A Whirlwind Tour of the World of $(\infty ,1)$-categories

Abstract

This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty ,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.

Added details to discussion of tmf; added references for definitions of 3- and 4-categories; corrected many typos

References in corpus (13)

• On the algebraic K-theory of higher categories
• Higher and derived stacks: a global overview
• Derived Algebraic Geometry III: Commutative Algebra
• On the Unicity of the Homotopy Theory of Higher Categories
• Stable Infinity Categories
• Segal topoi and stacks over Segal categories
• Spans in 2-Categories: A monoidal tricategory
• n-Relative Categories
• A note on the cotangent complex in derived algebraic geometry
• Theories homotopiques de Quillen combinatoires et derivateurs de Grothendieck
• A classification of Taylor towers of functors of spaces and spectra
• $\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories
• Approximation in K-theory for Waldhausen Quasicategories