Stacky Filtered Circle: A Differentiable Stacks Example

• The stacky filtered circle is a differentiable stack defined as [ℝ/ℤ] with ℤ acting by integral translations, endowing it with a Lie group structure.
• It provides a concrete model for strictification by equivalently representing a connected étale stacky Lie group as the crossed module (ℤ→ℝ).
• The construction bridges classical topological invariants with Lie algebra data, highlighting the role of discrete fundamental groups alongside continuous Lie groups.

The stacky filtered circle $S_1$ is the differentiable-stacky Lie group defined as the quotient stack

$S_1=[\mathbb{R}/\mathbb{Z}],$

where $\mathbb{Z}$ acts on $\mathbb{R}$ by integral translations. In the specialization of the strictification theorem of Trentinaglia–Zhu, this object provides a concrete example of a connected étale stacky Lie group that is equivalent to the crossed module $(H \to G)=(\mathbb{Z}\to\mathbb{R})$, with quotient stack $[\mathbb{R}/\mathbb{Z}]$ recovering $S_1$ itself (Trentinaglia et al., 2010).

1. Definition as a quotient stack

By definition, the stacky filtered circle is the quotient stack

$S_1=[\mathbb{R}/\mathbb{Z}],$

with $\mathbb{Z}$ acting on $\mathbb{R}$ by integral translations. As a group stack, $S_1=[\mathbb{R}/\mathbb{Z}],$0 inherits the induced group structure from the additive group $S_1=[\mathbb{R}/\mathbb{Z}],$1 (Trentinaglia et al., 2010).

This presentation fixes the object at the level of differentiable stacks rather than only at the level of ordinary topological spaces. A plausible implication is that the terminology “circle” should be read here in the stack-theoretic sense encoded by the quotient construction, not merely as the underlying topological orbit space.

2. Connectedness and étaleness

The stack $S_1=[\mathbb{R}/\mathbb{Z}],$2 is connected, in the sense that the underlying stack is path-connected. It is also étale, because the presentation $S_1=[\mathbb{R}/\mathbb{Z}],$3 is an étale groupoid (Trentinaglia et al., 2010).

These two properties are structurally decisive in the Trentinaglia–Zhu theorem. The theorem applies to every connected étale stacky Lie group, and the stacky filtered circle is introduced precisely as a case in which the theorem becomes completely explicit. This suggests that $S_1=[\mathbb{R}/\mathbb{Z}],$4 serves as a model example of how a differentiable-stacky group can be strictified into crossed-module data.

3. Fundamental group and infinitesimal data

For the stacky filtered circle, the fundamental group is identified as

$S_1=[\mathbb{R}/\mathbb{Z}],$5

By classical topology, the ordinary topological fundamental group of the circle is $S_1=[\mathbb{R}/\mathbb{Z}],$6, and one checks that the stacky $S_1=[\mathbb{R}/\mathbb{Z}],$7 agrees with the usual one in this case. The group $S_1=[\mathbb{R}/\mathbb{Z}],$8 is then regarded as a discrete Lie group (Trentinaglia et al., 2010).

Infinitesimally, $S_1=[\mathbb{R}/\mathbb{Z}],$9 has Lie algebra

$\mathbb{Z}$0

the standard $\mathbb{Z}$1-dimensional abelian Lie algebra. The unique connected, simply connected Lie group integrating $\mathbb{Z}$2 is

$\mathbb{Z}$3

The identification of $\mathbb{Z}$4 with $\mathbb{Z}$5 and of the simply connected integrator with $\mathbb{Z}$6 is the essential input for the crossed-module presentation. In this example, the topological and infinitesimal invariants are both elementary, which is why the strictification can be written in closed form.

4. The crossed module $\mathbb{Z}$7

The crossed module associated with the stacky filtered circle has boundary map

$\mathbb{Z}$8

Thus $\mathbb{Z}$9 is the inclusion $\mathbb{R}$0. Since $\mathbb{R}$1 is discrete and $\mathbb{R}$2 is connected, the natural left action of $\mathbb{R}$3 on $\mathbb{R}$4 by automorphisms must be trivial: $\mathbb{R}$5 The crossed-module axioms are then checked as follows (Trentinaglia et al., 2010): $\mathbb{R}$6

$\mathbb{R}$7

In this case the verification is immediate because $\mathbb{R}$8 is abelian and $\mathbb{R}$9 is abelian. The equivariance condition becomes

$(H \to G)=(\mathbb{Z}\to\mathbb{R})$0

and the Pfeiffer identity becomes

$(H \to G)=(\mathbb{Z}\to\mathbb{R})$1

This crossed module is therefore entirely determined by two simple ingredients: inclusion of integer periods into the additive real line, and the trivial $(H \to G)=(\mathbb{Z}\to\mathbb{R})$2-action on $(H \to G)=(\mathbb{Z}\to\mathbb{R})$3.

5. The strict Lie-$(H \to G)=(\mathbb{Z}\to\mathbb{R})$4-group and the quotient stack

From any crossed module $(H \to G)=(\mathbb{Z}\to\mathbb{R})$5, one builds the strict Lie-$(H \to G)=(\mathbb{Z}\to\mathbb{R})$6-group whose underlying groupoid is the action groupoid

$(H \to G)=(\mathbb{Z}\to\mathbb{R})$7

with source and target maps

$(H \to G)=(\mathbb{Z}\to\mathbb{R})$8

and arrow multiplication

$(H \to G)=(\mathbb{Z}\to\mathbb{R})$9

(Trentinaglia et al., 2010).

For the stacky filtered circle, $[\mathbb{R}/\mathbb{Z}]$0 and $[\mathbb{R}/\mathbb{Z}]$1. The action is translation,

$[\mathbb{R}/\mathbb{Z}]$2

The corresponding quotient stack is therefore

$[\mathbb{R}/\mathbb{Z}]$3

which recovers $[\mathbb{R}/\mathbb{Z}]$4.

The same presentation is summarized by a pull-back square of stacks: $S_1$8 Here $[\mathbb{R}/\mathbb{Z}]$5 is the group homomorphism $[\mathbb{R}/\mathbb{Z}]$6, $[\mathbb{R}/\mathbb{Z}]$7 is the unit-section of the quotient stack, and the upper-right square is Cartesian.

This realizes the stacky filtered circle as the quotient associated with a strict $[\mathbb{R}/\mathbb{Z}]$8-group. A plausible implication is that the example makes the abstract strictification theorem computationally transparent: every constituent of the strict model can be written down explicitly.

6. Position within strictification theory

The general theorem stated in this context is that every connected étale stacky Lie group is equivalent to a crossed module $[\mathbb{R}/\mathbb{Z}]$9. For the filtered circle, this yields the concrete crossed module

$S_1$0

whose quotient stack is exactly $S_1$1 (Trentinaglia et al., 2010).

The broader result defines stacky Lie groups as group objects in the $S_1$2-category of differentiable stacks and shows that every connected and étale stacky Lie group is equivalent to a crossed module of the form $S_1$3, where $S_1$4 is the fundamental group of the given stacky Lie group and $S_1$5 is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. The paper notes that this result is closely related to a strictification result of Baez and Lauda.

Within that framework, the stacky filtered circle functions as a canonical one-dimensional example. It exhibits, in the simplest nontrivial setting, how a quotient stack with group structure is converted into strict crossed-module data, how the fundamental group appears as the discrete part $S_1$6, and how the simply connected integrator of the Lie algebra appears as the continuous part $S_1$7.