* 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 discusses the algebraic/homotopy theoretic prerequisites for derived algebraic geometry.
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
part 1 – the sheaf of elliptic cohomology ring spectra
We will talk about a lifting problem that will lead to the formulation of tmf. This requires E-infinity rings and derived algebraic geometry.
An -spectrum is a sequence of pointed topological spaces and base-point preserving maps that are weak homotopy equivalences.
( is the loop space of ).
if is an -spectrum, define (homotopy classes of continuous maps). Then this is a generalized (Eilenberg-Steenrod) cohomology theory.
It should be noted that all our spaces are based and is a reduced cohomology theory. Define . are the coefficients (i.e. the cohomology over the point of the corresponding unreduced theory) of .
Brown’s representability theory: Any reduced cohomology theory on CW-complexes is represented by an -spectrum.
singular cohomology with coefficients in : the Eilenberg-MacLane spectrum .
complex K-theory: for even and otherwise
Le be the moduli stack of all elliptic curves, then .
(we will construct this more rigorously later)
If is a map that is a flat morphism, then we obtain an elliptic cohomology theory called .
This assignment is a presheaf of cohomology theories.
To get a single cohomology theory from that we want to take global sections, but there is no good way to say what a global section of a cohomology-theoy valued functor would be. One reason is that there is not a good notion to say what a sheaf of cohomology theorys is.
But if we had an (infinity,1)-category valued functor, then Higher Topos Theory would provide all that technology. So that’s what we try to get now.
goal find lift
Hopkins-Miller: use the multiplicative nature of cohomology theories to solve this, i.e. instead look for a more refined lift
theorem There exists a symmetric monoidal model category of spectra such that the homotopy category is the stable homotopy category as a symmetric monoidal category.
This and the following is described in more detail at symmetric monoidal smash product of spectra.
Definition An A-infinity ring is an ordinary monoid in and an E-infinity ring is an ordinray commutative monoid there.
So an -ring is an honest monoid with respect to the funny smash product that makes spectra a symmetric monoidal category, but it is just a monoid up to homotopy with respect to the ordinary product of spaces.
For more on this see (for the time being) the literature referenced at stable homotopy theory.
Let be an A-infinity ring spectrum.
the -monoidal structure on the spectrum induces a multiplicative cohomology theory.
is a commutative ring
is a module over .
Definition For an E-infinity ring, with a map such that the obvious diagrams commute is a module for that E-infinity ring.
Proposition is a graded module over .
Definition for an E-infinity ring and an -module, we have that is flat module if
is flat over in the ordinary sense
is an isomorphism of -modules
definition a morphism of E-infinity rings is flat if regarded as an -module using this morphism is flat.
A lift as indicated in the GOAL above (multiplicative version) does exists and is unique up to homotopy equivalence.
The tmf-spectrum is the global sections of this:
this is not elliptic (its not even nor has period 2), but is a multiplicative spectrum and hence defines a cohomology theory.
The spectrum tmf is obtained in the same manner by replacing by its Deligne-Mumford compactification.
part 2 - the stable symmetric monoidal -category of spectra
recall that we want global sections of the presheaf
(on the left we have something like the etale site of the moduli stack )
but there is no good notion of gluing in CohomologyTheories (lack of colimits) hence no good notion of sheaves with values in cohomology theories. is the homotopy category of some other category, to be identified, and passage to homotopy categories may destroy existence of useful colimits. The category of CohomologyTheories “is” the stable homotopy category.
A simple example:
in the (infinity,1)-category Top we have the homotopy pushout
but in the homotopy category the pushout is instead
The result is not even homotopy equivalent. In the homotopy category the pushout does not exist.
So we want to refine to the cateory of spectra that they come from by the Brown representability theorem.
In fact, we want to lift to that of E-infinity ring-spectra.
should be that of taking the homotopy category of an (infinity,1)-category.
Approach A (modern but traditional stable homotopy theory) choose a symmetric monoidal simplicial model category whose homotopy category is the stable homotopy category and whose tensor product is the smash product of spectra. For instance use the symmetric monoidal smash product of spectra.
Then define E-infinity ring spectra to be ordinary monoid objects in this symmetric monoidal model category of spectra.
Approach B (Jacob Lurie: be serious about working with (infinity,1)-category instead of just model category theory) .
define (infinity,1)-category (chapter 1 of HTT)
in this framework we’ll have a stable (infinity,1)-category of spectra, let’s call that
show that is a symmetric monoidal (infinity,1)-category
show that the homotopy category of an (infinity,1)-category of is the stable homotopy category, where the tensor product goes to the smash product of spectra
define an E-infinity ring to be a commutative monoid in an (infinity,1)-category in .
These two approaches are equivalent is a suitable sense. See Noncommutative Algebra, page 129 and Commutative Algebra, Remark 0.0.2 and paragraph 4.3.
derived algebraic geometry categorifies algebraic geometry
E-infinity ring categoriefies commutative ring
(infinity,1)-category catgeorifies category
Definition An (infinity,1)-category is (for instance modeled by)
use homotopy coherent nerve to go from a simplicially enriched category to its corresponding quasi-category
definition homotopy category of an (infinity,1)-category (see there)
definition morphism of (infinity,1)-categories is, when regarded as a quasi-category, just a morphism of simplicial sets.: this is an (infinity,1)-functor.
There is an (infinity,1)-category of (infinity,1)-functors between two (infinity,1)-categories
why simplicial sets?
because they provide a convenient calculus for doing homotopy coherent category theory.
suppose some (infinity,1)-category and its homotopy category .
A commutative-up-to-homotopy diagram in is a functor
for some diagram category.
to get a homotopy coherent diagram instead take the nerve of and then map .
The nerve automatically encodes the homotopy coherence. See Higher Topos Theory pages 37, 38 (but the general idea is well known from simplicial model category theory).
Now let be an (infinity,1)-category. Suppose that it has a zero object , i.e. an object that is both an initial object and a terminal object.
Assume that admits kernels and cokernels, i.e. all homotopy pullbacks and pushouts with in one corner.
Then from this we get loop space objects and delooping objects in (called suspension objects in this context).
in particular a loop space object is the kernel of the 0-map,while the suspension is the cokernel
One example of this is the (infinity,1)-category of pointed topological spaces.
definition a prespectrum object in an (infinity,1)-category with the properties as above is a (infinity,1)-functor
such that for is zero object 0.
(everything filled with 2-cells aka homotopies)
since we have cokernels we get maps from the universal property
and analogously maps
now is a spectrum object if the are equivalences, for all . (We don’t require to be equivalences.)
so to each (infinity,1)-category we get another (infinity,1)-category , the full subcategory on the spectrum objects.
In particular, we set
the stable (infinity,1)-category of spectra is the stabilization of the (infinity,1)-category Top of topological spaces.
I think we need pointed topological spaces here?
Fact: has an essentially unique structure of a symmetric monoidal (infinity,1)-category.
This monoidal structure is uniquely characterized by the following two properties:
preserves limits and colimits.
the sphere spectrum is the monoidal unit?/tensor unit wrt .
definition A symmetric monoidal (infinity,1)-category structure on an (infinity,1)-category is given by the following data:
- another (infinity,1)-category with an (infinity,1)-functor that is a coCartesian fibration
where is Segal's category with objects finite pointed sets and morphisms basepoint preserving functions between sets.
where is the fiber over , i.e. the pullback
here should go some pictures that illustarte this. But see the first few pages of Noncommutative Algebra for the intuition and motivation.
so let now be a symmetric monoidal (infinity,1)-category.
definition A commutative monoid in is a section of the structure map mentioned above .
The monoid object itself is the image of under , . (Sort of. I think the whole point is that we don’t ever say something like “this particular is the monoid object”. Rather, the picture should roughly be that we have all of the standard diagrams describing a commutative monoid object, except that the various objects in the diagrams are not necessarily the same object. However, these a priori different objects will be a fortiori homotopy equivalent, so that in particular the usual picture will reappear in the homotopy category. Moreover, of course, these diagrams will not be strictly commutative, but commutative up to coherent homotopy, so that in particular the usual strict commutativity reappears after passage to the homotopy category.)
There is one more condition on , though.
definition an E-infinity ring spectrum is a commutative monoid in an (infinity,1)-category in the stable (infinity,1)-category of spectra .
-rings themselves form an (infinity,1)-category. And this has all limits and colimits (see DAG III 2.1, 2.7), so we can talk about sheaves of rings!
part 3 - brave new schemes
Now the theory of schemes and derived schemes, but not over simplicial commutative rings, but over E-infinity rings.
So we are trying to guess the content of the not-yet-existsting
Let be an E-infinity ring.
Define its spectrum of an E-infinity ring? as the ringed space whose underlying topological space is the ordinary spectrum of the degroo-0 ring
and where is given on Zariski-opens for any by
Here is characterized by the following equivalent ways:
-rings the induced map is a homotopy equivalence of the left hand side with the subspace of the right hand side which takes to an invertible element of .
This geometry over E-infinity rings is in spectral algebraic geometry/brave new algebraic geometry?.
The analog for simplicial commutative rings instead of is what is discussed at derived scheme.
theorem (Jacob Lurie)
If s a space and a sheaf of E-infinity rings then is a classical scheme and is a quasicoherent -module.
theorem there exists a derived Deligne-Mumford stack such that is the ordinary DM- moduli stack of elliptic curves.