nLab invariant

Idea

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

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

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

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

The connected components of this is equivalently the group cohomology of $G$ with coefficients in the $\infty$-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

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

Examples

General

• invariant polynomial

• invariant differential form

• invariant metric

• invariant measure

• j-invariant

• knot invariant, link invariant

  • Milnor mu-bar invariant

  • Vassiliev knot invariant (universal)

• • topological invariant, homeomorphism invariant, homotopy invariant

  • k-invariant

  • Adams e-invariant

  • f-invariant

  • Hopf invariant

  • Kervaire invariant

  • Casson invariant

  • Chern-Simons invariant

  • Gromov-Witten invariant

  • Donaldson-Thomas invariant

  • eta invariant

• Dickson invariant

Invariant cocycles on principal $\infty$-bundles

Some illustration of the notion of invariants of $\infty$-actions from above.

Consider:

• $\mathbf{H}$ an $\infty$-topos,

• $G \in Grp(\mathbf{H})$ an $\infty$-group object in $\mathbf{H}$,

• $G \curvearrowright P \in G Act(\mathbf{H})$ an $\infty$-action,

• $X \coloneqq P \sslash G$ its homotopy quotient.

Note that the quotient coprojection is a $G$-principal $\infty$-bundle:

and every $G$-principal $\infty$-bundle is of this form (cf. EPBv2 Prop. 1.2.1, p. 15).

Further consider:

• $\mathcal{A} \in \mathbf{H}$ any object,

  to be thought of as a moduli stack for some kind of structure, for instance for differential forms or differential cohomology,

• $G \curvearrowright \mathcal{A} \coloneqq G \curvearrowright \ast \times \mathcal{A} \in G Act(\mathbf{H})$ its incarnation as a trivial $G$-action,

• $G \curvearrowright [P, \mathcal{A}] \coloneqq \big[G \curvearrowright P,\, G \curvearrowright \mathcal{A} \big] \in G Act(\mathbf{H})$ the internal hom of $G$-actions, being the mapping space $[P,\mathcal{A}]$ equipped with the induced conjugation action,

• $[P, \mathcal{G}]^G$ its $G$-fixed locus

(cf. DCCTv1, Def. 3.6.232; p. 339; DCCTv2 Def. 5.1.278, p. 443).

Proposition

We have a natural equivalence:

$$[P, \mathcal{A}]^G \;\simeq\; [X, \mathcal{A}] \,.$$

In words: “$G$-Invariant $\mathcal{A}$-cocycles on the total space $P$ of a $G$-principle bundle are equivalently plain $\mathcal{A}$-cocyles on the base space $X$.”

Proof

Recall the equivalence (EPBv2 Prop. 1.2.1, p. 15):

under which forming fixed loci is right base change to the point: $(-)^G \leftrightarrow \prod_{\mathbf{B}G}$.

Hence in our case:

$$[P,\mathcal{A}]^G \;\;\leftrightarrow\;\; \big[P \sslash G, \mathcal{A}\sslash G\big]_{\mathbf{B}G} \,\coloneqq\, \prod_{\mathbf{B}G} \big[ p_P, p_{\mathcal{A}} \big] \mathrlap{\,.}$$

Now observe that have have a homotopy pullback square of this form (EPBv2 Lem. 4.2.69 with Def. 4.2.66, p. 121):

where we used in the top row that the $G$-action on $\mathcal{A}$ being trivial (by assumption) means equivalently that its homotopy quotient is given by the product with the delooping $\mathbf{B}G$ of $G$:

But the internal hom $\big[P \sslash G,-\big]$ (being a right adjoints) preserves limits, whence

$$\big[ P \sslash G ,\, \mathcal{A} \times \mathbf{B}G \big] \,\simeq\, \big[ P \sslash G ,\, \mathcal{A} \big] \times \big[ P \sslash G ,\, \mathbf{B}G \big] \,,$$

and since pullbacks commute with products, the above pullback square is seen to be equivalent to:

Finally, since $P \sslash G \,\simeq\, X$, this proves the claim.

Related concepts

• invariant theory, geometric invariant theory

• group averaging, norm map

• coinvariant

• stabilizer subgroup

• gauge invariance

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation 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)