symmetric monoidal (∞,1)-category of spectra
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.
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 consdider its immediate generalizations to (infinity,1)-topos theory and its formalization in homotopy type theory.
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.
The sections in question are diagrams in Grpd of the form
hence the groupoid which they form is equivalently the hom-groupoid
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
These $\sigma$ now are manifestly functors that are the identiy on the group labels of the morphisms
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:
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
hence the groupoid of sections is the slice hom-groupoid
By the defining universal property of the homotopy pullback in this proposition.
Taken together this means that invariants of group actions are equivalently the sections of the corresponding universal associated bundle.
For $\mathbf{H}$ an (∞,1)-topos, $G \in Grp(\mathbf{H})$ an ∞-group and
an ∞-action of $G$ on $V \in \mathbf{H}$, the type of invariants is the absolute dependent product
The connected components of this is equivalently the group cohomology of $G$ with coefficients in the infinity-module $V$.
(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:
Since the ground field has characteristic zero, group averaging exists and provides a linear map
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
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:
It is clear that there is an inclusion
so it only remains to see that this is also a surjection.
To that end, consider any
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
and hence that
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 | ∞-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 $\mathbf{B}G \to \mathbf{B}H$ | |
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) |