nLab presheaf of groupoids

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 $\mathcal{C}$ an ordinary site, each covering family $\{U_i \overset{\iota_i}{\to} X\}$ induces a Čech groupoid, which is really a presheaf of groupoids

$$C(\{U_i\}) \in [\mathcal{C}^{op}, Grpd] \,.$$

Related concepts

• model structure on simplicial groupoids

References

A model category-structure on categories 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

• Sharon Hollander, A homotopy theory for stacks, Israel Journal of Mathematics, 163(1), 93-124 (arXiv:math.AT/0110247)