The theory complex orientations connects two objects: the infinite complex projective space CP∞ and the complex cobordism spectrum MU.
It’s a sort of algebraic number theory over S. Examples are
”Balmer’s interpretation of thick subcategory theorem”: the construction of the Balmer spectrum (and generalization like the constructible Balmer spectrum, as in chromatic nullstellensatz). How does this use complex orientation?
Quillen orientations on formal and p-divisible groups.
”Topological langlands”
Complex Orientations
Definition
A complex orientation on an E1 ring spectrum A∈Alg(1)(Sp) is data:
A map of spectra θ:Σ+∞CP∞→Σ2A.
The composite
\Sigma^2 \mathbb S \simeq\Sigma_+^\infty\mathbb CP^1 \xrightarrow{\Sigma^\infty_+ \iota} \Sigma^\infty_+\mathbb CP^\infty \xrightarrow{\theta} \Sigma^2 \underline A $$
as an element of the anima
$$Map_{Sp}(\Sigma^2 \mathbb S, \Sigma^2 A) \simeq Map_{Sp}(\mathbb S, A) \simeq \Omega^\infty A$$
is isomorphic to the multiplicative unit
$$1: \mathbb S \to A \in \Omega^\infty A$$
**Remark**
Here we're asking for the *condition* "there exists an isomorphism". Hence, this definition only depends on the underlying "homotopy ring spectrum" of $A$:
$$h(A) \in Alg_{E_1^\heartsuit}(h(Sp))$$
This also suggests a new notion
**Definition**
A *complex framing* on $A$ is data
- $\theta \in \Sigma^2 \underline A^{\mathbb CP^\infty}$ (as before)
- DATA of an isomorphism: $1 \to \iota^*\theta$
**Proposition**
A complex orientation on $R$ is a ring map $MU \to R$.
This provides a natural notion of morphisms and higher morphisms between complex orientations. We define
**Definition**
$$\mathbb Cor(R) := Map_{CAlg(Sp)}(MU, R)$$
That is, $Spec(MU)$ is the moduli of complex orientations. Lurie calls these Quillen-orientations. It's the spectral version of a formal group law over $Spec(R)$.
### Strict Elements
**Definition** The anima of $n$-(strict-)elements of a spectrum $X$ is given by
$$nElem(X):= Map_{Alg^{(n)}(S)}(\mathbb Z, \Omega^\infty X)$$
**Remark: Koszul-dual description**
The right hand side can be rewritten using toposic Koszul duality, namely that
$$Map_{Alg^{n}(S)}(\mathbb Z, \Omega^n \Omega^\infty\Sigma^n(X)) \simeq Map_{S}(B^n \mathbb Z, \Omega^{\infty} \Sigma^nX)$$
**Example: 1 and 2 elements**
There are two instances of this calculation that will be important
- $1Elem(X) \simeq Map_S(B\mathbb Z, \Omega^\infty \Sigma X) \simeq Map_{Sp}(\Sigma \mathbb S, \Sigma X) \simeq \Omega^\infty X$
- $2Elem(X) \simeq Map_S(B^2\mathbb Z, \Omega^\infty \Sigma^2 X) \simeq Map_{Sp}(\Sigma^\infty_+ \mathbb CP^\infty, \Sigma^2 X)$
That is, they're the spectra respectively encoding $X$-cohomology at $*$ and at $\mathbb CP^\infty$.
**Remark: forgetful maps**
We have functors
$$U_n:nElem(X) \to {(n-1)}Elem(X)$$
giving by forgetting $E_n$ maps to $E_{n-1}$.
**Definition**
A $E_2$ strictification of $x \in 1Elem(X)$ is a lift along $U_2$ to $\tilde x \in 2Elem(X)$.
**Remark: where this is going**
Combing with the example calculation, the main result of the paper of Grossman-Naples is that the map $$U_1: X^{\Omega^2 \mathbb CP^\infty} \to X$$
is the one induced by restriction along $\mathbb CP^1 \to \mathbb CP^\infty$.
**Remark**
Doing this in families, we can get $E_n$-stacks that are moduli of $n$-elements. The local geometry on these stacks is "strict deformation theory".
## Moduli
**Proposition**
The functor
$$CAlg(Sp) \xrightarrow{U} Sp \xrightarrow{\Omega^n}Sp\xrightarrow{nElem}S$$
is corepresented by the spherical group ring $\mathbb S[B^n\mathbb Z] \in CAlg(Sp)$. That is
$$Spec(\mathbb S[B^n\mathbb Z]) \simeq nElem(\Omega^n\underline{(-)})$$
**Proof**: by unpacking the adjunctions.
It turns out these are loop stacks of the flat affine line
**Proposition**
$$Spec(\mathbb S[B^n \mathbb Z]) \simeq \Omega^n_0Spec(\mathbb S[\mathbb N]):= \Omega^n\mathbb A$$
**Example** Let $V^\otimes$ be a symmetric monoidal stable $\infty$-category
$$Elem(Pic(V)) \simeq Elem(\Omega Br(V)) \simeq \Omega^n\mathbb A(Br(V))$$
That is, the strict picard group is the anima of points $\{x: Spec(Br(V)) \to \Omega^n\mathbb A \}$.
**Example** the element $1 \in \Omega^\infty MU$ (this discussion could explain the appearance of Wilson spaces)