On the Unicity of the Homotopy Theory of Higher Categories

Details On the Unicity of the Homotopy Theory of Higher Categories On the Unicity of the Homotopy Theory of Higher Categories

2011-11-30

arxiv.org/abs/1112.0040v2

Mathematics

Algebraic Topology

41 pages, 2 figures. General reformatting. Minor corrections to axiom (C.3) and section 5. Repaired outdated references and co

Scientific paper

We propose four axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of $(\infty,n)$-categories. This axiomatization requires that a homotopy theory of $(\infty,n)$- categories, when equipped with a small amount of extra structure, satisfies a simple, yet surprising, universal property. We further prove that the space of such quasicategories is homotopy equivalent to $(RP^\infty)^n$. In particular, any two such quasicategories are equivalent. This generalizes a theorem of To\"en when n = 1, and it verifies two conjectures of Simpson. We also provide a large class of examples of models satisfying our axioms, including those of Joyal, Kan, Lurie, Rezk, and Simpson.

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