Here’s something I had never really thought about before. Any group has a unique action on the one-element set , since there is a unique morphism . Abstractly, this is because is a terminal object in the category of sets. Now, any group also has a unique action on the empty set , since there is also a unique morphism (the “empty function”). Abstractly, this is because is an initial object in the category of sets.
This action is quite strange. There are two definitions (that I’m aware of) of what it means for an action of a group on a set to be transitive, and they disagree for this action:
- An action of on is transitive if, for every , there exists such that . This is vacuously true when is empty.
- An action of on is transitive if it has one orbit. This is false when is empty; there are zero orbits.
Wikipedia takes the stance that a transitive -set must be nonempty, which I suppose corresponds to supporting the second definition. There is a great reason for doing this: you want transitive -sets to correspond to conjugacy classes of subgroups of , and this only works if you can take stabilizers. And you can only take stabilizers if the -set is non-empty. Instead of using stabilizers you can use the kernel of the action, but then there are two actions – the unique action on , and the unique action on – that both correspond to the entire group .
Edit: Perhaps an even better reason to declare that the empty -set is not transitive is that otherwise, the decomposition of a -set into a coproduct of transitive -sets is not unique. (As I mention in the comments, the empty -set itself is the empty coproduct.)
These aren’t entirely idle thoughts; there are some major theorems identifying certain categories with the category of certain types of actions of a group , and as far as I can tell these theorems are wrong as stated in the literature because they don’t take into account the empty case.
It’s likely I’ve said several things on this blog which are false because I didn’t take into account the empty case. Hopefully I’ll do this less in the future.
Exercise: Define the topology on the empty topological space . Is it Hausdorff? Compact? Metrizable? Connected? Path-connected? Simply connected? What is its fundamental group? What does the algebra of continuous functions look like?