nLab Calculus of fractions and homotopy theory

This entry is about the book

• Pierre Gabriel, Michel Zisman:

  Calculus of fractions and homotopy theory

  Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35,

  Springer (1967)

  doi:10.1007/978-3-642-85844-4

  pdf

on homotopy theory, notably simplicial homotopy theory, and specifically the calculus of fractions for constructing homotopy categories.

This book first introduced a general localization and reflective localization of a category at an arbitrary class of morphisms (nowadays sometimes called the Gabriel-Zisman localization (see also Gabriel localization)); however the size issues are not discussed and the formalism of universes is understood as an excuse. The special case of categories of fractions is treated in detail.

The book has large historical importance for a clean and innovative formalism treating the interaction of category theory (including adjoint functors, Kan extensions, strict 2-categories), simplicial methods and homotopy theory. An important version of a definition of a homotopy category by the abstract categorical localization by the class of weak equivalences is introduced.

It has been proved that the homotopy categories of CW complexes and of simplicial sets are equivalent. The notion of a fundamental category of a simplicial set (now sometimes also called the homotopy category), refining the notion of fundamental groupoid, is defined using adjointness.

An important method for studying cofibrations of simplicial sets, namely the notion of an anodyne extension, is introduced for the first time and effectively used.

The book is written in a recognizable abstract, clean and precise language with economic, rather short and formal, formulations.

category: reference