nLab
presheaf of groupoids

Contents

Context

Category theory

Enriched category theory

(,1)(\infty,1)-Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The idea of presheaf of groupoids may refer to one of the following distinct but closely related concepts:

  1. a functor from some opposite category to the 1-category Grpd of all small groupoids;

  2. a Grpd-enriched functor from the opposite of a Grpd-enriched category to Grpd canonically regarded as enriched over itself, all with respect to its cartesian closed category-structure;

  3. a pseudofunctor from the opposite of a plain category to the 2-category (in fact: (2,1)-category) Grpd of all small groupoids;

  4. a 2-functor from the opposite 2-category of some 2-category to the 2-category Grpd of all small groupoids.

In the last two cases one also speaks of pre-stacks of groupoids or (2,1)-presheaves.

The first two of these carry structures of homotopical categories, even of model categories, that make them presentable (infinity,1)-category-presentations of the latter two.

Under the simplicial nerve-constructions, presheaves of groupoids in the sense of the first two items form a full subcategory of simplicial presheaves, and their model category-structures enhance to the corresponding model structures on simplicial presheaves, which present (∞,1)-presheaves of ∞-groupoids, hence pre-∞-stacks.

Examples

  • For 𝒞\mathcal{C} an ordinary site, each covering family {U iι iX}\{U_i \overset{\iota_i}{\to} X\} induces a Cech groupoid, which is really a presheaf of groupoids
C({U i})[𝒞 op,Grpd]. C(\{U_i\}) \in [\mathcal{C}^{op}, Grpd] \,.

References

A model category-structure on catgegories of presheaves of groupoids, modeling (2,1)-categories of stacks the way that the model structure on simplicial presheaves model (∞,1)-categories of ∞-stacks is discussed in

Last revised on June 13, 2018 at 15:49:18. See the history of this page for a list of all contributions to it.