Groupoid actions on spaces

Question

The action of a group $G$ on a topological space $X$ can be viewed as a functor $F: G \to \mathcal{Top}$ with $F(*)=X$. (Here I'm viewing a group as a category with one object, $ * $, and the morphisms are isomorphisms labeled by the group elements.)

We can extend this idea and define the action of a groupoid $\mathcal{G}$ on a space to be a functor $F:\mathcal{G} \to\mathcal{Top}$.

Are there any naturally occurring examples of a groupoid action on a space? (Other than the ones where the groupoid is actually a group.)

Answer 1

Perhaps the most natural example is given by universal covers?

Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of $X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths $[0,1]\to X$ which start at $x$ and end at $y$, where we are taking homotopy classes relative to $\lbrace0,1\rbrace$).

Let $\pi$ be the fundamental groupoid of $X$. Then there is a functor $\pi\to \text{Top}$ given on objects by $x\mapsto \tilde X_x$. On morphisms of $\pi$ from $x$ to $y$, the functor is given by the map $\tilde X_x \to \tilde X_y$ that is induced by concatenating paths.

Answer 2

I suggest that you google "actions of groupoids" and browse through the displayed pages... You will observe that actions of locally compact groupoids have been well studied, and people let groupoids act on objects like locally compact spaces, Hilbert spaces, $C^*$-algebras, etc...

A good starting point is probably the thesis by Jean Renault, {\it A groupoid approach to $C^*$-algebras}, Springer Lect. Notes in Math. 793, 1980.

Answer 3

Actions of a Lie groupoid are defined on p. 34 of K.C.H. Mackenzie, "General Theory of Lie Groupoids and Lie Algebroids" LMS Lecture Notes Series no 213, 2005.

Note that in general for a groupoid action of $G$ on sets it is often convenient to follow C. Ehresmann and to insist on having a function $f: E \to Ob(G)$ so that $g \in G(x,y)$ maps the fibre of $f$ over $x$ to the fibre over $y$. In fact a standard equivalence is between actions on sets in this sense; functors $G \to Sets$; and covering morphisms of the groupoid $G$. But the point of the first definition is that this easily transcribes to the case $E$ is a topological space, as in Mackenzie's book. A full exposition of covering space theory based on covering morphisms of groupoids, rather than actions, is given in the book now called "Topology and Groupoids", and was in the 1968 edition.

John Klein is also right to emphasise the covering space example. This leads to the idea that for the cellular homology of the universal cover of a CW-complex you actually need chain complexes with a groupoid of operators, rather than the usual group of operators. This idea was developed in a paper with Higgins (Proc Camb. Phil. Soc. (1990)) and is explained in the book Nonabelian algebraic topology, see for example Section 8.4.

Edit May 19: A simple and basic example of a groupoid acting on spaces generalises the case of a topological group acting on itself by left multiplication. A groupoid $G$ acts on the families of stars $St_G(x), x \in Ob(G)$, where $St_G(x)$ is the union of the sets $G(x,y)$ for all $y \in Ob(G)$, using the convention that if $g: z \to x, h:x \to y$ then $gh: z \to y$. If $G$ is a topological groupoid, then we get an action of $G$ on topological spaces.

An example of this is the case the space $X$ admits a universal cover. Then the fundamental groupoid $\pi_1 X$ may be topologised making it a topological groupoid. (R. Brown and G. Danesh-Naruie, ``The fundamental groupoid as a topological groupoid'', Proc. Edinburgh Math. Soc. 19 (1975) 237-244.) The star of $\pi_1 X$ at $x$ is of course the universal cover of $X$ based at $x$. (See also Topology and Groupoids 10.5.8, which deals with the case $(\pi_1 X)/N$ for $N$ a totally disconnected normal subgroupoid of $\pi_1 X$. )

Answer 4

There is a nice overview of Moerdijk and Mrcun in the proceedings of the PQR 2003 conference on groupoids and their actions and stuff. So this might provide quite a number of examples from (differential) geometry. They mainly investigate Lie groupoids.

In the setting of Lie groupoids, there are also more refined notions of "actions" and many examples. Maybe you take a look there.

Personally, I have a nice (simple?) example of a groupoid action: take a bunch of algebras (associative) over a common field (or ring...) and consider the "isomorphism groupoid": the objects are the algebras, the arrows the isomorphisms between them. Then you have an obvious "action".

Slightly more interesting is the "Picard groupoid" which also acts on the algebras. Now the arrows are (iso-classes) of Morita equivalences. You can act with these two groupoids on all kind of stuff like the K-theories or the lattices of ideas of the algebras and so on...

Answer 5

So far no one has mentioned two names: Ehresmann and Ronnie Brown. The first developed a lot of the theory of groupoid actions in his work on connections on fibre bundles. The second used groupoid actions extensively in the various forms of his book (now called Topology and Groupoids) and has developed the theory in various new directions with his coauthors.

Another class of examples comes from 'local systems' These were developed way back by Reidemeister (possibly even earlier, see the nLab entry on their history), rediscovered by Steenrod, but are most naturally seen as functors from the fundamental groupoid of a space to a suitable category such as vector spaces, Abelian groups or whatever. Taking classifying spaces of the stalks gives one examples of the groupoid acting on spaces. This includes the example of covering spaces, of course.

Answer 6

You're are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is

• Buss, A., Meyer, R., & Zhu, C., A higher category approach to twisted actions on $C^*$-algebras, Proceedings of the Edinburgh Mathematical Society, 56(2) (2013) pp 387-426, doi:10.1017/S0013091512000259, arXiv:0908.0455

wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer–Buss–Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced in:

• Paul S. Muhly, Mark Tomforde, Topological Quivers, International Journal of Mathematics 16 No. 7, (2005) pp. 693-755, doi:10.1142/S0129167X05003077, arXiv:math/0312109.

Then an action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.

Answer 7

Observation about groupoid-action on a unique object.

Let $G_0$ be the set of half-lines of the plane with common origin $O$, it has a topology induced by the usual biiection with the circle $S^1$. Consider the structure of indiscrete order on $G_0$, this is a groupoid $G$ , a morphism of $G$ is a pair $(r, s)$ of half-lines and is defined the angle from $r$ to $s$ (we fix the counterclockwise direction as positive), then $G$ acts on $G_0$ by geometric rotation.

In other way, if you have a (topological) group $G_0$, define the small indiscrete groupoid $G$ with object class $G_0$, and $G_0(x, y)= \{yx^{-1}\}$ (this is the comma $*\downarrow G_0$, where $\star$ is the unique object of $G_0$), $G$ act on $G_0$ in obvious way (composition).

It seem that this make a functor form the groups to the groupoid (transitive faithful)-actions on a pointed set (the special point is the unity), we have a obvious reverse functor too.