# nLab invariant

Contents

### Context

category theory

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

For $A$ a monoid equipped with an action on an object $V$, an invariant of the action is an element of $V$ which is taken by the action to itself, hence a fixed point for all the operations in the monoid.

## Definitions

A robust definition of invariants that generalizes to homotopy theory is via the expression of actions as action groupoids regarded as sitting over delooping groupoids, as discussed at infinity-action and at geometry of physics – representations and associated bundles.

We describe how the ordinary concept of invariants is recovered from this perspective and then consider its immediate generalizations to (infinity,1)-topos theory and its formalization in homotopy type theory.

### Via sections of action groupoid projections

###### Proposition

For $G$ a discrete group, $\rho$ a $G$-action on some set $S$, then the set of invariants of that action is equivalent to the groupoid of sections of the action groupoid projection of this proposition, corresponding to the action via this proposition.

###### Proof

The sections in question are diagrams in Grpd of the form

$\array{ \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{id}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,,$

hence the groupoid which they form is equivalently the hom-groupoid

$Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho) \in Grpd$

in the slice of Grpd over $\mathbf{B}G$. As in the proof of this proposition, with the fibrant presentation $(p_\rho)_\bullet$ of this proposition, this is equivalently given by strictly commuting diagrams of the form

$\array{ (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet \\ & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\rho)_\bullet}} \\ && (\mathbf{B}G)_\bullet } \,.$

These $\sigma$ now are manifestly functors that are the identity on the group labels of the morphisms

$\sigma_\bullet \;\colon\; \left( \array{ \ast \\ \downarrow^{\mathrlap{g}} \\ \ast } \right) \;\; \mapsto \;\; \left( \array{ \sigma(\ast) \\ \downarrow^{\mathrlap{g}} \\ \sigma(\ast) & = \rho(\sigma(\ast)(g)) } \right) \,.$

This shows that they pick precisely those elements $\sigma(\ast) \in S$ which are fixed by the $G$-action $\rho$.

Moreover, since these functors are identity on the group labels, there are no non-trivial natural isomorphisms between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

More generally, we may consider sections of these groupoid projections after pulling them back along some cocycle:

###### Proposition

Given an associated bundle $P \times_G V\to X$ modulated, as in this proposition, by a morphism of smooth groupoids of the form $g \colon X \longrightarrow \mathbf{B}G$, then its set of sections is equivalently the groupoid of diagrams

$\array{ X && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{g}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,,$

hence the groupoid of sections is the slice hom-groupoid

$\Gamma_X(P\times_G V) \simeq Grpd_{/\mathbf{B}G}(g, p_\rho) \,.$
###### Proof

By the defining universal property of the homotopy pullback in this proposition.

###### Remark

Taken together this means that invariants of group actions are equivalently the sections of the corresponding universal associated bundle.

### Invariants of $\infty$-group actions

For $\mathbf{H}$ an (∞,1)-topos, $G \in Grp(\mathbf{H})$ an ∞-group and

$* : \mathbf{B} G \vdash : V(*) : Type$

an ∞-action of $G$ on $V \in \mathbf{H}$, the type of invariants is the absolute dependent product

$\vdash \prod_{* : \mathbf{B}G} V(*) : Type \,.$

The connected components of this is equivalently the group cohomology of $G$ with coefficients in the infinity-module $V$.

## Properties

###### Proposition

(in characteristic zero, invariants for finite group are compatible with chain homology)

Let $(V_\bullet, \partial)$ be a chain complex over a ground field of characteristic zero, equipped with an action by a finite group $G$. Then taking $G$-invariants commutes with passing to chain homology:

$H_\bullet((V_\bullet,\partial)^G) \;\simeq\; H_\bullet((V_\bullet,\partial))^G \,.$
###### Proof

Since the ground field has characteristic zero, group averaging exists and provides a linear map

$\array{ V_\bullet & \overset{p}{\longrightarrow} & V_\bullet^G \\ x &\mapsto& \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} g(x) }$

onto the $G$-invariants.

Now for a chain homology-class $[x] \in H_\bullet((V_\bullet,\partial))$ being $G$-invariant means that $g[x] \coloneqq [g(x)] = [x]$ for all $g \in G$, which implies that $[x] = [p(x)]$. This means that each invariant homology class has an invariant representative, hence that the map from invariant cycles to invariant chain homology-classes

$Z((V_\bullet^G,\partial)) \longrightarrow H_\bullet((V_\bullet,\partial))$

is an epimorphism.

Next consider the kernel of this map, which a priori is $Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))$. It is now sufficient to show that this coincides with the space of $G$-invariant boundaries:

$Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial)) \;\simeq\; B((V_\bullet^G, \partial)) \,.$

It is clear that there is an inclusion

$B((V_\bullet^G, \partial)) \hookrightarrow Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))$

so it only remains to see that this is also a surjection.

To that end, consider any

$x \in Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial)) \,.$

Since in particular $x \in B((V_\bullet,\partial))$, there is $y \in V_\bullet$ with $x = \partial y$; and since moreover $x \in V_\bullet(G)$, the above implies that

$x = p(x) = p(\partial y) = \partial(p y)$

and hence that

$x \in B((V_\bullet^G,\partial)) \,.$
homotopy type theoryrepresentation theory
pointed connected context $\mathbf{B}G$∞-group $G$
dependent type on $\mathbf{B}G$$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$trivial representation
dependent product along $\mathbf{B}G \to \ast$homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$coinduced representation
spectrum object in context $\mathbf{B}G$spectrum with G-action (naive G-spectrum)

Last revised on August 5, 2023 at 08:34:25. See the history of this page for a list of all contributions to it.