An Overview of Higher Category Theory

This chapter is intended as a general introduction to higher category theory. We begin with what we feel is the most intuitive approach to the subject usingtopological categories. This approach is easy to understand but difficult to work with when one wishes to perform even simple categorical constr...

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Bibliographic Details
Published inHigher Topos Theory Vol. 170; pp. 1 - 52
Main Author Lurie, Jacob
Format Book Chapter
LanguageEnglish
Published United States Princeton University Press 06.07.2009
Subjects
Abstract spaces
Algebraic topology
Applied mathematics
Applied statistics
Category theory
Dimensional analysis
Dimensionality
Equivalence relation
Euclidean geometry
Formal logic
Functors
Geometry
Homotopy theory
Judicial system
Law
Legal motions
Legal objections
Legal proceedings
Logic
Logical topics
Mathematical logic
Mathematical relations
Mathematical set theory
Mathematics
Morphisms
Nerve of a covering
Philosophy
Plane geometry
Pure mathematics
Set theory
Statistical physics
Statistics
Topological spaces
Topological theorems
Topology
Trial court proceedings
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Summary:This chapter is intended as a general introduction to higher category theory. We begin with what we feel is the most intuitive approach to the subject usingtopological categories. This approach is easy to understand but difficult to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more suitable formalism of∞-categories(calledweak Kan complexesin [10] andquasi-categoriesin [43]), which provides a more convenient setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why they are equivalent to one
ISBN:0691140499
9780691140490
DOI:10.1515/9781400830558.1