Localization of categories

Localization is the process of freely inverting a collection of morphisms in a category (nLab , Stacks , Wiki ). The motivating examples from algebraic topology are localizing away from homotopy equivalences, weak homotopy equivalences, or quasi-isomorphisms in a chain complex. The theory has its origins in the localization of a commutative ring.

The localization of a category is characterized by a universal property and can, for any collection of morphisms, be constructed by a suitable colimit in Cat. In general, the morphisms in the localized category are zig-zags of arbitrary length. In order to obtain a simpler description, conditions are imposed on the morphism collection, beginning with it forming a subcategory and including other closure conditions. Under certain conditions, the localized category admits a calculus of fractions , also called a category of fractions.

Literature

Books and surveys

  • Borceux, 1994: Handbook of categorical algebra, Vol. 1, Ch. 5: Categories of fractions
    • Thorough and self-contained treatment of different aspects of localization
  • Kashiwara & Schapira, 2006: Categories and sheaves, Ch. 7: Localization (doi)
  • Gelfand & Manin, 2003: Methods of homological algebra, Sec. III.2: Derived categories and their localization (doi)

Bicategorical analogues

The two-dimensional version of the category of fractions is the bicategory of fractions or bicalculus of fractions, introduced by Pronk.

  • Pronk, 1996: Etendues and stacks as bicategories of fractions (pdf)
  • Vitale, 2010: Bipullbacks and calculus of fractions (pdf)
  • Pronk & Scull, 2019: Bicategories of fractions revisited: towards small homs and canonical 2-cells (arxiv)
  • Vazquez, Pronk, Szyld, 2021: The three F’s for bicategories I: Localization by fractions is exact (arxiv)
    • Sec 3. gives alternative axioms for a bicalculus of fractions