nLab
enriched category theory

Contents

Idea

Enriched category theory is the category theory of enriched categories.

Pretty much every notion of ordinary category theory has its analog in enriched category theory, and, by taking the enriching category to be Set, enriched category theory subsumes all of the theory of locally small categories.

Enriched categories have a multitude of uses and applications, that makes studying their general theory quite worthwhile.

Applications

As a tool in higher category theory

Enriched categories may be used to model objects in higher category theory: if the objects of the enriching category VV behave themselves as (n,r)-categories, then a VV-enriched category behaves like a (possibly special) (n+1,r+1)(n+1,r+1)-category.

For instance

In practice it is often useful to handle enriched categories such that only certain enriched full subcategories of them are the the desired models for objects in higher category theory. For instance often it is useful to work with general sSet-categories and have a prescription for how to find a Kan complex-enriched full subcategory inside them.

This is achieved notably by combining enriched category theory with model category theory:

an enriched model category or more generally an enriched homotopical category is an enriched category with extra information on how it behaves as a model in higher category. Notably sSet-model categories serve as models for (∞,1)-category theory.

Literature

The standard monograph on enriched category is

  • Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64, 1982, Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (pdf)

Entries on Enriched Category Theory

Monoidal categories

Enriched functors

Generalizations of monoidal categories

Enriched categories

Universal constructions

Homotopical enrichment

Revised on October 30, 2012 13:41:26 by Urs Schreiber (131.174.40.126)