries [4]).”
Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates (2020, methodology): “It now follows from the main results of [4] that the derived mapping∞-groupoids Map sSetsc (δ2(−,−), (X, S))∶ Δ op × Δ ⟶ Gpd.”
On the algebraic K‐theory of higher categories (2012, background): “Indeed, there is a canonical equivalence between any two of these homotopy theories [9, 11, 38] (or any other homotopy theory that satisfies the axioms of [69] or of [6]), through which one can surely translate the main theorems here into theorems in the language of any other model.”
Dualizing cartesian and cocartesian fibrations (2014, background): “ce bηof h. The Yoneda lemma now implies that ηbis induced by an autoequivalence of Catd∞ itself. By the Unicity Theorem of Toën [7], Lurie [5, Th. 4.4.1], and the ﬁrst author and Chris Schommer-Pries [2], we deduce that bηis canonically equivalent either to id or to op, and considering the case S= ∆”
Symmetries (2014, unknown): “Indeed Toën’s theorem has been generalized by Barwick and Schommer-Pries [4].”
From operator categories to higher operads (2013, methodology): “plete Segal {1}-operad is a left ﬁbration X N∆op classiﬁed by a complete Segal space in the sense of Rezk [17]. (In particular, Operad{1} CSS is a homotopy theory of (∞,1)-categories, in the sense of [5].) For simplicity, we will, by a small abuse, simply call such left ﬁbrations complete Segal spac”
The cobordism hypothesis (2017, background): “Regarding other definitions of (∞, n)-categories, see [BS], [BR1], [Si], and the references therein.”
On exact $\infty$-categories and the Theorem of the Heart (2012, background): “There are very many models for the homotopy theory of ∞ -categories in this sense, and they are all equivalent in an essentially unique fashion, up to orientation — see To¨en [26] or Lurie [12] or Barwick–Schommer-Pries [5].”