Joyal's CatLab
The theory of quasi-categories

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**Higher category theory** ##Contents * Contributors * References * Introduction * The theory of quasi-categories * The theory of n-quasi-categories

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*** **Categorical mathematics** ##Contents * Set theory * Category theory * Homotopical algebra * Higher category theory * Theory of quasi-categories * Higher quasi-categories * Geometry * Elementary geometry * Differential geometry * Lie theory * Algebraic geometry * Homotopical algebraic geometry * Number theory * Elementary number theory * Algebraic number theory * Algebra * Universal algebra * Group theory * Rings and modules * Commutative algebras * Lie algebras * Representation theory * Operads * Homological algebra * Logic * Boolean algebra * First order theory * Model theory * Categorical logic * Topology * General topology * Algebraic topology * Homotopy theory * Theory of locales * Topos theory * Higher topos theory * Combinatorics * Combinatorial geometry * Enumerative combinatorics * Algebraic combinatorics

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The notion of quasi-category was introduced by Michael Boardman and Rainer Vogt in their book Homotopy Invariant Algebraic Structures in Topological Spaces. A Kan complex and the nerve of a category are basic example. A quasi-category is sometime called a weak Kan complex in the literature. We have introduced the term quasi-category in order to stress the similarity between category theory and the theory of quasi-categories. We shall often use the term quategory as an abreviation for quasi-category.

It turns out that essentially all of category theory can be extended to quasi-categories. The resulting theory has applications to homotopy theory, homotopical algebra, higher category theory and higher topos theory.

Basic Notions


Revised on May 10, 2013 at 09:45:35 by Tim Porter