nLab
double groupoid

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Internal categories

Contents

Idea

A double groupoid is, equivalently,

Equipped with the relevant extra stuff, structure, property one obtains notions of double topological groupoids, double Lie groupoids, etc.

Definition

Definition

Let H\mathbf{H} be a (2,1)-topos, hence a (2,1)-category whose objects may be thought of as groupoids equipped with some geometric structure (stacks). Then a double groupoid with that geometric structure is a groupoid object in an (2,1)-category in H\mathbf{H}, hence a simplicial object

𝒢 H Δ op \mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}

which satisfies the groupoidal Segal conditions.

In the literature, the following special cases of def. 1 are often taken to be the default notion of “double groupoid”.

Example

The archetypical special case of def. 1 is that where H=\mathbf{H} = Grpd is the (2,1)-category of bare (geometrically discrete) groupoids.

Example

A special case of example 1, in turn, are bare double groupoids in the image of the embedding Grpd 1 Δ opGrpd Δ opGrpd_1^{\Delta^{op}} \to Grpd^{\Delta^{op}}, where Grpd 1Grpd_1 is the 1-category of groupoids (suppressing the 2-morphisms given by natural isomorphisms). A groupoid object in Grpd 1Grpd_{1} is equivalently a pair of groupoids 𝒢 1\mathcal{G}_1 and 𝒢 0\mathcal{G}_0 equipped with functors s,t:𝒢 1𝒢 0s,t \colon \mathcal{G}_1 \to \mathcal{G}_0, i:𝒢 0𝒢 1i \colon \mathcal{G}_0 \to \mathcal{G}_1 and :𝒢 1× 𝒢 0𝒢 1𝒢 1\circ \colon \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 \to \mathcal{G}_1 that satisfy the usual axioms of a small category groupoid without any non-trivial natural isomorphisms weakening them. This is called a strict double groupoid.

Remark

If one writes out the structure functors

𝒢 1 s t 𝒢 0 \array{ \mathcal{G}_1 \\ {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ \mathcal{G}_0 }

of a double groupoid 𝒢 \mathcal{G}_\bullet themselves in components, one obtains a square diagram of the form

𝒢 1,1 𝒢 1,0 𝒢 0,1 𝒢 0,0 \array{ \mathcal{G}_{1,1} & \stackrel{\to}{\to} & \mathcal{G}_{1,0} \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{G}_{0,1} & \stackrel{\to}{\to} & \mathcal{G}_{0,0} }

(where now we are notationally suppressing the degeneracy maps/identity assigning maps, for readability). In this form double groupoid are presented in traditional literature.

Example

For H=\mathbf{H} = SmoothGrpd, double groupoids in H\mathbf{H} which are in the inclusion of LieGrpdΔ op{\Delta}^{op} \to SmoothGrpd Δ op{}^{\Delta^{op}} are called double Lie groupoids.

Remark

More generally, one can consider double groupoids in an arbitrary (∞,1)-topos H\mathbf{H}, to be a 3-coskeletal groupoid object in an (∞,1)-category consisting degreewise of 1-truncated objects. The realization map

lim:H Δ opH \underset{\to}{\lim} \colon \mathbf{H}^{\Delta^{op}} \to \mathbf{H}

restricted to such double groupoids is a presentation of 2-truncated objects in H\mathbf{H}.

References

Double Lie groupoids are discussed (usually for the strict case) in

  • Ronnie Brown, Kirill Mackenzie, Determination of a double Lie groupoid by its core diagram. J. Pure Appl. Algebra 80 (1992), no. 3, 237–272

  • Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge (2005)

Some homotopical aspects of double groupoids and their relationship to homotopy 2-types are explored in

Revised on April 25, 2013 12:33:17 by Urs Schreiber (82.169.65.155)