weighted limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The notion of _weighted limit_ (also called _indexed limit_ or _mean cotensor product_ in older texts) is naturally understood from the point of view on [[limits]] as described at _[[representable functor]]_:

Weighted limits make sense and are considered in the general context of $V$-[[enriched category theory]], but restrict attention to $V=$ [[Set]] for the moment, in order to motivate the concept.

Let $K$ denote the [[small category]] which indexes [[diagrams]] over which we want to consider limits and eventually weighted limits. Notice that for

$$
  F \colon K \to Set
$$

a [[Set]]-valued [[functor]] on $K$, the [[limit]] of $F$ is canonically identified simply with the [[set]] of [cones](cone#ConesOverADiagram) with tip the [[singleton]] set $pt = \{\bullet\}$:

$$
  lim F \;=\; [K,Set](\Delta pt, F)
  \,.
$$

This means, more generally, that for

$$
  F \,\colon\, K \to C
$$

a functor with values in an arbitrary [[category]] $C$, the [[object]]-wise limit of the functor $F$ under the [[Yoneda embedding]]

$$
  C\big(-,F(-)\big)
  \,\colon\,
  K
    \overset{F}{\longrightarrow}
  C
    \overset{Y}{\longrightarrow}
  Set^{C^{op}}
$$

can be expressed by the right hand side of:

\[
  \label{ConicalLimitInWeightedLimitForm}
  \underset{
    \underset{k \in K}{\longleftarrow}
  }{lim}
  C\big(-,F(k)\big)
  \;=\;
  [K,Set]\Big(
    \Delta pt
    ,\,
    C\big(-,F(-)\big)
  \Big)
  \,.
\]

(This is the limit over the diagram $C\big(-,F(-)\big) \,\colon\, K \to Set^{C^{op}}$ which, if [[representable functor|representable]], defines the desired limit of $F$, see [this example](representable+functor#ExampleLimits) at *[[representable functor]]*).

The **idea** of weighted limits is to:

1. allow in the formula (eq:ConicalLimitInWeightedLimitForm) the particular functor $\Delta pt$ to be replaced by any other functor $W \,\colon\, K \to Set$;

2. to generalize everything straightforwardly from the [[Set]]-[[enriched category|enriched]] context to arbitrary $V$-enriched contexts (see below).

The idea is that the weight $W \colon K \to V$ encodes the way in which one generalizes the concept of a [cones](cone#ConesOverADiagram) over a diagram $F$ (that is, something with just a tip from which morphisms are emanating down to $F$) to a more intricate structure over the diagram $F$. For instance in the application to [[homotopy limits]] discussed below, with $V$ being [[SimpSet]], the weight is such that it ensures that not only [[1-morphisms]] are emanating from the tip, but that any triangle formed by these is filled by a [[2-morphism|2-cell]], every tetrahedron by a [[3-morphism|3-cell]], etc.

## Definition ##

Let $V$ be a [[closed category|closed]] [[symmetric monoidal category]]. All categories in the following are $V$-[[enriched category|enriched categories]], all functors are $V$-[[enriched functors]].

A **weighted limit** over a functor

$$
  F \,\colon\, K \to C
$$

with respect to a _weight_ or _indexing type_ functor

$$
  W \,\colon\, K \to V
$$

is, if it exists, the object $lim^W F \in C$ which [[representable functor|represents]] the functor (in $c \in C$)

$$
  [K,V]\Big(W, C\big(c,F(-)\big)\Big)
  \;\colon\;
  C^{op} \to V
  \,,
$$

i.e. such that for all objects $c \in C$  there is an isomorphism

$$
  C\big(c, lim^W F\big)
  \simeq
  [K,V]\Big(W(-), C\big(c,F(-)\big)\Big)
$$

[[natural isomorphism|natural]] in $c$.

(Here $[K,V]$ denotes the $V$-[[enriched functor category]], as usual.)

In particular, if $C = V$ itself, then we get the direct formula
$$
  lim^W F \;\simeq\; [K,V](W,F)
  \,.
$$

This follows from the above by the [[end]] manipulation

$$
  \begin{aligned}
    [K,V](W(-),C(c,F(-)))
    &:=
    \int_{k \in K} V(W(k),V(c,F(k)))
    \\
    &  \simeq
      \int_{k \in K} V(c,V(W(k),F(k))
    \\
    &  \simeq
      V(c, \int_{k \in K} V(W(k),F(k))
    \\
    & =:
      V(c, [K,V](W,F))
    \,.
  \end{aligned}
$$

### Weighted limits for $V = Set$

Let us spell out what a weighted limit looks like in ordinary category theory, to give intuition for the difference between weighted limits and ordinary limits.

Given a weight $W : K \to Set$ and a diagram $F : K \to C$, a weighted limit comprises an object $L$ together with a projection $\pi_{k, w} : L \to F(k)$ for each $k \in K$ and $w \in W(k)$ such that the following diagram commutes for $k, k \in K$, $w \in W(k)$ and $\kappa : k \to k'$:
\begin{tikzcd}
	L & {F(k)} \\
	& {F(k')}
	\arrow["{F(\kappa)}", from=1-2, to=2-2]
	\arrow["{\pi_{k, w}}", from=1-1, to=1-2]
	\arrow["{\pi_{k', W(k)(w)}}"', from=1-1, to=2-2]
\end{tikzcd}
This is required to be universal in the sense that given every such diagram as above with domain $C$, there is a unique morphism $C \to L$ making the diagrams commute.

It is clear that when $W$ is the constant functor sending everything to a singleton set, this recovers the usual notion of limit for $F$.

## Motivation from enriched category theory

Let $V$ be a [[monoidal category]].

Imagine you're tasked to write down the definition of _[[limit]]_ in a category $C$ [[enriched category|enriched]] over $V$. You would start saying there is a [[diagram]] $F \colon K \to C$ and a limit is a [[universal construction|universal]] [[cone]] over it, i.e. it's the universal choice of an [[object]] $c$ together with an arrow $f_k \colon c \to F(k)$ for each object $k$ of $K$.

Here's where you stop and ask yourself: what is 'an arrow' in $C$? $C$ has no *[[hom-sets]]* — it has *[[hom-objects]]* — hence what's 'an element' of $C(c, F(k))$ in $V$?

There are two ways to specify an element of an object $X$ in a [[monoidal category]] $(V, I, \otimes)$:

1. Give an arrow $I \to X$ (think of sets, where elements of $X$ are indeed the same thing as arrows $\{*\} \to X$. These are called [[global elements]] of $X$, and are more often than not a misbehaved notion of element, since often $I$ is 'too big' to thoroughly probe $X$ (on the other hand, notice the [[enriched category#passage_between_ordinary_categories_and_enriched_categories|underlying category of an enriched category]] is defined by taking global elements of the hom-objects)
2. Give *any* arrow into $X$. These are called [[generalized elements]], and the existence of the [[Yoneda embedding]] assures us they completely capture the categorical structure of $V$.

Hence you now say: a cone over $F$ is a choice of a *generalized element* $f_k$ of $C(c, F(k))$, for every $k$ in $K$. This means specifying an arrow $W_k \to C(c, F(k))$ in $V$, for each $k$. It's now quite natural to ask for the functoriality of this choice in $k$, hence we end up defining a 'generalized cone' over $F$ as an element
$$
  [K, V](W(-), C(c, F(-)))
$$
Hence $W$ is simply a uniform way to specify the sides of a cone.
A confirmation that this is indeed the right definition of limit in the enriched settings come from the fact that conical completeness (a [[conical limit]] is one where $W = \Delta I$, hence we pick only global element) is an inadequate notion, see for example Section 3.9 in [Kelly's book](#Kelly) (aptly named *The inadequacy of conical limits*).

## Examples  ##

### Homotopy limits ###

For $V$ some category of higher structures, the _local_ definition of [[homotopy limit]] over a diagram $F : K \to C$ replaces the ordinary notion of [[cone]] over $F$ by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.

One can convince oneself that for the choice of [[SimpSet]] for $V$ this is realized in terms of the weighted limit $lim^W F$ with the weight $W$ taken to be

$$
  W : K \to \Simp\Set
$$
$$
  W : k \mapsto N(K/k)
   \,,
$$

where $K/k$ denotes the [[over category]] of $K$ over $k$ and $N(K/k)$ denotes its [[nerve]].

This leads to the classical definition of homotopy limits in $\Simp\Set$-enriched categories due to

* A.K. Bousfield and D.M. Kan, _Homotopy limits, completions, and localizations_

See for instance also

*  [[Jean-Marc Cordier]] and [[Timothy Porter]], _Homotopy Coherent Category Theory_, Trans. Amer. Math. Soc. 349 (1997) 1-54, ([pdf](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf))

* [[Nicola Gambino]], _Weighted limits in simplicial homotopy theory_ ([pdf](http://www.crm.cat/Publications/08/Pr790.pdf) or [pdf](http://www.math.unipa.it/%7Engambino/Research/Papers/weighted.pdf))

In some nice cases the weight $N(K/-)$ can be replaced by a simpler weight; an example is discussed at [[Bousfield-Kan map]].

### Homotopy pullback ###

For instance in the case that
$K = \{r \to t \leftarrow s\}$ is the shape of [[pullback]] diagrams we have

$$
  W(r) = \{r\}
$$
$$
  W(s) = \{s\}
$$
$$
  W(t) = N( \{r \to t \leftarrow s\} )
$$
and $W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}$ injects the vertex $r$ into $\{r \to t \leftarrow s\}$ and similarly for $W(s \to t)$.

This implies that for $F : K \to C$ a pullback diagram in the [[SimpSet]]-enriched category $C$, a $W$-weighted [[cone]] over $F$ with tip some object $c  \in C$, i.e. a natural transformation

$$
  W \Rightarrow C(c, F(-))
$$

is

* over $r$ a "morphism" from the tip $c$ to $F(r)$ (i.e. a vertex in the Hom-simplicial set $C(c,F(r))$);

* similarly over $s$;

* over $t$ three "morphisms" from $c$ to $F(t)$ together with 2-cells between them (i.e. a 2-[[horn]] in the Hom-simplicial set $C(c,F(t))$)

* such that the two outer morphisms over $t$ are identified with the morphisms over $r$ and $s$, respectively, postcomposed with the morphisms $F(r \to t)$ and $F(s \to t)$, respectively.

So in total such a $W$-weighted cone looks like

$$
  \array{
     &&& c
     \\
     & \swarrow &\Rightarrow& \downarrow
      &\Leftarrow& \searrow
     \\
     F(r)
      &&
     \stackrel{F(r \to t)}{\to}
      &
     F(t)
      &
     \stackrel{F(s \to t)}{\leftarrow}
     &&
     F(s)
  }
$$

as one would expect for a "homotopy cone".

# Related pages

* [[strict 2-limit]]

* [[saturated class of limits]]

* [[weighted colimit]]

## References

### General

The notion of weighted limits was introduced (under the name "*mean cotensor product*") in:

* [[Francis Borceux]], p. 10 of: *Une notion de $V$-limite*, in *Colloque sur l'algèbre des catégories. Amiens 1973. Résumés des conférences*, Cahiers de topologie et géométrie différentielle **14** 2 (1973) 153-223 [[numdam:CTGDC_1973__14_2_153_0](http://www.numdam.org/item/CTGDC_1973__14_2_153_0)]

* [[Francis Borceux]], [[Gregory Maxwell Kelly]], *A notion of limit for enriched categories*, Bulletin of the Australian Mathematical Society **12** 1 (1975) 49-72 [[doi:10.1017/S0004972700023637](https://doi.org/10.1017/S0004972700023637)]

and, independently, (under the name "*Hom (formel)*") by:

* C. Auderset, _Adjonction et monade au niveau des 2-categories_, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. ([numdam](http://www.numdam.org/item/CTGDC_1974__15_1_3_0/))

Textbook accounts:

* {#Kelly} [[Max Kelly]], [section 3.1, p. 37](http://www.emis.de/journals/TAC/reprints/articles/10/tr10.pdf#page=37) in:  _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories, **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* [[Francis Borceux]], §6.6 in: *[[Handbook of Categorical Algebra]]*, Vol. 2:  *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

In

* [[Emily Riehl]], _Weighted limits and colimits_ (2008) ([pdf](http://www.math.jhu.edu/~eriehl/weighted.pdf))

is given an account of lectures by [[Mike Shulman]] on the subject. The definition appears there as [definition 3.1, p. 4](http://www.math.jhu.edu/~eriehl/weighted.pdf#page=4) (in a form a bit more general than the one above).

### Presenting homotopy limits

On weighted limits as presentations of [[homotopy limits]]:

* {#Hirschhorn02} [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

To compare with the above discussion notice that

* The functor
$$
  W \;\coloneqq\; N(K/-)
$$
is discussed there in definition 14.7.8 on p. 269.

* the $V$-enriched hom-category $[K,V]$ which on $V$-functors $S,T$ is the [[end]] $[K,V](S,T) = \int_{k \in K} V(S(k), T(k))$ appears as $hom^K(S,T)$ in definition 18.3.1 (see bottom of the page).

* for $V$ set to [[SimpSet]] the above definition of homotopy limit appears in example 18.3.6 (2).

* [[Emily Riehl]], §6.6 and Chapter 7 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

Discussion of weighted [[(infinity,1)-limit|$(\infty,1)$-limits]]:

* [[Martina Rovelli]], _Weighted limits in an (∞,1)-category_ (2019) [[arxiv:1902.00805](https://arxiv.org/abs/1902.00805)]

[[!redirects weighted limits]]
[[!redirects indexed limit]]
[[!redirects indexed limits]]
[[!redirects mean cotensor product]]
[[!redirects mean cotensor products]]


Weil algebra

> There are two different concepts called _Weil algebra_. This entry is about the notion of Weil algebra in [[Lie theory]]. For the notion in [[infinitesimal object|infinitesimal]] geometry see [[infinitesimally thickened point]]/[[local Artin algebra]].

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The notion of _Weil algebra_ is ordinarily defined for a [[Lie algebra]] $\mathfrak{g}$. It may be understood as the [[Chevalley-Eilenberg algebra]] of the tangent [[Lie 2-algebra]] $T \mathfrak{g}$ or $inn(\mathfrak{g})$ of $\mathfrak{g}$, generalizing the notion of [[tangent Lie algebroid]] $T X$ from a 0-[[truncated]] Lie algebroid $X$ (a [[smooth manifold]]) to the one-object [[Lie algebroid]] $\mathfrak{g}$.

Generally, for every [[Lie-∞-algebroid]] $\mathfrak{a}$ one may define the corresponding tangent Lie-$\infty$-algebroid $T \mathfrak{a}$, whose Chevalley-Eilenberg algebra may be called the Weil algebra of $\mathfrak{a}$:

$$
  W(\mathfrak{a})
  =
  CE(T \mathfrak{a})
  \,.
$$

### Weil algebra of a Lie algebra

Let $\mathfrak{g}$ be a finite-dimensional [[Lie algebra]]. The **Weil algebra** $W(\mathfrak{g})$ of $\mathfrak{g}$ is

* the graded [[Grassmann algebra]] generated from the dual [[vector space]] $\mathfrak{g}^*$ together with another copy of $\mathfrak{g}^*$ shifted in degree

  $$
    W(\mathfrak{g})
    \coloneqq
    \wedge^\bullet
   (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])
  $$

* equipped with a [[derivation]] $d : W(\mathfrak{g}) \to W(\mathfrak{g})$ that makes this a [[dg-algebra]], defined by the fact that on $\mathfrak{g}^*$ it acts as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ plus the degree shift morphism $\mathfrak{g}^* \to \mathfrak{g}^*$.

This Weil algebra has trivial [[chain homology and cohomology|cohomology]] everywhere (except in degree 0 of course) and sits in a sequence

$$
  CE(\mathfrak{g}) \leftarrow W(\mathfrak{g})
  \leftarrow inv(\mathfrak{g})
$$

with the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and its algebra of [[invariant polynomial]]s on $\mathfrak{g}$. This may be understood as a model for the sequence of algebras of differential forms on the [[generalized universal bundle|universal G-bundle]]

$$
  G \to \mathcal{E}G \to \mathcal{B}G
  \,.
$$

As such, the Weil algebra plays a crucial role in the study of the [[Lie algebra cohomology]] of $\mathfrak{g}$.

## Definition

We first consider Weil algebras of [[L-∞ algebras]], then more generally of [[L-∞ algebroid]]s.

We use the notation and grading conventions that are described in detail at [[Chevalley-Eilenberg algebra]].

### For $L_\infty$-algebras

#### Plain Weil algebra

Let $\mathfrak{g}$  be an [[L-∞ algebra]] of [[finite type]]. By our grading conventions this means that the [[graded vector space]] $\mathfrak{g}^*$ obtained by degreewise dualization is in non-negative degree, and $\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1]$ is its shift up into positive degree.

A quick abstract way to characterize the Weil algebra of $\mathfrak{g}$ is as follows. Notice that there is a [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  (F \dashv U)
  \colon
  dgAlg
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\longrightarrow}}
  Vect[\mathbb{Z}]
$$

between the [[category]] [[dgAlg]] of [[dg-algebras]] and the category of $\mathbb{Z}$-graded [[vector space]]s (all over some fixed [[field]]). Notice that a free object is unique _up to [[isomorphism]]_ .

\begin{definition}\label{PlainWeilAlgebraOfLInfinityAlgebra}
**(plain Weil algebra of $L_\infty$-algebra)**
\linebreak
The **Weil algebra** $W(\mathfrak{g})$ of an $L_\infty$-algebra is the unique representative of the [[free functor|free]] [[dg-algebra]] on $\wedge^1 \mathfrak{g}^*$ for which the projection of graded vector spaces $\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^*$ extends to a [[dg-algebra]] [[homomorphism]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$

\end{definition}

We discuss below in the [Properties](#Properties) section that this is equivalent to the following component-wise definition

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

The **Weil algebra** $W(\mathfrak{g})$ is the [[semi-free dga]] whose underlying graded-commutative algebra is the [[exterior algebra]]

$$
  \wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])
$$

on $\mathfrak{g}^*$ and a shifted copy of $\mathfrak{g}^*$,
and whose [[differential]] is the sum

$$
  d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d}
$$

of two graded [[derivations]] of degree +1 defined by

* $\mathbf{d}$ acts by degree shift $\mathfrak{g}^* \to \mathfrak{g}^*[1]$ on elements in $\mathfrak{g}^*$ and by 0 on elements of $\mathfrak{g}^*[1]$;

* $d_{CE(\mathfrak{g})}$ acts on unshifted elements in $\mathfrak{g}^*$ as the differential of the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$ and is extended uniquely to shifted generators by graded-commutativity

  $$
    [d_{CE(\mathfrak{g})}, \mathbf{d}] = 0
  $$

  with $\mathbf{d}$:

  $$
    d_{CE(\mathfrak{g})} \mathbf{d} \omega
    \coloneqq
    - \mathbf{d} d_{CE(\mathfrak{g})} \omega
  $$

  for all $\omega \in \wedge^1 \mathfrak{g}^*$.

=--

#### Adjusted Weil algebras
 {#AdjustedWeilAlgebras}

For some purposes of [[schreiber:L-infinity algebra connections|$L_\infty$-algebra valued connection]], the [above](#WeilForLInfinitityAlgebra) definition of Weil algebra of an $L_\infty$-algebra is not quite appropriate. While Def. \ref{WeilForLInfinitityAlgebra} gives the Weil algebra up to compatible [[isomorphism]], in applications there is in fact the freedom to choose it up to compatible [[quasi-isomorphism]], and this freedom allows to find better representatives.

For more on this see at *[[adjusted Weil algebra]]*.

### For $L_\infty$-algebroids

Where the [[Chevalley-Eilenberg algebra]] of an [[L-∞ algebra]] has in degree 0 the ground field, that of an [[L-∞ algebroid]] has more generally an [[algebra over a Lawvere theory|algebra over]] a [[Lawvere theory]]. For [[L-∞ algebroid]]s over [[smooth manifold]]s this is the algebra of [[smooth function]]s on a manifolds, regarded as a [[smooth algebra]] ($C^\infty$-ring).

So let $T$ be a [[Fermat theory]]. Write $T Alg$ for the corresponding [[category]] of [[algebra over a Lawvere theory|algebra]]. There is a [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  (F \dashv U) :
   T Alg
    \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  CRing
$$

to the category [[CRing]] of commutative [[Ring]]s.

We need the facts that

* a [[module]] over a $T$-algebra $A$ is uniquely specified by its underlying module over $U(A)$;

* the universal [[derivation]] on a $T$-algebra $A$ is the  [[de Rham differential]]

  $$
    d_{dR} : A \to \Omega^1(A)
  $$

 with values in the $A$-module of $T$-[[Kähler differential]]s.

See the corresponding entries for more details. The second point means that for $v : A \to N$ any $T$-[[derivation]] on $A$, there is a unique $A$-[[module]] [[homomorphism]]

$$
  \Omega^\bullet(A) \to N
$$

such that the diagram

$$
  \array{
    && \Omega^\bullet(A)
    \\
    & {}^{\mathllap{d_{dR}}}\nearrow & \downarrow^{\mathrlap{v}}
    \\
    A &\stackrel{v}{\to}& N
  }
$$

commutes.

Let now $\mathfrak{a}$ be an [[L-∞ algebroid]] with [[Chevalley-Eilenberg algebra]] considered as the following data;

1. a graded commutative [[semifree dga]] $CE(\mathfrak{a})$ over the ground field;

1. the structure of a $T$-[[algebra over a Lawvere theory|algebra]] on the [[associative algebra]] $A \coloneqq CE(\mathfrak{a})_0$ (over the ground field)

   such that $d_{CE(\mathfrak{a})} : CE(\mathfrak{a})_0 \to CE(\mathfrak{a})_1$ is a [[derivation]] of $T$-algebra modules.

By [[semifree dga|semi-freeness]] there exists a $\mathbb{N}$-[[graded vector space]] $(\mathfrak{a}^*)^\bullet$ and an [[isomorphism]]

$$
  CE(\mathfrak{a})
   \simeq
  (\wedge^\bullet_{A} (\mathfrak{a}^*), d_{CE(\mathfrak{a})})
  \,.
$$

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

The **Weil algebra** $W(\mathfrak{a})$ of the $L_\infty$-algebroid $\mathfrak{a}$ is the Chevalley-Eilenberg algebra of the $L_\infty$-algebroid defined as follows

* the $T$-algebra $A$ in degree 0 is the same as that of $\mathfrak{A}$;

* the underlying graded algebra is the [[exterior algebra]] on $\mathfrak{a}^*$ and a shifted copy $\mathfrak{a}^*[1]$ as well as one copy of the [[Kähler differential]] module $\Omega^1$ in lowest degree (though of as the shifted copy of $A$ itself)

  $$
    \wedge^\bullet (\Omega^1(A) \oplus  (\mathfrak{a}^*) \oplus \mathfrak{a}^*[1])
    \,.
  $$

* the [[differential]] is the sum

  $$
    d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}
  $$

  of two degree +1 graded derivations, where $d_{CE(\mathfrak{a})}$ and $\mathbf{a}$ are defined on $\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1]$ as [above](#WeilForLInfinitityAlgebra) for $L_\infty$-algebras and on $A$ itself $d_{CE(\mathfrak{a})}$ vanishes and $\mathbf{d}$ acts as the universal derivation

  $$
    \mathbf{d}|_A = d_{\mathrm{dR}} : A \to \Omega^1(A)
    \,.
  $$

=--

## Properties
  {#Properties}

### Free property
 {#FreeProperty}

The main point of the definition is that the differential restricted to the original (unshifted) generators is the original differential plus the shift:

$$
  d_{W(\mathfrak{a})} |_{\mathfrak{a}^*} =
   d_{CE(\mathfrak{a})} + \mathbf{d}
  \,.
$$

By solving the condition $d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0$ and using that $d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0$ this already fixes uniquely the differential $d_{W(\mathfrak{a})}$. To see this we only need to show that the value of $d_{W(\mathfrak{a})}(x)$ on a generator $x=\sigma(t) \in \mathfrak{a}^*[1]$ is completely determined by $d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}$. One computes:

$$
  \begin{aligned}
    0 & =
    d_{W(\mathfrak{a})}(d_{W(\mathfrak{a})} t)
    \\
    & =
    d_{W(\mathfrak{a})}(d_{CE(\mathfrak{a})}t + \sigma t)
    \\
    & = \sigma d_{CE(\mathfrak{a})} t + d_{W(\mathfrak{a}) } x
  \end{aligned}
$$

and hence

$$
  d_{W(\mathfrak{a})} x = - \sigma d_{CE(\mathfrak{a})} \sigma^{-1} (x)
  \,.
$$

This implies the following universal [[free functor|freeness property]]:

+-- {: .num_prop}
###### Proposition

Let $\mathfrak{g}$ be an $L_\infty$-algebra. Morphisms of $dg$-algebras $W(\mathfrak{g}) \to A$ are in natural bijection to morphisms of [[graded vector space]]s $\mathfrak{g}^* \to A$.

=--

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

Forgetting the differential, $W(\mathfrak{g})$ is the free graded-commutative algebra generated by (a shifted copy of) $\mathfrak{g}^*$ and $\mathfrak{g}^*[1]$. Therefore,
$$
Hom_{dgca}(W(\mathfrak{g}),A)\subseteq Hom_{gca}(W(\mathfrak{g}),A)=Hom_{grVect}(\mathfrak{g}^*,A)\oplus Hom_{grVect}( \mathfrak{g}^*[1],A).
$$
Projecting down to $Hom_{grVect}(\mathfrak{g}^*,A)$, one obtains a natural map
$$
Hom_{dgca}(W(\mathfrak{g}),A)\to Hom_{grVect}(\mathfrak{g}^*,A),
$$
which is a bijection.

To prove injectivity, we just have to show that the restriction of a dgca morphism $f:W(\mathfrak{g})\to A$ to $\mathfrak{g}^*$ determines the restriction of $f$ to $\mathfrak{g}^*[1]$. One has, for any $x=\sigma(t)\in \mathfrak{g}^*[1]$,
$$
\begin{aligned}
f(x)&=f(\sigma(t))=f(d_{W(\mathfrak{g})}t-d_{CE(\mathfrak{g})}t)\\
&=d_A f(t)- f(d_{CE(\mathfrak{g})}t).
\end{aligned}
$$
Since $d_{CE(\mathfrak{g})}(t)$ lies in the sub-gca of $W(\mathfrak{g})$ generated by $\mathfrak{g}^*$, the element $f(d_{CE(\mathfrak{g})}(t))$, and therefore $f(x)$, is determined by $f\vert_{\mathfrak{g}^*}$.

Next we show surjectivity, i.e. that every morphism of graded vector spaces $\phi:\mathfrak{g}^*\to A$ can be extended to a dgca morphism $f:W(\mathfrak{g})\to A$. Denote by $f_0: \wedge^\bullet \mathfrak{g}^*\to A$ the extension of $\phi$ to a graded commutative algebra morphism, and let $\psi:\mathfrak{g}^*[1]\to A$ be the graded vector space morphism defined by
$$
\psi(x)=d_A \phi(t)-f_0d_{CE(\mathfrak{g})}(t),
$$
for any $x=\sigma(t)\in \mathfrak{g}^*[1]$.
The graded vector space morphism $\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A$ extends to a commutative graded algebra $f:W(\mathfrak{g})\to A$, whose restriction to $\mathfrak{g}^*$ is $\phi$. We want to show that $f$ is actually a dgca morphism. We only need to test commutativity with the differentials on generators $t\in \mathfrak{g}^*$ and $x=\sigma(t)\in \mathfrak{g}^*[1]$. We have
$$
d_A f(t)=d_A\phi(t)=\psi(\sigma(t))+f_0d_{CE(\mathfrak{g})}(t)=f(\sigma(t))+
f d_{CE(\mathfrak{g})}(t)=f d_{W(\mathfrak{g})}(t),
$$
which in particular implies that $d_A f\vert_{\wedge^\bullet \mathfrak{g}^*}=f d_{W(\mathfrak{g})}\vert_{\wedge^\bullet \mathfrak{g}^*}$,
and
$$
d_A f(x)= d_A \psi(x) = -d_A f_0d_{CE(\mathfrak{g})}(t)=-d_A f (d_{CE(\mathfrak{g})}(t)).
$$
Since $d_{CE(\mathfrak{g})}(t)\in \wedge^\bullet \mathfrak{g}^*$, we obtain
$$
d_A f(x)= -f d_{W(\mathfrak{g})} (d_{CE(\mathfrak{g})}(t))=
-f d_{W(\mathfrak{g})}(d_{W(\mathfrak{g})}(t)-x)=f d_{W(\mathfrak{g})}(x).
$$

=--

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

For $A=CE(\mathfrak{g})$ the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$, the inclusion $\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g})$ induces a canonical surjective dgca morphism $W(\mathfrak{g})\to CE(\mathfrak{g})$. This is the identity on the unshifted generators, and 0 on the shifted generators.

=--

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

For $A = \Omega^\bullet(X)$ the [[de Rham complex]] of a [[smooth manifold]] $X$, we have that

$$
  Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(X))
  =
  (\Omega^\bullet(X) \otimes \mathfrak{g})^1
$$

is the collection of total degree 1 [[differential form]]s with values in the $\infty$-Lie algebra $\mathfrak{g}$.

A morphism of

$$
  (A, F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X)
$$

sends the unshifted generators $t^a$ to differential forms $A^a$, which one thinks of as local connection forms, and sends the shifted generators $\sigma t^a$ to their [[curvature]]. The respect for the differential on the shifted generators is the [[Bianchi identity]] on these curvatures.

A morphism $W(\mathfrak{g}) \to \Omega^\bullet(X)$ encodes a collection of _flat_ $L_\infty$-algebra valued forms precisely if it factors by the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$ from above through the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$.

=--

The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete [[isomorphism]] to the free dg-algebra on $\mathfrak{g}^*$.

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

The _canonical free dg-algebra_ on $\mathfrak{g}^*$ is

$$
  F(\mathfrak{g}) \coloneqq \wedge^\bullet( \mathfrak{g}^* \oplus \mathfrak{g}^*[1], d_F )
$$

where the differential $d_f$ is on the unshifted generators $t \in \mathfrak{g}^*$ the shift isomorphism $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ extended as a derivation and vanishes on the shifted generators

$$
  d_F : t \mapsto \sigma(t)
  \,,
$$

$$
  d_F : \sigma(t) \mapsto 0
  \,.
$$

=--

Or in other words, if $\bar \mathfrak{g}$ is the $\infty$-Lie algebra whose underlying graded vector space is that of $\mathfrak{g}$, but all whose brackets vanish, then

$$
  F(\mathfrak{g}) = W(\bar \mathfrak{g})
  \,.
$$

Notice the evident

+-- {: .num_lemma}
###### Observation

The [[cochain cohomology]] of $F(\mathfrak{g})$ vanishes in positive degree.

=--

To see this, let $K \coloneqq \sigma^{-1} : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the degree down-shift isomorphism $\mathfrak{g}^*[1] \to \mathfrak{g}^*$ extended as a graded derivation of degree -1, then

$$
  [d_{F(\mathfrak{g})}, K] = Id : F(\mathfrak{g}) \to F(\mathfrak{g})
$$

and hence for any $\omega \in F(\mathfrak{g})$ such that $d_{F(\mathfrak{g})} \omega = 0$ we have $\omega = d_{F(\mathfrak{g})} K \omega$.

+-- {: .num_prop}
###### Lemma

Given $\mathfrak{g}$, there is an [[isomorphism]] of [[dg-algebra]]s

$$
  f : F(\mathfrak{g}) \to W(\mathfrak{g})
$$

given by

$$
  f : t \mapsto t
$$

$$
  f : \sigma(t) \mapsto d_{W(\mathfrak{g})}  t  = d_{CE(\mathfrak{g})} t + \sigma(t)
  \,.
$$

=--

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

It is clear that $f$ is a dg-algebra homomorphism. The inverse dg-algebra morphism is given on generators by

$$
  f^{-1} : t \mapsto t
$$

$$
  f^{-1} : \sigma(t) \mapsto \sigma(t) - d_{CE(\mathfrak{g})}(t)
  \,.
$$
Note that the isomorphism $f$ is precisely the dgca isomorphism induced between $W(\overline\mathfrak{g})$ and  $W(\mathfrak{g})$ by the identity of $\mathfrak{g}^*$ as a graded vector spaces morphism $\overline{\mathfrak{g}}^*\to\mathfrak{g}^*$.
=--

+-- {: .nun_cor}
###### Corollary

The [[cochain cohomology]] of the Weil algebra of an $L_\infty$-algebra is trivial.

=--

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

This means that [[homotopy theory|homotopy-theoretically]] the Weil algebra is the point. Dually, the $\infty$-Lie algebra $inn(\mathfrak{g})$ is a model for the point. In fact, one can see that $inn(\mathfrak{g})$ is the [[universal principal ∞-bundle]] over $\mathfrak{g}$ in the canonical [[model category|model]] for the [[(∞,1)-topos]] [[SynthDiff∞Grpd]]. In fact, it is a [[groupal model for universal principal ∞-bundles]]. This is discussed at [[∞-Lie algebra cohomology]].

=--

### Characterization in the smooth $\infty$-topos
 {#CharacterizationInSmoothTopos}

The Weil algebra of a Lie algebra is naturally identified with the
de Rham algebra of differential forms on the "universal $G$-principal bundle with connection" in its stacky incarnation ([Freed-Hopkins 13](#FreedHopkins13)):

Write $\mathbf{B}G_{conn}\simeq \mathbf{\Omega}(-,\mathfrak{g})//G$ for the universal [[moduli stack]] of $G$-[[principal connections]] (as discussed there), a [[smooth groupoid]]. The quotient projection may be regarded as the universal $G$-connection:

$$
  \array{
     && \mathbf{\Omega}_{flat}(-,\mathfrak{g})
     \\
     && \downarrow
     \\
     \mathbf{E}G_{conn} &\coloneqq & \mathbf{\Omega}(-,\mathfrak{g})
     \\
     \downarrow && \downarrow
     \\
     \mathbf{B}G_{conn} &\coloneqq &\mathbf{\Omega}(-,\mathfrak{g})//G
  }
$$

(After forgetting the connection/form data this is just the [[universal principal bundle]] $\mathbf{E}G \to \mathbf{B}G$)

The differential $k$-forms on a [[smooth groupoid]] $X$ are just homs $X \to \mathbf{\Omega}^k(-)$ into the sheaf of $k$-forms. (See at [[geometry of physics -- differential forms]]). These $\Omega^k(X)$ inherit the de Rham differential and hence form the de Rham complex of the stack. (Notice that this is very different from the hom of $X$ into a shift of the full de Rham complex regarded as a sheaf of complexes. The latter is instead a model for the real [[ordinary cohomology]] of $X$, see at _[[smooth infinity-groupoid -- structures]]_ for more on this).

{#FreedHopkinsResult} One finds ([Freed-Hopkins 13](#FreedHopkins13)) that the de Rham complex, in this sense, of $\mathbf{E}G_{conn}$ is the Weil algebra:

$$
  \Omega^\bullet(\mathbf{E}G_{conn})
  \coloneqq
  \Omega^\bullet( \mathbf{\Omega}(-,\mathfrak{g}) )
  \simeq
  W(\mathfrak{g})
  \,.
$$

[[!include Weil algebra abstractly -- table]]

Turning this around, this motivates to algebraically _define_ the [[connection on a principal ∞-bundle]], [via Lie integration](connection+on+a+smooth+principal+infinity-bundle#ByLieIntegration), as discussed there.

### Relation to Cartan model for equivariant de Rham cohomology

The Weil algebra may be identified with the
[[Cartan model]] for [[equivariant de Rham cohomology]] for the special
case of the Lie group $G$ acting on itself by right multiplication. Concersely, the [[Cartan models]] form a generalization of the Weil algebra. See at _[equivariant de Rham cohomology -- Cartan model](equivariant+de+Rham+cohomology#TheCartanModel)_ for more.

## As the CE-algebra of the $L_\infty$-algebra of inner derivations {#AsInnerDer}

By the discussion at [[∞-Lie algebra]] and [[Chevalley-Eilenberg algebra]], we may _identify_ the [[full subcategory]] of the [[opposite category]] [[dgAlg]] on commutative [[semi-free dga]]s in non-negative degree with that of [[∞-Lie algebra]]s/[[∞-Lie algebroid]]s.

That means that the Weil algebra $W(\mathfrak{g})$ of some [[L-∞ algebra]] $\mathfrak{g}$ is the Chevalley-Eilenberg algebra of _another_ $\infty$-Lie algebra.

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

For any $\infty$-Lie algebra $\mathfrak{g}$ write $inn(\mathfrak{g})$ for the $\infty$-Lie algebra whose CE-algebra is $W(\mathfrak{g})$:

$$
  CE(inn(\mathfrak{g}))
  \coloneqq
  W(\mathfrak{g})
  \,.
$$

=--

In the following we discuss these _inner automorphism $\infty$-Lie algebras_ in more detail. (See section 6 of ([SSSI](#SSSI))).

### For an ordinary Lie algebra

+-- {: .num_lemma}
###### Observation

For $\mathfrak{g}$ an ordinary [[Lie algebra]] the [[inner derivation Lie 2-algebra]] is the [[strict Lie 2-algebra]] given by the [[dg-Lie algebra]]

$$
  inn(\mathfrak{g}) = ( \mathfrak{g} \stackrel{d}{\to} \mathfrak{g}, [-,-])
$$

whose

* elements in degree -1 are the elements $x \in \mathfrak{g}$, thought of as inner degree-(-1) [[derivation]]s

  $\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$

  given by contraction with $x$;

* elements in degree 0 are the derivations of degree 0 that are of the form

  $ \mathcal{L}_X
    \coloneqq
    [d_{CE(\mathfrak{g})}, \iota_x]
    \colon
    CE(\mathfrak{g}) \to CE(\mathfrak{g})$;

* the differential $d = [d_{CE}, -] : \mathfrak{g} \to \mathfrak{g}$ is the commutator of derivations with the differential $d_{CE(\mathfrak{g})}$;

* the bracket is the graded commutator of derivations.

=--

Equivalently this is identified with the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the [[adjoint action]] of $\mathfrak{g}$ on itself.

One checks that for all $x, y \in \mathfrak{g}$ we have in $inn(\mathfrak{g})$ the brackets

* $[\iota_x, \iota_y] = 0$

* $[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$

* $[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$

and of course

* $ \mathcal{L}_x = [d, \iota_x] $.

These identities are known as [[Cartan calculus]]. In this context $\mathcal{L}_x$ is called a [[Lie derivative]].

In this sense one may understand $inn(\mathfrak{g})$ for general $\infty$-Lie algebras $\mathfrak{g}$ as providing an $\infty$-version of [[Cartan calculus]].

## Relation to other concepts

### $\infty$-Lie algebra valued differential forms {#LieAlgValuedForms}

For $\mathfrak{g}$ an [[∞-Lie algebra]],  $X$ a [[smooth manifold]], an
[[∞-Lie algebra valued differential form]] is a morphism

$$
  \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A
$$

of [[dg-algebra]]s, from the Weil algebra into the [[de Rham complex]] of $X$.

The image of the unshifted generators $A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X)$ are the forms themselves, the image of the shifted generators $F_A : \wedge^1 \mathfrak{g}^*[1]$ are the corresponding [[curvature]]s. The respect for the differential on the shifted generators are the [[Bianchi identity]] on the curvatures.

Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

$$
  (F_A = 0)
  \;\;\Leftrightarrow
  \;\;
  \left(
  \array{
     && CE(\mathfrak{g})
     \\
     & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
     \\
     \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})
  }
  \right)
  \,.
$$

### Invariant polynomials and Chern-Simons elements

A [[cocycle]] in the [[∞-Lie algebra cohomology]] of the [[∞-Lie algebra]] $\mathfrak{g}$ is a closed element in the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$.

An [[invariant polynomial]] $\langle -\rangle$ on $\mathfrak{g}$ is a closed element in the Weil algebra $\langle -\rangle \in W(\mathfrak{g})$, subject to the additional condition that it its entirely in the shifted copy of $\mathfrak{g}$, $\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1])$.

$$
  \langle -\rangle  \in \wedge^\bullet( \mathfrak{g}^*[1] )
$$

$$
  d_{W(\mathfrak{g})} \langle -\rangle = 0
  \,.
$$

For $x \in \mathfrak{g}$ an element of the $\infty$-Lie algebra, let

$$
  \iota_x : W(\mathfrak{g}) \to W(\mathfrak{g})
$$

the evident operation of contraction with $x$

$$
  \iota_x : t \mapsto t(x)
$$

$$
  \iota_x : \sigma(t) \mapsto 0
$$

extended as a graded derivation. Then the [[Lie derivative]]

$$
  \mathcal{L}_x
   \coloneqq ad_x
   \coloneqq
   [d_{W(\mathfrak{g})}, \iota_x]
   \colon
   W(\mathfrak{g}) \to W(\mathfrak{g})
$$

encodes the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. By the above definition of an [[invariant polynomial]] $\langle - \rangle$, we have

$$
  \iota_x \langle - \rangle = 0
$$

and

$$
  d_{W(\mathfrak{g})} \langle - \rangle = 0
$$

and hence

$$
  ad_x \langle -\rangle  = 0
  \,.
$$

Since the cohomology of $W(\mathfrak{g})$ is trivial, there is necessarily for each invariant polynomial an element $cs_{\langle -\rangle}$ such that

$$
  d_{W(\mathfrak{g})} cs_{\langle -\rangle} =
  \langle -\rangle
  \,.
$$

This is the [[Chern-Simons element]] of the invariant polynomial. Notice, crucially, that this is ingeneral _not_ restricted to the shifted part $\wedge^\bullet (\mathfrak{g}^*[1])$ Its restriction

$$
  \mu_{\langle -\rangle}
  \coloneqq
  cs_{\langle - \rangle}|_{\wedge^\bullet \mathfrak{g}^*}
$$

to the unshifted copy, hence to the [[Chevalley-Eilenberg algebra]], is the cocycle that is in transgression with $\langle - \rangle$.

For

$$
  (A,F_A) \colon W(\mathfrak{g}) \to \Omega^\bullet(X)
$$

a collection of $\mathfrak{g}$-valued differential forms (as [above](LieAlgValuedForms)) and $\langle -\rangle  : CE(b^{n-1}\mathbb{R}) \to W(\mathfrak{g})$ an [[invariant polynomial]], the composite

$$
  \langle F_A\rangle
  :
  CE(b^{n-1}\mathbb{R})
  \stackrel{\langle - \rangle}{\to}
  W(\mathfrak{g})
  \stackrel{(A,F_A)}{\to}
  \Omega^\bullet(X)
$$

is the corresponding [[curvature characteristic form]], a closed $n$-form on $X$. For $(\langle - \rangle, cs) : W(b^{n-1}) \to W(\mathfrak{g})$ the corresponding [[Chern-Simons element]] we have that $cs(A,F_A)$ is the corresponding [[Chern-Simons form]] on $X$.

## Examples

### Weil algebra of a Lie algebra {#WeilofLieAlg}

Let $\mathfrak{g}$ be a finite dimensional [[Lie algebra]]. This Lie algebra regarded as a [[Lie algebroid]] has as base manifold the point, $X_0 = pt$. Its algebra of functions is accordingly the ground field, and the algebra $\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^*$ is just a [[Grassmann algebra]].

The [[Chevalley-Eilenberg algebra]] is

$$
  CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}})
  \,,
$$

where the differential acts on the elements of $\mathfrak{g}^*$ in degree 1 by the linear dual of the Lie bracket.

$$
  d \mathfrak{g}|_{\mathfrak{g}^*} = [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*
  \,.
$$

The corresponding Weil algebra is obtained by adding another copy of $\mathfrak{g}^*$ in degree 2

$$
  W(\mathfrak{g}) = (\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W(\mathfrak{g})})
$$

where with $\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1]$ the degree shift isomorphism, the differential acts as

$$
  d_{W(\mathfrak{g})}|_{\mathfrak{g}^*}
  :
  [-,-]^* + \sigma
$$

$$
  d_{W(\mathfrak{g})}|_{\mathfrak{g}^*[1]}
  :
  \sigma \circ d_{CE(\mathfrak{g})} \circ \sigma^{-1}
  \,.
$$

For illustration, we spell this out in a basis.

Let $\{t_a\}_a$ be a basis for the underlying vector space of $\mathfrak{g}$ and let $\{C^a{}_{b c}\}$ be the corresponding structure constants of the Lie bracket

$$
  [t_b, t_c] = C^a{}_{b c} t_a
  \,.
$$

Then the Chevalley-Eilenberg algebra is generated on generators $\{t^a\}$ of degree 1, on which the differential acts as

$$
  d_{CE(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
  \,.
$$

The Weil algebra in turn is generated from these generators $\{t^a\}$ in degree 1 and generators $\{r^a\}$ in degree 2, with differential given by

$$
  d_{W(\mathfrak{g})} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$

$$
  d_{W(\mathfrak{g})} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c
  \,.
$$

### Weil algebra of a 0-Lie algebroid

A 0-[[truncated]] Lie algebroid is one for which the chain complex of modules over the $T$-algebra in degree 0 vanishes:

$$
  CE(\mathfrak{a}) = (\wedge^\bullet_A (\mathfrak{a}^*), d_{CE(\mathfrak{a})})
  = (A, d = 0)
  \,.
$$

For instance for $T$=[[CartSp]] the theory of [[smooth algebra]]s, any [[smooth manifold]] $X$ regarded as an
[[L-∞ algebroid]] is a 0-Lie algebroid with $CE(X) = C^\infty(X)$ the [[smooth algebra]] of [[smooth function]]s on $X$.

+-- {: .num_prop}
###### Observation

The Weil algebra of a 0-Lie algebroid $X$ is the [[Kähler differential|Kähler]] [[de Rham complex]] of $A = CE(X)$:

$$
  W(\mathfrak{a}) =
  (\Omega^\bullet(A), d_{dR})
  \,.
$$

=--

This Weil algebra is the Chevalley-Eilenberg algebra of the
[[tangent Lie algebroid]] $T X$ of $X$, which is the [[de Rham algebra]]
$\Omega^\bullet(X)$ of $X$:

$$
  W(X) = CE(T X) = (\Omega^\bullet(X), d_{dR})
  \,.
$$

## Related concepts

* [[∞-Lie algebra cohomology]]

* [[Chevalley-Eilenberg algebra]]

* **Weil algebra**

* [[invariant polynomial]]

* [[Sullivan model of free loop space]]

## References

Among the original references on Weil algebras for ordinary Lie algebras is

* [[Henri Cartan]], _Cohomologie réelle d'un espace fibré principal diffrentielle_ I, II, Séminaire Henri Cartan,
1949/50, pp. 19-01 – 19-10 and 20-01 – 20-11, CBRM, (1950).

and

* {#Cartan50} [[Henri Cartan]], *Notions d'algébre différentielle; application aux groupes de Lie et aux variétés ou opère un groupe de Lie* , Colloque de topologie (espaces fibrs), Bruxelles, (1950), pp. 15–27.

This also explains the use of the Weil algebra in the calculation of the [[equivariant cohomology|equivariant]] [[de Rham cohomology]] of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book

A clasical textbook account of standard material is in chapter VI, vol III of

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)
 {#GHV}

Some remarks on the notation there as compared to ours: our $d_W$ is their $\delta_W$ on p. 226 (vol III). Their $\delta_E$ is our $d_{CE}$. Their $\delta_\theta$ is our $d_\rho$ ($\theta$/$\rho$ denoting the representation)..

In the context of [[equivariant de Rham cohomology]]:

* {#AtiyahBott84} [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology 23, 1 (1984) (<a href="https://doi.org/10.1016/0040-9383(84)90021-1">doi:10.1016/0040-9383(84)90021-1</a>, [pdf](https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))

* {#Kalkman93} [[Jaap Kalkman]], _BRST model applied to symplectic geometry_, Ph.D. Thesis, Utrecht, 1993 ([arXiv:hep-th/9308132](https://arxiv.org/abs/hep-th/9308132), [cds:9308132](http://cds.cern.ch/record/568522), [euclid:1104252784](http://projecteuclid.org/euclid.cmp/1104252784))

and with an eye towards [[supersymmetry]]:

* {#Miettinen96} Mauri Miettinen, _Weil Algebras and Supersymmetry_ ([arXiv:hep-th/9612209](https://arxiv.org/abs/hep-th/9612209), [cds:317377](http://cds.cern.ch/record/317377), [spire:427720](http://inspirehep.net/record/427720))

* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))

The (obvious but conceptually important) observation that [[Lie algebra-valued 1-forms]] regarded as morphisms of graded vector spaces $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the Weil algebra $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the _universal $\mathfrak{g}$-connection_ appears in early articles for instance highlighted on p. 15 of

* Franz W. Kamber; Philippe Tondeur, _Semisimplicial Weil algebras and characteristic classes for foliated bundles in Cech cohomology_ , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283--294. Amer. Math. Soc., Providence, R.I., (1975).

A survey of Weil algebras for Lie algebras is also available at

* [[eom|Encyclopedia of Mathematics]]: [Weil algebra of a Lie algebra](http://eom.springer.de/W/w130050.htm)

Weil algebra for [[L-infinity algebra]]s and their role in defining [[invariant polynomial]]s and [[Chern-Simons element]]s on $\infty$-Lie algebras from [[infinity-Lie algebra cohomology|L-infinity algebra cocycle]] are considered in

* {#SSSI} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]

The abstract characterization is due to

* {#FreedHopkins13} [[Daniel Freed]], [[Michael Hopkins]], _Chern-Weil forms and abstract homotopy theory_, Bull. Amer. Math. Soc. 50 (2013), 431-468 ([arXiv:1301.5959](http://arxiv.org/abs/1301.5959))

Further discussion of Weil algebras for the [[string Lie 2-algebra]]:

* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 ([arXiv:1903.02873](https://arxiv.org/abs/1903.02873))

[[!redirects Weil algebras]]


Weil algebra abstractly -- table

**[[Chevalley-Eilenberg algebra]] CE $\leftarrow$ [[Weil algebra]] W $\leftarrow$ [[invariant polynomials]] inv**

[[differential forms]] on [[moduli stack]] $\mathbf{B}G_{conn}$ of [[principal connections]] ([Freed-Hopkins 13](Weil+algebra#FreedHopkins13)):

$$
  \array{
    CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G)
    \\
    \uparrow && \uparrow
    \\
    W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g}))
    \\
    \uparrow && \uparrow
    \\
    inv(\mathfrak{g})
    &\simeq&
    \Omega^\bullet(\mathbf{B}G_{conn})
    & \simeq &
   \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G)
  }
$$


Weil cohomology theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Weil cohomology theory_ is defined to be a [[cohomology theory]] on a suitable class of [[projective varieties]] which satisfies some natural set of [[axioms]] among which is notably [[Poincaré duality]] and the existence of a [[Lefschetz fixed point theorem]].

These axioms are named after [[André Weil]], who noticed that the existence of such a [[cohomology theory]] would already imply the [[Weil conjectures]] about the behaviour of the number of points in [[algebraic varieties]].

Examples of Weil cohomology theories, hence of [[cohomology theories]] satisfying these axioms, are the variants of [[étale cohomology]] known as
_[[l-adic cohomology]]_ or better _[[pro-étale cohomology]]_.

## Related concepts

* [[basics of étale cohomology]]

* [[motive]], [[motivic cohomology]]

* [[Weil conjecture]], [[standard conjectures on algebraic cycles]]

## References

Reviews are in

* Mircea Mustaţă, _Weil cohomology theories and the Weil conjectures_ [pdf](http://www.math.lsa.umich.edu/~mmustata/lecture5.pdf)

* [[Alain Connes]], [[Matilde Marcolli]], Section 8.1 of _Noncommutative Geometry, Quantum Fields and Motives_  ([pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf))

* Wikipedia, _[Weil cohomology theory](http://en.wikipedia.org/wiki/Weil_cohomology_theory)_

[[!redirects Weil cohomology theories]]

[[!redirects Weil cohomology]]


Weil conjecture on Tamagawa numbers

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$.  The **mass** of $q$ is a weighted count of the number of quadratic forms in the [[genus]] of $q$, up to isomorphism (weighted by multiplicity).  The [[Smith-Minkowski-Siegel mass formula]] gives a (complicated but computable) formula for the mass of $q$.

Over [[number fields]], ideas of Tamagawa and Weil allow a reformulation of this formula as the statement that the [[Tamagawa number]] of a certain [[algebraic group]] associated to $q$ is equal to 1.  _Weil's conjecture_ is then the statement, now a theorem of [[Robert Langlands]], K. F. Lai and Robert Kottwitz, that the [[Tamagawa number]] of any semisimple simply-connected [[algebraic group]] is equal to 1.

There is [[function field analogy|analogue]] of the conjecture for [[function fields]], and it has been proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]].

## The statement

### Number field case
 {#NumberFieldCase}

Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$.

+-- {: .num_defn}
###### Definition
The **mass** of $q$ is the sum
  \[ \sum_{q'} \frac{1}{|O_{q'}(\mathbf{Z})|} \]
taken over the positive-definite [[quadratic forms]] $q'$ in the genus of $g$.
=--

Let $\mathbf{A}$ be the ring of [[adeles]], a [[locally compact]] [[commutative ring]] containing $\mathbf{Q}$ as a [[discrete]] subring.

+-- {: .num_defn}
###### Definition
Let $O_q(\mathbf{A})$ denote the [[automorphism group]] of $q_\mathbf{A}$, the [[base change]] of $q$ to $\mathbf{A}$.  Let $SO_q(\mathbf{A}) \subset O_q(\mathbf{A})$ denote the subgroup of [[automorphisms]] with [[determinant]] 1.
=--

$SO_q(\mathbf{A})$ is a [[locally compact]] [[topological group]] containing $SO_q(\mathbf{Q})$ as a [[discrete]] subgroup and $SO_q(\hat{\mathbf{Z}} \times \mathbf{R})$ as a [[compact]] [[open]] subgroup.

+-- {: .num_theorem}
###### Theorem
**(Tamagawa-Weil reformulation of Siegel mass formula)**.
Let $\mu_{\mathrm{Tam}}$ denote the [[Tamagawa measure]].  Then
  \[ \mu_{\mathrm{Tam}}(SO_q(\mathbf{Q}) \backslash SO_q(\mathbf{A})) = 2. \]
Equivalently,
  \[ \mu_{\mathrm{Tam}}(Spin_q(\mathbf{Q})\backslash Spin_q(\mathbf{A})) = 1, \]
where $\Spin_q$ is the 2-fold universal cover of $SO_q$.
=--

+-- {: .num_theorem}
###### Theorem
**(Langlands-Lai-Kottwitz, "[[Weil conjecture]]")**.
Let $G$ be a semisimple simply-connected [[algebraic group]] over $\mathbf{Q}$.  Then
  \[ \mu_{\mathrm{Tam}}(G(\mathbf{Q})\backslash G(\mathbf{A})) = 1. \]
=--

### Function field case
 {#FunctionFieldCase}

Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$.  Let $K_X$ denote the [[function field]] of $X$.  For $x \in X$, write $O_x$ for the [[completion]] of the [[local ring]] at $x$ and $K_x$ for its [[fraction field]].

+-- {: .num_defn}
###### Definition
The **[[ring of adeles]] of $K_X$** is defined as
  \[ \mathbf{A}_X = \product^{res}_x K_x \subset \product_x K_x, \]
i.e. the subgroup consisting of elements $\{g_x\}_{x \in X}$ such that $g \in G(O_x)$ for all but finitely many $x$.
=--

$\mathbf{A}_X$ is a [[locally compact]] [[commutative ring]] with [[discrete]] subring $K_X \subset \mathbf{A}_X$.

Let $G_0$ be a semisimple simply-connected linear [[algebraic group]] over $K_X$.  Then $G_0(K_X) \subset G_0(\mathbf{A})$ is a [[discrete]] subgroup of the [[locally compact]] group $G_0(\mathbf{A})$.  One defines a [[Tamagawa measure]] on $G(\mathbf{A})$ in a similar way as usual, i.e. by choosing a [[differential form]] and multiplying the forms on $G_0(K_x)$ ($x \in X$).  Then the [[function field]] version of [[Weil's conjecture]] is

+-- {: .num_theorem}
###### Theorem
**(Gaitsgory-Lurie, "Weil conjecture for function fields")**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$.  Then
  \[ \mu_{\mathrm{Tam}}(G_0(K_X)\backslash G_0(\mathbf{A}_X)) = 1. \]
=--

This was proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]].  They reformulated the conjecture as a statement about the [[cohomology]] of the [[moduli stack of G-principal bundles]] $Bun_G(X)$ on $X$, in view of the [[function field analogy]]:

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

Under the [[function field analogy]], a [[global field]] $K_X$ such as a [[function field]] or a [[number field]] is interpreted as the field of [[global sections]] of the [[rational functions]] on an [[arithmetic curve]] $X$ over a [[finite field]] $\mathbb{F}_q$ or "over $\mathbb{F}_1$" (the would-be [[field with one element]]), respectively. Moreover, under this [[analogy]]

* the [[ring of adeles]] $\mathbb{A}_X$ is the ring of functions on all punctured [[formal disks]] in $X$ subject to the condition that all but at most finitely many of them extend to the un-punctured disk;

* accordingly $G(\mathbb{A}_X)$ is the group of $G$-valued such functions;

* the [[quotient]] $K_{X}\backslash \mathbb{A}_{X}$ is hence the quotient of such functions on punctured formal disks around finitely many points by the functions on $\Sigma$ with these finitely many points removed; and similarly $G(K_X)\backslash G(\mathbb{A}_X)$ is the quotient of group-valued such function;

* the ring $\mathcal{O}$ is the ring of functions on all formal disks in $\Sigma$;

* hence the further double [[quotient stack]]

  $$
    Bun_G(X) = G(K_X)\backslash G(\mathbb{A}_X)//G(\mathcal{O})
  $$

  is the [[groupoid]] of [[Cech cohomology|Cech cocycles]] with Cech coboundaries between them for $G$-[[principal bundles]] relative to [[covers]] of $\Sigma$ with patches being the complement of finitely many points and the formal disks around these points.

For more on this see at _[moduli space of bundles -- over curves](http://ncatlab.org/nlab/show/moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.

=--

First they proved a [[Grothendieck-Lefschetz trace formula]] for $Bun_G(X)$, generalizing work of [[Kai Behrend]]:

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

**(Gaitsgory-Lurie, "Grothendieck-Lefschetz trace formula for $Bun_G(X)$")**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$.  Then
  \[ \frac{|Bun_G(X)(\mathbf{F}_q)|}{q^{\dim(Bun_G(X))}} = \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell)  \]
where $\overline{Bun}_G(X)$ denotes the [[base change]] of $Bun_G(X)$ to the [[algebraic closure]] of $\mathbf{F}_q$, where ${\vert -\vert}$ denotes [[groupoid cardinality]], and where $Frob : \overline{Bun}_G(X) \to \overline{Bun}_G(X)$ denotes the [[Frobenius map]].

=--

Then they proved the following result, via [[nonabelian Poincaré duality]] which provides a [[local-global principle]].

+-- {: .num_theorem}
###### Theorem
**(Gaitsgory-Lurie)**.
Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$.  Then
  \[ \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell) = \prod_{x \in X} \frac{q^{d \cdot \deg(x)}}{|G(\kappa(x))|} \]
=--

## References

* Wikipedia, _[Weil conjecture on Tamagawa numbers](https://en.wikipedia.org/wiki/Weil_conjecture_on_Tamagawa_numbers)_

### Work of Gaitsgory-Lurie

A proof of the [[function field]] case is discussed in

* {#GaitsgoryLurie} [[Dennis Gaitsgory]], [[Jacob Lurie]], _Weil's conjecture for function fields_ (2014-2017) [["second draft" pdf](http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf), ["first volume of expanded account" pdf](https://www.math.ias.edu/~lurie/papers/tamagawa-abridged.pdf)]

The proof was announced in

* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincaré Duality_, talk at  [FRG Chern-Simons workshop](http://people.mpim-bonn.mpg.de/teichner/Older/FRG-3.html), Jan. 15-17, 2011

and is outlined in the lecture notes

* [[Jacob Lurie]], _Tamagawa Numbers via Nonabelian Poincare Duality (282y)_, lecture notes, 2014 ([web](http://www.math.harvard.edu/~lurie/282y.html))

See also the shorter lecture notes

* [[Jacob Lurie]], _Tamagawa numbers via nonabelian Poincare duality_, 5 lectures at [Young Topologists Meeting 2014](http://www.math.ku.dk/english/research/conferences/2014/ytm2014), notes taken by [[Aaron Mazel-Gee]], [pdf](http://math.berkeley.edu/~aaron/livetex/lurie-tamagawa-poincare.pdf).

### Previous work

The idea of the relationship between [[Tamagawa numbers]] and [[moduli spaces]] of [[vector bundles]] goes back to [[Günter Harder]], who primarily considered the case $G = SL_n$.

* [[Günter Harder]], _Eine Bemerkung zu einer Arbeit von P. E. Newstead._, Journal für die reine und angewandte Mathematik, 242 (1970): 16-25, [eudml](http://eudml.org/doc/151010).

* [[Günter Harder]], M. S. Narasimhan, _On the cohomology groups of moduli spaces of vector bundles on curves_, Mathematische Annalen, 212 (1975), Issue 3, pp 215-248.

The [[moduli stack]] of [[principal bundles]] was studied in more generality in

* [[Kai Behrend]], Ajneet Dhillon, _On the Motive of the Stack of Bundles_, 2005,  [arXiv](http://arxiv.org/abs/math/0512640).

* [[Kai Behrend]], Ajneet Dhillon, _Connected components of moduli stacks of torsors via Tamagawa numbers._, Canad. J. Math, 2009, [arXiv](http://arxiv.org/abs/math/0503383).

See also

* Aravind Asok, Brent Doran, Frances Kirwan. _Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics._ Bulletin of the London Mathematical Society (2008), [arXiv](http://arxiv.org/abs/0801.4733).

[[!redirects Weil conjecture on Tamagawa number]]

[[!redirects Weil conjectures on Tamagawa numbers]]


Weil conjectures

[[!redirects Weil conjecture]]

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

#Contents#
* table of contents
{:toc}

## Idea

The _Weil conjectures_ are a sequence of [[conjectures]] about counting the number of points of [[algebraic varieties]] $X$ over [[finite fields]]; $\mathbb{F}_p$ and extensions thereof. That is, the number of [[homomorphisms]]

$$
  \operatorname{Spec} \mathbb{F}_{p^n} \longrightarrow X
  \,
$$
for fixed $p$ but all $n \geq 1$.

Effectively the conjecture says that the [[generating function]] for the number of points as $n$ varies -- the [[Weil zeta function]] -- is a [[rational function]] with some nice properties.

It was realized that the all except one of the conjectures (the [[Riemann hypothesis]]) would follow formally from the existence of a suitable [[cohomology theory]] on [[algebraic varieties]] which behaves in essential aspects like [[ordinary cohomology]] of [[topological spaces]] and which in particular satisfies a [[Lefschetz fixed point theorem]] - what is now called a [[Weil cohomology theory]].

Later [[Alexander Grothendieck]] found that the relevant [[cohomology]] theory is [[étale cohomology]] of [[schemes]].

## Related concepts

* [[number theory]]

* [[Riemann zeta function]]

* [[Frobenius endomorphism]]

* [[Weil conjecture on Tamagawa number]]

## References

* [[James Milne]], section 26 of _[[Lectures on Étale Cohomology]]_

* [[Sophie Morel]] _The Weil conjectures, from Abel to Deligne_  ([IAS video](http://video.ias.edu/members/2013/1014-SophieMorel)). Note that the title was chosen as a joke by Morel; she clarifies that there is no known connection between Abel and the Weil conjectures.

* Wikipedia, _[Weil conjectures](http://en.wikipedia.org/wiki/Weil_conjectures)_

* [[Dennis Gaitsgory]],  [[Jacob Lurie]], _Weil's Conjecture for Function Fields_ ([pdf](http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf))

[[!redirects Weil conjectures]]

[[!redirects Weil's conjecture]]
[[!redirects Weil's conjectures]]


Weil divisor

#Contents#
* table of contents
{:toc}

## Idea

An [[algebraic cycle]] of [[codimension]] 1. For more see at _[[divisor (algebraic geometry)]]_.

## References

* [[James Milne]], section 13 of _[[Lectures on Étale Cohomology]]_

[[!redirects Weil divisors]]


Weil group

> Not to be confused with [[Weyl group]].

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $F$ be a [[p-adic field]], with [[residue field]] denoted $\kappa$.

The *Weil group* $W_F$ is the [[subgroup]] of the [[Galois group]] $\mathrm{Gal}(\overline{F}/F)$ defined as the [[inverse image]] of [[Frobenius automorphisms]] $\mathrm{Frob}^{\mathbb{Z}}\subset \mathrm{Gal}(\overline{\kappa}/\kappa)$ under the [[surjective]] map $\mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)$.

## Related entries

* [[Weil-Deligne representation]]

* [[local Langlands conjecture]]

## References

* {#Tate77} [[John Tate]], Section 1 in: *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])

See also:

* Wikipedia, *[Weil group](https://en.wikipedia.org/wiki/Weil_group)*

[[!redirects Weil groups]]


Weil reciprocity law

#Contents#
* table of contents
{:toc}

## Idea

The __Weil reciprocity law__ is a theorem of [[André Weil]] about the [[function field]] $K(C)$ of an [[algebraic curve]] $C$ over an [[algebraically closed field]] $K$.

* Askold Khovanskii, _Logarithmic functional and the Weil reciprocity law_, [pdf](http://www.math.toronto.edu/askold/Khovanskii-zima.pdf)
* enclyclo.co.uk [Weil reciprocity law](http://www.encyclo.co.uk/define/Weil%20reciprocity%20law)

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## References

The following two articles make parallel between some notions of [[QFT]] and of number theory and in particular about the analogy between the Weil reciprocity law for [[function field]]s and the [[Takahashi-Ward identities]] of [[quantum field theory]]:

* [[Leon Takhtajan]], _Quantum field theories on algebraic curves and A. Weil reciprocity law_, [arxiv/0812.0169](http://arxiv.org/abs/0812.0169)

* Leon Takhtajan, _Quantum field theories on an algebraic curve_, [pdf](http://www.math.sunysb.edu/~leontak/Takhtajan-00.pdf), 2000

[[!redirects Weil reciprocity]]


Weil uniformization theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _uniformization theorem_ for [[principal bundles]] over [[algebraic curves]] $X$ (going back to [[André Weil]]) expresses the [[moduli stack of principal bundles]] on $X$ as a double [[quotient stack]] of the $G$-valued [[Laurent series]] around finitely many points by the product of the $G$-valued [[formal power series]] around these points and the $G$-valued functions on the complement of theses points.

If a single point $x$ is sufficient and if $D$ denotes the [[formal disk]] around that point and $X^\ast, D^\ast$ denote the complements of this point, respectively then the theorem says for suitable [[algebraic group]] $G$ that there is an equivalence of [[stacks]]

$$
   [X^\ast, G] \backslash [D^\ast, G] / [D,G]
   \simeq
   Bun_X(G)
 \,,
$$

between the double [[quotient stack]] of $G$-valued functions ([[mapping stacks]]) as shown on the left and the [[moduli stack of G-principal bundles]] over $X$, as shown on the right.

The theorem is based on the fact that $G$-bundles on $X$ trivialize on the complement of finitely many points  and that this double quotient then expresses the $G$-[[Cech cohomology]] with respect to the cover given by the complement of the points and the [[formal disks]] around them.

For details see at _[moduli stack of bundles -- over curves](http://ncatlab.org/nlab/show/moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.

## Applications

The theorem is a key ingredient in the [[function field analogy]] where for $K$ a [[global field]] the nonabelian generalization of quotients of the [[idele class group]] by [[integral adeles]]

$$
  GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K)
$$

are analogous to the moduli stack of $G$-bundles. This motivates notably the [[geometric Langlands correspondence]] as a geometric analog of the number-theoretic [[Langlands correspondence]].

## Related concepts

* [[differential cohesion and idelic structure]]

## References

Review is for instance

* {#Sorger99} [[Christoph Sorger]], _Lectures on moduli  of principal $G$-bundles over algebraic curves_, 1999 ([pdf](http://users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf))

See also

* [[Jochen Heinloth]], _Uniformization of $\mathcal{G}$-Bundles_ ([pdf](https://dare.uva.nl/search?identifier=35a7a6fe-9e92-4e37-8c2c-4b357709cc6f))

For more references see at _[[moduli stack of bundles]]_.

[[!redirects Weil uniformization]]


Weil zeta function

#Contents#
* table of contents
{:toc}

## Idea

The _Weil zeta function_ is a [[zeta function]] for [[arithmetic varieties]] over [[finite fields]] which, under the [[function field analogy]], is [[analogy|analogous]] to the [[Dedekind zeta function]] for [[number fields]] $K$ (and hence of the [[Riemann zeta function]], for $K = \mathbb{Q}$).

The [[Weil conjectures]], and their proof, concern the properties of the Weil zeta function.

## Properties

### Relation to other zeta-, theta- and L-functions

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

### Function field analogy

[[!include function field analogy -- table]]

## References

Introductions and lecture notes include

* [[Marc Hindry]], section 2.3 of _Introduction to zeta and $L$-functions from arithmetic geometry and some applications_  ([pdf](http://www.math.jussieu.fr/~hindry/Notes_rev_Brasilia.pdf))

* Daqing Wan, _Lectures on zeta functions over finite fields_, 2007 ([pdf](http://www.math.uci.edu/~dwan/gottingen.pdf))

* E. Kowalski, section 1.6 in _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

[[!redirects Weil zeta functions]]


Weil-Deligne representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let

* $F$ be a [[p-adic field]] with $\kappa$ denoting its [[residue field]];

* $\Vert\sigma\Vert$ denotes the [[valuation]] of the corresponding element of $F^{\times}$ under the isomorphism of local [[class field theory]].

\begin{definition}\label{WeilDeligneRepresentation}
**(Weil-Deligne representation)**\linebreak
A *Weil-Deligne representation* is a [[pair]] $(\rho_{0},N)$ where

* $\rho_{0} \colon W_{F} \to GL_{n}(\mathbb{C})$ is a [[linear representation]] of the [[Weil group]] of $F$

and

* $N$ is a nilpotent _monodromy operator_

satisfying

$$
  \rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}
   \;=\;
  \left\Vert \sigma \right\Vert N
$$

for all $\sigma\in W_{F}$.

\end{definition}

## Related entries

* [[local Langlands correspondence]]

## References

* {#Tate77} [[John Tate]], Section 4 in: *Number theoretic background*, in: *Automorphic forms, representations and L-functions*, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI ([ISBN:978-0-8218-3371-1](https://bookstore.ams.org/pspum-33-2), [pdf](http://www.math.ucsd.edu/~csorense/teaching/math205/Tate_Weil.pdf), [[TateNumberTheory.pdf:file]])

* [[Robin Zhang]], *Weil-Deligne Representations I -- Local Langlands seminar* ([pdf](http://math.columbia.edu/~rzhang/files/Weil-Deligne%20Representations%20I.pdf), [[ZhangWeilDeligneRepresentations.pdf:file]])

[[!redirects Weil-Deligne representations]]


Weil-Petersson metric

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

#Contents#
* table of contents
{:toc}

## Idea

A natural [[Riemannian metric]] on a [[moduli space of curves]] or more generally a [[moduli space of varieties]], such as a [[moduli space of Calabi-Yau spaces]].

## References

* Wikipedia, _[Weil-Petersson metric](https://en.wikipedia.org/wiki/Weil%E2%80%93Petersson_metric)_

[[!redirects Weil-Petersson metrics]]


Weil-étale site

[[!redirects Weil-étale topology for arithmetic schemes]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Grothendieck topology]] (conjectured) for [[arithmetic schemes]] ([Lichtenbaum](#Lichtenbaum)).

Comparison to the standard [[étale site]] is in ([Morin 11](#Morin11)).

## References

* {#Lichtenbaum} [[Stephen Lichtenbaum]], _The Weil-étale topology for Number Rings_, Ann. of Math

* {#Morin11} [[Baptiste Morin]], _On the Weil-étale cohomology of number fields_, Trans. Amer. Math. Soc. 363 (2011), 4877-4927_ ([pdf](http://www.math.uni-muenster.de/reine/u/baptiste.morin/papers/Flask-topos.pdf))


[[!redirects Weil-étale topology for arithmetic schemes]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Grothendieck topology]] (conjectured) for [[arithmetic schemes]] ([Lichtenbaum](#Lichtenbaum)).

Comparison to the standard [[étale site]] is in ([Morin 11](#Morin11)).

## References

* {#Lichtenbaum} [[Stephen Lichtenbaum]], _The Weil-étale topology for Number Rings_, Ann. of Math

* {#Morin11} [[Baptiste Morin]], _On the Weil-étale cohomology of number fields_, Trans. Amer. Math. Soc. 363 (2011), 4877-4927 ([pdf](http://www.math.uni-muenster.de/reine/u/baptiste.morin/papers/Flask-topos.pdf))


Weimin Chen

* [webpage](https://www.math.umass.edu/directory/faculty/weimin-chen)

## Selected writings

Introducing [[Chen-Ruan cohomology]] for [[orbifolds]]:

* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))

On [[mapping stacks]] between [[orbifolds]]:

* [[Weimin Chen]], _On a notion of maps between orbifolds, I. Function spaces_, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 ([doi:10.1142/S0219199706002246](https://doi.org/10.1142/S0219199706002246))

On [[symplectic orbifolds]]:

* [[Weimin Chen]], _Resolving symplectic orbifolds with applications to finite group actions_, Journal of Gökova Geometry Topology **12** (2018), 1-39 ([arXiv:1708.09428](https://arxiv.org/abs/1708.09428))

category: people


Weinstein symplectic category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

When [[symplectic geometry]] is used to model [[mechanics]] in [[physics]], then a [[symplectic manifold]] $(X,\omega)$ encodes the [[phase space]] of a [[mechanical system]] and a [[symplectomorphism]]

$$
  \phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)
$$

encodes a process undergone by this system, for instance the time evolution induced by a [[Hamiltonian vector field]].

Now the [[graph]] of a [[symplectomorphism]] $\phi$ is a [[Lagrangian submanifold]] of the [[Cartesian product]] space $X_1 \times X_2$ regarded as a [[symplectic manifold]] with symplectic form $p_1^\ast \omega_1 - p_2^\ast \omega_2$. In other words, a symplectomorphism $\phi$ as above constitutes a [[Lagrangian correspondence]] between $(X_1,\omega_1)$ and $(X_2, \omega_2)$. See for instance ([Cattaneo-Mnev-Reshetikhin 12](#CattaneoMnevReshetikhin12)) for a review.

This suggests that instead of the [[category]] whose [[objects]] are [[symplectic manifolds]] and whose [[morphisms]] are [[symplectomorphisms]], one might consider a kind of [[category of correspondences]] whose objects are symplectic manifolds, and whose morphisms include [[Lagrangian correspondences]], so that [[composition]] is given by forming the [[fiber product]] along adjacent legs of [[correspondences]].

[[Alan Weinstein]] called this would-be category the _symplectic category_ and suggested that it is the natural [[domain]] for [[geometric quantization]].

However, take at face value, symplectic manifolds with [[Lagrangian correspondences]] between them do not quite form a [[category]], since the usual [[composition]] is only well-defined when the intersection of $L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times X_3$ is [transverse](http://ncatlab.org/nlab/show/transversal+maps).

Proposals for how to rectify this are in [Wehrheim & Woodward](#WehrheimWoodward) and in [Kitchloo](#Kitchloo) (by turning this into an [[(infinity,1)-category]]).

## Refinements

### Prequantum correspondences
 {#PrequantumCorrespondences}

A refinement of the symplectic category to [[prequantum geometry]] is the following (see [S 13](#SyntheticQFT)).

Write $\mathbf{B}U(1)_{conn}$ for the [[moduli stack]] of smooth [[circle group]]-[[principal connections]]. Write [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]], and $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$ for the corresponding [[slice (∞,1)-topos]]. Finally write

$$
  Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn})
$$

for the [[(∞,1)-category of correspondences]] in $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$.

An object in here is a [[prequantum geometry]] $(X,\nabla)$ given by a map

$$
  \array{
    X
    \\
    \downarrow^{\mathrlap{\nabla}}

    \\
    \mathbf{B}U(1)_{conn}
  }
  \,.
$$

Under the [[curvature]] map $F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl}$ this maps to a [[presymplectic structure]]

$$
  (X,\omega) = (X, F_{\nabla})
  \,.
$$

If here $\omega$ is non-degenerate, this is a symplectic structure as in Weinstein's symplectic category.

Moreover, a [[morphism]] $(X_1,\nabla_1) \to (X_2,\nabla_2)$ is a [[diagram]] of the form

$$
  \array{
    && Z
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1
    && \swArrow_{\eta} &&
    X_2
    \\
    & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}}
    \\
    && \mathbf{B}U(1)_{conn}
  }
  \,,
$$

hence a [[correspondence]] space (smooth $\infty$-groupoid) $Z$ over $X$ and $Y$ together with an [[equivalence in an (∞,1)-category]]

$$
  \eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1
  \,.
$$

On the underlying [[curvatures]] this implies that

$$
  i_2^\ast \omega_2 = i_1^\ast \omega_1
  \,.
$$

Hence if $Z \to X \times Y$ is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.

### Motivic stabilization

[[Nitu Kitchloo]] defines the [[stable (infinity,1)-category|stable]] symplectic category $\mathbb{S}$, which has as [[objects]] [[symplectic manifolds]], and [[morphisms]] are certain [[Thom spectra]]
associated to [[Lagrangian correspondences]] $\overline{M} \times N$, where $\overline{M}$ denotes the conjugate with symplectic form $-\omega$.  One can view this as a category of symplectic [[motives]].

Considering an oriented version of the category $\mathbb{S}$, there is a canonical [[fiber functor]] $F : M \mapsto \mathbb{S}(pt, M)$, and one may consider the [[motivic Galois group]] $G$ of monoidal [[automorphisms]] of $F$.  ([Kitchloo 12, question 8.6, p. 19](#Kitchloo)).

It turns out to have a natural subgroup which is isomorphic to the quotient of the [[Grothendieck-Teichmüller group]].

## Related concepts

* [[Fukaya category]]

* [[Lagrangian correspondences and category-valued TFT]]

## References
 {#References}

The way that [[Lagrangian correspondences]] encode [[symplectomorphisms]] in [[symplectic geometry]] and hence evolution in [[mechanics]] is reviewed (and put in the broader context of [[BV-BRST formalism]]) in

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical and quantum Lagrangian field theories with boundary_ ([arXiv:1207.0239](http://arxiv.org/abs/1207.0239))
 {#CattaneoMnevReshetikhin12}

In his work on Fourier integral operators,

* [[Lars Hörmander]], _Fourier Integral Operators I._, Acta Math. 127 (1971) 79–183.
14

following

* [[Victor Maslov]], _Theory of Perturbations and Asymptotic Methods_, (in Russian) Moskov. Gos. Univ., Moscow (1965).

observed that, under a transversality assumption, the set-theoretic composition of two [[Lagrangian submanifolds]] is again a Lagrangian submanifold, and that this composition is a "[[classical limit]]" of the composition of certain [[linear operators]].

Shortly thereafter,

* [[Jędrzej Śniatycki]], W.M. Tulczyjew, *Generating forms of Lagrangian submanifolds*, Indiana Univ. Math. J. **22** (1972) 267-275

defined symplectic relations as [[isotropic submanifolds]] of products and showed that this class of relations was closed under "clean" composition.  Following in part some (unpublished) ideas of [[Alan Weinstein]],

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Some problems in integral geometry and some related problems in microlocal analysis_, Amer. J. Math. 101 (1979), 915–955 ([JSTOR](http://www.jstor.org/stable/2373923))

observed that the linear canonical relations (i.e., lagrangian subspaces of products of [[symplectic vector spaces]]) could be considered as the morphisms of a category, and they constructed a
partial [[quantization]] of this category (in which lagrangian subspaces are enhanced by halfdensities.)
The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally

* [[Alan Weinstein]], _Symplectic Manifolds and Their Lagrangian Submanifolds_, Advances in Math. **6** (1971) 329346 \[doi:10.1016/0001-8708(71)90020-X\]

* [[Alan Weinstein]], _Symplectic geometry_, Bulletin Amer. Math. Soc. **5** (1981) 1-13 [[doi:10.1090/S0273-0979-1981-14911-9](http://dx.doi.org/10.1090/S0273-0979-1981-14911-9)]

See also:

* W.M.Tulczyjew, S.Zakrzewski, _The category of Fresnel kernels_, J. Geom. Phys. 1:3, 1984, 79--120 doi

Lecture notes reviewing these developments include

* {#Weinstein09} [[Alan Weinstein]], _Symplectic Categories_, proceedings of Geometry Summer School, Lisbon, July 2009 ([arXiv:0911.4133](http://arxiv.org/abs/0911.4133))

from the introduction of which parts of the commented list of references above is taken. Further review includes

* {#Canez11} Santiago Canez, _Double Groupoids, Orbifolds, and the Symplectic Category_ ([arXiv:1105.2592](http://arxiv.org/abs/1105.2592))

Further refinements in [[higher category theory]]:

* {#WehrheimWoodward} [[Katrin Wehrheim]], [[Chris T. Woodward]], _Functoriality for Lagrangian correspondences in Floer theory_, Quantum Topology **1** 2 (2010) 129-170 [[doi:10.4171/qt/4](https://doi.org/10.4171/qt/4), [arXiv:0708.2851](https://arxiv.org/abs/0708.2851)]

* {#Kitchloo} [[Nitu Kitchloo]], _The Stable Symplectic Category and Geometric Quantization_ [[arXiv:1204.5720](http://arxiv.org/abs/1204.5720)]

* [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012 [[arxiv:1212.6905](http://arxiv.org/abs/1212.6905)]

* [[Damien Calaque]], _Three lectures on derived symplectic geometry and topological field theories_, Indagationes Mathematicae **25** 5 (2014) 926–947 [[doi:10.1016/j.indag.2014.07.005](https://doi.org/10.1016/j.indag.2014.07.005)]

* [[Rune Haugseng]], _Iterated spans and classical topological field theories_, Mathematische Zeitschrift **289** 3 (2018) 1427–1488 [[arXiv](https://arxiv.org/abs/1409.0837), [doi:10.1007/s00209-017-2005-x](https://doi.org/10.1007/s00209-017-2005-x)]

A closed [[symmetric monoidal category]] version of the symplectic category and the observation that this hence is a [[categorical semantics]] for [[quantum logic]] qua [[linear logic]] is in

* {#Slavnov05} [[Sergey Slavnov]], _From proof-nets to bordisms: the geometric meaning of multiplicative connectives_, Mathematical Structures in Computer Science __15__:06  (2005) 1151--1178

* [[Sergey Slavnov]], _Geometrical semantics for linear logic (multiplicative fragment)_, Theoretical Computer Science 357, no. 1--3 (2006) 215--229 [doi](https://doi.org/10.1016/j.tcs.2006.03.020)

Remarks about refinements to correspondences of smooth $\infty$-groupoids in the slice over prequantum moduli is in

* {#SyntheticQFT} [[Urs Schreiber]], _[[schreiber:Synthetic Quantum Field Theory]]_, Talk at [CMS Summer Meeting 2013](http://cms.math.ca/Events/summer13/)

String diagrams for the linear and affine Weinstein category using graphical linear algebra

* Cole Comfort, [[Aleks Kissinger]], _A Graphical Calculus for Lagrangian Relations_, In Proceedings ACT 2021. [doi](https://doi.org/10.4204/EPTCS.372.24)

[[!redirects stable symplectic category]]

[[!redirects symplectic category]]

[[!redirects motivic symplectic category]]


Weinstein, the geometry of Lubin-Tate spaces

category: reference

This entry in a note concerning the text

* [[Jared Weinstein]], the geometry of Lubi-Tate spaces,  Lecture 1: Formal groups and formal modules, [pdf](http://www.math.ias.edu/~jaredw/FRGLecture.pdf)

## Table of contents
* Motivation: the local Kronecker-Weber theorem.

* Formal groups and formal $O_F$ -modules

* The invariant differential form, and the logarithm.

* Formal groups: Categorical definition.

* The de Rham complex, and the module $D(G /A)$.

* The crystalline nature of $H^1_{dR}$

* The action of Frobenius.

* The Dieudonné module.

* More on Dieudonné theory (but not quite enough).

* The Grothendieck-Messing crystal.


Weiping Zhang

* [GoogleScholar page](https://scholar.google.com/citations?user=Jx6VIsMAAAAJ)

## Selected writings

On the [[Witten genus]] in the case of [[string^c structure]]:

* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

On the [[Green-Schwarz mechanism]] in [[heterotic string theory]] and [[Hořava-Witten theory]]:

* [[Fei Han]], [[Kefeng Liu]], [[Weiping Zhang]], _Anomaly Cancellation and Modularity. II: $E_8 \times E_8$ case_, Sci. China Math. 60, 985–994 (2017) ([arXiv:1209.4540](https://arxiv.org/abs/1209.4540), [doi:10.1007/s11425-016-9034-1](https://doi.org/10.1007/s11425-016-9034-1))

On [[gravitino]] [[anomaly cancellation]] in [[D=11 N=1 supergravity]]:

* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]

category: people


Weiss topology

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

#Contents#
* table of contents
{:toc}

## Idea

A _Weiss topology_ is a type of [[Grothendieck topology]] that provides a [[site]] of definition for [[(infinity,1)-toposes]] as they appear in [[manifold calculus]].

## Related concepts

* [[Goodwillie calculus]]

* [[manifold calculus]]

## References

* {#BriWei} [[Pedro Boavida de Brito]], [[Michael Weiss]], _Manifold calculus and homotopy sheaves_, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 ([arXiv:1202.1305](http://arxiv.org/abs/1202.1305))

* [MO discussion](http://mathoverflow.net/a/230293/381)


Weitzenböck connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

On a ([[pseudo-Riemannian manifold|pseudo]])[[Riemannian manifold]] the _Weitzenböck connection_ is the [[affine connection]] on the [[tangent bundle]] which has vanishing [[curvature]] but possibly non-vanishing [[torsion]].

This is in contrast to the [[Levi-Civita connection]], for which it is the other way around.

[[!redirects Weitzenboeck connection]]

[[!redirects Weitzenböck connections]]
[[!redirects Weitzenboeck connections]]


Weizhe Zheng

* [webpage](https://server.mcm.ac.cn/~zheng/)

## Selected writings

On gluing (descent) of [[pseudofunctors]] ([[2-functors]]):

* {#Zheng12} [[Weizhe Zheng]], prop. 1.6 of: _Gluing pseudofunctors via $n$-fold categories_, J. Homotopy Relat. Struct. **12** 189–271 (2017) ([arXiv:1211.1877](http://arxiv.org/abs/1211.1877), [doi:10.1007/s40062-016-0126-2](https://doi.org/10.1007/s40062-016-0126-2))

category: people


well group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In the context of [[persistent homology]], the _well group_ ([EMP 11](#EMP11)) of a [[continuous function]] $f$ into a [[metric space]] is a [[homology|homological]] measure of how robust the [[level sets]] of $f$ are against [[deformations]] of $f$. Concretely, the Well group at [[radius]] $r$ of a given point in the [[codomain]] consists of all those [[Čech homology]]-classes of the domain which are reflected in the [[level sets]] of _every_ [[continuous function]] whose values differ from those of $f$ at most by $r$.

As the level varies, the collection of well groups form a [[zigzag persistence module]], also called a *well module* ([EMP 11, Sec. 3](#EMP11)).

Since a well group becomes [[trivial group|trivial]] as soon as one of the [[level sets]] is [[empty set|empty]], the non-triviality of a well group proves that the existence of a [[inhabited set|non-empty]] [[level set]] of $f$ is *robust* within deformations of size $\lt r$. In [[topological data analysis]] this may be used to detect if there are guaranteed to be any data points at all meeting a certain target of indicator values, known with limited precision, see [there for more](topological+data+analysis#StrategyOfPersistentCohomotopy).

But well groups are known not to resolve all relevant cases and are not known to be [[computability|computable]] in all relevant cases ([Franek & Krčál 2016](#FranekKRcal16)). An enhancement of well groups (from [[ordinary homology|homology]] to [[cohomotopy]]) which fixes these problems is [[persistent cohomotopy]] ([Franek & Krčál 2017](#FranekKRcal17), [2018](#FranekKRcal18)).

## Definition

Given a [[continuous function]] $f$ to a [[Euclidean space]] and a choice of [[topological subspace]] $A$ of the latter

$$
  \array{
    f^{-1}(A) &\subset & X
    \\
    \downarrow && \big\downarrow {}^{\mathrlap{f}}
    \\
    A &\subset& \mathbb{R}^n
  }
$$

the _well groups_ at [[radius]] $r \in (0,\infty)$ are the [[intersections]] of the [[Čech homology]] [[homology groups|groups]] of the [[pre-images]] $g^{-1}(A) \subset X$ for all [[continuous functions]] $X \overset{g}{\to} \mathbb{R}^n$ whose [[maximum|maximal]] [[distance]] from $f$ is $\left \vert g-f\right \vert \leq r$.

$$
  W_\bullet(f,r)
  \;\coloneqq\;
  \underset{
    {X \overset{g}{\to} \mathbb{R}^n}
    \atop
    {\left \vert g-f\right \vert \leq r}
  }{\bigcap}
  image
  \Big(
    H_\bullet\big(
      g^{-1}(A)
    \big)
    \overset{
      H_\bullet\big(  g^{-1}(A) \subset X \big)
    }{\longrightarrow}
    H_\bullet
    \big(
      X
    \big)
  \Big)
$$

(e.g. [Franek-Krčál 16, p. 2](#FranekKrcal16))

\begin{remark}\label{NeedForCechHomology}
  Since the [[preimages]] $g^{-1}(A)$ need not be [[CW-complexes]], it is important to use [[Čech homology]] in the above definition. With [[singular homology]] the definition would trivialize ([FK16, p. 3 and Sec. 2](#FranekKrcal16)).
\end{remark}

## Related concepts

* [[persistence module]]

* [[zigzag persistence]]

* [[stability of persistence diagrams]]

## References

The concept of [[well groups]] was introduced in

* {#EMP11} [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Quantifying Transversality by Measuring the Robustness of Intersections*, Foundations of Computational Mathematics, **11** 3  (2011) 345–361 $[$[arXiv:0911.2142](https://arxiv.org/abs/0911.2142)$]$

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *The Robustness of Level Sets*, In: M. de Berg, U. Meyer (eds.) _Algorithms – ESA 2010_. ESA 2010. Lecture Notes in Computer Science **6346** Springer (2010)  $[$[doi:10.1007/978-3-642-15775-2_1](https://doi.org/10.1007/978-3-642-15775-2_1)$]$

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Homology and Robustness of Level and Interlevel Sets*, Homology, Homotopy and Applications, **15** (2013) 51-72 $[$[euclid:1383943667](https://projecteuclid.org/euclid.hha/1383943667)$]$

Review in:

* [[Sara Kališnik]], Section 4.2.2 of: _Persistent homology and duality_ (2013) ([pdf](http://www.matknjiz.si/doktorati/2013/Kalisnik-14521-4.pdf), [[KalisnikPersistent.pdf:file]])

Review, computational analysis and discussion of ([[persistent Cohomotopy|persistent]]) [[Cohomotopy]] as an improvement over homology well groups:

* {#FranekKrcal16} [[Peter Franek]], [[Marek Krčál]], _On Computability and Triviality of Well Groups_, Discrete Comput Geom (2016) 56: 126 ([arXiv:1501.03641](https://arxiv.org/abs/1501.03641), [doi:10.1007/s00454-016-9794-2](https://doi.org/10.1007/s00454-016-9794-2))

* [[Peter Franek]], [[Marek Krčál]], _Persistence of Zero Sets_, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))

* [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology (2018) 1: 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))

Survey:

* [[Peter Franek]], [[Marek Krčál]], _Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory_, talk at [ACAT meeting 2015](https://www2.ist.ac.at/acat) ([pfd slides](https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf))

[[!redirects well groups]]

[[!redirects well module]]
[[!redirects well modules]]


well-connected space

A *well-connected* [[topological space]] is one that satisfies sufficiently strong local connectivity assumptions. These are usually defined in terms of a basis of open sets or a basis of neighbourhoods of the space. Note that this term is not universally defined, but is a placeholder for possibly more complicated adjectives that depend on the situation at hand.

## Examples ##

* A [[locally path-connected space]] has the nice property that path-components are the same as components. It is also true in this case that $X^I \to X\times X$ is an open map.

* A locally path-connected and [[semi-locally simply-connected space]] admits a [[universal covering space]].

* The path fibration $P X \to X$ of a [[path-connected space]] admits local sections if and only if $X$ is [[locally contractible space|locally relatively contractible]].

* [[CW complex]]es are the \'most\' well-connected spaces, having a basis of open sets which are themselves contractible as spaces.

[[!redirects well-connected space]]
[[!redirects well-connected topological space]]
[[!redirects well connected space]]
[[!redirects well connected topological space]]
[[!redirects well-connected spaces]]
[[!redirects well-connected topological spaces]]
[[!redirects well connected spaces]]
[[!redirects well connected topological spaces]]