A Survey of Elliptic Cohomology - equivariant cohomology

<div class="rightHandSide toc">
[[!include cohomology - contents]]

***

[[!include higher algebra - contents]]

</div>

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]].

See there for background and context.

This entry considers [[equivariant cohomology]] from the perspective of algebraic geometry.

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

Next:

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

>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**

***

#Contents#

* toc
{:toc}

# Introduction #

The slogan is: for $A$ a [[cohomology theory]], finding [[equivariant cohomology]] $A$-theory corresponds to finding [[group scheme]] over $A({*})$.

The main example we'll be looking at here is complex [[K-theory]].

# Notions of equivariant cohomology #

## Borel equivariant cohomology ##

**Remark** if $A_G$ is an [[equivariant cohomology]] theory and if $Y \to X$ is a $G$-[[principal bundle]] then we want that $A_G(Y) \simeq A(X)$.
So whenever we have a $G$-space where the $G$-[[action]] is free enough.

Let $X$ be a $G$-space, form the [[Borel construction]] $\mathcal{E}G \times_G X$ with $\mathcal{E}G \to \mathcal{B}G$ the [[universal principal bundle]].

Then we can _define_

$$
  A_G^{Borel}(X) := A(\mathcal{E}G \times_G X)
  \,.
$$

> notice that $\mathcal{E}G \times_G X$ is the realization of the [[action groupoid]] $X//G$. This Borel equivariant cohomology theory is what is discussed currently at the entry [[equivariant cohomology]]. The following will actually define a refinement of the discussion currently at [[equivariant cohomology]].

**Problem** If the cohomology theory is given by a geometric model, such as [[topological K-theory]] in terms of [[vector bundle]]s or [[elliptic cohomology]] potentially in a [[geometric model for elliptic cohomology]], then the above notion of equivariant cohomology need not coincide with the cohomology theory given by the equivariant version of these geometric models. In particular, equivariant [[vector bundle]]s are geometric cocycles of equivariant [[K-theory]] $K_G$ and there is a morphism

$$
  K_G(X) \to K_G^{Borel}(X)
$$

but it is not an [[isomorphism]]. Instead, $K_G$ is a _completion_ of $K_G^{Bor}$.

Here $K_G(X)$ is the [[Grothendieck group]] of [[equivariant vector bundle]]s over the [[G-space]] $X$ (say $G$ is a compact [[Lie group]]).

## Grothendieck ring of equivariant vector bundles ##

**Definition** An [[equivariant vector bundle]] over $X$ is

* a [[G-space]] $E$ and $G$-equivariant map $p : E \to X$ such that this is a (complex, here) [[vector bundle]] of finite rank

* for each $g \in G$ the map $g : E_x \to E_{g x}$ is linear.

Morephism are the obvious $G$-equivariant morphisms of [[vector bundle]]s.

*examples**

1. $X$ a trivial $G$-space,then a $G$-equivariant vector bundle is a family of complex representations;

1. for $E \to X$ a [[vector bundle]] the $k$th tensor power of $E$ is a $\Sigma_k$-equivariant vector bundle;

1. if $G$ acts smoothly on $X$ then the complexified [[tangent bundle]] $T X \otimes \mathbb{C} \to X$ is a $G$-equivariant vector bundle.

**remark** The [[category]] of $G$-[[equivariant vector bundle]], has

* [[direct sum]] $\oplus$

* [[tensor product]] $\otimes$

And we can pull back $Vect^G$ \to $Vecg^H$ along any group homomorphism $\phi : H \to G$

So we are entitled to say

**definition** the [[Grothendieck group]] of $Vect^G(X)$ is

$$
  K_G(X) := Groth(E \to X, \oplus)
  \,.
$$

With the remaining [[tensor product]] $\otimes$ this yields a commutative [[ring]].

**proposition**

1. if $X = pt$ then $K_G(X) \simeq Rep(G)$ is the [[representation ring]] of $G$.

   1. in general, $K_G(X)$ is an algebra over $Rep(G)$.

1. if $G$ [[free action|acts freely]] on $X$, then $K_G(X) \simeq K(X/G)$.

So in particular

$$
  K(\mathcal{E}G \times_G X)
  \simeq
  K_G(\mathcal{E}G \times X)
$$

so we get a map

$$
  \alpha : K_G(X) \to K_G(\mathcal{E}G \times X)
  \simeq
  K(\mathcal{E}G \times_G X)
  := K_G^{Borel}(X)
$$

**theorem** (Atiyah-Segal) This $\alpha$ induces an [[isomorphism]]

$$
  \hat K_G(X) \simeq K_G^{Borel}(X)
$$

where

$$
  \hat K_G(X) := \lim_\leftarrow K_G(X)/I_G^n K_G(X)
$$

where $I_G = ker(Rep(G) \simeq K_G({*}) \to K_G(\mathcal{E}G) \simeq K(\mathcal{B}G)
  \stackrel{\epsilon}{\to} \mathbb{Z})$

**consider** $X = {*}$, $G = S^1$, $\mathbb{C}P^\infty \simeq \mathcal{B}S^1$

$$
  \alpha : Rep(G)
  \to
  K(\mathcal{B}G)
  = K(\mathbb{C}P^\infty)
  \simeq
  \mathbb{Z}[ [ t ] ]
$$

1. since $S^1$ is an abelian group, every [[irreducible representation]] is 1-dimensional

   $$
     \phi : S^1 \to \mathbb{C}^\times
   $$

1. $\chi : S^1 \hookrightarrow \mathbb{C}^times$

$$
  Rep(G) \simeq \mathbb{Z}[\chi, \chi^{-1}]
  \,.
$$

## algebraic interpretation ##

**goal now** find an algebraic interpretation of $\alpha$ such that

$$
  Rep(S^1) = \mathcal{o}_{G_m}
$$

and

$$
  Rep(\mathbb{C}P^\infty) = \mathcal{o}_{\hat G_m}
$$

adopt the [[functor of points]] perspective

$$
  X : CRings \to Set
$$

a [[functor]].

For $A \in $ [[CRing]] get a spectrum

$$
  Spec A : R \mapsto CRing(A,R)
$$

for $X$ a functor, it is an [[affine scheme]] if it is a [[representable functor]] in that there is $A$ with $X \simeq Spec A$.

**examples**

1. $\mathbf{A}^n(R) := R^n$

1. $\hat \mathbf{A}^n(R) := Nil(R)^n$

1. $G_m(R) := R^\times \hookrightarrow \mathbf{A}^1(R)$

1. $\mathbb{P}^n(R) := R^{n+1}/\sim$, where $\sim$ is multiplication by $R^\times$

**proposition**

$G_m$ is affine.

**proof** $A = \mathbb{Z}[x,x^{-1}]$, let $u \in Spec(\mathbf{A}(R))$ a map $u A \to R$, then define

$$
  \phi : Spec A \to G_m
$$

by

$$
  u \mapsto u(x)
$$

Conversely, given $v \in G_m(R) = R^\times$, define

$$
  \Psi : G_m \to Spec A
$$

by

$$
  v \mapsto \Psi(v)
$$

with

$$
 \Psi(v)(\sum_k a_k x^k) := \sum_k a_k v^k
$$

**endofproof**

similarly, $\mathbf{A}^n \simeq Spec \mathbb{Z}[x_1, \cdots, x_n]$

# group schemes #

Given a functor $X : CRing \to Set$ define the ring of funtions $\mathcal{o}_X$ as

$$
  \mathcal{o}_X := Hom_{Func(CRing,Set)}(X, \mathbf{A}^1)
$$

in the [[functor category]].

> notice notation: this is global sectins of the [[structure sheaf]], not the structure sheaf itself, properly speaking

we have

$$
  \mathcal{o}_{Spec A}
  \simeq
  A
$$

so that in particular

$$
  \mathcal{o}_{G_m} \simeq \mathbb{Z}[x,x^{-1}]
  \,.
$$

**definition**

Let $X$ be an [[affine scheme]] and $Y : CRing \to Set$ a [[functor]] with a [[natural transformation]] $p : Y \to X$. A **system of formal coordinates** is a sequence of maps

$$
  X_i : Y \to \hat \mathbf{A}^1
$$

such that

$$
  a \mapsto (x_1(a), \cdots, x_n(a), p(a))
  \in
  \hat \mathbf{A}^n \times X
$$

is an [[isomorphism]]. A $Y$ that admits a system of formal coordinates is a **[[formal scheme]]** over $X$.

> **warning** very restrictive definition. See [[formal scheme]]

A **[[formal group]]** $G$ over a [[scheme]] $X$ is a one-dimensional [[formal scheme]] with specified abelian [[group]] structure on each [[fiber]] $p^{-1}\{x\}$.

This means that there is a natural map

$$
  \sigma : G \times_X G \to G
$$

and a natural map $\zeta : X \to G$ which maps $x \mapsto 0 \in p^{-1}\{x\}$.

**definition (formal multiplicative group)**

define $\hat G_m$ on each $R \in CRing$ by

$$
  \hat G_m(R)
  =
  \left\{
    1+n | n \in Nil(R)
  \right\}
$$

which is a group under multiplication.

there is an [[isomorphism]] of underlying [[formal scheme]]s

$$
  \hat G_m  \simeq \hat \mathbf{A}^1
$$

We compute $\mathcal{o}_{\hat \mathbf{A}^1} \simeq \mathcal{o}_{\hat G_m}$ in two ways:

1. Recall that $\hat \mathbf{A}^1$ can be defined as $Spf \; \mathbb{Z} [t]$, so the global sections of the structure sheaf (which is what we have been calling $\mathcal{o}$) is
$$\mathcal{o}_{\hat \mathbf{A}^1} = \lim_\rightarrow \mathbb{Z}[t]/ (t^n) = \mathbb{Z} [[t]] .$$

1. We can also see this in the functor of points perspective.  Consider the functor $\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n)$, then for any ring $R$ $$\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$$
By the universal property of colimits we have
$$\mathrm{Nat} (\hat \mathbf{A}^1 , \mathbf{A}^1 ) \simeq \lim_\leftarrow \mathrm{Nat} (\mathrm{Spec} \; \mathbb{Z}[t]/ (t^n) , \mathbf{A}^1) \simeq \mathbb{Z} [[t]].$$

$$
  \mathcal{o}_{\hat G_m} \simeq \mathbb{Z}[ [ t] ]
  \,.
$$

**recall** we have a morphism $\alpha : Rep(S^1) \to K(\mathbb{C}P^\infty)$

such that

$$
  \mathcal{o}_{G_m}
  \simeq
   \mathbb{Z}[x,x^{-1}]
   \simeq
   Rep(S^1)
  \to
  K(\mathbb{C}P^\infty) \simeq
  \mathbb{Z}[ [ t ] ]
  \simeq
  \mathcal{o}_{\hat G_m}
$$

is the canonical inclusion

$$
  \mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
$$

**exercise**

$$
  x \in \mathbb{Z}[x,x^{-1}] \simeq \mathcal{o}_{G_m}
$$

is the natural transformation $R^\times \to R$

and

$$
  t \in \mathbb{Z}[ [t] ] \simeq
  \mathcal{o}_{\hat G_m}
$$

is the natural transformation

$$
  \{1 + Nil(R)\}
  \to
  Nil(R)
  \to
  R
$$

so that we get a map

$$
  \mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
$$

by sending $x$ to $1 + t$, and this corresponds to taking the [[germ]] of functions at $1 \in G_m$

#lesson #

given $G$ an algebraic group such that the [[formal spectrum]] $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gives a completion map

$$
  A_{S^1}({*}) \to A(\mathbb{C}P^\infty) =
  A^{Bor}_{S^1}({*})
$$


A Survey of Elliptic Cohomology - formal groups and cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

This entry disscusses basics of [[formal group law]]s arising from [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory|cohomology theories]]

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

Next:

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

## rough notes from a talk##

>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs **somebody to go through it and polish it**

**Formal groups and elliptic cohomology.**

In all of the following, all [[cohomology theory|cohomology theories]] are [[multiplicative cohomology theory|multiplicative]] and all [[formal group law]]s are one-dimensional (and commutative).

**[[A Survey of Elliptic Cohomology - cohomology theories|Last time]].** we saw that orienting a [[periodic cohomology theory|periodic]] [[even cohomology theory]] gives a [[formal group law]]
over the [[cohomology ring]] $A^0(\bullet)$.  (Note: $A^0$ and not $A^\bullet$ because of the periodicity property.)

**Today** we discuss a generalization of the above statement:
orienting a [[weakly periodic cohomology theory|weakly periodic]] [[even cohomology theory]] $A$ gives a [[formal group]] over $A^0(\bullet)$.
In particular, [[elliptic cohomology]] theories give [[elliptic curve]]s over $A^0(\bullet)$.

##Formal group laws and Landweber's criterion##

[[formal group law|Formal group law]]s of dimension $1$ over $R$ are classified by [[morphism]]s from the [[Lazard ring]] to $R$.
We can define $A_f^n(X)=MP^n(X)\otimes_{MP(\bullet)}R$.
Here $MP$ denotes [[complex cobordism cohomology theory|complex cobordism]], in particular $MP(\bullet)$ is isomorphic to [[Lazard ring|Lazard's ring]].

**Definition.**  A sequence $v_0,\ldots,v_n$ of elements of $R$ is regular
if [[endomorphism]]s of $R/(v_0,\ldots,v_{k-1})$ given by multiplication by $v_k$
are injective for all $0\le k\le n$.

**[[Landweber criterion]]**  Let $f(x,y)$ be a formal group law and $p$ a prime,
$v_i$ the coefficient of $x^{p^i}$ in $[p]_f(x)=x+_f\cdots+_fx$.
If $v_0,\ldots,v_i$ form a regular sequence for all $p$ and $i$ then $f(x,y)$ gives a
[[cohomology theory]] via the formula with [[tensor product]] above.

**Example.**  $g_a(x,y)=x+y$, $[p]_a(x)=px$, $v_0=p$, $v_i=0$ for all $i\ge1$;
regularity condtions imply that
the zero map $R/(p)\to R/(p)$ must be injective.  The last statement implies that
$R$ contains the rational numbers as a subring.

Note that $HP^*(X,R)=\prod_k H^{n+2k}(X,R)$ is a [[cohomology theory]] over any [[ring]] $R$.

**Example.**  $g_m(x,y)=xy$, $[p]_m(x)=(x+1)^p-1$, $v_0=p$, $v_1=1$, $v_i=0$ for all $i \gt 1$.
The regularity conditions are trivial.
Hence we know that $K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z}$ is a cohomology theory.

## Formal groups from formal group laws##

 Given a commutative topological
$R$-algebra $A$ and a [[formal group law]] $f(x,y)$ if $f(a,b)$ converges for all $a,b\in A$ and the formula giving an inverse to $a$ converges for all $a\in A$,
we get an abelian [[group]] $(A,+_f)$, where $a+_f b=f(a,b)$.

**Example.**  For any $A$ the pair $(N(A),+_f)$ is an abelian group, where
$N(A)$ denotes the set of nilpotent elements of $A$.

**Example.**  Let $A$ be an oriented complex oriented cohomology theory.
Then computing [[Chern class]]es of [[line bundle]]s is the same as evaluating the
formal group law of $A$ on some algebra.
Recall that [[line bundle]]s on $X$ are [[classifying space|classified]] by maps from $X$ to $\mathbb{C}P^\infty$,
pairs of line bundles are classified by maps to $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$,
and tensor product of line bundles gives a map $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$.
Now apply cohomology functor to the sequence $X\to \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$.
We have a degree 0 element $t$ in the cohomology of $\mathbb{C}P^\infty$.
Its image in the cohomology of $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$ is a formal group law.
The image of this formal group law in the cohomology of $X$ makes sense
if $X$ is a finite cell complex so that $A^0(X)$ is a nilpotent algebra.

**Question:** When do two formal group laws yield isomorphic groups?

**Definiton.**  A homomorphism of formal group laws $f$ and $g$ over $A$
is a formal power series $\phi\in A[x]$ such that $\phi(f(x,y))=g(\phi(x),\phi(y))$.
(The constant term of~$\phi$ is zero.)
Hence formal group laws form a category.

**Example.**  If $R$ contains rational numbers as a subring,
then we have two canonical homomorphisms.  The first one is $\exp\colon g_a\to g_m$, where
$\exp(x)=\sum_{k \gt 0}x^k/k!$.  Its inverse is $\log\colon g_m\to g_a$, where
$\log(x)=\sum_{k \gt 0}(-1)^{k+1}x^k/k$.
This shows up in cohomology as Chern character.
(Isomorphism from $K^n(X)\otimes_{\mathbb{Z}} \mathbb{Q}$ to $\prod_kH^{n+2k}(X,\mathbb{Q})$.

**Formal groups.**  A [[formal group]] is a group in the [[category]] of [[formal scheme]]s.

A [[formal scheme]] $\hat{Y}$ is defined for any [[closed subscheme|closed immersion]] of [[scheme]]s $Y \hookrightarrow X$.
Intuitively the [[formal scheme]] $\hat Y$ is the $\infty$-jet bundle in the normal direction of $Y$ inside of $X$.

**Definition.**  The locally [[ringed space]] $\hat Y$ is defined as the [[topological space]] $Y$
with structure sheaf $lim O_X/{\mathcal{I}}^n$, where $\mathcal{I}$ is the defining sheaf of ideals of the closed immersion $Y\hookrightarrow X$.
(Where $Y$ is a closed subscheme of $X$.)

**Examples.**  $X=\hat Y$ when $Y=X$.  $\mathrm{Spec} k[t]=X$, $Y=V(t)$, $\hat X=k[t,t^{-1}]$.

In fact not every locally noetherian formal scheme can be obtained as a completion of a single noetherian scheme in another scheme; such formal schemes are called *algebraizable*.

**Definition. (formal spectrum)** The [[formal spectrum]] $\mathrm{Spf} R$ of a commutative noetherian ring $R$ with a specified ideal $I \subset R$ whose powers define a local basis of a topology around $0$ which is Hausdorff, is the locally [[ringed space]] with the underlying [[topological space]] $\mathrm{Spec} R/I$ whose global sections of the [[structure sheaf]] are the [[limit]]

$$
  O_{\mathrm{Spf} R}(\mathrm{Spf} R)=\lim_n (R/I^n)
  \,.
$$

(This is incomplete description, one needs to talk sheaves of ideals instead)

### formal group laws from elliptic curve ###

Recall from the above that a given a [[formal group law]] $F(x,y) \in R[ [x,y] ]$ we get te structure of a [[formal group]] on the [[formal spectrum]] $Spf$ by taking the product to be given by

$$
  \array{
    Spf R[[x,y]] \simeq
    Spf[[x]] \times Spf R[[y]]
    &\to& Spf R[[z]]
    \\
    f(x,y)&\leftarrow |& z
  }
$$

Isomorphic [[formal group law]]s give [isomorphism|isomorphic]] (of [[formal group]]s) if $G$ a [[formal group]] has $G \simeq Spf R[ [z] ]$; we must choose such an iso to get a [[formal group law]].

Now we get [[formal group]]s from [[elliptic curve]]s over $R$

**Definition** An **[[elliptic curve]]** over a commutative [[ring]] $R$ is a [[group object]] in the [[category]] of [[scheme]]s over $R$ that is a relative 1-dimensional, , [[smooth scheme|smooth curve]], [[proper scheme|proper]] curve over $R$.

This implies that it has [[genus]] 1. (by a direct argument of the Chern class of the tangent bundle.)

Given an [[elliptic curve]] over $R$, $E \to Spec R$, we get a [[formal group]] $\hat E$ by completing $D$ along its identity [[section]] $\sigma_0$

$$
  E \to Spec(R) \stackrel{\sigma_0}{\to} E
$$

(the one dual to the map that maps everything to $0 \in R$), we get a [[ringed space]] $(\hat E, \hat O_{E,0})$

**example** if $R$ is a [[field]] $k$, then the [[structure sheaf]] $\hat O_{E,0} \simeq k[ [z] ]$

then

$$
  \hat O_{E \times E, (0,0)} \simeq
  \hat O_{E,0}
  \hat \otimes_k
  \hat O_{E,0}
  \simeq
  k[[x,y]]
$$

**example** **(Jacobi quartics)**

$$
  y^2 = 1- 2 \delta x^2 + \epsilon x^4
$$

defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.

The corresponding [[formal group law]] is **Euler's formal group law**

$$
  f(x,y) = \frac{x\sqrt{1- 2 \delta y^2 + \epsilon y^4}
   + y \sqrt{1- 2 \delta x^2 + \epsilon x^4}}
  {1- \epsilon x^2 y^2}
$$

if $\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0$ then this is a non-trivial elliptic curve.

If $\Delta = 0$ then $f(x,y) \simeq G_m, G_a$ (additive or multiplicative formal group law corresponding to [[ordinary cohomology]] and [[topological K-theory]] [[KU]], respectively).

## weakly periodic cohomology theories and formal groups ##

A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A$ is **[[weakly periodic cohomology theory|weakly periodic]]** if the natural map

$$
  A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to}
  A^{n+2}({*})
$$

is an [[isomorphism]] for all $n \in \mathbb{Z}$.

Compare with the notion of a [[periodic cohomology theory]].

# Relation to formal groups #

One reason why weakly periodic cohomology theories are of interest is that their [[cohomology ring]] over the space $\mathbb{C}P^\infty$ defines a [[formal group]].

To get a [[formal group]] from a [[weakly periodic cohomology theory|weakly periodic]], [[even cohomology theory|even]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism

$$
  i_0 : {*} \to \mathbb{C}P^\infty
$$

and take the kernel

$$
  J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))
$$

to be the [[ideal]] that we complete along to define the [[formal scheme]] $Spf A^0(\mathbb{C}P^\infty)$ (see there for details).

Notice that the map from the point is unique only up to [[homotopy]], so accordingly there are lots of chocies here, which however all lead to the same result.

The fact that $A$ is weakly periodic allows to reconstruct the [[cohomology theory]] essentially from this [[formal scheme]].

To get a [[formal group law]] from this we proceed as follows: if the [[Lie algebra]] $Lie(Spf A^0(\mathbb{C}P^\infty))$ of the [[formal group]]

$$
  Lie(Spf A^0(\mathbb{C}P^\infty))
  \simeq
  ker(i_0^*)/ker(i_0^*)^2
$$

is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an [[isomorphism]]

$$
  Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])
$$

if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ "forgets the $t$-coordinate".

**Definition** An **elliptic cohomology theory** over $R$ is

* a commutative [[ring]] $R$

* an [[elliptic curve]] $E/R$

* a [[weakly periodic cohomology theory|weakly periodic]], [[multiplicative cohomology theory|multiplicative]], [[even cohomology theory|even]] [[cohomology theory]] $A^\bullet$

* [[isomorphism]]s $A^0({*}) \simeq R$ and $\hat E \simeq Spf(A^0(\mathbb{C}P^\infty))$.

So we have on one side

$$
  \array{
    \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty)
    \\
    \downarrow && \downarrow
    \\
    Spec R &\stackrel{\simeq}{\to}&
    Spec A^0({*})
    \\
    \downarrow^{\sigma_0} && \downarrow
    \\
    \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty)
  }
$$

We can check that the [[Landweber exactness criterion]] is satisfied for the [[formal group law]] of the [[Jacobi quartic]], i.e. for [[Euler's formal group law]] over $\mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]$, so this provides an example of an [[elliptic cohomology]] theory.

$$
  A^n_G(X) = M P (X) \otimes_{M P({*})} \mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]
$$


A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

>**Abstract** We sketch some basic ideas ([[Jacob Lurie]]'s ideas, that is) about [[higher geometry]] motivated from the [[higher category theory|higher]] version of the [[moduli stack]] of [[elliptic curve]]s: the derived moduli stack of [[derived elliptic curve]]s. We survey aspects of the theory of [[generalized scheme]]s and then sketch how the derived moduli stack of derived elliptic curves is an example of a generalized scheme modeled on the formal dual of [[E-∞ ring]]s.

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

=--

For fully appreciating the details of  the main theorem here the material discussed in the previous sessions (and a little bit more) is necessary, but our exposition of [[generalized scheme]]s is meant to be relatively self-contained (albeit necessarily superficial).

This are the entries on the previous sessions:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

***

# the derived moduli stack of derived elliptic curves #
* automatic table of contents goes here
{:toc}

## Motivation and Statement

In  the context of [[elliptic cohomology]] one assigns to every [[elliptic curve]] $\phi$ over a [[ring]] $R$ a [[cohomology theory]] [[Brown representability theorem|represented]] by an [[E-∞ ring]] [[spectrum]] $E_\phi$.

Since, by definition, we may identify the [[elliptic curve]] $\phi$ over $R$ with a patch $\phi : Spec R \to \mathcal{M}_{1,1}$ of the [[moduli stack]] $\mathcal{M}_{1,1}$ of elliptic curves, this assignment

$$
  \mathcal{O}^{Der} : \phi \mapsto E_\phi
$$

looks like an [[E-∞ ring]] valued [[structure sheaf]] on $\mathcal{M}_{1,1}$.

There is a very general theory of [[higher geometry]] for [[nLab:generalized scheme|generalized spaces]] with generalized [[nLab:structure sheaf|structure sheaves]]. Using this one may regard the pair

$$
  (\mathcal{M}_{1,1}, \mathcal{O}^{Der})
$$

as a [[structured (infinity,1)-topos|structured space]] that is a "derived" [[Deligne-Mumford stack]].

The central theorem about [[elliptic cohomology]] of [[Jacob Lurie]], refining the Goerss-Hopkins-Miller theorem says that

+-- {: .standout}

**the central theorem, first version**

The [[moduli stack]] $\mathcal{M}_{1,1}$ of [[elliptic curve]]s equipped with the [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{Der}$ may be regarded as the _derived moduli stack_ of [[derived elliptic curve]]s in that for any [[E-∞ ring]] $R$ the space of derived stack morphisms

$$
  (Spec R, R) \to (\mathcal{M}_{1,1}, \mathcal{O}^{Der})
$$

is equivalent to the space of [[derived elliptic curve]]s over $R$.

=--

After we have looked at some concepts in [[higher geometry]] a bit more closely below, we will restate this in slightly nicer fashion.

## References

A sketch of what this theorem means and how it is proven is part of the content of

* [[Jacob Lurie]], [[A Survey of Elliptic Cohomology]].

and goes back to Jacob Lurie's PhD thesis (listed [[Jacob Lurie|here]]).

The general theory for the context of [[higher geometry]] invoked here has later been spelled out in

* [[Jacob Lurie]], [[Structured Spaces]]

The special case of the general theory that is needed here, where the coefficient objects of structure sheaves are [[E-∞ ring]]s, is described in

* [[Jacob Lurie]], [[Spectral Schemes]],

while the general theory of [[E-∞ ring]]s themselves, in the [[(∞,1)-category]] theory context needed here, is developed in

* [[Jacob Lurie]], [[higher algebra|Commutative geometry]].

## Notions of Space {#NotionsOfSpace}

The statement that we are after really lives in the context of [[higher geometry]] (often called "derived geometry"). Here is an outline of the central aspects.

The **central ingredient** which we choose at the beginning to get a theory of [[higher geometry]] going is an [[(∞,1)-category]] $\mathcal{G}$ whose objects we think of as **model spaces** : the simplest objects exhibiting the geometric structures that we mean to consider.

+-- {: .standout}

**Examples for categories of model spaces**

* with smooth structure

  * $\mathcal{G} = $ [[Diff]], the category of smooth [[manifold]]s;

  * $\mathcal{G} = \mathbb{L}$, the category of [[smooth locus|smooth loci]];

* without smooth structure

  * $\mathcal{G} = (C Ring^{fin})^{op}$, the formal dual of [[CRing]]: the category of (finitely generated) algebraic [[affine scheme]]s;

  * $\mathcal{G} = (sC Ring^{fin})^{op}$, the formal dual of [[simplicial object]]s in  [[CRing]];

  * $\mathcal{G} = (E_\infty Ring^{fin})^{op}$, the formal dual of [[E-∞ ring]]s: the category of (finitely generated) algebraic derived [[affine scheme]]s.

=--

These [[(∞,1)-category|(∞,1)-categories]] $\mathcal{G}$ are naturally equipped with the structure of a [[site]] (and a bit more, which we won't make explicit for the present purpose). Following [[Jacob Lurie]] we call such a $\mathcal{G}$ a **[[geometry (for structured (infinity,1)-toposes)|geometry]]** .

We want to be talking about generalized spaces _modeled on_ the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:

+-- {: .standout}

**Hierarchy of generalized spaces modeled on $\mathcal{G}$**

$$
  \array{
    \mathcal{G}
    &\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}&
    Sch(\mathcal{G})
    &\hookrightarrow&
    \mathcal{L}Top(\mathcal{G})^{op}
    &\hookrightarrow&
    Sh_{(\infty,1)}(Pro(\mathcal{G}))
    \\
    \\
    model spaces
    &&
    spaces locally like model spaces
    &&
    concrete spaces coprobeable by model spaces
    &&
    spaces probeable by model spaces
    \\
    \\
    affine\;\mathcal{G}-schemes
    &&
    \mathcal{G}-schemes
    &&
    \mathcal{G}-structured\;(\infty,1)-toposes
    &&
    \infty-stacks\;on\;\mathcal{G}
    \\
    \stackrel{tame\;but\;restrictive}{\leftarrow} &
    &&&&
    &
    \stackrel{versatile\;but\;possibly\;wild}{\to}
  }
$$

=--

We explain what this means from right to left.

### spaces probeable by model spaces: $\infty$-stacks

An object $X$ _probeable_ by objects of $\mathcal{G}$ should come with an assignment

$$
  X : (U \in \mathcal{G}) \mapsto (X(U) \in \infty Grpd)
$$

of an [[∞-groupoid]] of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{G}$

$$
   X \in PSh(\mathcal{G})
   =
   Funct(\mathcal{G}^{op}, \infty Grpd)
$$

But for $X$ to be _consistently_ probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the [[coverage|topology]] of $\mathcal{G}$. More precisely, this should mean that the [[(∞,1)-presheaf]] $X$ is actually an object in an [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$

$$
  X \in
  Sh(\mathcal{G}) \stackrel{}{\hookrightarrow}
  PSh(\mathcal{G})
  \,.
$$

Such objects are called [[∞-stack]]s on $\mathcal{G}$. The [[(∞,1)-category]] $Sh(\mathcal{G})$ is called an [[∞-stack]] [[(∞,1)-topos]].

A supposedly pedagogical discussion of the general philosophy of [[∞-stacks]] as probebable spaces is at [[motivation for sheaves, cohomology and higher stacks]].

The [[∞-stack]]s on $\mathcal{G}$ that are used in the following are those that satisfy [[descent]] on [[?ech cover]]s. But we will see [[(∞,1)-topos]]es of [[∞-stack]]s that may satisfy different descent conditions, in particular with respect to [[hypercover]]s. Every [[∞-stack]] [[(∞,1)-topos]] has a [[hypercompletion]] to one of this form.

For concretely working with [[hypercomplete (∞,1)-topos]]es it is often useful to use [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].

+-- {: .standout}

$$
  \array{
    Sh^{hc}_{(\infty,1)}(C)
   &\stackrel{\stackrel{\;\;\;\;\;lex\;\;\;\;\;\;}{\leftarrow}}
  {\hookrightarrow}&
     PSh_{(\infty,1)}(C)
    && \text{abstract nonsense def of (∞,1)-topos}
    \\
    \uparrow^{\simeq} &&
    \uparrow^{\simeq}
    && \text{Lurie's theorem}
    \\
    ([C^{op}, SSet]_{loc})^\circ
    &\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}}{\to}&
    ([C^{op}, SSet]_{glob})^\circ
    &&
    \text{model category of simplicial presheaves}
  }
$$

=--

+-- {: .un_remark}
###### Warning

This discussion here is glossing over all set-theoretic size issues. See [[Structured Spaces|StSp, warning 2.4.5]].

=--

### concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes

Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a _concrete underlying space_ , even for general definitions of what _that_ might mean.

**(Counter-)Example** For $\mathcal{G} = $ [[Diff]], for every $n \in \mathbb{N}$ we have the [[∞-stack]] $\Omega_{cl}^n(-)$ (which happens to be an ordinary [[sheaf]]) that assigns to each manifold $U$ the set of closed [[differential form|n-form]]s on $U$. This is important as a generalized space: it is something like the rational version of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}, n)$. But at the same time this is a "wild" space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.

In the classical theory for instance of [[ringed space]]s or [[diffeological space]]s a _concrete underlying space_ is taken to be a [[topological space]]. But this in turn is a bit _too_ restrictive for general purposes: a topological space is the same as a [[localic topos]]: a [[category of sheaves]] on a [[category of open subsets]] of a [[topological space]]. The obvious generalization of this to [[higher geometry]] is: an [[n-localic (∞,1)-topos]] $\mathcal{X}$.

This makes us want to say and make precise the statement that

+-- {: .standout}

An **concrete [[∞-stack]]** $X$ is one which has an _underlying_ [[(∞,1)-topos]] $\mathcal{X}$:

the collection of $U$-probes of $X$ is a [[subobject]] of the collection of [[(∞,1)-topos]]-morphisms from $U$ to $\mathcal{X}$:

$$
  X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})
$$

=--

We think of $\mathcal{X}$ as the [[(∞,1)-topos]] of [[∞-stack]]s on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the [[terminal object]] ${*}_{\mathcal{X}} \in \mathcal{X}$.

To say that $\mathcal{X}$ is _modeled on $\mathcal{G}$_ means that among all the [[∞-stack]]s on the would-be space a [[structure sheaf]] of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a [[structure sheaf]] $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.

This yields an [[(∞,1)-functor]]

$$
  \mathcal{O} : \mathcal{G} \to \mathcal{X}
  \,.
$$

We think of $X$ as being a concrete space _co-probebale_ by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).

Such an $\mathcal{O}$ is a _consistent_ collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.

More precisely:

+-- {: .un_def}
###### Definition
**($\mathcal{G}$-structure, [[Structured Spaces|StSp, def. 1.2.8]])**

An [[(∞,1)-functor]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$  is a **$\mathcal{G}$-valued structure sheaf** on the [[(∞,1)-topos]] if

* it preserves finite [[limit]]s

* and sends covering coproducts $(\coprod_i U_i) \to U$ to [[effective epimorphism]]s.

A pair $(\mathcal{X}, \mathcal{O})$ of an [[(∞,1)-topos]] $\mathcal{X}$ equipped with $\mathcal{G}$-valued [[structure sheaf]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a [[structured (∞,1)-topos]].

=--

In summary:

+-- {: .standout}

A **concrete [[∞-stack]] $X$ modeled on $\mathcal{G}$** is

* an [[(∞,1)-topos]] $\mathcal{X}$ ("of $\infty$-stacks on $X$")

* equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.

=--

The fundamental **example** for [[structured (∞,1)-topos]]es
are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.

+-- {: .un_theorem}
###### Theorem
**([[Structured Spaces|StSp, thm. 2.1.1]])**

Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of [[geometry (for structured (infinity,1)-toposes)|geometries]],
then the obvious [[(∞,1)-functor]] $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a [[left adjoint]]

$$
  f^*
  :
  \mathcal{L}Top(\mathcal{G}')
   \stackrel{\leftarrow}{\to}
  \mathcal{L}Top(\mathcal{G})
  :
  Spec_{\mathcal{G}}^{\mathcal{G}'}
$$

called the **relative spectrum functor**.

=--

For $\mathcal{G}$ any [[geometry (for structured (infinity,1)-toposes)|geometry]], write $\mathcal{G}_{disc}$ for the [[geometry (for structured (infinity,1)-toposes)|geometry]] obtained from this by forgetting its [[coverage|Grothendieck topology]] and instead using the discrete topology where only equivalences cover.

Notice
that we may identify $\mathcal{G}_{disc}$-structures on
the archetypical [[(∞,1)-topos]] [[∞Grpd]], being finite [[limit]]-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$
with [[ind-object]]s in $\mathcal{G}^{op}$, hence with the
opposite of [[pro-object]]s in $\mathcal{G}$. This gives a canonical inclusion

$$
  Pro(\mathcal{G})
    \hookrightarrow
  \mathcal{L}Top(\mathcal{G})^{op}
  \,.
$$

+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.1.2]])**

The composite [[(∞,1)-functor]]

$$
  Spec^{\mathcal{G}}
  :
  Pro(\mathcal{G})^{op}
  \hookrightarrow
  \mathcal{L}Top(\mathcal{G}_{disc})
  \stackrel{Spec_{\mathcal{G}}^{\mathcal{G}_{disc}}}{\to}
  \mathcal{L}Top(\mathcal{G})
$$

we call the **absolute spectrum functor**

=--

This [[category theory|abstract nonsense]] is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:

+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.2.9]])**

For every $U \in \mathcal{G}$ there is naturally induced a [[coverage|topology]] on the [[over category]] $Pro(\mathcal{G})/U$. Define the [[(∞,1)-topos]]

$$
  Spec U := Sh_{(\infty,1)}(Pro(\mathcal{G})/U)
  \,,
$$

naturally to be thought of as the [[(∞,1)-topos]] of [[∞-stack]]s _on $U$_ .

This is canonically equipped with a [[(∞,1)-functor]]

$$
  \mathcal{O}_{Spec X}
  :
  \mathcal{G} \to Spec X
  \,.
$$

=--

And this is indeed the concrete underlying space produced by the absolute spectrum functor:

+-- {: .un_theorem}
###### Theorem
**[[Structured Spaces|StSp, prop. 2.2.11, thm. 2.2.12]])**

For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a [[structured (∞,1)-topos]] and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.

=--

**Example** For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard [[coverage|topology]] we have that 0-localic $\mathcal{G}$-structured spaces are _[[locally ringed space]]s_ , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary [[ringed space]]s. Applying the above machinery to this situaton gives a spectrum functor that takes a [[ring]] $R$ first to the [[ringed space]] $({*,R})$ and this then to the [[locally ringed space]] $(Spec R, R)$.

### Spaces locally like model spaces: generalized schemes

We have seen that $\mathcal{G}$-[[structured (∞,1)-topos]]es are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an "underlying topological structure" in the form of an underlying [[(∞,1)-topos]]. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.

In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.

But in many situations one would like to regard $\mathcal{G}$-[[structured (∞,1)-topos]]es that are not globally but _locally_ equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.

+-- {: .un_def}
###### Definition
**[[Structured Spaces|StSp, def. 2.3.9]]

A [[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O})$ is a **$\mathcal{G}$-generalized scheme** if

* there exists a collection $\{V_i \in \mathcal{X}\}$

* such that

  * this covers $\mathcal{X}$ in that the canonical morphism

    $$
      (\coprod_i V_i) \to {*}_{\mathcal{X}}
    $$

    to the [[terminal object]] in $\mathcal{X}$
    is an [[effective epimorphism]]

  * the [[structured (∞,1)-topos]]es
    $(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by
    the $V_i$ are model spaces in that there exists
    $\{U_i \in \mathcal{G}\}$ and equivalences

    $$
      (\mathcal{X}/V_i, \mathcal{O}|_{V_i})
      \simeq
      Spec^{\mathcal{G}} U_i
    $$

=--

**Examples**

> **warning** the following statements really pertain to pregeometries, not geometries. for the moment this here is glossing over the difference between the two. See [[geometry (for structured (∞,1)-toposes)]] for the details.

* ordinary smooth [[manifold]]s are [[n-localic (infinity,1)-topos|0-localic]] [[Diff]]-[[generalized scheme]]s ([Structured Spaces|StSp, ex. 4.5.2]])

* ordinary [[schemes]] are those $(CRing^{fin})^{op}$-[[generalized scheme]]s whose underlying [[(∞,1)-topos]] is
[[n-localic (infinity,1)-topos|0-localic]] and whose [[structure sheaf]] is [[n-truncated object of an (infinity,1)-category|0-truncated]]  ([Structured Spaces|StSp, prop. 4.2.9]])

* [[Deligne-Mumford stack]]s are 1-localic $(CRing^{fin})_{et}^{op}$-[[generalized scheme]]s ([Structured Spaces|StSp, prop. 4.2.9]])

* This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-[[generalized scheme]] a **derived Deligne-Mumford stack

* Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in [[Spectral Schemes]] a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by [[E-∞ ring]]s) is gonna be a 1-localic  $(E_\infty Ring^{fin})^{op}$-[[generalized scheme]].

## The derived moduli space of elliptic curves

With the above machinery for [[higher geometry]] in hand, we now set out to describe the particular application that we are interested in: the study of the derived [[moduli stack]] of [[derived elliptic curve]]s.

### Derived elliptic curves

* [[derived elliptic curve]]

$$
  A \mapsto E(A)
$$

### The derived moduli stack

Lurie's discussion of the derived moduli stack $(\mathcal{M}_{1,1}, \mathcal{O}^{Der})$  is more than a re-interpretation of the Goerss-Miller-Hopkins theorem. It is in particular a re-derivation of this result, from the following perspective

+-- {: .standout}

**the central statement, conceptually**

**Input** We have the $(E_\infty Ring)^{op}$-probeable space

$$
  (E : R \mapsto \{derived\;elliptic\;curves\;over\;R\})
  \in Sh(E_\infty Ring^{op})
  \,.
$$

**Question**: Does this happen to even be a $E_\infty Ring^{op}$-[[generalized scheme]]?

**Answer** Yes. It is actually a [[derived Deligne-Mumford stack]].

=--

Let $\mathcal{M}_{1,1}$ be the ordinary [[moduli stack]] of [[elliptic curve]]s.

Using constructions in [[elliptic cohomology]] we may associate to each [[elliptic curve]] over $R$, i.e. each morphism $\phi : Spec R \to \mathcal{X}$, an [[E-infinity ring]] $E_\phi$ -- the multiplicative spectrum that represents the elliptic cohomology theory given by $T$.

This gives an $E_\infty$-ring valued structure sheaf

$$
  \mathcal{O}^{Der} : (\phi : Spec R \to \mathcal{X}) \mapsto
  E_\phi
  \,.
$$

**Question** What, if anything, is this derived stack a derived [[moduli stack]] of?

### the classification of derived elliptic curves ###

The big theorem is that the derived space $(\mathcal{X}, \mathcal{O}^{Der})$ classifies derived elliptic curves over $E_\infty$-rings

This is the theorem that we said above we wanted to consider, stated now a little bit more precisely.

+-- {: .un_theorem }
###### Theorem
**(J. Lurie)**

For

* $A$ any [[E-∞ ring]]

* and $E(A)$ is the space of [[derived elliptic curve]]s over $A$ (the realization of the topological category of elliptic curves over $A$).

we have an equivalence

$$
  Hom(Spec A, (\mathcal{X}, \mathcal{O}^{Der}))
  \simeq
  E(A)
$$

naturally in $A$.

=--

+-- {: .proof}
###### Proof

Jacob Lurie writes that the proof proceeds alonmg these steps. Details will be discussed in the next session.

1. consider the presheaf of preoriented ellitptic curves $E'(A)$ first

1. observe that this restricted to ordinary rings produces the ordinary moduli stack

1. notice that every oo-stack with good deformation theory that restricts this way is a derived Deligne-Mumford stack $(\mathcal{X}, \mathcal{O}')$ that assigns connective $E_\infty$-rings over affines

1. let $\omega$ be the line bundle on $\mathcal{M}_{1,1}$ regarded as a coheren sheaf. There is then from the preorientation of the universal curve over $(\mathcal{M}, \mathcal{O}')$ a morphism

   $$
     \beta : \omega \to \pi_2 \mathcal{O}'
   $$

1. let $\mathcal{O}$ be the sheaf obtained from $\mathcal{O}'$ by inverting $\beta$

1. show that

   1. for $n = 2 k $ we have an isomorphism $\omega^k \to \pi_{2 k}\mathcal{O}$

   1. for $n = 2 k + 1$ we have an isomorphism $0 \to \pi_{2k +1}\mathcal{O}$

      strategy: reduce to neighbourhood of a point

1. notice that this implies the desired statement

=--

[[!redirects derived moduli stack of derived elliptic curves]]


A Survey of Elliptic Cohomology - towards a proof

<div class="rightHandSide toc">

[[!include higher algebra - contents]]

</div>

>**Abstract** This entry attempts to give an outline of a proof of Lurie's main theorem.

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

=--

Here are the entries on the previous sessions:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

***

# Towards a proof #
* automatic table of contents goes here
{:toc}

Recall the main theorem.

+-- {: .un_theorem }
###### Theorem
**(J. Lurie)**

For

* $A$ any [[E-∞ ring]]

* and $E(A)$ is the space of oriented [[derived elliptic curve]]s over $A$ (the realization of the topological category of elliptic curves over $A$).

There exists a derived Deligne-Mumford stack $M^{Der} = (M, O^{Der})$ such that we have an equivalence

$$
  Hom(Spec A, M^{Der})
  \simeq
  E(A)
$$

natural in $A$.

And $O^{Der}$ provides the lift of Goerss-Hopkins-Miller.

=--

##Preliminaries##

Recall that for $A$ an $E_\infty$-ring a [[derived elliptic curve]] $F$ is a commutative [[derived group scheme]] over $A$ such that $F_0$ over $\pi_0 A$ is an elliptic curve.

Denote by $E' (A)$ the space of preoriented (derived) elliptic curves (so equipped with a map $\mathbb{C} P^\infty \to F ( \mathrm{Spec} A )$. And $E(A)$ the space of oriented elliptic curves.

Note that a map $\mathrm{Spec} A \to M^{Der}$ is a map
$\mathrm{Spec} \pi_0 A \to M_{1,1}$ and a map of rings
$O (\mathrm{Spec} \pi_0 A) \to A$.

##Representability##

In his thesis, Lurie proves the following.

**Proposition.** Let $F$ be a functor from connective $E_\infty$-ring spectra to spaces s.t.

1. The restriction of $F$ to discrete rings is represented by a (classical) DM-stack $X$, i.e. $F(R) \simeq \mathrm{Nerv} X(R)$;

1. $F$ is a sheaf with respect to the etale topology;

1. $F$ has a good deformation theory.

Then there exists a derived DM-stack $(X , \hat O )$ representing $F$ s.t. $\hat O (U)$ is connective for $U$ affine.

**Examples.**

1. The functor $E'$: observe that every classical elliptic curve over a discrete $R$ has a unique preorientation.  Hence $E'$ is represented by DM-stack $(M, O' )$.

1. The functor $E$: the theorem doesn't apply as a discrete ring cannot be weakly periodic.

**Claim.** $(M, O')$ represents $E'$ for all $E_\infty$-rings, so we dropped connectivity.

_Proof._ Recall the map $A \to \tau_{\ge 0} A$ to the connected cover.

1. Let $A$ be an $E_\infty$-ring, then we have an equivalence

$$\mathrm{Hom} (O' (\mathrm{Spec} \pi_0 \tau_{\ge 0} A) , \tau_{\ge 0} A ) \to \mathrm{Hom} (O' (\mathrm{Spec} \pi_0 A) , A).$$

1. $E' ( \tau_{\ge 0} A \simeq E' (A)$ since every elliptic curve is by definition flat.

We need the following to prove the claim.

**Proposition.** The functor $M \mapsto M \otimes_{\tau_{\ge 0} A} A,$ from flat modules over $\tau_{\ge 0} A$ to flat modules over $A$ is an equivalence.

_Proof of proposition (sketch)._
Let $M,N$ be $A$-modules then there is a spectral sequence

$$\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M, \pi_* N )_q \Rightarrow \pi_{p+q} (M \otimes_A N ) .$$

Suppose $M$ is flat, then

$$\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M , \pi_* N) = \pi_* N \otimes_{\pi_0 A} \pi_0 M$$

if $p=0$ and 0 otherwise.  Thus,

$$\pi_* (M \otimes_A N ) \simeq \pi_* N \otimes_{\pi_0 A} \pi_0 M .$$

So we have

$$ F: \mathrm{Mod}^{flat} (\tau_{\ge 0} A ) \leftrightarrow \mathrm{Mod}^{flat} (A) : G .$$

This is an equivalence and $F$ respects the monoidal structure, hence the equivalence extends to the categories of algebras and the proposition (and hence claim) is proved.

We need to know that $\pi_i O'$ are coherent sheaves over $M_{1,1}$, where a [[coherent sheaf]] is an assignment $\mathrm{Spec} R \to M_{1,1}$ which behaves well under finite limits.  Let $\omega$ be the line bundle of invariant differentials on $M_{1,1}$ so that is

$$ \omega = e^* \Omega_{F |_{\mathrm{Spec} R}}. $$

Recall that a preorientation determines a map $\beta : \omega \to \pi_2 (O' )$.  So define a sheaf of $E_\infty$-rings $O$ as $O' [ \beta^{-1} ]$ which is characterized (maybe) by

$$ \pi_n O = \lim ( \pi_{n+2k} O' \otimes_{O_{1,1}} \omega^{-k} ) .$$

**Remark.**

* This formula comes from a much simpler situation...Let $R$ be (an honest to God) ring, $x \in R$ then

$$ R[x^{-1}] \simeq \lim (R \stackrel{x}{\to} R  \stackrel{x}{\to} \dots )$$

* Suppose $U \to M_{1,1}$ with $U$ affine, $\omega$ restricted to $U$ is trivialized then $O' (U) =A$ and $\beta \in \pi_2 A$.  Hence, $O' [\beta^{-1} ] (U) = A [ \beta^{-1} ]$ and we get the formula above $\pi_i (A [ \beta^{-1} ]) = (\pi_i A) [\beta^{-1} ]$.

More generally, for $U \to M_{1,1}$ we have that $O(U)$ is weakly periodic, so

$$ \pi_2 O (U) \otimes_{\pi_0 O(U)} \pi_n O(U) \to \pi_{n+2} O (U) $$

is an equivalence.

*  $(M, O)$ classifies oriented elliptic curve.  Let $F$ over $U = \mathrm{Spec} B$ be a preoriented elliptic curve over $B$, so we have the classifying map $f: O' (U) = A \to B$ and this is an orientation iff

$$ \pi_n B \otimes_{\pi_0 B} \omega \to \pi_{n+2} B $$

is an isomorphism.  This map can be identified with $\times \beta$, so the preorientation is an orientation iff there is a unique factorization through $O(U) = A [ \beta^{-1} ]$.

###Reduction###

**Claim.** To prove the theorem it is enough to show

1. $O_{1,1} = \pi_0 O' \to \pi_0 O$ is an isomorphism;

1. For $n$ odd, the sheaf $\pi_n O =0$.

_Proof._  Suppose (1) and (2) hold.  Let $f: \mathrm{Spec} R \to M_{1,1}$ be etale for $R$ discrete.  We must show that $O (\mathrm{Spec} R)$ is an elliptic cohomology theory associated to $f$. Condition (1) ensures $\pi_0 A \simeq R$, (2) guarantees evenness and from above we have weakly periodic.  We must show that $\mathrm{Spf} A^{0} (\mathbb{C}P^\infty ) \simeq \hat E_f$ which follows from having an orientation.

###Reducing to a Local Calculation###
We wish to show that $\pi_n O = 0$ for $n$ odd.  From above, it suffices to show that

$$ f_k : \pi_{n+2k} O' \otimes \omega^{-k} \to \pi_n O $$

is zero for all $k$. Note that $\mathrm{im} (f_k )$ is a quotient of $\pi_{n+2k} O' \to \mathrm{im} (f_k )$ which is coherent.  Suppose $p$ is an etale cover of $M_{1,1}$ then
$\mathrm{im} (f_k ) = 0$ iff $p^{i} \mathrm{im} (f_k )=0$. We can find an etale cover by a disjoint union of level 3 and level 4 modular forms denoted $\mathrm{Spec} R$.

That $M:= p^{i} \mathrm{im} (f_k ) =0$ is equivalent to $M \otimes_R R/m$ for all $m \in R$.  It is not difficult to show that all residue fields $R/m$ are finite in this case.

Now it is enough to show condition (2) formally locally as $\hat R_m /m \simeq R_m / m$.

##Key Ingredients##
In the previous section we had a moduli stack preoriented elliptic curves $(M,O')$.  The structure sheaf of $O'$ took values in _connected_ $E_\infty$-rings.  We had a refinement $(M, O)$ which was a moduli stack for oriented elliptic curves.  From the orientation condition we deduced that the structure sheaf took values in _weakly periodic_ $E_\infty$-rings.

Further, we showed how to reduce the main theorem to a (formal) local computation.  That is, we only need to consider $\hat R_m$, the completion of a ring localized at a maximal ideal.

We delay the completion of the proof until later, but now we introduce the key technical tool: $p$-divisible groups.

###$p$-divisible Groups###
Let $R$ be a complete, local ring (e.g. the $p$-adic integers $\mathbb{Z}_p$) and $E_0$ an [[elliptic curve]] over $R_0 := R/M = \mathbf{F}_q$.  What do we need in order to lift $E_0$ to an elliptic curve over $R$?

Let $p$ be a prime (say the characteristic of $\mathbf{F}_q$) and $E$ an [[elliptic curve]] over $R$.  Using the _multiplication by_ $p$ map $p^n \colon E \to E$ we can define a sheaf of Abelian groups
$$ E[p^\infty] := \mathrm{colim} \; E[p^n],$$
where $E[p^n]$ is the kernel of the map $p^n$.  That is, $E[p^n]$ corresponds to the $p$-torsion points of $E$.

**Definition.** A $p$-divisible group $\mathfrak{I}$ over $R$ is a sheaf of Abelian groups on the flat site of schemes over $R$ such that

1. $p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;

1. $\mathfrak{I} = \mathrm{colim} \; \mathfrak{I} [p^n]$ where $\mathfrak{I} = \mathrm{ker} \; (p^n \colon \mathfrak{I} \to \mathfrak{I})$.

1. $\mathfrak{I}[p^n]$ is a finite, flat, commutative $R$-group scheme.  Note that finite means that $\mathfrak{I}$ is affine and whose global sections is a finite $R$-module.

For instance, the constant sheaf $\underline{\mathbb{Z}_p}$ is a $p$-divisible group.

###Serre-Tate Theory###

Now let $R$, in addition to above, be Noetherian with residue field $\mathbf{F}_q$ for $q=p^n$.

**Theorem (Serre-Tate).**
Let $\overline{E}$ be an elliptic curve over $\mathbf{F}_q$, then there is an equivalence of categories between elliptic curves over $R$ that restrict to $\overline{E}$ and the category of $p$-divisible groups $\mathfrak{I}$ over $R$ such that the restriction of $\mathfrak{I}$ to $\mathbf{F}_q$ is $\overline{E} [p^\infty ]$.

The theorem is somewhat surprising as, _a priori_, the latter category sees only torsion phenomena of the elliptic curves.

###Derived Serre-Tate Theory###

**Definition.** Let $A$ be an $E_\infty$-ring.  A functor $\mathfrak{I}$ from commutative $A$-algebras to topological Abelian groups is a $p$-divisible group if

1. $B \mapsto \mathfrak{I} (B)$ is a sheaf;

1. $p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;

1. $(\mathrm{ho})\mathrm{colim} \; \mathfrak{I} [p^n] \simeq \mathfrak{I}$, where as above $\mathfrak{I} [p^n] := (\mathrm{ho}) \mathrm{ker} \; p^n$;

1. $\mathfrak{I}[p^n]$ is a derived commutative group scheme over $A$ which is finite and flat.

If $\mathfrak{I}$ is a $p$-divisible group over $\mathbf{F}_q$, then $\mathfrak{I} [p]$ is a finite $\mathbf{F}_q$-module of dimension $r$ called the _rank_ of $\mathfrak{I}$.

**Proposition.** If $\mathfrak{I} = E [p^\infty]$, for $E$ an elliptic curve, then $\mathfrak{I}$ has rank 2.

One can verify the proposition over $\mathbb{C}$ pretty easily; it is more subtle over a finite field.  We have a derived version of the Serre-Tate theorem.

**Theorem (Serre-Tate).** Let $A$ be an $E_\infty$-ring such that $\pi_0 A$ is a complete, local, Noetherian ring and $\pi_i A$ are finitely generated $\pi_0 A$-modules.  Let $E_0$ be a (derived) elliptic curve over $\pi_0 A / M$, then there is an equivalence of $\infty$-categories: elliptic curves over $A$ that restrict to $E_0$ and $p$-divisible groups over $A$ that restrict to $E_0 [p^\infty]$.

###Proof of the Classical Theorem###
We give a proof of the classical result, however the proof is quite formal and should carry over to the derived setting.

Let $R$ be a ring as in the theorem such that $N \cdot R =0$ for some $N \in \mathbb{N}$ and let $I \subset R$ be a nilpotent ideal, so $I^{r+1} =0$, and set $R_0 = R/I$.  Let $\mathfrak{I} \colon R-\mathrm{alg} \to \mathrm{Ab}$ and
$$ \mathfrak{I}_I (A) := \mathrm{ker} \; ( \mathfrak{I} (A) \to \mathfrak{I} (A / I \cdot A ) .$$
Further, define
$$ \overline{\mathfrak{I}} (A) := \mathrm{ker} \; (\mathfrak{I}(A) \to \mathfrak{I} (A / \mathrm{nil}R \cdot A)) \subset \mathfrak{I}_I (A) .$$

**Lemma.** If $\mathfrak{I}$ is a formal group over $R$, then $\mathfrak{I}_I$ is annihilated by $N^r$.

**Proposition.** Let $\mathfrak{I}$, $\mathfrak{H}$ be elliptic curves or $p$-divisible groups over $R$ and let $N=p^n$.  Denote by $\mathfrak{I}_0$ and $\mathfrak{H}_0$ the restriction of $\mathfrak{I}$ and $\mathfrak{H}$ to $R_0$-algebras.  Then

1. $\mathrm{Hom}_{R-grp} (\mathfrak{I}, \mathfrak{H})$ and $\mathrm{Hom}_{R_0-grp} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ have no $N$-torsion;

1. $\mathrm{Hom} (\mathfrak{I} , \mathfrak{H}) \to \mathrm{Hom} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ is injective;

1. For $f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0$ there is a unique homomorphism $N^r f \colon \mathfrak{I} \to \mathfrak{H}$ which lifts $N^r \cdot f_0$;

1. $f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0$ lifts to $f \colon \mathfrak{I} \to \mathfrak{H}$ if and only if $N^r f$ annihilates $\mathfrak{I}[N^r] \subset \mathfrak{I}$.

Note that if $E$ is an elliptic curve over $R$ then the $E_0$ above is given by
$$E_0 = E \times_{\mathrm{Spec} \; R} \mathrm{Spec} \; R/I .$$

Using the previous results one can prove the following alternate version of the Serre-Tate theorem.

**Theorem (alternative Serre-Tate).**
Let $R$ be a ring with $p$ nilpotent and $I \subset R$ a nilpotent ideal.  Let $R_0 = R/I$.  We have a categorical equivalence: elliptic curves over $R$ and the category of triples $\{ (E, E_0 [p^\infty] , \epsilon )\}$; where $E$ is an elliptic curve over $R_0$, $E_0 [p^\infty]$ is a $p$-divisble group over $R$, and $\epsilon \colon E_0 [p^\infty] \to E[p^\infty]_0$ is a natural isomorphism.

###Adding in Completions###
We really want to consider elliptic curves over $R$ completed by an ideal $m$, this is the $\hat{R}_m$ from far above.  We can reduce this problem to that of elliptic curves over $R$ and a system of $p$-divisible groups over $R/m^n$ by combining the Serre-Tate theorem and the following theorem of Grothendieck.

**Theorem (Formal GAGA).** Let $X, Y$ be exceedingly nice schemes over $R$ and $\hat X$, $\hat Y$ be their formal completions, then there is a bijection
$$\mathrm{Hom}_{R} (X,Y) \leftrightarrow \mathrm{Hom}_{\hat R} (\hat X, \hat Y).$$

##Deformation Theory##
Fix a morphism $\mathrm{Spec} \; \mathbf{F}_q \to M_{1,1}$, that is an [[elliptic curve]] $E_0 /k$.  Let $\mathbf{O}_k$ be the sheaf over $M_{1,1}$ that classifies deformations of $E_0$ to oriented elliptic curves over $A$ where $A$ is an $E_\infty$-ring with $\pi_0 A$ a complete, local ring.

Let us assume for the moment that $M_{1,1} = \mathrm{Spec} \; R$ (more generally we pass to an affine cover).  One can show that $\mathbf{O}_k$ moreover classifies oriented $p$-divisible groups which deform $E_0 [p^\infty]$.

###Deformations of FGLs and Lubin-Tate Spectra###
Recall that for $F,G$ [[formal group law]]s over $R$ a morphism is $f \in R [ [x] ]$ with $f(0) =0$ such that $f(X +_F Y) = f(X) +_G f(Y)$.  Now, let $k$ be a field of characteristic $p$ and $F,G$ formal group laws over $k$.  Let $f: F \to G$ be a morphism, then
$$f = x^{p^n} + \dots$$
where $n$ is the [[height]] of $f$.  For $F$ a formal group law the height of $F$, $\mathrm{ht} \; F$, is the height of $[p]F$, that is the multiplication by $p$ map.

Let $\Gamma$ be a formal group law over $k$, then a deformation of $F$ consists of a complete local ring $B$ ($B/M = k$) and $F$ a formal group law over $B$ such that $p_* F = \Gamma$, where $p: B \to B/M$ is the canonical surjection. To such deformations $F_1 , F_2$ are isomorphic if there is an isomorphism $f: F_1 \to F_2$ which induces the identity on $p_* F_1 = p_* F_2 = \Gamma$.

**Theorem (Lubin-Tate).** Let $k, \Gamma$ as above with $\mathrm{ht} \; \Gamma \lt \infty$ then there exists a complete local ring $E(k, \Gamma )$ with residue field $k$ and a formal group law over $E(k, \Gamma)$, $F^{univ}$ which reduces to $\Gamma$ such that there is a bijection of sets
$$\mathrm{Hom} (E(k, \Gamma) , R) \to \mathrm{Deform} (\Gamma, R) .$$

If $k = \mathbf{F}_p$, then $E(k, \Gamma) \simeq \mathbb{Z}_p [ [ u_1 , \ldots , u_{n-1} ] ]$ where $\mathrm{ht} \; \Gamma =n$.  More generally, if $k= \mathbf{F}_q$, then $E(k, \Gamma) \simeq W\mathbf{F}_q [ [ u_1 , \ldots , u_{n-1} ] ]$, that is, the Witt ring.

Let $E(k, \Gamma), F^{univ}$ as in the theorem, then we define $\overline{F}^{univ}$ over $E(k, \Gamma) [u^\pm]$, degree $u=2$, by
$ \overline{F}^{univ} (X,Y) = u^{-1} F^{univ} (uX , uY). $  We then define a homology theory as
$$( E_{k,\Gamma})_* (X) = E(K,\Gamma)[ [u^\pm] ] \otimes_{MU_*} MU_* (X).$$

Define a category (really a stack and let $p$ and $n$ vary) $\mathbf{FG}$ of pairs $(k,\Gamma)$ where $\mathrm{char} \; k =p$ and $\mathrm{ht} \; \Gamma =n$.  By associating to any such pair its Lubin-Tate theory we get a functor from $\mathbf{FG}$ to multiplicative cohomology theories.

**Theorem (Hopkins-Miller I).** This functor lifts to $E_\infty$-rings.

**The philosophy is that there should be a sheaf of $E_\infty$-rings on the stack of formal groups $\mathbf{FG}$ with global sections the sphere spectrum.  Then tmf and taf are low height approximations.**

Let $E_{\infty}^{LT}$ be the subcategory of $E_\infty$-rings such that the associated cohomology theory is isomorphic to some $E_{k,\Gamma}$.

**Theorem (Hopkins-Miller II).** $\pi : E_{\infty}^{LT} \to \mathbf{FG}$ is a weak equivalence of topological categories.  That is, the lift above is pretty unique.

This implies Hopkins-Miller I by taking a Kan extension $\mathbf{FG} \to E_\infty$ along the inclusion $E_{\infty}^{LT} \hookrightarrow E_\infty$.

##The Final Sprint##
Let us sketch how to proceed...Let $\mathfrak{I}$ be a $p$-divisible group over $A$ with $\pi_0 A$ complete and local.  Then $\mathfrak{I}$ fits in an exact sequence
$$ 0 \to \hat \mathfrak{I} \to \mathfrak{I} \to \mathfrak{I}_{et} \to 0 .$$
There are two cases: either the underlying elliptic curve $E_0$ is super singular ($\mathrm{ht} \; E_0 =2$), else $E_0$ is ordinary.

###The Supersingular Case###
In this case $\widehat{E_0} [p^\infty] = E_0 [p^\infty]$, so one can show $\widehat{\mathfrak{I}} = \mathfrak{I}$ for every deformation.  Now an orientation means
$$\widehat{\mathfrak{I}} \simeq \mathrm{Spf} \; A^{\mathbb{C}P^\infty}[p^\infty],$$
so $E_{k, \widehat{E_0}}$ classifies deformations to oriented $p$-divisible groups, hence $\mathbf{O}_k \simeq E_{k, \widehat{E_0}}$. By construction $E_{k,\widehat{E_0}}$ is even.

###The Ordinary Case###
This is more subtle (see DAG IV 3.4.1 for some hints).

First we can analyze the formal part of the exact sequence and see that deformations to formal $p$-divisible groups of height 1 are classified by $E_{k, \widehat{E_0}} [ p^\infty]$.  By analyzing extensions etale $k$-algebras we see that $\mathbf{O}_k \simeq E_{k,\widehat{E_0}}^{\mathbb{C}P^\infty}$.  The homotopy groups are then $E_{k , \widehat{E_0}} [ [t] ]$ and are odd as the degree of $t$ is 0.


A-brane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[branes]]  of the [[A-model]] [[topological string]], hence a [[TQFT]]-shadow of the [[D-branes]] of the [[superstring]]. Form the [[Fukaya category]] of [[target space|target]] [[spacetime]].

## Related concepts

* [[2d TQFT]], [[TCFT]]

* [[Kapustin-Witten TQFT]]

* [[B-brane]], [[homological mirror symmetry]]

[[!include table of branes]]

## References

* [[Anton Kapustin]], [[Dmitri Orlov]], _Remarks on A-branes, Mirror Symmetry, and the Fukaya category_, J. Geom. Phys. 48 (2003), no. 1, 84--99 ([arXiv:hep-th/0109098](http://arxiv.org/abs/hep-th/0109098))

[[!redirects A-branes]]


A-hat genus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### In terms of an operator index

For $X$ a [[smooth manifold]] of even [[dimension]] and with [[spin structure]], write $\mathcal{S}(X)$ for the [[spin bundle]] and

$$
  \mathcal{S}(X) \simeq \mathcal{S}^+(X) \oplus \mathcal{S}^-(X)
$$

for its decomposition into [[chiral spinor]] bundles. For $(X,g)$ the [[Riemannian manifold]] structure and $\nabla$ the corresponding [[Levi-Civita connection|Levi-Civita]] [[spin connection]] consider the map

$$
  c \circ \nabla
  \;\colon\;
  \Gamma(\mathcal{S}^+(X))
  \to
  \Gamma(\mathcal{S}^-(X))
$$

given by [[composition|composing]] the action of the [[covariant derivative]] on [[sections]] with the [[symbol map]]. This is an [[elliptic operator]]. The [[index of a Dirac operator|index]] of this operator is called the **$\hat A$-[[genus]]**.

### In terms of the universal $Spin$-orientation of $KO$

More abstractly, there is the universal [[orientation in generalized cohomology]] of [[KO]] over [[spin structure]], known as the [[Atiyah-Bott-Shapiro orientation]], which is a [[homomorphism]] of [[E-∞ rings]] of the form

$$
  M Spin \longrightarrow KO
$$

from the universal [[spin structure]] [[Thom spectrum]]. The $\hat A$-genus

$$
  \Omega_\bullet^{SO}\longrightarrow \pi_\bullet(KO)\otimes \mathbb{Q}
$$

is the corresponding homomorphism in [[homotopy groups]].

## Properties

### Characteristic series

The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the $\hat A$-genus is

$$
  \begin{aligned}
    K_{\hat A}(e)
     & =
    \frac{z}{e^{z/2} - e^{-z/2}}
    \\
     &=
    \exp\left(
       - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!}
     \right)
  \end{aligned}
  \,,
$$

where $B_k$ is the $k$th [[Bernoulli number]] ([Ando-Hopkins-Rezk 10, prop. 10.2](#AndoHopkinsRezk10)).

### Relation to the Todd genus
  {#RelationToTheToddGenus}

On an [[almost complex manifold]] $M_{\mathrm{U}}$, the [[Todd class]] coincides with the [[A-hat genus|A-hat class]] up to the [[exponential]] of half the [[first Chern class]]:

$$
  Td(M_{\mathrm{U}})
   \;=\;
  \big(e^{c_1/2} \hat A\big)(M_{\mathrm{U}})
  \,.
$$

(e.g. [Freed 87 (1.1.14)](#Freed87)).

In particular, on manifolds $M_{S\mathrm{U}}$ with [[special unitary group|SU]]-[[G-structure|structure]], where $c_1 = 0$, the Todd class is actually equal to the [[A-hat genus|A-hat class]]:

$$
  Td(M_{S\mathrm{U}})
   \;=\;
  \hat A(M_{S\mathrm{U}})
  \,.
$$

\linebreak

Given the [[complexification]] of a  [[real vector bundle]] $\mathcal{X}$ to a [[complex vector bundle]] $\mathcal{E} \otimes \mathbb{C}$, the $\hat A$-class of $\mathcal{E}$ is the [[square root]] of the [[Todd class]] of $\mathcal{E} \otimes \mathbb{C}$ (e.g. [de Lima 03, Prop. 7.2.3](#deLima03)).

### As a Rozansky-Witten invariant

+-- {: .num_prop #RozanskyWittenWilsonLoopOfUnknotIsSquareRootOfAHat}
###### Proposition
**([[Rozansky-Witten Wilson loop of unknot is A-hat genus|Rozansky-Witten Wilson loop of unknot is square root of A-hat genus]])**

For $\mathcal{M}^{4n}$ a [[hyperkähler manifold]] (or just a [[holomorphic symplectic manifold]]) the [[Rozansky-Witten invariant]] [[Wilson loop observable]] associated with the [[unknot]] in the [[3-sphere]] is the [[square root]] $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the [[A-hat genus]] of $\mathcal{M}^{4n}$.

=--

This is [Roberts-Willerton 10, Lemma 8.6](Rozansky-Witten+Wilson+loop+of+unknot+is+A-hat+genus#RobertsWillerton10), using the [[Wheels theorem]] and the [[Hitchin-Sawon theorem]].

## Related entries

* [[Todd genus]]

* [[index of an elliptic complex]]

[[!include genera and partition functions - table]]

## References

Review:

* {#Freed87} [[Daniel Freed]], Sections 1.1, 1.2 of: _Geometry of Dirac operators_, 1987 ([pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]])

The $\hat A$-genus as the index of the spin complex is discussed for instance in:

* {#Gilkey} [[Peter Gilkey]], Section 3 of: _The Atiyah-Singer Index Theorem -- Chapter 5_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/gilkey3.pdf))

* {#deLima03} Levi Lopes de Lima, _The Index Formula for Dirac operators: an Introduction_, 2003 ([pdf](https://impa.br/wp-content/uploads/2017/04/PM_10.pdf))

The relation of the characteristic series to the [[Bernoulli numbers]] is made explicit for instance in prop. 10.2 of

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

A construction via a [[1-dimensional Chern-Simons theory]] is in

* [[Owen Gwilliam]], [[Ryan Grady]], _One-dimensional Chern-Simons theory and the $\hat A$-genus_ ([arXiv:1110.3533](http://arxiv.org/abs/1110.3533))

[[!redirects A-hat genera]]


A-infinity algebra > history

< [[A-infinity algebra]]
[[!redirects A-infinity algebra -- history]]


A-infinity category > history

< [[A-infinity category]]
[[!redirects A-infinity category -- history]]


A-infinity operad

# The $A_\infty$ operad
* table of contents
{: toc}

##Idea

An __$A_\infty$ operad__ is an [[operad]] over some [[enriched category theory|enriching category]] $C$ which is a (free) [[homological resolution|resolution]] of the standard [[associative operad]] enriched over $C$ (that is, the operad whose algebras are [[monoids]]).

Important examples, to be discussed below, include:

* The topological operad of Stasheff associahedra.

* The little $1$-cubes operad.

* The standard dg-$A_\infty$ operad.

* The standard categorical $A_\infty$ operad.

An $A_\infty$ operad, like the standard associative operad, can be defined to be either a _symmetric_ or a _non-symmetric_ operad.  On this page we assume the non-symmetric version.  When regarded as a symmetric operad, an $A_\infty$ operad may also be called an $E_1$ operad.

An [[algebra over an operad]] over an $A_\infty$ operad is called an __$A_\infty$-object__ or [[A-∞ algebra]], where -object is often replaced with an appropriate noun; thus we have the notions of **$A_\infty$-[[A-infinity-space|space]]**, **$A_\infty$-[[A-infinity-algebra|algebra]]**, and so on.  In general, $A_\infty$-objects can be regarded as 'monoids up to coherent homotopy.'  Likewise, a [[category over an operad|category]] over an $A_\infty$ operad is called an $A_\infty$-[[A-infinity-category|category]].

Some authors use the term '$A_\infty$ operad' only for a _particular_ chosen $A_\infty$ operad in their chosen ambient category, and thus use '$A_\infty$-object' and '$A_\infty$-category' for algebras and categories over this particular operad.  The $A_\infty$ operads discussed below are common choices for this 'standard' $A_\infty$ operad.

## The topological Stasheff associahedra operad

### Definition

Let $\{K(n)\}$ be the sequence of [[associahedron|Stasheff associahedra]].  This is naturally equipped with the structure of a (non-symmetric) operad $K$ enriched over [[Top]] called the **topological Stasheff associahedra operad** or simply the **Stasheff operad**.  Since each $K(n)$ is [[contractible space|contractible]], $K$ is an $A_\infty$ operad.

The original article that defines associahedra, and in which the operad $K$ is used implicitly to define $A_\infty$-[[topological space]]s, is ([Stasheff](#Stasheff)).

A textbook discussion (slightly modified) is in [MarklShniderStasheff, section 1.6](#MarklShniderStasheff)

### Properties

Stasheff's $A_\infty$-operad is the relative [[Boardman-Vogt resolution]] $W([0,1], I_* \to Assoc)$ where $I_*$ is the operad for [[pointed object]]s [BergerMoerdijk](#BergerMoerdijk).

## The little $1$-cubes operad

Let $\mathcal{C}_1(n)$ denote the configuration space of $n$ disjoint intervals linearly embedded in $[0,1]$.  Substitution gives the sequence $\{\mathcal{C}_1(n)\}$ an operad structure, called the **little 1-cubes operad**; it is again an $A_\infty$ operad.  This is a special case of the [[little n-cubes operad]] $\mathcal{C}_n$, which is in general an $E_n$ [[E-n operad|operad]].

The little $n$-cubes operads (in their symmetric version) were among the first operads to be explicitly defined, in the book that first explicitly defined operads: [The geometry of iterated loop spaces](#May).

##The standard dg-$A_\infty$ operad

The **standard dg-$A_\infty$ operad** is the dg-operad (that is, operad enriched in [[cochain complex]]es $Ch^\bullet(Vect)$)

* freely generated from one $n$-ary operation $f_n$ for each $n \geq 1$, taken to be in degree
$2 - n$;

* with the differential of the $n$th generator given by
$$ -
\sum_{j+p+q = n}^{1 \lt p \lt n} (-1)^{j p + q} a_{p,j,n} ,$$
where $a_{p,j,n}$ is $f_p$ attached to the $(j+1)$st input of $f_{n-p+1}$.

This can be shown to be a standard free resolution of the linear associative operad in the context of dg-operads; see [Markl 94](http://arxiv.org/abs/hep-th/9411208), [proposition 3.3](http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=13); therefore it is an $A_\infty$ operad.

It can also be shown to be _isomorphic_ to the operad of top-dimensional (cellular) chains on the topological Stsheff associahedra operad.  This is discussed on [pages 26-27](http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=26) of [Markl 94](http://arxiv.org/abs/hep-th/9411208)

In the dg-context it is especially common to say '$A_\infty$-algebra' and '$A_\infty$-category' to mean specifically algebras and categories over _this_ operad.  The explicit description of this operad given above means that such $A_\infty$-algebras and categories can be given a fairly direct description without explicit reference to operads.

In addition to

* Martin Markl, _Models for operads_ ([arXiv](http://arxiv.org/abs/hep-th/9411208))

another reference is section 1.18 of

* Yu. Bespalov, V. Lyubashenko, O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))

A relation of the linear dg-$A_\infty$ operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.

## The standard categorical $A_\infty$ operad

Let $O$ be the operad in [[Set]] freely generated by a single binary operation and a single nullary operation.  Thus, the elements of $O(n)$ are ways to associate, and add units to, a product of $n$ things.  Let $B(n)$ be the [[indiscrete category]] on the set $O(n)$; then $B$ is an $A_\infty$ operad in [[Cat]].  $B$-algebras are precisely (non-strict, biased) [[monoidal category|monoidal categories]], and $B$-categories are precisely (biased) [[bicategory|bicategories]].

If instead of $O$ we use the $Set$-operad freely generated by a single $n$-ary operation for every $n$, we obtain a $Cat$-operad whose algebras and categories are _unbiased_ monoidal categories and bicategories.

## Related concepts

* **$A_\infty$-operad**

* [[E-∞ operad]]

* [[L-∞ operad]]

## References

* [[Jim Stasheff]], _Homotopy associative H-spaces I_, _II_, Trans. Amer. Math. Soc. 108 (1963), 275-312

* [[Peter May]], _The Geometry of Iterated Loop Spaces_, Springer 1972 ([doi:10.1007/BFb0067491](https://doi.org/10.1007/BFb0067491), [pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))

* [[Martin Markl]], Steve Shnider, [[Jim Stasheff]], _Operads in Algebra, Topology and Physics_ ([web](http://books.google.de/books?id=fMhZjT9lQo0C&pg=PA56&lpg=PA56&dq=Stasheff+associahedra&source=bl&ots=ZuGXjT4zbp&sig=V-taGG2LHS0msHK-PTxmUXXCvEY&hl=de#PPP1,M1))

* [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))
{#BergerMoerdijk}

[[!redirects A-infinity-operad]]

[[!redirects A-∞ operad]]
[[!redirects A-∞-operad]]

[[!redirects A-∞ operads]]
[[!redirects A-∞-operads]]

[[!redirects E1-operad]]
[[!redirects E1-operads]]


A-infinity operad > history

< [[A-infinity operad]]
[[!redirects A-infinity operad -- history]]


A-infinity-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _$A_\infty$-algebra_ is a [[monoid]] internal to a [[homotopical category]] such that the [[associativity]] law holds not as an equation, but only up to higher [[coherent]] [[homotopy]].

## Definition

+-- {: .num_defn}
###### Definition

An **$A_\infty$-algebra** is an [[algebra over an operad]] over an [[A-∞ operad]].

=--

## Realizations

### In chain complexes

Let here $\mathcal{E}$ be the [[category of chain complexes]]
$\mathcal{Ch}_\bullet$. Notice that often in the literature this choice of $\mathcal{E}$ is regarded as default and silently assumed.

An $A_\infty$-algebra in chain complexes is concretely the following data.

A chain $A_\infty$-algebra is the structure of a degree 1 [[coderivation]]

$$
  D : T^c V \to T^c V
$$

on the reduced tensor coalgebra $T^c V = \oplus_{n\geq 1} V^{\otimes n}$ (with the standard noncocommutative coproduct, see [[differential graded Hopf algebra]]) over a [[graded vector space]] $V$ such that
$$
  D^2 = 0
  \,.
$$

Coderivations on free coalgebras are entirely determined by their "value on cogenerators", which allows one to decompose $D$ as a sum:

$$
  D = D_1 + D_2 + D_3 + \cdots
$$

with each $D_k$ specified entirely by its action

$$
  D_k : V^{\otimes k} \to V
  \,.
$$
which is a map of degree $2-k$ (or can be alternatively understood as a map $D_k : (V[1])^{\otimes k}\to V[1]$ of degree $1$).

Then:

* $D_1 : V\to V$ is the _differential_ with $D_1^2 = 0$;

* $D_2 : V^{\otimes 2} \to V$ is the _product_ in the algebra;

* $D_3 : V^{\otimes 3} \to V$ is the _associator_ which measures the failure of $D_2$ to be associative;

* $D_4 : V^{\otimes 4} \to V$ is the _pentagonator_ (or so) which measures the failure of $D_3$ to satisfy the pentagon identity;

* and so on.

One can also allow $D_0$, in which case one talks about weak $A_\infty$-algebras, which are less understood.

There is a resolution of the operad $\mathrm{Ass}$ of associative algebras (as operad on the category of chain complexes) which is called the $A_\infty$-operad; the algebras over the $A_\infty$-[[A-infinity operad|operad]] are precisely the $A_\infty$-algebras.

A **morphism of $A_\infty$-algebras** $f : A\to B$ is a collection $\lbrace f_n\rbrace_{n\geq 1}$ of maps
$$
f_n : (A[1])^{\otimes n}\to B[1]
$$
of degree $0$ satisfying
$$
\sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j})
= \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots \otimes f_{i_r}).
$$
For example, $f_1\circ D_1 = D_1\circ f_1$.

#### Rectification

+-- {: .num_theorem }
###### Theorem
**(Kadeishvili (1980), Merkulov (1999))**

If $A$ is a [[dg-algebra]], regarded as a strictly associative $A_\infty$-algebra, its [[chain homology and cohomology|chain cohomology]] $H^\bullet(A)$, regarded as a [[chain complex]] with trivial differentials, naturally carries the structure of an $A_\infty$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_\infty$-algebra.

=--

More details are at _[[Kadeishvili's theorem]]_.

+-- {: .num_remark }
###### Remark

This theorem provides a large supply of examples of $A_\infty$-algebras: there is a natural $A_\infty$-algebra structure on all cohomologies such as

* [[de Rham cohomology]]

* [[Hochschild cohomology]]

etc.

=--

### In Topological space

An $A_\infty$-algebra in [[Top]] is also called an _[[A-∞ space]]_ .

#### Examples

Every [[loop space]] is canonically an [[A-∞ space]]. (See there for details.)

#### Rectification

+-- {: .num_theorem }
###### Theorem

Every $A_\infty$-space is [[weak homotopy equivalence|weakly homotopy equivalent]] to a topological [[monoid]].

=--

This is a classical result by ([Stasheff 1963](#Stasheff63), [BoardmanVogt](#BoardmanVogt)). It follows also as a special case of the more general result on rectification in a [[model structure on algebras over an operad]] (see there).

### In spectra

See [[ring spectrum]] and [[algebra spectrum]].

## Related concepts

* **$A_\infty$-algebra**, [[A-∞-category]]

  * [[augmented A-∞ algebra]]

  * [[curved A-∞ algebra]]

* [[An-space]]

* [[A-n algebra]]

* [[E-∞ algebra]]

* [[L-∞ algebra]], .

[[!include k-monoidal table]]

[[!include deformation quantization - table]]

## References

A survey of $A_\infty$-algebras in chain complexes is in

* [[Bernhard Keller]], _A brief introduction to $A_\infty$-algebras_ ([pdf](http://people.math.jussieu.fr/~keller/publ/IntroAinfEdinb.pdf))

Classical articles on $A_\infty$-algebra in topological spaces are

* {#Stasheff63a} [[Jim Stasheff]], *Homotopy associativity of H-spaces I*, Trans. Amer. Math. Soc. **108** 2 (1963) 275-292 [[doi:10.2307/1993608](https://doi.org/10.2307/1993608)]

* {#Stasheff63b} [[Jim Stasheff]], *Homotopy associativity of H-spaces II* **108** 2 (1963) 293-312 [[doi:10.2307/1993609](https://doi.org/10.2307/1993609), [doi:10.1090/S0002-9947-1963-0158400-5](https://doi.org/10.1090/S0002-9947-1963-0158400-5)]

* [[Michael Boardman]] and [[Rainer Vogt]], _Homotopy invariant algebraic structures on topological spaces_ , Lect. Notes Math. 347 (1973).
{#BoardmanVogt}

A brief survey is in section 1.8 of

* [[Martin Markl]], Steve Shnider, [[Jim Stasheff|James D. Stasheff]], _Operads in algebra, topology and physics_, Math. Surveys and Monographs __96__, Amer. Math. Soc. 2002.

The 1986 thesis of [[Alain Prouté]] explores the possibility of obtaining analogues of [[minimal model]]s for $A_\infty$ algebras. It was published in TAC much later.

* [[Alain Prouté]], _Algèbres différentielles fortement homotopiquement associatives ($A_\infty$-algèbres)_, thesis, available as [Reprints in Theory and Applications of Categories, No. 21, 2011, pp. 1–99](http://www.logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf)

[[!redirects A-infinity-algebras]]
[[!redirects A-infinity algebra]]
[[!redirects A-infinity algebras]]
[[!redirects A-∞ algebra]]
[[!redirects A-∞ algebras]]
[[!redirects A? algebra]]
[[!redirects A? algebras]]
[[!redirects A-∞-algebra]]

[[!redirects A-∞-algebras]]


A-infinity-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _$A_\infty$-category_ is a kind of [[category]] in which the associativity condition on the [[composition]] of [[morphism]]s is relaxed "up to higher coherent homotopy".

The "A" is for Associative and the "${}_\infty$" indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.

In the most widespread use of the word $A_\infty$-categories are _linear_ categories in that they have [[hom-object]]s that are [[chain complex]]es. These are really models/presentations for [[stable (∞,1)-category|stable (∞,1)-categories]].

If higher coherences in a linear $A_\infty$-category happen to be equal to identity morphisms
(which is encoded by the vanishing of the maps $m_n$ for $n\ge3$, defined below), it becomes the same as a [[dg-category]]. In fact, every linear $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]]. In this way, we have that $A_\infty$-categories related to [[dg-category|dg-categories]] as models for [[stable (∞,1)-category|stable (∞,1)-categories]] in roughly the same way as [[quasi-category|quasi-categories]] relate to [[simplicially enriched category|simplicially enriched categories]] as models for [[(∞,1)-category|(∞,1)-categories]]: the former is the general incarnation, while the latter is a [[semi-strict infinity-category|semi-strictified]] version.

### Ordinary linear $A_\infty$-categories

In what is strictly speaking a restrictive sense -- which is however widely and conventionally understood in [[homological algebra]] as the standard notion of $A_\infty$-category (see references below) -- the [[hom-space]]s of an $A_\infty$-category are taken to be linear spaces, i.e. [[module]]s over some [[ring]] or [[field]], and in fact [[chain complex]]es of such modules.

Therefore an $A_\infty$-category in this standard sense of [[homological algebra]] is a category which is in some way [[homotopical enrichment|homotopically enriched]] over a [[category of chain complexes]] $Ch$. Since a category which is [[enriched category|enriched]] in the ordinary sense of [[enriched category theory]] is a [[dg-category]], there is a close relation between $A_\infty$-categories and [[dg-category|dg-categories]].

$A_\infty$-categories in this linear sense are a [[horizontal categorification]] of the notion of [[A-infinity-algebra]]. As such they are to [[A-infinity-algebra]]s as [[Lie infinity-algebroid]]s are to [[L-infinity-algebra]]s. For this point of view see [Konsevich-Soibelman 08](#KonsevichSoibelman08).

+-- {: .num_defn}
###### Definition

A category $C$ such that

1. for all $X,Y$ in $Ob(C)$ the [[hom-set|Hom-set]]s $Hom_C(X,Y)$ are finite dimensional [[chain complex]]es of $\mathbf{Z}$-graded modules

2. for all objects $X_1,...,X_n$ in $Ob(C)$ there is a family of linear composition maps (the higher compositions)
$m_n : Hom_C(X_0,X_1) \otimes Hom_C(X_1,X_2) \otimes \cdots \otimes Hom_C(X_{n-1},X_n) \to Hom_C(X_0,X_n)$ of degree $n-2$ (homological grading convention is used) for $n\geq1$

3. $m_1$ is the differential on the chain complex $Hom_C(X,Y)$

4. $m_n$ satisfy the quadratic $A_\infty$-associativity equation for all $n\geq0$.
=--

$m_1$ and $m_2$ will be [[chain complex|chain map]]s but the compositions $m_i$ of higher order are not chain maps, nevertheless they are [[Massey product]]s.

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of $A_\infty$-categories and $A_\infty$-functors. Many features of $A_\infty$-categories and $A_\infty$-functors come from the fact that they form a symmetric closed [[multicategory]], which is revealed in the language of [[comonad|comonads]].

From a higher dimensional perspective $A_\infty$-categories are weak $\omega$-categories with all morphisms invertible. $A_\infty$-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.

### Examples (and remarks) ##

* Every [[dg-category]] may be regarded as a special case when there are no higher maps (trivial homotopies) of an $A_\infty$-category.

* Every $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]].

  * This is a corollary of the $A_\infty$-categorical [[Yoneda lemma]].

  * beware that this statement does not imply that the notion of
  $A_\infty$-categories is obsolete (see section 1.8 in Bespalov et al.):
  in practice it is often easier to work with a given naturally
  arising $A_\infty$-category than constructing its equivalent [[dg-category]]

    * for instance when dealing with a [[Fukaya A-infinity-category|Fukaya]] $A_\infty$-category;

    * or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly $A_\infty$-categories instead of dg-categories.

* The [[path space ]] of a topological space $X$

* The [[Fukaya category]] $Fuk(X)$ of a topological space $X$ -- a [[Calabi-Yau A-∞ category]]

* $A_\infty$-[[A-infinity-algebra|algebras]] are the $A_\infty$-categories with one object.

  * For example, the delooping $\mathbf{B}\Omega{X}$ of [[loop space]] $\Omega{X}$ of a topological space $X$

### More general $A_\infty$-categories ##

In the widest sense, $A_\infty$-category may be used as a term for a category in which the [[composition]] operation constitutes an algebra over an [[operad]] which resolves in some sense the associative operad $Ass$.

One should be aware, though, that this use of the term is not understood by default in the large body of literature concerned with the above linear notion.

A less general but non-linear definition is fairly straight forward in any category in which there is a notion of _homotopy_ with the usual properties.
+-- {: .un_defn}
###### Definition

An $A_\infty$-category is a [[category over an operad|category over]]
the $A_\infty$-[[A-infinity operad|operad]]: e.g. the free resolution
in the context of dg-operads of the linear associative operad.
=--

+-- {: .num_example}
###### Examples

* [[Trimble n-category|Trimble n-categories]];

* also the classical notion of [[bicategory]] can be interpreted as an $A_\infty$-category in [[Cat]] for a suitable Cat-operad.
=--

## Related concepts

* [[A-∞ space]], [[associahedron]]

* [[A-∞ algebra]]

* [[stable (∞,1)-category]]

## References #

### For $A_\infty$-categories in the sense of homological algebra

For a short and precise introduction see

* B. Keller, _Introduction to $A_\infty$-algebras and modules_ (dvi, ps) and Addendum (ps), Homology, Homotopy and Applications 3 (2001), 1-35;

* B. Keller, _$A_\infty$ algebras, modules and functor categories_, (pdf, ps).

and for a [[Fukaya category]]-oriented introduction see chapter 1 in

* P. Seidel,  _Fukaya category and Picard-Lefschetz theory_

A very detailed treatment of $A_\infty$-categories is a recent book

* Yu. Bespalov, [[Volodymyr Lyubashenko]], O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))

  * notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are [here](http://www.math.ksu.edu/~lub/cmcMoCor.pdf).
* Oleksandr Manzyk, _A-infinity-bimodules and Serre A-infinity-functors_, dissertation [pdf](https://kluedo.ub.uni-kl.de/files/1910/dissertation.pdf), [djvu](https://kluedo.ub.uni-kl.de/files/1910/dissertation.djvu);  _Serre $A_\infty$ functors_, talk at Categories in geometry and math. physics, Split 2007, slides, [pdf](http://www.irb.hr/korisnici/zskoda/manzyukslides.pdf), work with [[Volodymyr Lyubashenko]]

The relation of $A_\infty$-categories to [[differential graded algebra]]s is emphasized in the introduction of

* {#KonsevichSoibelman08} [[Maxim Kontsevich]], [[Yan Soibelman]], _Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry. I_, in [[Anton Kapustin]], [[Maximilian Kreuzer]], [[Karl-Georg Schlesinger]] (eds.) _Homological mirror symmetry -- New developments and perspectives_, Springer 2008 ([arXiv:math/0606241](http://arxiv.org/abs/math/0606241), [doi:10.1007/978-3-540-68030-7](https://www.springer.com/de/book/9783540680291))

which mostly discusses just [[A-infinity-algebra]]s, but points out a generalizations to $A_\infty$-categories, see the overview on [p. 3](http://arxiv.org/PS_cache/math/pdf/0606/0606241v2.pdf#page=3)

Essentially the authors say that an $A_\infty$-category should be a non(-graded-)commutative [[dg-manifold]]/[[L-infinity-algebroid]].

More [[category theory]] and [[homotopy theory]] of $A_\infty$-categories is discussed in

* {#LefevreHasegawa03} Kenji Lefèvre-Hasegawa, _Sur les A-infini catégories_ ([arXiv:math/0310337](http://arxiv.org/abs/math/0310337))

* [[Bruno Valette]], _Homotopy theory of homotopy algebras_, Annales de l'Institut Fourier __70__:2 (2020)  683--738 [doi](https://10.5802/aif.3322) [Zbl:1335.18001](https://zbmath.org/?q=an:1335.18001) [arXiv:1411.5533](https://arxiv.org/abs/1411.5533)

See also

* [[Yong-Geun Oh]], [[Hiro Lee Tanaka]], _$A_\infty$-categories, their $\infty$-categories, and their localizations_ ([arXiv:2003.05806](https://arxiv.org/abs/2003.05806))

### For $A_\infty$-categories in the wider sense ##

If one understands $A_\infty$-category as "operadically defined higher category", then relevant references would include:

* Eugenia Cheng, _Comparing operadic definitions of $n$-category_ ([arXiv](http://arxiv.org/abs/0809.2070))

With operads modeled by [[dendroidal sets]], [[n-categories]] for low $n$ viewed as objects with an $A-\infty$-composition operation are discussed in  section 5 of

* Andor Lucacs, _Cyclic Operads, Dendroidal Structures, Higher Categories_ ([pdf](http://igitur-archive.library.uu.nl/dissertations/2011-0211-200314/lukacs.pdf))

and

* Andor Lucacs, _Dendroidal weak 2-categories_ ([arXiv:1304.4278](http://de.arxiv.org/abs/1304.4278))

See also the references at _[[model structure on algebras over an operad]]_.

[[!redirects A-infinity category]]
[[!redirects A-infinity categories]]
[[!redirects A-∞ category]]
[[!redirects A-∞ categories]]
[[!redirects A? category]]
[[!redirects A? categories]]
[[!redirects A∞-category]]
[[!redirects A∞-categories]]
[[!redirects A-∞-category]]
[[!redirects A-∞-categories]]


A-infinity-cocategory

The concept _$A_\infty$-cocategory_ is the result of combining [[A-infinity-category]] with [[cocategory]].


A-infinity-module

## Idea

__$A_\infty$-module__ $M$ over an $A_\infty$-algebra $A$ is an object with an $\infty$-[[infinity-representation|representation]] of $A$, modelled in the same formalism as the $A_\infty$-algebras are (usually in $k$-linear context, most often in (∞,1)-categories of chain complexes). In particular, one considers $A_\infty$-modules over an associative algebra or a dg-algebra viewed as a special case of an $A_\infty$-algebra.

## Definition

...

## Literature

* [[Bernhard Keller]], _Introduction to A-infinity algebras and modules_, Homology Homotopy Appl. 3(1) (2001) 1--35 arXiv:[math.RA/9910179](https://arxiv.org/abs/math/9910179) addendum.[dvi](https://webusers.imj-prg.fr/~bernhard.keller/publ/addendum.dvi)
* [[Bernhard Keller]], _Bimodule complexes via strong homotopy actions_, Algebr. Represent. Theory 3(4) (2000) 357--376
* [[Yuri Berest]], Oleg Chalykh, _$A_\infty$-modules and Calogero-Moser spaces_, J. Reine Angew. Math. __607__ (2007) 69--112 [MR2009f:16019](http://www.ams.org/mathscinet-getitem?mr=2338121) [doi](https://doi.org/10.1515/CRELLE.2007.046)

category: algebra


A-infinity-ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An $A_\infty$-ring is a [[monoid in an (∞,1)-category]] in an additive (that is, [[stable (infinity,1)-category|stable]]) [[(∞,1)-category]]. Alternatively one can take a model which is a (non-homotopic) additive monoidal category, but the monoid is replaced by an algebra over a resolution of the associative operad.

For example there is a variant for the [[stable (∞,1)-category of spectra]]. Sometimes this is called an [[associative ring spectrum]].

This may be modeled equivalently as an ordinary [[monoid]] with respect to the [[symmetric monoidal smash product of spectra]].

Notice the difference to an ordinary [[ring spectrum]] which which is not necessarily coherently homotopy-associative.

$A_\infty$-rings play the role of [[ring]]s in [[higher algebra]].

The higher analog of a commutative ring is an [[E-∞ ring]].

## Related concepts

* [[ring]], [[ring groupoid]]

* [[ring spectrum]], [[functor with smash products]]

* [[symmetric ring spectrum]]

* [[cohomology theory]]

* [[periodic ring spectrum]]

## References

See the references at *[[ring spectrum]]*.

Another version of the $A_\infty$-ring is simply what is usually called the $A_\infty$-[[A-infinity-algebra|algebra]] in the case when the [[ground ring]] is the ring of integers. See

* Gerald Dunn, _Lax operad actions and coherence for monoidal $n$-Categories, $A_{\infty}$ rings and modules_, Theory Appl. Cat. 1997, n.4 ([TAC](http://www.emis.de/journals/TAC/volumes/1997/n4/3-04abs.html))

[[!redirects A-infinity-rings]]
[[!redirects A-infinity ring]]
[[!redirects A-infinity rings]]
[[!redirects A∞-ring]]
[[!redirects A∞-rings]]
[[!redirects A-∞-ring]]
[[!redirects A-∞ ring]]
[[!redirects A-∞ rings]]

[[!redirects ∞-ring]]
[[!redirects ∞-rings]]
[[!redirects infinity-ring]]
[[!redirects infinity-rings]]

[[!redirects A-∞ ring spectrum]]
[[!redirects A-infinity ring spectrum]]
[[!redirects A-∞ ring spectra]]
[[!redirects A-infinity ring spectra]]