Contents

group theory

# Contents

## Definition

### In sets

###### Definition
$(-)\cdot(-) \;\colon\; G \times X \to X$

of a group $G$ on a set $X$ is called free if for every $x \in X$, the equation $g \cdot x = x$ implies $g = \mathrm{e}$ (the neutral element), hence if only the action of the neutral element has fixed points

(1)$\underset{x \in X}{\exists} \;\; g \cdot x \,=\, x \;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\; g \,=\, \mathrm{e} \,.$

Equivalently, an action is free if and only if for any pair of elements $x,y \in X$, there is at most one group element $g \in G$ such that $g \cdot x = y$.

###### Remark

Beware the similarity to and difference of free actions with effective action: a free action is effective, but an effective action need not be free.

###### Remark

A free action that is also transitive is called regular (at least after passing to its linear permutation representation).

###### Proposition

A group action is free according to Def. iff its shear map is a monomorphism:

(2)$G \curvearrowright X \; \text{is free} \;\;\;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ G \times X & \xhookrightarrow{\;(pr_2, \cdot)\;} & X \times X \\ (g, x) & \mapsto & \big( x, g \cdot x \big) \,. }$

###### Proof

In one direction, assume that the action is free in the sense of Def. . Then given $(x,g\cdot x) = (x', g' \cdot x')$ it follows that $x = x'$ and thus that $g \cdot x = g' \cdot x$, which by (1) implies that $g = g'$. This implication means that the shear map is injective.

In the other direction, assume that the shear map is injective. Then given $g \cdot x = x$, hence equivalently $(x, g \cdot x) = (x, \mathrm{e} \cdot x)$ it follows that ($x = x$ and) $g = \mathrm{e}$. This is the required implication (1).

### In categories with finite products

In the form (2) the definition of free action makes sense more generally for action objects internal to any ambient category (with finite products):

###### Definition

Let $\mathcal{C}$ be a category with finite products, let $G \,\in\, Grp(\mathcal{C})$ be an internal group and $G \curvearrowright X \,\in\, G Act(\mathcal{C})$ an internal action:

$G \times X \xrightarrow{\;\; rho \;\;} X \,.$

Then this is a free action if its shear map is a monomorphism in $\mathcal{C}$:

(3)$\rho \; \text{is free} \;\;\;\; \Leftrightarrow \;\;\;\; G \times X \xhookrightarrow{\; (\rho, pr_2)\;} X \times X \,.$

### In an $\infty$-topos

Def. also makes sense in the generality of $(\infty,1)$-category theory, with monomorphisms understood to be (-1)-truncated morphisms.

###### Definition

(free $\infty$-action in an $\infty$-topos)
Let

we say that the $\infty$-action is free, if its shear map $shear_1$ is (-1)-truncated:

###### Lemma

(higher shear maps of free $\infty$-actions are $(-1)$-truncated)
If an $\infty$-action $G \curvearrowright X$ is free according to Def. and $X$ is inhabited, $X \twoheadrightarrow \ast$, then also all its higher shear maps $shear_n$ are (-1)-truncated:

###### Proof

Notice that for $X \twoheadrightarrow \mathcal{X}$ any effective epimorphism in the $\infty$-topos $\mathbf{H}$, we have for $n \in \mathbb{N}_+$ the following homotopy pullback pasting diagram:

Here the two bottom squares are homotopy Cartesian by definition of Cech nerves, and the top square is homotopy Cartesian since the Cech nerve is a groupoid object (see also at groupoid objects in an (∞,1)-topos are effective) which satisfies the groupoidal Segal conditions (by this Def.).

But this implies, for all $n \geq 1$, that the $n+1$st shear map is the homotopy fiber product in the arrow category $\mathbf{H}^{\Delta}$ of the 1st with the $n$th shear map over the identity morphism on $X$:

Now the claim follows by induction from the fact that (-1)-truncated morphisms are the right class in an orthogonal factorization system (namely the (n-connected, n-truncated) factorization system for $n = -1$) and such classes of morphisms are closed under all $\infty$-limits, in particular under homotopy pullbacks, in the arrow category (by this Prop.).

## Examples

• Any group $G$ acts freely on itself by multiplication $\cdot \colon G \times G \to G$, which is called the (left) regular representation of $G$.

• An action of $\mathbb{Z}/2\mathbb{Z}$ on a set $X$ corresponds to an arbitrary involution $i \colon X \to X$, but the action is free just in case $i$ is a fixed point-free involution.

• There is a rich structure in the classification of free group actions on n-spheres, see there for more.

• For any set $X$ equipped with a transitive action $* : G \times X \to X$, the automorphism group $Aut_G(X)$ of $G$-equivariant automorphisms of $X$ (i.e., bijections $\phi : X \to X$ commuting with the action of $G$) acts freely on $X$. In particular, suppose $\phi \in Aut_G(X)$ is such that $\phi(x) = x$ for some $x\in X$, and let $y\in X$ be arbitrary. By the assumption that $G$ acts transitively, there is a $g \in G$ such that $y = g*x$. But then $G$-equivariance implies that $\phi(y) = \phi(g*x) = g*\phi(x) = g*x = y$. Since this holds for all $y\in Y$, $\phi$ must be equal to the identity $\phi = id_X$, and therefore $Aut_G(X)$ acts freely on $X$.

• A combinatorial species $F : \mathbb{P} \to Set$ is said to be flat if all of the actions $S_n \times F(n) \to F(n)$ are free (see Combinatorial species and tree-like structures). For example, the species of linear orders is flat.

## Properties

### Homotopy colimits of free actions

###### Proposition

If this is a free $\infty$-action (Def. ) then the homotopy quotient

(4)$X \!\sslash\! G \,=\, \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim} X \times G^{\times_n} \;\;\; \in \; \mathbf{H}$

is equivalent to the plain quotient, namely to the coequalizer

(5)$\tau_0(X) / \tau_0(G) \,\coloneqq\, coeq \big( \tau_0(X) \times \tau_0(G) \overset{\phantom{---}}{\rightrightarrows} \tau_0(X) \big) \;\;\; \in \; \mathbf{H}_0$

of 0-truncated objects

(6) 

in that

(7)$X \!\sslash\! G \;\; \simeq \;\; i_0\big( \tau_0(X)/\tau_0(G) \big) \;\;\; \in \; \mathbf{H}_0 \xhookrightarrow{i_0} \mathbf{H} \,.$

In particular, if both $X$ and $G$ are already 0-truncated, then the action of $G$ on $X$ is free iff it is free in the 1-category theoretic sense (3), and then the homotopy quotient coincides with their ordinary quotient.

###### Proof

First we show that $X \!\sslash\! G$ is 0-truncated, in that for every $U \,\in\, \mathbf{H}$ and every $\infty$-group $K$ in the inverse image $Grpd_\infty \xrightarrow{ LConst } \mathbf{H}$ of the terminal geometric morphism, every morphism into it out of $U \times \mathbf{B}K$ factors through the projection onto $U$:

Here $\ast$ denotes the terminal object, which we adjoin, without changing the situation, to bring out the form of the lifting problem.

Incidentally, the left morphism above is an effective epimorphism as shown, hence is (-1)-connected. Threfore, if the right vertical morphism $X \!\sslash\! G \xrightarrow{\;\;} \ast$ were (-1)-truncated (hence if $X \!\sslash\! G$ were subterminal), then the (n-connected, n-truncated) factorization system would imply the required lift. While this would-be argument fails, as $X \!\sslash\! G$ is in general far from being subterminal, the following argument observes that with a suitable choice of atlases for all four $\infty$-stacks, their groupoid objects do form a lifting problem to which the (n-connected, n-truncated) factorization system does apply:

So consider extending the above square diagram to a square of augmented simplicial objects by considering atlases and their Cech nerves, as shown by the following solid arrows:

Observe that all the upper horizontal squares in this diagram have, as indicated:

1. a (-1)-connected morphisms $\twoheadrightarrow$ on the left, since the underlying $\infty$-groupoid $K \in Grpd_\infty$ of every discrete $\infty$-group is inhabited and since $LConst$, being an inverse image-functor, is a lex left adjoint and hence preserves Cech nerves and $\infty$-colimits and hence effective epimorphisms.

2. a (-1)-truncated morphism $\hookrightarrow$ on the right, by Lemma .

Since $n$-connected/$n$-truncated morphisms in $\infty$-categories of $\infty$-presheaves (here: of simplicial objects in $\mathbf{H}$) are detected objectwise (since they are characterized by categorical homotopy groups), this means that the entire square diagram of simplicial objects (i.e. disregarding the bottom square) has a (-1)-connected morphism on the left and a (-1)-truncated morphism in the right. Therefore, the (n-connected, n-truncated) factorization system implies that there exist compatible dashed lifts filling all the upper squares, as shown.

But then taking the $\infty$-colimit over simplicial objects and using that groupoid objects in an $\infty$-topos are effective, recovers the bottom square, but now also equipped with a dashed lift. This is the claimed factorization which shows that $X \!\sslash\! G$ is 0-truncated;

(8)$i_0 \circ \tau_0 \big( X \!\sslash\! G \big) \;\; \simeq \;\; X \!\sslash\! G \,.$

To conclude the proof of (7), use that $\tau_0$ is a left adjoint (6), hence preserves $\infty$-colimits, and also preserves products, i.e. homotopy products (by this Prop.). Here this implies that:

$\begin{array}{lll} X \!\sslash\! G & \;\simeq\; i_0 \circ \tau_0 \big( X \!\sslash\! G \big) & \text{ (8) } \\ & \;\simeq\; i_0 \circ \tau_0 \Big( \underset{\underset{ [n] \in \Delta^{op} }{\longrightarrow}}{\lim} \, X \times G^{\times_n} \Big) & \text{ (4) } \\ & \;\simeq\; i_0 \Big( \underset{\underset{ [n] \in \Delta^{op} }{\longrightarrow}}{\lim} \, \tau_0(X) \times \big(\tau_0(G)\big)^{\times_n} \Big) & \text{ (6) } \\ & \;\simeq\; i_0 \Big( coeq \big( \tau_0(X) \times \tau_0(G) \overset{\phantom{---}}{\rightrightarrows} \tau_0(X) \big) \Big) & \text{ (9) } \\ & \;=\; i_0 \big( \tau_0(X) / \tau_0(G) \big) & \text{ (5). } \end{array}$

In the second but last line we used (from this Example) that the inclusion of the diagram consisting of a pair of parallel morphisms into the opposite of the simplex category is final functor

(9)$\left(  \underoverset {d_0} {d_1} {\rightrightarrows}  \right) \;\xhookrightarrow{\phantom{---}}\; \Delta^{op} \;\;\;\;\;\; \text{is final} \,,$

meaning that the colimit over a simplicial object in a 1-category is equivalently the coequalizer of the first two face maps.

Last revised on November 3, 2021 at 03:25:34. See the history of this page for a list of all contributions to it.