* characteristic class
* universal characteristic class
* secondary characteristic class
* differential characteristic class
* fiber sequence
/long exact sequence in cohomology
* fiber ∞-bundle
, principal ∞-bundle
, associated ∞-bundle
* ∞-group extension
### Special and general types ###
* cochain cohomology
* ordinary cohomology
, singular cohomology
* group cohomology
, nonabelian group cohomology
, Lie group cohomology
* Galois cohomology
* groupoid cohomology
, nonabelian groupoid cohomology
* generalized (Eilenberg-Steenrod) cohomology
* cobordism cohomology theory
* integral cohomology
* elliptic cohomology
* abelian sheaf cohomology
* Deligne cohomology
* de Rham cohomology
* Dolbeault cohomology
* etale cohomology
* group of units
, Picard group
, Brauer group
* crystalline cohomology
* syntomic cohomology
* motivic cohomology
* cohomology of operads
* Hochschild cohomology
, cyclic cohomology
* string topology
* nonabelian cohomology
* principal ∞-bundle
* universal principal ∞-bundle
, groupal model for universal principal ∞-bundles
* principal bundle
, Atiyah Lie groupoid
* principal 2-bundle
* covering ∞-bundle
* (∞,1)-vector bundle
/ (∞,n)-vector bundle
* quantum anomaly
, Spin structure
, Spin^c structure
, String structure
, Fivebrane structure
* cohomology with constant coefficients
/ with a local system of coefficients
* ∞-Lie algebra cohomology
* Lie algebra cohomology
, nonabelian Lie algebra cohomology
, Lie algebra extensions
, Gelfand-Fuks cohomology
* bialgebra cohomology
### Special notions
* Čech cohomology
### Variants ###
* equivariant cohomology
* equivariant homotopy theory
* Bredon cohomology
* twisted cohomology
* twisted bundle
* twisted K-theory
, twisted spin structure
, twisted spin^c structure
* twisted differential c-structures
* twisted differential string structure
, twisted differential fivebrane structure
* differential cohomology
* differential generalized (Eilenberg-Steenrod) cohomology
* differential cobordism cohomology
* Deligne cohomology
* differential K-theory
* differential elliptic cohomology
* differential cohomology in a cohesive topos
* Chern-Weil theory
* ∞-Chern-Weil theory
* relative cohomology
### Extra structure
* Hodge structure
, in generalized cohomology
### Operations ###
* cohomology operations
* cup product
* connecting homomorphism
, Bockstein homomorphism
* fiber integration
* cohomology localization
* universal coefficient theorem
* Künneth theorem
* de Rham theorem
, Poincare lemma
, Stokes theorem
* Hodge theory
, Hodge theorem
nonabelian Hodge theory
, noncommutative Hodge theory
* Brown representability theorem
* hypercovering theorem
* Eckmann-Hilton-Fuks duality
## Algebraic theories
* algebraic theory
/ 2-algebraic theory
/ (∞,1)-algebraic theory
## Algebras and modules
* algebra over a monad
∞-algebra over an (∞,1)-monad
* algebra over an algebraic theory
∞-algebra over an (∞,1)-algebraic theory
* algebra over an operad
∞-algebra over an (∞,1)-operad
* associated bundle
, associated ∞-bundle
## Higher algebras
* monoidal (∞,1)-category
* symmetric monoidal (∞,1)-category
* monoid in an (∞,1)-category
* commutative monoid in an (∞,1)-category
* symmetric monoidal (∞,1)-category of spectra
* smash product of spectra
* symmetric monoidal smash product of spectra
* ring spectrum
, module spectrum
, algebra spectrum
* A-∞ algebra
* A-∞ ring
, A-∞ space
* C-∞ algebra
* E-∞ ring
, E-∞ algebra
, (∞,1)-module bundle
* multiplicative cohomology theory
* L-∞ algebra
* deformation theory
## Model category presentations
* model structure on simplicial T-algebras
/ homotopy T-algebra
* model structure on operads
model structure on algebras over an operad
## Geometry on formal duals of algebras
* Isbell duality
* derived geometry
* Deligne conjecture
* delooping hypothesis
* monoidal Dold-Kan correspondence
This is a sub-entry of
see there for background and context.
This entry contains a basic introduction to getting equivariant cohomology from derived group schemes.
the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish
Derived Elliptic Curves
Definition A derived elliptic curve over an (affine) derived scheme is a commutative derived group scheme (CDGS) such that is an elliptic curve.
Let be an -ring. Let denote the -groupoid of oriented elliptic curves over . Note that is in particular a space (we will return to this point later).
The point is to prove the following due to Lurie.
Theorem The functor is representable by a derived Deligne-Mumford stack . Further, is equivalent to the topos underlying and . Also, restricting to discrete rings, provides a lift in sense of Hopkins and Miller.
Extend to where is a trivial -space;
Define where is a compact abelian Lie group where is again a trivial -space;
Extend to for any (finite enough) -space;
Define for any compact Lie group.
To accomplish (1) we need a map
over . Then we can define . Such a map arises from a completion map
which we may interpret as a preorientation . Recall that such a map is an orientation if the induced map to the formal completion of is an isomorphism.
Recall two facts:
There is a bijection ;
Orientations of the multiplicative group associated to are in bijection with maps of -rings , where is the K-theory spectrum.
Theorem We can define equivariant -cohomology using if and only if is a -algebra.
The Abelian Lie Group Case for a Point
Fix oriented. Now let be a compact abelian Lie group. We construct a commutative derived group scheme over whose global sections give which is equipped with an appropriate completion map.
Definition Define the Pontryagin dual, of by .
, the -fold torus. Then as
If , then .
Pontryagin Duality If is an abelian, locally compact topological group then .
Definition Let be an -algebra. Define by
Further, is representable.
, so is final over , hence it is isomorphic to .
How do we get a completion map for all given an orientation ? By a composition: define
Proposition There exists a map such that the assignment factors as . That is the functor factors through the category of classifying spaces of compact Abelian Lie groups (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of .
Proof. That such a factorization exists defines on objects. Now by choosing a base point in we have
as spaces. Now we need a map
Because this map must be functorial in and we can restrict to the universal case where is trivial and then
is just a preorientation .
The Abelian Case for General Spaces
We will see that is the global sections of a quasi-coherent sheaf on .
Theorem Let be preoriented and a finite -CW complex. There exist a unique family of functors from finite -spaces to the category of quasi-coherent sheaves on such that
maps -equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;
maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;
If and then , where is the induced map;
The are compatible under finite chains of inclusions of subgroups .
Proof. Use (2) to reduce to the case where is a -equivariant cell, i.e. for some subgroup . Use (1) to reduce to the case where . Use (3) to conclude that . Finally, (4) implies that , where is specified by the preorientation.
For trivial actions there is no dependence on the preorientation.
is actually a sheaf of algebras.
If are -spaces then we have maps
Define relative version for by
and for all -spaces we have a map
Definition as an -ring (algebra).
We now verify loop maps on .
Recall that in the classical setting is represented by a space and we have suspension maps . Now we need to consider all possible -equivariant deloopings, that is -maps from .
Theorem Let be oriented, a finite dimensional unitary representation of . Denote by the unit sphere inside of the unit ball. Define . Then
is a line bundle on , i.e. invertible;
For all (finite) -spaces the map
is an isomorphism.
Proof for and . Then
As is contractible and by property (3) above for is the identity section. As is oriented, is smooth of relative dimension 1, so can be though of as the invertible sheaf of ideals defining the identity section of .
Suppose and are representations of then is an equivalence. So if is a virtual representation (i.e. ) then
Definition Let be a virtual representation of and define
The point is that in order to define equivariant cohomology requires functors for all representations of , not just the trivial ones. In the derived setting we obtain this once we have an orientation of .
The Non-Abelian Lie Group Case
Let be an -ring, an orientated commutative derived group scheme over , and a (not necessarily Abelian) compact Lie group.
Theorem There exists a functor from (finite?) -spaces to Spectra which is uniquely characterized by the following.
For , ;
maps homotopy colimits to homotopy limits;
If is Abelian, then is defined as above;
For all spaces the map
where is a -space characterized by the requirement that for all Abelian subgroups , is contractible and empty for not Abelian. Further, for Borel equivariant cohomology we require
If , then ;
is an isomorphism.
Proof. In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where has only Abelian stabilizer groups. Then via (3) we reduce to being a colimit of -equivariant cells for Abelian. Via homotopy equivalence (1) we reduce to . Using property (2) we see , so (4) yields