symmetric monoidal (∞,1)-category of spectra
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 a discrete group, a -action on some set , 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
These now are manifestly functors that are the identiy on the group labels of the morphisms
This shows that they pick precisely those elements which are fixed by the -action .
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:
hence the groupoid of sections is the slice hom-groupoid
|homotopy type theory||representation theory|
|pointed connected context||∞-group|
|dependent sum along||coinvariants/homotopy quotient|
|context extension along||trivial representation|
|dependent product along||homotopy invariants/∞-group cohomology|
|dependent product of internal hom along||equivariant cohomology|
|dependent sum along||induced representation|
|context extension along|
|dependent product along||coinduced representation|
|spectrum object in context||spectrum with G-action (naive G-spectrum)|