nLab
localizer

Contents

Context

Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Locality and descent

Contents

Idea

Given a category 𝒞\mathcal{C}, by a localizer in 𝒞\mathcal{C} (or on 𝒞\mathcal{C}) is often meant to be a class WMor(𝒞)W \subset Mor(\mathcal{C}) of morphisms in 𝒞\mathcal{C} such that (𝒞,W)(\mathcal{C},W) is a category with weak equivalences, intended as the input datum for the (simplicial) localization of 𝒞\mathcal{C} at WW. Typically there are some extra conditions imposed on what constitutes a localizer in a given context.

Examples

The term appears for instance in the discussion of Cisinski model structures, see that Cisinski model structure – Localizers.

It also appears as the notion of basic localizers on Cat.

Last revised on August 7, 2013 at 17:00:47. See the history of this page for a list of all contributions to it.