Group Action

Let $(G,\cdot,1,(-)^{-1})$ be a Group and $X$ be an H-Set. A function $\alpha:G\to X\to X$ is a group action if it satisfies the following equivalent conditions:

1. $\alpha(1,x)=x$ and $\alpha(g_{1}\cdot g_{2},x)=\alpha(g_{1},\alpha(g_{2},x))$
2. $\alpha$ induces a Group Homomorphism $G\to\mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ denotes the Symmetry Group of $X$.
3. $\alpha$ induces a Functor $\mathbf{B}G\to\mathrm{Sets}$ from the Delooping of $G$ to the Category of Sets. In other words, a group action is a Presheaf.

Left vs Right Group Actions

Typically, we need to distinguish between left vs. right actions, corresponding to the distinction between covariant and contravariant functors. This distinction is not required for groups; as we can turn a left action into a right action and vice versa by looking at $\alpha(g^{-1},x)$; this flips the order of operations due to the Shoes and Socks Principle.

In Displayed Group Theory

Recall that Presheaves are equivalent to Discrete Cartesian Fibrations via the Grothendieck Construction. This suggests that a group action ought to be a sort of a "discrete G-fibration" when viewed as a Displayed Group.

If take the Category of Elements of $\alpha:\mathbf{B}G\to\mathrm{Sets}$, we get the following Displayed Category:

• The type of objects over $\bullet:\mathbf{B}G$ is $X$, the set that $G$ acts upon.
• A morphism $x\to_{g}y$ is a proof that $\alpha(g,x)=y$.

This is also known as Action Groupoid.

There is a nicer definition in terms of Augmented Displayed Groups and Stabilizer Groups.

• The type of -1 elements is $X$
• The type of objects over $x$ and $g$ is a proof that $\alpha(g,x)=x$.
• The identity element over $x$ and $e$ is $\alpha(e,x)=x$
• The multiplication of $\alpha(g_{1},x)=x$ $\alpha(g_{2},x)=x$ is $\alpha(g_{1}g_{2},x)=x$
• The inverse of $\alpha(g,x)=x$ is the proof that $\alpha(g^{-1},x)=x$