**
cohomology**
*
cocycle,
coboundary,
coefficient
*
homology
*
chain,
cycle,
boundary
*
characteristic class
*
universal characteristic class
*
secondary characteristic class
*
differential characteristic class
*
fiber sequence/
long exact sequence in cohomology
*
fiber ∞bundle,
principal ∞bundle,
associated ∞bundle,
twisted ∞bundle
*
∞group extension
*
obstruction
### 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 (EilenbergSteenrod) cohomology
*
cobordism cohomology theory
*
integral cohomology
*
Ktheory
*
elliptic cohomology,
tmf
*
taf
*
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 2bundle/
gerbe
*
covering ∞bundle/
local system
*
(∞,1)vector bundle /
(∞,n)vector bundle
*
quantum anomaly
*
orientation,
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,
GelfandFuks cohomology,
*
bialgebra cohomology
### Special notions
*
Čech cohomology
*
hypercohomology
### Variants ###
*
equivariant cohomology
*
equivariant homotopy theory
*
Bredon cohomology
*
twisted cohomology
*
twisted bundle
*
twisted Ktheory,
twisted spin structure,
twisted spin^c structure
*
twisted differential cstructures
*
twisted differential string structure,
twisted differential fivebrane structure
* differential cohomology
*
differential generalized (EilenbergSteenrod) cohomology
*
differential cobordism cohomology
*
Deligne cohomology
*
differential Ktheory
*
differential elliptic cohomology
*
differential cohomology in a cohesive topos
*
ChernWeil theory
*
∞ChernWeil theory
*
relative cohomology
### Extra structure
*
Hodge structure
*
orientation,
in generalized cohomology
### Operations ###
*
cohomology operations
*
cup product
*
connecting homomorphism,
Bockstein homomorphism
*
fiber integration,
transgression
*
cohomology localization
### Theorems
*
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
*
EckmannHiltonFuks duality
***
**
higher geometry** / **
derived geometry**
## Ingredients ##
*
higher topos theory
*
higher algebra
## Concepts ##
* **geometric
little (∞,1)toposes**
*
structured (∞,1)topos
*
geometry (for structured (∞,1)toposes)
*
generalized scheme
* **geometric
big (∞,1)toposes**
*
cohesive (∞,1)topos
*
function algebras on ∞stacks
*
geometric ∞stacks
## Constructions
*
loop space object,
free loop space object
*
fundamental ∞groupoid in a locally ∞connected (∞,1)topos /
of a locally ∞connected (∞,1)topos
## Examples
*
derived algebraic geometry
*
étale (∞,1)site,
Hochschild cohomology of
dgalgebras
*
dggeometry
*
dgscheme
*
schematic homotopy type
*
derived noncommutative geometry
*
noncommutative geometry
* derived smooth geometry
*
differential geometry,
differential topology
*
derived smooth manifold,
dgmanifold
*
smooth ∞groupoid,
∞Lie algebroid
*
higher symplectic geometry
*
higher Klein geometry
*
higher Cartan geometry
## Theorems
*
Isbell duality
*
Jones' theorem,
DeligneKontsevich conjecture
*
Tannaka duality for geometric stacks
***
**
higher algebra**
universal algebra
## Algebraic theories
*
algebraic theory /
2algebraic theory /
(∞,1)algebraic theory
*
monad /
(∞,1)monad
*
operad /
(∞,1)operad
## 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
*
action,
∞action
*
representation,
∞representation
*
module,
∞module
*
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
*
∞module,
(∞,1)module bundle
*
multiplicative cohomology theory
*
L∞ algebra
*
deformation theory
## Model category presentations
*
model structure on simplicial Talgebras /
homotopy Talgebra
*
model structure on operads
model structure on algebras over an operad
## Geometry on formal duals of algebras
*
Isbell duality
*
derived geometry
## Theorems
*
Deligne conjecture
*
delooping hypothesis
*
monoidal DoldKan correspondence
Next:
rough notes from a talk
the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention
A complex oriented cohomology theory (meant is here and in all of the following a generalized (EilenbergSteenrod) cohomology) is one with a good notion of Thom classes, equivalently first Chern class for complex vector bundle
(this “good notion” will boil down to certain extra assumptions such as multiplicativity and periodicity etc. What one needs is that the cohomology ring assigned by the cohomology theory to $\mathbb{C}P^\infty \simeq \mathcal{B}U(1)$ is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).
ordinary Chern class lives in integral cohomology $H^*(,\mathbb{Z})$
or in Ktheory $K^*()$ where for a vector bundle $V$ we would set $c_1(V) := ([V]1)\beta$ where $\beta$ is the Bott generator.
In the first case we have that under tensor product of vector bundles the class behaves as
$c_1(V\otimes W) = c_1(V) + c_1(W)$
whereas in the second case we get
$c_1(V \otimes W) = c_1(V)c_1(W)\beta^{1} + c_1(V) + c_1(W)
\,.$
In general we will get that the Chern class of a tensor product is given by a certain power series $E^*(pt)$
not all formal group laws arises this way. the Landweber criterion gives a condition under which there is a cohomology theory
definition of complexorientation
there is an
$x \in \tilde E^2(\mathbb{C}P^\infty)$
such that under the map
$\tilde E^2(\mathbb{C}P^\infty)
\to
\tilde E^2(\mathbb{C}P^1)
\simeq
\tilde E^2(S^1)
\simeq E^0({*})$
induced by
$\mathbb{C}P^1 \to \mathbb{C}P^\infty$
we have $x \mapsto 1$
remark this also gives Thom classes since $\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma$ is a homotopy equivalence
$\tilde E^2((\mathbb{C}P^\infty)^\gamma)
\simeq
\tilde E^2((\mathbb{C}P^\infty))
\ni X$
Thom iso $\tilde H^{*+2}(X^\gamma) \simeq H^*(X)$
…
(here and everywhere the tilde sign is for reduced cohomology)
definition (Bott element and even periodic cohomology theory)
Periodic cohomology theories are complexorientable. $E^*(\mathbb{C}P^\infty)$ can be calculated using the AtiyahHirzebruch spectral sequence
$H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)$
notice that since $\mathbb{C}P^\infty$ is homotopy equivalent to the classifying space $\mathcal{B}U(1)$ (which is a topological group) it has a product on it
$\mathbb{C}P^\infty \times \mathbb{C}P^\infty
\to
\mathbb{C}P^\infty$
which is the one that induces the tensor product of line bundles classified by maps into $\mathbb{C}P^\infty$.
on (at least on even periodic cohomology theories) this induces a map of the form
$\array{
\mathbb{C}P^\infty \times \mathbb{C}P^\infty
&\to&
\mathbb{C}P^\infty
\\
E({*})[[x,y]] &\leftarrow& E(*)[[t]]
\\
f(x,y) &\leftarrow & t
}$
this $f$ is called a formal group law if the following conditions are satisfied

commutativity $f(x,y) = f(y,x)$

identity $f(x,0) = x$

associtivity f(x,f(y,z)) = f(f(x,y),z)
remark the second condition implies that the constant term in the power series $f$ is 0, so therefore all these power series are automatically invertible and hence there is no further need to state the existence of inverses in the formal group. So these $f$ always start as
$f(x,y) = x + y + \cdots$
The Lazard ring is the “universal formal group law”. it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$
$L = \mathbb{Z}[a_{i j}] / (relations 13 below)$
and relatins as follows

$a_{i j} = a_{j i}$

$a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$

the obvious associativity relation
the universal formal group law we get from this is the power series in $x,y$ with coefficients in the Lazard ring
$\ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]]
\,.$
remark the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements $x$. But the underlying formal group is independent of this choice and well defined.
For any ring $S$ with formal group law $g(x,y) \in power series in x,y with coefficients in S$ there is a unique morphism $L \to S$ that sends $\ell$ to $g$.
remark Quillen’s theorem says that the Lazard ring is the ring of complex cobordisms
some universal cohomology theories $M U$ is the spectrum for complex cobordism cohomology theory. The corresponding spectrum is in degree $2 n$ given by
$M U(2n) = Thom
\left(
standard associated bundle to universal bundle
\array{
E U(n) \\ \downarrow \\ B U(n)
}
\right)$
periodic complex cobordism cohomology theory is given by
$M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U$
we get a canonical orientation? from
$\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to}
M U(1)
\;\;\;\;
M U(\mathbb{C}P^\infty)$
this is the universal even periodic cohomology theory with orientation
Theorem (Quillen) the cohomology ring $M P(*)$ of periodic complex cobordism cohomology theory over the point together with its formal group law is naturally isomorphic to the universal Lazard ring with its formal group law $(L,\ell)$
how one might make a formal group law $(R,f(x,y))$ into a cohomology theory
use the classifying map $M P({*}) \to R$ to build the tensor product
$E^n(X) := M P^n(X) \otimes_{M P({*})} R$
this construction could however break the left exactness condition. However, $E$ built this way will be left exact of the ring morphism M P{{*}) \to R
is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger).