Banach bundle

# Banach bundles
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## Idea

A Banach bundle is a [[bundle]] in which every [[fibre]] is a [[Banach space]].  Certain other conditions apply.

## Definitions

A __Banach bundle__ is an [[open map|open]] (necessarily [[surjection|surjective]]) [[continuous map]] of [[Hausdorff topological spaces]] $p\colon Y\to B$, each of whose [[fibers]] carries a structure of a [[complex number|complex]] [[Banach space]], this structure being continuous in the base point (in other words, the global operations $\mathbb{C} \times Y \to Y$ of multiplication by a scalar, $Y \times_B Y \to Y$ of addition and $Y \to \mathbb{R}$ of taking the norm are continuous) and such that for every [[net]] $\{y_\alpha\}_{\alpha\in A}$, if ${\|y_{\alpha}\|} \to 0$ and $p(y_\alpha) \to b$, then $y_\alpha \to 0 = 0_b \in p^{-1}(b)$.

We distinguish a different concept of __Banach algebraic bundle__, where the base space $B$ is also a [[Banach algebra]] and the multiplication is defined as a map $\cdot\colon Y\times Y \to Y$ (not only $Y \times_B Y \to Y$), that is we can multiply the points in different fibers, and $p(a \cdot b) = p(a) \cdot p(b)$.

A Banach bundle is a __Hilbert bundle__ if each fiber is a [[separable space|separable]] [[Hilbert space]]. As usual, the inner product can be obtained by the polarization formula $(x,y) \coloneqq \frac{1}{4}({\|x+y\|^2} - {\|x-y\|^2})$ from the norm of a Banach space if the norm satisfies the parallelogram identity.  From this, we infer that for Hilbert bundles, the inner product is continuous as a map $Y \times_B Y \to \mathbb{C}$. Hilbert bundles are important in the study of [[induced representations]] of [[locally compact group]]s, and [[Mackey theory]] in particular; more recently their study is connected to the study of [[Hilbert module]]s.

A __morphism of Banach bundles__ $(p\colon Y \to B)\to (p'\colon Y' \to B)$ over the same base is a morphism of total spaces commuting with the projections, $\mathbb{C}$-linear in each fiber, and preserving the norm. A Banach bundle is sometimes said to be Hilbertizable if it is isomorphic to the underlying Banach bundle of a Hilbert bundle; structurally, there is no difference between a Hilbert bundle and a Hilbertizable Banach bundle (again using the polarisation formula to prove that being a Hilbert space is a [[property-like structure]]).

One also considers Banach $*$-[[star-algebra|algebraic]] bundles, where an antilinear [[involution]] $*$ preserving the norm is involved, is continuous as a global map $Y \to Y$ and is an [[antihomomorphism]] of algebras satisfying $p(y^\ast) = p(y^{-1})$.

## References

* Wikipedia: [Banach bundle](http://en.wikipedia.org/wiki/Banach_bundle_%28non-commutative_geometry%29)

For Banach bundles see ch. 13 in vol. 1 (from page 125; def. 13.4 on p. 127) and for Banach algebraic bundles see from 783 on in vol. 2 of

* J. M. G. Fell, R. S. Doran, _Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles_, Vol. 1. Basic representation theory of groups and algebras. Pure and Applied Mathematics, __125__, Academic Press 1988. xviii+746 pp. [MR90c:46001](http://www.ams.org/mathscinet-getitem?mr=936628) Vol. 2, Banach $*$-algebraic bundles, induced representations, and the generalized Mackey analysis. Pure and Applied Mathematics __126__, Acad. Press 1988. pp. i--viii and 747--1486, [MR90c:46002](http://www.ams.org/mathscinet-getitem?mr=936629)

[[!redirects Banach bundle]]
[[!redirects Banach bundles]]

[[!redirects Banach algebraic bundle]]
[[!redirects Banach algebraic bundles]]

[[!redirects Hilbert bundle]]
[[!redirects Hilbert bundles]]


Banach coalgebra

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# Banach coalgebras
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## Idea

Banach coalgebras (or cogebras) are like [[Banach algebras]], but [[coalgebras]].  The [[dual vector space|dual]] of a Banach coalgebra is a Banach algebra (but not conversely).  We can also consider Banach [[bialgebras]] (or bigebras).

## Definitions

A __Banach coalgebra__, or __Banach cogebra__, is a [[comonoid object]] in the [[monoidal category]] $Ban$ of [[Banach spaces]] with [[short linear maps]] and the [[projective tensor product]].  (Recall that a [[Banach algebra]] is a [[monoid object]] in $Ban$.)

Explicitly, we have:

1.  a [[Banach space]] $A$
1.  a [[short linear map]], the __comultiplication__:
$$ \Delta\colon A \to A {\displaystyle\hat{\otimes}_\pi} A $$
to the [[projective tensor product]];
1.  a short [[linear functional]], the __counit__:
    $$ \epsilon\colon A \to K ,$$
    where $K$ is the [[ground field]];
1.  an [[equation]], the __coassociativity__:
    $$ (\Delta {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = (\id_A {\displaystyle\hat{\otimes}_\pi} \Delta) \Delta x \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} A \cong A {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} A)$$
    for each $x\colon A$;
1.  an equation, the __left coidentity__:
    $$ (\epsilon {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = x \in K {\displaystyle\hat{\otimes}_\pi} A \cong A $$
    for each $x\colon A$;
1.  and an equation, the __right coidentity__:
    $$ (\id_A {\displaystyle\hat{\otimes}_\pi} \epsilon) \Delta x = x \in A {\displaystyle\hat{\otimes}_\pi} K \cong K $$
    for each $x\colon A$.

Technically, we\'ve defined a __counital coassociative Banach coalgebra__.  We can leave out (3,5,6) to get a __non-counital Banach coalgebra__, and (also) leave out (4) to get a __non-coassociative Banach coalgebra__.  Warning: these terms are examples of the [[red herring principle]].  Note that (3) is a [[property-like structure]] (and 4--6 are obviously just [[properties]]).

On the other hand, we can *add* the property of __cocommutativity__:

*  $\tau \Delta x = \Delta x$ for each $x\colon A$,

where the [[braiding]] $\tau\colon A {\displaystyle\hat{\otimes}_\pi} A \to A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $\tau (u \otimes v) = v \otimes u$.  Then we have a __cocommutative Banach coalgebra__.

To __freely adjoin a counit__ to a non-counital Banach coalgebra $A$, take the Banach space $A \oplus_1 K$ (using the $l^1$-[[l-1-direct sum|direct sum]]), let $\Delta_{A \oplus K} (x,c)$ be $(\Delta_A x, 0, 0, c) \in (A {\displaystyle\hat{\otimes}_\pi} A) \oplus_1 A \oplus_1 A \oplus_1 K \cong (A \oplus_1 K) {\displaystyle\hat{\otimes}_\pi} (A \oplus_1 K)$, and let $\epsilon_{A \oplus_1 K} (x,c)$ be $c$.  Then $A \oplus_1 K$ is a counital Banach coalgebra.  (Freely forcing coassociativity or cocommutativity ---or even freely adjoining $\Delta$ in the first place--- is harder.)

The [[category]] __$Ban Coalg$__ of Banach coalgebras has, as [[objects]], Banach coalgebras and, as [[morphisms]], [[short linear maps]] $f\colon A \to B$ with equations
$$ \Delta_B f x = (f {\displaystyle\hat{\otimes}_\pi} f) \Delta_A x $$
and (unless we are allowing non-counital coalgebras)
$$ \epsilon_B f x = \epsilon_A x $$
for all $x\colon A$.  Warning: the term 'homomorphism' is used more generally; see below.

If $A$ and $B$ are Banach coalgebras, then their __[[projective tensor product]]__ $A {\displaystyle\hat{\otimes}_\pi} B$ is a Banach coalgebra, generated by
$$ \Delta_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = \Delta_A x \otimes \Delta_B y \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} (B {\displaystyle\hat{\otimes}_\pi} B) \cong (A {\displaystyle\hat{\otimes}_\pi} B) {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} B) $$
and
$$ \epsilon_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = (\epsilon_A x) (\epsilon_B y) \in K .$$
Similarly (but more simply), the [[ground field]] $K$ is itself a Banach coalgebra, with $\Delta$ and $\epsilon$ both essentially the [[identity map]].  In this way, $Ban Coalg$ becomes a [[symmetric monoidal category]].

The [[full subcategory]] $Cocomm Ban Coalg$ of cocommutative Banach coalgebras becomes a [[cartesian monoidal category]] under the projective tensor product.  Actually, $K$ is the [[terminal object]] even in $Ban Coalg$ (with the unique coalgebra morphism to $K$ being $\epsilon$ itself), but the [[pairing]]
$$ (f,g)(x) \coloneqq (f {\displaystyle\hat{\otimes}_\pi} g) \Delta x \in A {\displaystyle\hat{\otimes}_\pi} B $$
(given $f\colon \Gamma \to A$, $g\colon \Gamma \to B$, and $x\colon \Gamma$) is a morphism of $Ban Coalg$ only when $A$ and $B$ are cocommutative.  (I believe that $Ban Coalg$ *does* have a [[product]], but it must be more complicated.)

## The dual algebras of a coalgebra

If $A$ is a Banach coalgebra, then the [[dual vector space]] $A^*$ is a [[Banach algebra]].  Actually, this is more general than $A^* = [A,K]$; if $B$ is any Banach algebra, then so is $[A,B]$ (the Banach space of [[bounded linear maps]] from $A$ to $B$).

This result is nothing special about Banach (co)algebras; it holds in any [[closed monoidal category]].  The multiplication operation in $[A,B]$ is given by
$$ (\lambda \mu) x = m (\lambda {\displaystyle\hat{\otimes}_\pi} \mu) \Delta x ,$$
where $m\colon B {\displaystyle\hat{\otimes}_\pi} B \to B$ is (generated by) the multiplication operation on $B$.  $[A,B]$ is associative, unital, or commutative if $A$ and $B$ are (with 'co‑' in the names of $A$\'s properties).  In particular, $A^*$ has one of these properties iff $A$ has the corresponding property.

Note that $[B,A]$ (or even $B^*$) is *not*, in general, a Banach coalgebra.  (That\'s because $Ban$ is closed, not [[coclosed category|coclosed]].)

## Operators between coalgebras

Let $A$ and $B$ be Banach coalgebras.

Of course, $A$ and $B$ are [[Banach spaces]], so we may consider the whole panoply of [[linear operators]] from $A$ to $B$.  In general, a linear operator is only a [[partial function]], defined on a [[linear subspace]] of $A$ (and otherwise only required to be a [[linear map]]); but in particular we consider the [[densely-defined operator]]s (each defined on a [[dense subspace|dense]] subspace of $A$), the [[linear mappings]] (each defined on all of $A$), the [[bounded operators]] (each defined on all of $A$ and [[bounded map|bounded]] or equivalently [[continuous map|continuous]]), and the [[short operators]] (each bounded with a [[norm]] at most $1$).

A __comultiplicative linear operator__ from $A$ to $B$ is a linear operator $T\colon A \to B$ such that the following hold for all $x \in \dom T$:
*  $\Delta x \in (\dom T) {\displaystyle\hat{\otimes}_\pi} (\dom T)$,
*  $\Delta_B T x = (T {\displaystyle\hat{\otimes}_\pi} T) \Delta_A x$ (which exists by the previous line), and
*  $\epsilon_B T x = \epsilon_A x$ (which always exists).

We can also consider __densely-defined comultplicative linear operators__.  A __coalgebra homomorphism__, or __cohomomorphism__, is a comultiplicative linear mapping; we can also consider __bounded homomorphisms__ and __short homomorphisms__.  The last of these are, as above, the [[morphisms]] in $Ban Coalg$; of course, any of these classes of operators (except the densely-defined ones, which are not closed under [[composition]]) could be taken to be morphisms of a different category with the same objects, but then we would have [[isomorphisms]] that are not [[isometries]].  (See also [[isomorphism of Banach spaces]].)

## Beyond coalgebras

A __Banach [[bialgebra]]__, or __Banach bigebra__, is a [[bimonoid]] in $Ban$: a Banach space $A$ equipped with the structures of both a [[Banach algebra]] and a Banach coalgebra, such that $\Delta$ and $\epsilon$ are both morphisms of Banach algebras, or equivalently such that the multiplication and unit of the Banach algebra are both morphisms of Banach coalgebras.  Explicitly, this requirement is:
*  $\Delta (x y) = (\Delta x) (\Delta y)$ (with the induced multiplication on $A {\displaystyle\hat{\otimes}_\pi} A$),
*  $\Delta 1 = 1 \otimes 1$ (which is the identity in $A {\displaystyle\hat{\otimes}_\pi} A$),
*  $\epsilon (x y) = (\epsilon x) (\epsilon y)$ (with the multiplication on the right in $K$), and
*  $\epsilon 1 = 1$ (with the $1$ on the right in $K$).

The [[category]] __$Ban Bialg$__ of Banach bialgebras has, as objects, Banach bialgebras and, as morphisms, short linear maps that are morphisms of both Banach algebras and Banach coalgebras.

A __Banach [[Hopf algebra]]__ is a [[Hopf object]] in $Ban$: a Banach bialgebra $A$ with a (necessarily unique) short linear map (the __antipode__) $S\colon A \to A$ such that
$$ m (S {\displaystyle\hat{\otimes}_\pi} \id) \Delta x, m (\id {\displaystyle\hat{\otimes}_\pi} S) \Delta x = 1 \epsilon x $$
(where $m$ is the multiplication with identity $1$) for all $x\colon A$.

The [[category]] __$Ban Hopf Alg$__ of Banach Hopf algebras has, as objects, Banach Hopf algebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve antipodes:
$$ f (S_A x) = S_B (f x) $$
for all $x\colon A$.

A __Banach $*$-[[star-algebra|coalgebra]]__ is a $*$-[[star-monoid object|monoid object]] in $Ban$: a Banach coalgebra $A$ equipped with an [[antilinear map]] (the __adjoint__) $x \mapsto x^*\colon A \to A$ such that
$$ \Delta x^* = (\tau \Delta x)^* $$
(where $\tau$ again is the [[braiding]] and $*$ on $A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $(x \otimes y)^* = x^* \otimes y^*$) for all $x\colon A$ and
$$ \epsilon x^* = \overline {\epsilon x} $$
(where a bar indicates [[complex conjugation]]) for all $x\colon A$.

The [[category]] __$Ban {*} Coalg$__ of Banach $*$-coalgebras has, as objects, Banach $*$-coalgebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve adjoints:
$$ f(x^*) = f(x)^* $$
for all $x\colon A$.

There are also $C^*$-[[C-star-coalgebra|coalgebras]], which have their own page.

## Examples

It's well known that the [[sequence space]] $l^1$ of absolutely summable [[infinite sequences]], thought of as $l^1(\mathbb{Z})$ (where $\mathbb{Z}$ is the [[abelian group]] of [[integers]] under addition), is a [[Banach algebra]] under [[convolution]]; however, it is also a Banach coalgebra, and these structures together make it a Banach bialgebra, in fact a Banach Hopf $*$-algebra.  Since $l^1$ is a Banach coalgebra, its [[dual vector space|dual space]] $l^\infty$ (the sequence space of absolutely bounded sequences) is a Banach algebra (which is also well known); and although there is no guarantee that it should work, in this case $l^\infty$ is also a Banach coalgebra, and indeed a Banach Hopf $*$-algebra too.

Explicitly:  The projective tensor square $l^1 {\displaystyle\hat{\otimes}_\pi} l^1$ is the space of absolutely summable infinite [[matrix|matrices]]; convolution takes the matrix $(a_{i,j})_{i,j}$ to the sequence
$$ (\sum_{i + j = k} a_{i,j})_k $$
(summing along antidiagonals); comultiplication takes $(a_k)_k$ to the [[diagonal matrix]]
$$ \Delta a = (\sum_{i = j = k} a_k)_{i,j} $$
(which is not quite the origin of the symbol '$\Delta$' but might as well be).  The tensor square $l^\infty {\displaystyle\hat{\otimes}_\pi} l^\infty$ is the space of infinite matrices with absolutely bounded entries;
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the dual multiplication on $l^\infty$ takes the matrix $(a_{i,j})_{i,j}$ to the sequence
$$ (\sum_{i = j = k} a_{i,j})_k = (a_{k,k})_k $$
of its diagonal entries; the dual comultiplication (which part of me wants to call 'nvolution', but let\'s say coconvolution instead) takes $(a_k)_k$ to
$$ (\sum_{i + j = k} a_k)_{i,j} = (a_{i + j})_{i,j} $$
(so each antidiagonal is constant).

We are lucky that coconvolution exists, since the dual of a Banach algebra need not be a Banach coalgebra; but arguably coconvolution is easier to describe than convolution, so let us shift perspective and take coconvolution as basic.  Then convolution necessarily exists on the dual of $l^\infty$, but (at least in [[classical mathematics]]) $l^1$ is only a subspace of that.  So from this perspective, what's lucky is that $l^1$ is closed under convolution.  (In [[dream mathematics]], $l^1$ is the entire dual of $l^\infty$, so no luck is required.)  Of course, $l^1$ *is* the dual of $c_0$ (the space of sequences with limit $0$), but $c_0$ is *not* closed (coclosed?) under coconvolution (try any non-zero example), so we are still lucky.

[[!redirects Banach coalgebra]]
[[!redirects Banach coalgebras]]
[[!redirects Banach cogebra]]
[[!redirects Banach cogebras]]

[[!redirects BanCoalg]]
[[!redirects Ban Coalg]]

[[!redirects Banach bialgebra]]
[[!redirects Banach bialgebras]]
[[!redirects Banach bigebra]]
[[!redirects Banach bigebras]]

[[!redirects Banach Hopf algebra]]
[[!redirects Banach Hopf algebras]]
[[!redirects Banach-Hopf algebra]]
[[!redirects Banach-Hopf algebras]]
[[!redirects Banach–Hopf algebra]]
[[!redirects Banach–Hopf algebras]]
[[!redirects Banach--Hopf algebra]]
[[!redirects Banach--Hopf algebras]]

[[!redirects Banach *coalgebra]]
[[!redirects Banach *coalgebras]]
[[!redirects Banach *cogebra]]
[[!redirects Banach *cogebras]]
[[!redirects Banach *-coalgebra]]
[[!redirects Banach *-coalgebras]]
[[!redirects Banach *-cogebra]]
[[!redirects Banach *-cogebras]]
[[!redirects Banach * coalgebra]]
[[!redirects Banach * coalgebras]]
[[!redirects Banach * cogebra]]
[[!redirects Banach * cogebras]]
[[!redirects Banach star-coalgebra]]
[[!redirects Banach star-coalgebras]]
[[!redirects Banach star-cogebra]]
[[!redirects Banach star-cogebras]]
[[!redirects Banach star coalgebra]]
[[!redirects Banach star coalgebras]]
[[!redirects Banach star cogebra]]
[[!redirects Banach star cogebras]]


Banach fixed-point theorem

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## Statement

The **Banach fixed-point theorem** or **contraction mapping theorem** states:

Let $(X, \rho)$ be a [[sequentially Cauchy complete]] [[metric space]] with a point $x_0:X$ and a [[rational number]] $C : \mathbb{Q}$ such that for all $x:X$ and $y:X$, $\rho(x, y) \leq C$. Let $T : X \to X$ be an [[endomap]] with a [[rational function|rational]] [[Lipschitz constant]] $0 \lt c \lt 1$. Then $X$ has a unique [[fixed point]], a point $x$ with $\rho(T (x), x) = 0$,
such that for any $y : X$ with $\rho(T (y), y) = 0$, $x = y$.

## Related concepts

* [[inverse function theorem]]

* [[ordinary differential equation]]

## References

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))

See also:

* Wikipedia, _[Banach fixed-point theorem](https://en.wikipedia.org/wiki/Banach_fixed-point_theorem)_

[[!redirects Banach fixed point theorem]]
[[!redirects contraction mapping theorem]]


Banach lattice

Banach lattice
(From https://www.encyclopediaofmath.org/index.php/Banach_lattice)

With minor editing:

A vector lattice that is at the same time a Banach space with a norm which satisfies the monotonicity condition:

                            |x|</=|y|=>||x||</=||y||

A Banach lattice is also called a KB-lineal, whereas an arbitrary normed lattice, i.e. a vector lattice with a monotone norm, is called a KN-lineal. When completing a normed lattice in norm, the order relations may be extended to the resulting Banach space so that it becomes a Banach lattice. If it is possible to introduce in a lattice a Banach topology which converts it to a Banach lattice, such a topology is unique. The simplest example of a Banach lattice is the space C(Q) of continuous functions on an arbitrary compact topological space Q with the natural (pointwise) order and with the ordinary (uniform) norm. Other examples of Banach lattices include Lp  spaces and [[Orlicz spaces]] (cf. Orlicz space). In Banach lattices convergence in norm is (o)-convergence for convergence with a regulator. This is not true of normed lattices.

An important special case is a Banach lattice of bounded elements. If a lattice X contains a strong unit 1, i.e. xEX if for each  there exists a /\ such that x|</=/\1 , then the smallest /\  for which this inequality is valid is taken as |x|. The normed lattice thus obtained is called a normed lattice of bounded elements; if it is complete in norm, it is called a Banach lattice of bounded elements. In a Banach lattice (and even in a normed lattice) of bounded elements convergence in norm is identical with convergence with a regulator, while the boundedness of a set of elements in norm is identical with order boundedness. If a normed lattice of bounded elements is conditionally o^-complete, it is complete in norm.

The space C(Q) is a Banach lattice of bounded elements in which the function x(q)E1 is taken as the unit. For any Banach lattice  of bounded elements there exists a compact Hausdorff space  such that  is algebraically and lattice isomorphic to the space . This is an abstract characterization of the Banach lattice of continuous functions on a compact Hausdorff space.

In any normed lattice an additive functional that is continuous in norm is regular and, moreover, is representable as the difference of two additive functionals which are continuous in norm. In a Banach lattice each positive additive functional is continuous in norm, which means that the classes of regular functionals and additive functionals which are continuous in norm coincide. The space X' which is dual in the sense of Banach to the normed lattice X is a conditionally complete Banach lattice. In a normed lattice the Hahn–Banach theorem may be strengthened as follows: For any x0>0  there exists a positive additive functional f , which is continuous in norm, such that f(x0)=||x0||,||f||=1 .

References

[1]	B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)

[2]	M.M. Day, "Normed linear spaces" , Springer (1958)


Banach manifold

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## Idea

A notion of [[infinite-dimensional manifold]].
A __Banach manifold__ is a [[manifold]] modelled on [[Banach spaces]].  By default, transition maps are taken to be [[smooth map|smooth]].

## Properties

### As absolute neighbourhood retracts

\begin{example}\label{ParacompactBanachManifoldsAreANRs}
Every [[paracompact topological space|paracompact]] Banach manifold is an [[absolute neighbourhood retract]].
\end{example}
By [Palais 1966, Cor. to Thm. 5 on p. 3](#Palais66).

### Embedding into the category of diffeological spaces

The category of smooth Banach manifolds has a [[full and faithful functor]] into the category of [[diffeological spaces]]. In terms of Chen smooth spaces this was observed in ([Hain](#Hain)). For more see at _[Fréchet manifold -- Relation to diffeological spaces](#Fréchet+manifold#RelationBetweenDeffeologicalAndFrechetStructure)_.

## Related concepts

* [[Hilbert manifold]]

* [[ILH manifold]]

* [[Frechet manifold]]

* [[convenient manifold]]

## References

For general references see at _[[infinite-dimensional manifold]]_.

Aspects of the [[homotopy theory]] of Banach manifolds:

* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

The [[full subcategory]] embedding into the category of [[diffeological spaces]]:

* {#Hain} [[Richard Hain]], _A characterization of smooth functions defined on a Banach space_,  Proc. Amer. Math. Soc. 77 (1979), 63-67 ([web](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html), [pdf](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf))

[[!redirects Banach manifolds]]


Banach module

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## Definition

For $A$ a [[Banach algebra]], then a _Banach module_ is a [[module]] over $A$ in the [[category]] of [[Banach spaces]], hence a [[Banach space]] $V$ equipped with a [[continuous function|continuous]] [[bilinear map]] $V\times A\to V$ satisfying the [[action]] property.

## Properties

* [[Tate's acyclicity theorem]]

[[!redirects Banach modules]]


Banach ring

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## Idea

A Banach ring is a [[complete space|complete]] [[normed ring]], hence a [[commutative monoid]] in the [[monoidal category]] of complete [[normed groups]] (with [[short map|short]] group homomorphisms and the [[projective tensor product]]).

If not just in complete normed groups but in complete [[normed vector spaces]] ([[Banach spaces]]), then this is a [[Banach algebra]].

The [[Berkovich spectrum]] of a Banach ring $R$ is the [[topological space]] of multiplicative [[seminorms]] on $R$ that are bounded by the norm on $R$.

## Examples

* The [[integers]] $\mathbb{Z}$ equipped with their [[absolute value]] [[norm]] ${\vert- \vert_\infty}$ are a Banach ring.

* The integers with the $p$-adic norm ${|-|_p}$ are an incomplete normed ring whose completion is the Banach ring $\mathbb{Z}_p$ of $p$-[[adic integers]].

## Related concepts

[[!include analytic geometry ingredients -- table]]

## References

A quick review is in

* [[Sarah  Brodsky]], _Non-archimedean geometry_ ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf))

A standard textbook account in the context of [[rigid analytic geometry]] is

* {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], section 1.2.4 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

A set of lecture notes in the context of [[Berkovich spaces]] is

* [[Vladimir Berkovich]], section 1.2 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

Discussion from a more [[topos theory|topos-theoretic]] point of view is in

* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 6.5 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

category: analysis

[[!redirects Banach ring]]
[[!redirects Banach rings]]

[[!redirects complete normed ring]]
[[!redirects complete normed rings]]


Banach space

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### Context
#### Functional analysis
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# Banach spaces
* table of contents
{: toc}

## Idea

A Banach space $\mathcal{B}$ is both a [[vector space]] (over a [[normed field]] such as $\mathbb{R}$) and a [[complete space|complete]] [[metric space]], in a compatible way. Hence a complete [[normed vector space]].

A source of simple Banach spaces comes from considering a [[Cartesian space]] $\mathbb{R}^n$ (or $K^n$ where $K$ is the normed field) with the norm:

$$ {\|(x_1,\ldots,x_n)\|_p} \coloneqq \root p {\sum_{i = 1}^n {|x_i|^p}} $$

where $1 \leq p \leq \infty$ (this doesn't strictly make sense for $p = \infty$, but taking the limit as $p \to \infty$ and reading $\mathbb{R}^\infty = \underset{\longrightarrow}{\lim}_n \mathbb{R}^n$ as the [[direct limit]] (as opposed to the [[inverse limit]]) we arrive at the formula ${\|(x_1,\ldots,x_n)\|_\infty} \coloneqq \max_i {|x_i|}$).

However, the theory of these spaces is not much more complicated than that of finite-dimensional vector spaces because they all have the same underlying topology.  When we look at infinite-dimensional examples, however, things become trickier.  Common examples are [[Lebesgue spaces]], [[Hilbert spaces]], and [[sequence spaces]].

In the literature, one most often sees Banach spaces over the field $\mathbb{R}$ of [[real numbers]]; Banach spaces over the field $\mathbb{C}$ of [[complex numbers]] are not much different, since they are also over $\mathbb{R}$.  But people do study them over [[p-adic numbers]] too.  _Unless otherwise stated, we assume $\mathbb{R}$ below._

## Definitions

Let $V$ be a [[vector space]] over the field of [[real number]]s.  (One can generalise the choice of [[field]] somewhat.)  A **pseudonorm** (or **[[seminorm]]**) on $V$ is a function
$$ {\| - \|}\colon V \to \mathbb{R} $$
such that:

1.  $ {\|0\|} \geq 0 $;
2.  $ {\|r v\|} = {|r|} {\|v\|} $ (for $r$ a [[scalar]] and $v$ a vector);
3.  $ {\|v + w\|} \leq {\|v\|} + {\|w\|} $.

It follows from the above that ${\|v\|} \geq 0$; in particular, ${\|0\|} = 0$.  A **[[norm]]** is a pseudonorm that satisfies a converse to this: $v = 0$ if ${\|v\|} = 0$.

A [[norm]] on $V$ is **complete** if, given any infinite [[sequence]] $(v_1, v_2, \ldots)$ such that
\[ \label{Cauchy} \lim_{m,n\to\infty} {\left\| \sum_{i=m}^{m+n} v_i \right\|} = 0 ,\]
there exists a (necessarily unique) **sum** $S$ such that
\[ \label{converge} \lim_{n\to\infty} {\left\| S - \sum_{i=1}^n v_i \right\|} = 0 ;\]
we write
$$ S = \sum_{i=1}^\infty v_i $$
(with the right-hand side undefined if no such sum exists).

Then a **Banach space** is simply a vector space equipped with a complete norm.  As in the real line, we have in a Banach space that
$$ {\left\| \sum_{i=1}^\infty v_i \right\|} \leq \sum_{i=1}^\infty {\|v_i\|} ,$$
with the left-hand side guaranteed to exist if the right-hand side exists as a finite real number (but the left-hand side may exist even if the right-hand side diverges, the usual distinction between [[absolute convergence]] and [[conditional convergence]]).

If we do not insist on the space being complete, we call it a **normed (vector) space**.  If we have a [[topological vector space]] such that the topology comes from a norm, but we do not make an actual choice of such a norm, then we talk of a **normable space**.

### Banach spaces as metric spaces

The three axioms for a pseudonorm are very similar to the three axioms for a [[pseudometric]].

Indeed, in any pseudonormed vector space, let the **distance** $d(v,w)$ be
$$ d(v,w) = {\|w - v\|} .$$
Then $d$ is a pseudometric, which is __translation-invariant__ in that
$$ d(v+x,w+x) = d(v,w) $$
always holds.  Conversely, given any translation-invariant pseudometric $d$ on a vector space $V$, let ${\|v\|}$ be
$$ {\|v\|} = d(0,v) .$$
Then ${\|-\|}$ satisfies the axioms (1--3) for a pseudonorm, except that it may satisfy (2) only for $r = 0, \pm 1$.  (In other words, it is only a [[G-pseudonorm]].)  It will actually be a pseudonorm iff the pseudometric satisfies a homogeneity rule:
$$ d(r v,r w) = {|r|} d(v,w) .$$
Thus pseudonorms correspond precisely to homogeneous translation-invariant pseudometrics.

Similarly, norms correspond to homogenous translation-invariant metrics and complete norms correspond to complete homogeneous translation-invariant metrics.  Indeed, (eq:Cauchy) says that the sequence of partial sums is a [[Cauchy sequence]], while (eq:converge) says that the sequence of partial sums converges to $S$.

Thus a Banach space may equivalently be defined as a vector space equipped with a complete homogeneous translation-invariant metric.  Actually, one usually sees a sort of hybrid approach: a Banach space is a normed vector space whose corresponding metric is complete.

### Maps between Banach spaces
{#morphisms}

If $V$ and $W$ are pseudonormed vector spaces, then the **norm** of a linear function $f\colon V \to W$ may be defined in either of these equivalent ways:

*  $ {\|f\|} = \sup \{ {\|f v\|} \;|\; {\|v\|} \leq 1 \} $;
*  $ {\|f\|} = \inf \{ r \;|\; \forall{v},\; {\|f v\|} \leq r {\|v\|} \} $.

(Some other forms are sometimes seen, but these may break down in degenerate cases.)

For finite-dimensional spaces, any linear map has a well-defined finite norm.  In general, the following are equivalent:

*  $f$ is [[continuous map|continuous]] (as measured by the pseudometrics on $V$ and $W$) at $0$;
*  $f$ is continuous (everywhere);
*  $f$ is [[uniformly continuous map|uniformly continuous]];
*  $f$ is [[Lipschitz map|Lipschitz continuous]];
*  ${\|f\|}$ is finite (and, in [[constructive mathematics]], [[located real number|located]]);
*  $f$ is [[bounded map|bounded]] (as measured by the [[bornologies]] given by the pseudometrics on $V$ and $W$).

In this case, we say that $f$ is **bounded**.  If $f\colon V \to W$ is not assumed to be linear, then the above conditions are no longer equivalent.

The bounded linear maps from $V$ to $W$ themselves form a pseudonormed vector space $\mathcal{B}(V,W)$.  This will be a Banach space if (and, except for degenerate cases of $V$, only if) $W$ is a Banach space.  In this way, the category $Ban$ of Banach spaces is a [[closed category]] with $\mathbb{R}$ as the unit.

The clever reader will note that we have not yet defined $\mathbf{Ban}$ as a category! (surprisingly in the _nLab_) There are many (nonequivalent) ways to do so.

In [[functional analysis]], the usual notion of '[[isomorphism]]' for Banach spaces is a bounded bijective linear map $f\colon V \to W$ such that the [[inverse function]] $f^{-1}\colon W \to V$ (which is necessarily linear) is also bounded. In this case one can accept all bounded linear maps between Banach spaces as morphisms.  Analysts sometimes refer to this as the "isomorphic category".

Another natural notion of isomorphism is a surjective linear isometry. In this case, we take a morphism to be a **[[short map|short]]** [[linear map]], or [[short linear map|linear contraction]]: a linear map $f$ such that ${\|f\|} \leq 1$.  This category, which is what category theorists generally refer to as $\mathbf{Ban}$, is sometimes referred to as the "isometric category" by analysts. Note that this makes the 'underlying set' (in the sense of $\mathbf{Ban}$ as a [[concrete category]] like any closed category) of a Banach space its (closed) **unit ball**
$$ Hom_Ban(\mathbb{R},V) \cong \{ v \;|\; {\|v\|} \leq 1 \} $$
rather than the set of all vectors in $V$ (the underlying set of $V$ as a vector space).

+-- {: .query}
Yemon Choi: This is really here to remind myself how to make query boxes. But while I'm at it, is it really OK to refer to the "unit ball functor" as "taking the underlying set"? I notice that on the discussion about internal homs at [[internal hom]] it is claimed that "Every closed category is a [[concrete category]] (represented by $I$), and the underlying set of the internal hom is the external hom" which seems to require "underlying set" to be interpreted in this looser sense.

_Toby_:  Sure, but the point of putting 'underlying set' in scare quotes is precisely to point out that the category-theoretic underlying set is not what one would normally expect.
=--

+-- {: .query}
Mark Meckes: I've expanded this section in part to be consistent with analysts' terminology.  I've made some assumptions about category theorists' conventions which might not be correct.  (If I find time I might write about other categories of Banach spaces that analysts think about.)

_Toby_:  Looks good to me!
=--

From a category-theorist\'s perspective, the isomorphic category is really the [[full image]] of the [[inclusion functor]] from $Ban$ to $TVS$ (the category of [[topological vector spaces]]), which may be denoted $Ban_{TVS}$.  If you\'re working in $Ban_{TVS}$, then you only care about the topological linear structure of your space (although you do also care that it can be derived from some metric); if you\'re working in $Ban$, then you care about all of the structure on the space.

## Examples

Many examples of Banach spaces are parametrised by an exponent $1 \leq p \leq \infty$.  (Sometimes one can also try $0 \leq p \lt 1$, but these generally don\'t give Banach spaces.)

*  The [[Cartesian space]] $\mathbb{R}^n$ is a Banach space with
   $$ {\|(x_1,\ldots,x_n)\|_p} = \root p {\sum_i {|x_i|^p}} .$$
   (We can allow $p = \infty$ by taking a limit; the result is that ${\|x\|_\infty} = \max_i {|x_i|}$.)  Every finite-dimensional Banach space is isomorphic to this for some $n$ and $p$; in fact, once you fix $n$, the value of $p$ is irrelevant up to isomorphism.

*  The [[sequence space]] $l^p$ is the set of infinite [[sequence]]s $(x_1,x_2,\ldots)$ of real numbers such that
   $$ {\|(x_1,x_2,\ldots)\|_p} = \root p {\sum_i {|x_i|^p}} $$
   exists as a finite real number.  (The only question is whether the sum converges.  Again $p = \infty$ is a limit, with the result that ${\|x\|_\infty} = \sup_i {|x_i|}$.)  Then $l^p$ is a Banach space with that norm.  These are all versions of $\mathbb{R}^\infty$, but they are no longer isomorphic for different values of $p$.  (See [[isomorphism classes of Banach spaces]].)

*  More generally, let $A$ be any [[set]] and let $l^p(A)$ be the set of [[function]]s $f$ from $A$ to $\mathbb{R}$ such that
   $$ {\|f\|_p} = \root p {\sum_{x: A} {|f(x)|^p}} $$
   exists as a finite real number.  (Again, ${\|f\|_\infty} = \sup_{x\colon A} {|f(x)|}$.)  Then $l^p(A)$ is a Banach space.  (This example includes the previous examples, for $A$ a countable set.)

*  On any [[measure space]] $X$, the [[Lebesgue space]] $\mathcal{L}^p(X)$ is the set of measurable almost-everywhere-defined real-valued functions on $X$ such that
   $$ {\|f\|_p} = \root p {\int {|f|^p}} $$
   exists as a finite real number.  (Again, the only question is whether the integral converges.  And again $p = \infty$ is a limit, with the result that ${\|f\|_\infty}$ is the [[essential supremum]] of ${|f|}$.)  As such, $\mathcal{L}^p(X)$ is a complete pseudonormed vector space; but we identify functions that are equal almost everywhere to make it into a Banach space.  (This example includes the previous examples, for $X$ a set with counting measure.)

*  Any [[Hilbert space]] is Banach space; this includes all of the above examples for $p = 2$.

## Operations on Banach spaces

The category $Ban$ of Banach spaces is [[complete category|small complete]], [[cocomplete category|small cocomplete]], and [[symmetric monoidal closed category|symmetric monoidal closed]] with respect to its standard internal hom (described at [[internal hom]]). Some details follow.

* The category of Banach spaces admits small [[product]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its product in $Ban$ is the subspace of the vector-space product
$$\prod_{\alpha \in A} X_\alpha$$
consisting of $A$-tuples $\langle x_\alpha \rangle$ which are _uniformly_ bounded (i.e., there exists $C$ such that $\forall \alpha \in A: {\|x_\alpha\|} \leq C$), taking the least such upper bound as the norm of $\langle x_\alpha \rangle$. This norm is called the $\infty$-norm; in particular, the product of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^{\infty}(A)$.

* The category of Banach spaces admits [[equalizer]]s. Indeed, the equalizer of a pair of maps $f, g: X \rightrightarrows Y$ in $Ban$ is the [[kernel]] of $f-g$ under the norm inherited from $X$ (the kernel is closed since $f-g$ is continuous, and is therefore complete). In fact every equalizer is even a [[section]] by the [[Hahn-Banach theorem]]. Every [[extremal monomorphism]] is even already an equalizer (and a section): Let $f\colon X \to Y$ be an extremal monomorphism, $\iota\colon \Im(f) \to Y$ the embedding of $Im(f)$ into the codomain of $f$ and $f\prime \colon X \to Im(f)$ $f$ with restricted codomain. Since $f\prime$ is an epimorphism, $f=\iota f\prime$, and $f$ extremal, $f\prime$ is an isomorphism, thus $f$ is an embedding.

* The category of Banach spaces admits small [[coproduct]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its coproduct in $Ban$ is the completion of the vector space coproduct
  $$\bigoplus_{\alpha \in A} X_\alpha$$
  with respect to the norm given by
  $$ {\left\| \bigoplus_{s \in S} x_s \right\|} = \sum_{s \in S} {\|x_s\|} ,$$
  where $S \subseteq A$ is finite and ${\|x_s\|}$ denotes the norm of an element in $X_s$. This norm is called the $1$-norm; in particular, the coproduct of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^1(A)$.

* {#Coequalizers} The category of Banach spaces admits [[coequalizers]]. Though one may expect the coequalizer of a pair of maps $f,g: X \rightrightarrows Y$ to be the [[cokernel]] of $f-g$ under the quotient norm (in which the norm of a coset $y + C$ is the minimum norm attained by elements of $y + C$; here $C$ is the [[image]] $(f-g)(X)$), these spaces are not Banach spaces in general, as the image of a map $f$ need not be closed (indeed, the inclusion $i: \ell_1 \hookrightarrow c_0$ has dense image in $c_0$), and so the quotient space may not be complete. However, the quotient by the closure of $(f-g)(X)$ suffices.

* To describe the tensor product $X \otimes_{Ban} Y$ of two Banach spaces (making $Ban$ symmetric monoidal closed with respect to its usual internal hom), let $F(X \times Y)$ be the free vector space generated by the set $X \times Y$, with norm on a typical element defined by
$$ {\left\| \sum_{1 \leq i \leq n} a_i (x_i \otimes y_i) \right\|} = \sum_{1 \leq i \leq n} {|a_i|} {\|x_i\|} \cdot {\|y_i\|}.$$
Let $\overline{F}(X \times Y)$ denote its completion with respect to this norm. Then take the cokernel of $\overline{F}(X \times Y)$ by the closure of the subspace spanned by the obvious bilinear relations. This quotient is $X \otimes_{Ban} Y$.

In the literature on Banach spaces, tensor product above is usually called the __projective tensor product__ of Banach spaces; see other [[tensor product of Banach spaces]].  The product and coproduct are considered __direct sums__; see other [[direct sums of Banach spaces]].

To be described:

*  duals ($p + q = p q$);
*  completion ($Ban$ is a [[reflective subcategory]] of $PsNVect$ (pseudo-normed vector spaces)).
*  $Ban$ as a (somewhat larger) category with duals.

## Integration in Banach spaces

This paragraph describes some aspects of integration theory in Banach spaces that are relevant to understand the literature about [[AQFT]]. In the given context, elements of a Banach space $\mathcal{B}$ are sometimes called vectors, a function or measure taking values in $\mathcal{B}$ are therefore called vector functions and vector measures. Functions and measures taking values in the [[field]] that the Banach space is defined upon as a vector space are called scalar functions and scalar measures.

We will consider two types of integrals:

* integrals of vector functions with respect to a scalar measure, specifically the Bochner integral,

* integrals of scalar functions with respect to a vector measure, specifically the spectral integral of a normal operator on a Hilbert space.

### Bochner integral

The Bochner integral is a direct generalization of the Lesbegue integral to functions that take values in a Banach space. Whenever you encounter an integral of a function taking values in a Banach space in the [[AQFT]] literature, it is safe to assume that it is meant to be a Bochner integral. Two points already explained by Wikipedia are of interest:

1. A version of the dominated convergence theorem is true for the Bochner integral.
2. There are theorems that are not valid for the Bochner integral, notably the Radon-Nikodym theorem does not hold in general.

* [Wikipedia] (http://en.wikipedia.org/wiki/Bochner_integral)

_reference_: Joseph Diestel: "Sequences and Series in Banach Spaces" ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0542.46007&format=complete)), chapter IV.

### The spectral integral

The integral with respect to the spectral measure of a bounded normal operator on a Hilbert space is an example of a Banach space integral with respect to a vector measure.
In this paragraph we present a well known, but somewhat less often cited result, that is of use in some proofs in some approaches to [[AQFT]], it is the version of the dominated convergence theorem for the given setting.

Let A be a bounded normal operator on a Hilbert space and E be it's spectral measure (the "resolution of identity" in the terms of Dunford and Schwartz). Let $\sigma(A)$ be the spectrum of A. For a bounded complex Borel function f we then have
$$
           f(A) \coloneqq \int_{\sigma(A)} f(\lambda) E(d\lambda)
$$

+-- {: .un_theorem}
###### Theorem (dominated convergence)
If the uniformly bounded sequence $\{f_n\}$ of complex Borel functions converges at each point of $\sigma(A)$ to the function $f$, then $f_n(A) \to f(A)$ in the strong operator topology.
=--

+-- {: .proof}
See Dunford, Schwartz II, chapter X, corollary 8.
=--

## Properties

### Relation to bornological spaces

Every [[inductive limit]] of [[Banach spaces]] is a [[bornological vector space]]. ([Alpay-Salomon 13, prop. 2.3](#AlpaySalomon13))

Conversely, every [[bornological vector space]] is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete ([Schaefer-Wolff 99](#SchaeferWolff99))

## Related concepts

* [[reflexive Banach space]]
* [[projective Banach space]]

* [[Banach analytic space]]

[[!include analytic geometry ingredients -- table]]

## References

Named after [[Stefan Banach]].

* Walter Rudin, _Functional analysis_
* Dunford, Nelson; Schwartz, Jacob T.: "Linear operators. Part I: General theory." ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47001&format=complete)), "Linear operators. Part II: Spectral theory, self adjoint operators in Hilbert space." ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47002&format=complete))

* Z. Semadeni, _Banach spaces of continuous functions_, vol. I, Polish scientific publishers. Warszawa 1971

* {#AlpaySalomon13} Daniel Alpay, Guy Salomon, _On algebras which are inductive limits of Banach spaces_ ([arXiv:1302.3372](http://arxiv.org/abs/1302.3372))

* {#SchaeferWolff99} H. H. Schaefer with M. P. Wolff, _Topological vector spaces_, Springer 1999

* [[Jiří Rosický]], _Are Banach spaces monadic?_, ([arXiv:2011.07543](https://arxiv.org/abs/2011.07543))

category: analysis

[[!redirects Banach space]]
[[!redirects Banach spaces]]
[[!redirects Banach vector space]]
[[!redirects Banach vector spaces]]

[[!redirects Ban]]

[[!redirects norm isomorphism]]
[[!redirects norm-isomorphic]]
[[!redirects norm isomorphic]]


Banach-Alaoglu theorem

# The Banach--Alaoglu Theorem
* table of contents
{: toc}

## Idea

The closed unit ball of the double dual of a [[Banach space]] is weak* compact. This theorem is equivalent to the [[Tychonoff theorem]] for [[compact space|compact]] [[Hausdorff topological space|Hausdorff]] [[topological space|spaces]] (eg [Rossi 2009](#Rossi09)), and as such is not constructive in general. However, the restriction of the statement to _separable_ Banach spaces _is_ constructive, and is used in PDE theory to find solutions. A constructive proof can be found e.g. in [BridgesVita](#BridgesVita).

## Localic version

As usual, when formulating the theorem using [[locales]] instead of [[topological spaces]] the theorem holds constructively; see [Pelletier](#Pelletier) and [Henry](#Henry).

## Related entries

The following is a generalisation to [[locally convex topological vector space|locally convex]] spaces:

* [[Bourbaki-Alaoglu theorem|Bourbaki–Alaoglu theorem]]

## References

* Wikipedia, [Banach–Alaoglu theorem](https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem)

* {#Rossi09} Stefano Rossi, _The Banach-Alaoglu theorem is equivalent to the Tychonoff theorem for compact Hausdorff spaces_, arXiv:[0911.0332](http://arxiv.org/abs/0911.0332)

* {#Pelletier} Joan Wick Pelletier, _Locales in functional analysis_ [DOI](http://dx.doi.org/10.1016/0022-4049%2891%2990013-R)

* {#Henry} Simon Henry, _Localic Metric spaces and the localic Gelfand duality_ [arxiv](https://arxiv.org/pdf/1411.0898)

* {#BridgesVita} Bridges and Vita, _Techniques of Constructive Analysis_ [link](http://www.springer.com/in/book/9780387336466)

[[!redirects Banach-Alaoglu theorem]]
[[!redirects Banach–Alaoglu theorem]]
[[!redirects Banach--Alaoglu theorem]]
[[!redirects Banach-Alaoglu Theorem]]
[[!redirects Banach–Alaoglu Theorem]]
[[!redirects Banach--Alaoglu Theorem]]


Banach-Steinhaus theorem

* [uniform boundedness principle](https://en.wikipedia.org/wiki/Uniform_Boundedness_Principle) -  article on Banach–Steinhaus theorem in WIkipedia


Banach-Tarski paradox

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## Idea

What is known as the _Banach-Tarski paradox_ is the [[theorem]] ([Banach-Tarski 24](#BanachTarski24)) that the [[axiom of choice]] implies that any two [[bounded subsets]] in [[Euclidean space]] of [[dimension]] $d \geq 3$ may be partitioned by a [[finite number]] of  pairwise congruent [[subsets]].

This is perceived as a [[paradox]] due to its counter-intuitive interpretation, which becomes particularly vivid if one takes one of the two bounded subsets to be the [[disjoint union]] of two copied of the other: In this case the theorem says, intuitively, that it is possible to break up any shape in 3d [[Euclidean space]] into a [[finite number]] of pieces, such that re-assembling these pieces suitably yields not just the original shape, but that and an entire other copy of it.

It has been pointed out that it is not just the use of the [[axiom of choice]] that is responsible for this perceived [[paradox]], but also the point-based concept of [[topological spaces]] as such, see the discussion _In point-free topology_ [below](#InPointFreeTopology).

## In point-free topology
 {#InPointFreeTopology}

It is argued in ([Simpson 12](#Simpson12)) that the Banach-Tarski paradox disappears if one works in [[point-free topology]], hence with [[locales]] instead of just [[topological spaces]]:

> We view spaces of interest as [[locales]], and the notion of "part" is given by the standard notion of [[sublocale]], $[\cdots]$. Every [[topological space]] determines a [[locale]] $[\cdots]$.  However, when a space is viewed as a locale, the notion of [[sublocale]] gives rise to new "parts" of spaces that are not merely subsets, and need  not  be  determined by their points.

> The usual contradictions are avoided $[$this way$]$. The different pieces in  the partitions defined by Vitali and by Banach and Tarski are deeply intertangled with  each  other.  According  to  our  notion  of "part",  two  such  intertangled pieces are not disjoint from each other, so additivity does not apply. An intuitive explanation for the failure of disjointness is that, although two such pieces share no point in common, they nevertheless overlap on the topological “glue” that bonds neighbouring points together.

## References

The original article:

* {#BanachTarski24} [[Stefan Banach]], [[Alfred Tarski]], _Sur la décomposition des ensembles de points en parties respectivement congruentes_  Fundamenta Mathematicae (in French). 6: 244–277, 1924 ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf))

Textbook account:

* [[Cornelia Druţu]], [[Michael Kapovich]], Chapter 17 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

See also:

* Wikipedia, _[Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox)_

Discussion in [[point-free topology]]:

* {#Simpson12} [[Alex Simpson]], _Measure, randomness and sublocales_, Annals of Pure and Applied Logic Volume 163, Issue 11, November 2012, Pages 1642-1659 ([pdf](http://homepages.inf.ed.ac.uk/als/Research/Sources/mrs.pdf), [doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))


bananaspace

*[bananaspace](https://www.bananaspace.org/wiki/%E9%A6%96%E9%A1%B5)* (香蕉空间) is a wiki on mathematics in Chinese created by Chenjing Bu, somewhat akin in style to (and apparently inspired by) the [[nLab]].

The name "bananaspace" is an inside-joke on a typo for *[[Banach space]]* that occurred in the presentation of a friend of the founders ([here](https://twitter.com/jasonchen0325/status/1526583860191825920), cf. [Miller 2011, p. 2](http://www-personal.umich.edu/~asnowden/teaching/2011/18.904/lec/lec-03-02a.pdf#page=2)).

## See also

* [[Stacks Project]]

* [[Kerodon]]

category: reference


Banch-Tarski theorem


Baptiste Morin

* [webpage](https://www.math.u-bordeaux.fr/~bmorin/)

##Publications
*  _Utilisation d'une cohomologie étale équivariante en topologie arithmétique_
Compositio Math. 144 (2008), no. 1, 32-60.

category: people


bar > history

< [[bar]]


bar and cobar construction

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#Contents#
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## Bar  and cobar constructions

* There is a brief entry at [[bar construction]] together with a blog link

   * [[Todd Trimble]], _On the Bar Construction_ ([blog](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html))

* There is some discussion of the bar-cobar adjointness as it relates to [[twisting cochain|twisting cochains]], at that entry.

* Here we will concentrate on the bar-cobar adjointness itself and start exploring the links with other parts of differential algebra.

One of the earliest examples of a pair of [[adjoint functor|adjoint functors]] studied in [[algebraic topology]] was that giving the relationship between the functors for [[reduced suspension]] and [[loop space object|based loop space]]. If we consider a [[pointed topological space|pointed]] [[connected topological space]] $(X,x_0)$, then its [[reduced suspension]] $\Sigma X$ is obtained by taking  the cylinder $I\times X$ and identifying the subspace $\{0,1\}\times X\cup I\times \{x_0\}$ to a point.  (Think of crushing the two ends of the cylinder and the line through the base point to a point.) This can also be thought of as forming $S^1\wedge X$ the  [[smash product]] of the circle with $X$.

Adjoint to $\Sigma$ is the [[based loop space]] functor: $\Omega Y$ is the space of pointed maps from $S^1$ to $Y$. This has a monoid structure (up to homotopy) given by concatenation of loops.  (Back in $S^1$, we have a comonoid structure with respect to the pointed coproduct $S^1\to S^1\vee S^1$ as described at [[interval object]]. This in some sense  is _'subdivision as an inverse for composition'_.)

(perhaps: Picture to go here?)

Using ordinary (co)homology to study spaces such as CW-complexes, we naturally use the complexes of (cellular) chains on spaces. The structure of chains on the suspension is  easy to work out using the obvious cellular structure, but that on the loop space is much harder as $\Omega X$ is given the [[compact-open topology]] and only has the homotopy type of a CW-complex, so no nice cellular structure is given us 'on a plate'.  The idea is thus to start with  a chain complex model, $C_*(X)$, for a CW-complex, $X$, (usually the complex of cellular chains on $X$),  and we try to construct from $C_*(X)$ a 'model' for the chain complex of  the loop space $\Omega X$ of $X$. Adams' cobar construction was such a method (see below). This was adjoint to a bar construction defined by Eilenberg and MacLane.

Both directions use an abstract algebraic model of concatenation of paths and so their construction is linked to that of free monoids, and through those to [[monads]], [[operads]] and related abstract machinery to handle concatenation and its higher categorical analogues in categorical contexts.

The chain complex $C_*(X)$ has a rich coalgebraic structure induced by a cellular diagonal approximation on $X$ so the *cobar* construction will start with a [[differential graded coalgebra|dg-coalgebra]] as 'input' and as output we will hope for both a coalgebra structure (reflecting the chain coalgebra idea) and an algebra structure (coming from modelling the concatenation of loops). We therefore might hope for, and in fact do get, a [[differential graded Hopf algebra]].

Going the other way, we start with a [[differential graded algebra]] and use 'coconcatenation' or 'subdivision'  to get a coalgebra structure. In fact, once again, this is a Hopf algebra.

These topologically motivated constructions can be applied in much greater generality as we will see both here and elsewhere:

## Definitions

### The bar construction

(due originally Eilenberg-MacLane) Remember this goes from 'algebras' to Hopf algebras in general.

\[B :pre \varepsilon CDGA \to pre CDGHA\]

Let $(A,d,\varepsilon) $ be a commutative, augmented differential $\mathbb{Z}$-graded algebra, $d(A_n)\subseteq A_{n-1}$, $\overline{A} = Ker \varepsilon$.

The **bar construction** $B(A,d,\varepsilon)$ is given by
$$B(A,d,\varepsilon) = (T(s\overline{A}), D),$$
where

*  $T(s\overline{A})$ is the commutative [[differential graded Hopf algebra]] generated by $s\overline{A}$, $s$ being the _suspension_ (shift, translation, etc) operator discussed in [[graded vector space|graded vector spaces]],

*  $D = d_I + d_E$, where

$$d_I(s a_1\otimes \ldots\otimes s a_n) = -\sum_{i = 1} ^n\eta(i-1)s a_1\otimes \ldots \otimes s a_{i-2}\otimes s d a_{i-1}\otimes\ldots s a_n,$$

and

$$d_E(s a_1\otimes \ldots\otimes s a_n) = -\sum_{i = 1} ^n\eta(i-1)s a_1\otimes \ldots \otimes s a_{i-2}\otimes s a_{i-1}.a_i\otimes \ldots s a_n,$$

with $\eta(i) = (-1)^{\sum_{k=1}^i |s a_k|}$.

**Note**

1. _the image of a 1-connected cdga is a connected commutative Hopf algebra_.

1. The construction uses the _suspension_ operator on the graded vector spaces.  This mirrors the reduced suspension at the cell complex level.

* The construction uses a tensor algebra construction.  This from one point of view handles the formal concatenation aspect, but has also a rich structure of a coalgebraic structure with reduced diagonal, given by

  $$\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n),$$

  (see [[differential graded coalgebra]]).

  This can be interpreted as looking at how a formal concatenation can be 'subdivided' into its various parts.

### The Cobar construction

(due to J. F. Adams, see [Felix-Halperin0Thomas 92](#FelixHalperinThomas92))

We define a functor:

\[F :pre \eta  CoDGC \to pre CoDGHA\]

so essentially from cocommutative [[differential graded coalgebra|differential graded coalgebras]] to  cocommutative [[differential graded Hopf algebra|differential graded Hopf algebras]] (with frills attached in the way of coaugmentations, etc).

Let $(C,\partial,\eta)$ be a cocommutative differential $\mathbb{Z}$-graded coaugmented coalgeba:

$$\partial(C_n) \subseteq C_{n-1},  \quad \overline{C} = C/\eta(k), \quad \overline{\Delta} : \overline{C} \to \overline{C}\otimes \overline{C}.$$

The _Cobar construction_ $F(C,\partial, \eta)$ is the cocommutative pre-dgha defined by

* $F(C,\partial,\eta) = (T(s^{-1}\overline{C}), \delta)$,
 where $\delta = \partial_I + \partial_E$.

Here

*  $T(s^{-1}\overline{C})$ is the cocommutative Hopf algebra generated by $s^{-1}\overline{C}$, as before(in [[differential graded coalgebra]]) $\overline{C}$ is the cokernel of the coaugmentation, $\eta$)

*   $$\partial_I (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)s^{-1}c_1\otimes \ldots\otimes s^{-1}c_{i-1}\otimes s^{-1}\partial c_i\otimes \ldots s^{-1}c_n,$$

and

*  $$\partial_E (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)\sum_\mu (-1)^{|c'_{i\mu}| +1} (s^{-1}c_1\otimes \ldots\otimes s^{-1}c'_{i\mu}\otimes  s^{-1}c^{\prime\prime}_{i\mu}\otimes \ldots\otimes s^{-1}c_n),$$
with $\overline{\Delta}c_i = \sum_\mu c'_{i\mu}\otimes  c^{\prime\prime}_{i\mu}$; $\eta(i) = (-1)^{ \sum^i_{k=1}|s^{-1}c_k|}.$

The image of a 1-connected cdgc is a connected cocommutative dgha.

If $C$ is of finite type, $\#F(C,\partial,\eta)$ is isomorphic to $B\#(C,\partial,\eta)$ as a differential $\mathbb{Z}$-graded Hopf algebra.

If $A$ is not (graded) commutative, the differential $d_E$ of $B(A,d,\varepsilon)$ does not respect the shuffle product on $T(s\overline{A})$; $B(A,d,\varepsilon)$ thus becomes merely a differential $\mathbb{Z}$-graded coalgebra.  Similarly if $C$ is not (graded) cocommutative $F(C,\partial,\eta)$ is merely a differential $\mathbb{Z}$-graded algebra.

In particular, let

*  $\varepsilon-DGA$ be the category of augmented differential graded algebras, ($A = \oplus_{p\geq 0}A_p$).

* $DGC_0$, the category of connected differential graded coalgebras,

then the Bar and Cobar constructions yield functors

$$B: \varepsilon DGA\to DGC_0$$

$$F : DGC_0\to  \varepsilon DGA.$$

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

(Husemoller-Moore-[[Jim Stasheff|Stasheff]])

$B$ is right adjoint to $F$.

For any objects $(A,d)$ in $\varepsilon-DGA$, and $(C,\partial)$ of $DGC_0$, the natural adjunction morphisms

$$\hat{\alpha} : FB(A,d) \to (A,d)$$

$$\hat{\beta} : (C,\partial) \to BF(C,\partial)$$

are weak equivalences / [[quasi-isomorphism|quasi-isomorphisms]].

=--

These latter morphisms are defined by

* $\hat{\alpha} : T(s^{-1}\overline{T(s\overline{A})}), \delta)\to (A,d)$ is the zero mapping on $s^{-1}T^{\geq 2}(s\overline{A})$ and the natural isomorphism $s^{-1}s\overline{A} \stackrel{\simeq}{\to} \overline{A}$ on $s^{-1}s\overline{A}$.

* $\hat{\beta} : (C,\partial) \to (T(\overline{sT(s^{-1}\overline{C}}),D)$ is the unique lifting of
$$C\to s^{-1}\overline{C} \to \overline{T(s^{-1}\overline{C})}\to \overline{sT(s^{-1}\overline{C})}.$$

## Related concepts

* [[Koszul duality]]

## References

The source used for the above was

* D. Tanré, _Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan_, Lecture Notes in Maths No. 1025, Springer, 1983.

This was augmented with material from

* H. J. Baues, _Geometry of loop spaces and the cobar construction_, Mem. Amer.
Math. Soc. 25 (230) (1980) ix+171.

See also:

* {#FelixHalperinThomas92} [[Yves Felix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Adams' Cobar Equivalence_, Transactions of the American Mathematical Society, Vol. 329, No. 2 (1992), pp. 531-549 ([jstor:2153950](https://www.jstor.org/stable/2153950))

Review:

* [[Dev Sinha]], Section 1 in: _Koszul duality in algebraic topology - an historical perspective_, J. Homotopy Relat. Struct. (2013) 8: 1 ([arXiv:1001.2032](https://arxiv.org/abs/1001.2032))
* [[Mathieu Anel]], [[André Joyal]], _On the bar-cobar duality for algebras and operads_, slides 2012, [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Boston-Operades.pdf)
* [[Mathieu Anel]], [[André Joyal]], _Sweedler theory for (co)algebras and the bar-cobar constructions_, [arXiv:1309.6952](https://arxiv.org/abs/1309.6952)

* [[Kathryn Hess]], _The cobar construction: a modern perspective_, lectures 2007, [pdf](https://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf)

Identifying the cobar construction on [[singular homology|singular chains]] on a [[topological space]] with the dg-algebra of chains on its [[loop space]]:

* [[John Frank Adams]], *On the cobar construction*,  Proc Natl Acad Sci USA **42** 7 (1956) 409–412 [[doi:10.1073/pnas.42.7.409](https://doi.org/10.1073%2Fpnas.42.7.409)]

* [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Cubical rigidification, the cobar construction, and the based loop space*, Algebr. Geom. Topol. **18** (2018) 3789-3820 [[arXiv:1612.04801](https://arxiv.org/abs/1612.04801), [doi:10.2140/agt.2018.18.3789](https://doi.org/10.2140/agt.2018.18.3789)]

* [[Manuel Rivera]], *Adams' cobar construction revisited*, Documenta Math. **27** (2022) 1213-1223 [[arxiv:1910.08455](https://arxiv.org/abs/1910.08455)]

Generalization of the bar-cobar constructions to [[dg-Hopf algebras]]:

* [[Benoit Fresse]], _The universal Hopf operads of the bar construction_ ([arXiv:math/0701245](https://arxiv.org/abs/math/0701245))

* [[Murray Gerstenhaber]], [[Alexander Voronov]], Section 3.2 of: _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))

* Justin Young, _Brace bar-cobar duality_ ([arXiv:1309.2820](https://arxiv.org/abs/1309.2820))

[[!redirects cobar construction]]
[[!redirects cobar constructions]]
[[!redirects co-bar construction]]
[[!redirects co-bar constructions]]

[[!redirects bar-cobar duality]]
[[!redirects bar-cobar dualities]]

[[!redirects bar-cobar construction]]
[[!redirects bar-cobar constructions]]


bar construction

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

#Contents#
* table of contents
{:toc}

## Idea

The **bar construction** takes a [[monad]] $(T, \mu, \epsilon)$ equipped with an [[algebra over a monad|algebra-over-a-monad]] $(A, \rho)$ to the ([[augmented simplicial set|augmented]]) [[simplicial object]] whose structure maps are given by the structure maps of the monad and its action on its algebra:

$$
  \mathrm{B}(T,A)
  \coloneqq
  \left(
    \cdots
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    T T A
    \stackrel{\stackrel{\mu \cdot Id_A}{\longrightarrow}}{\stackrel{T \cdot \rho}{\longrightarrow}}
    T A \stackrel{\rho}{\longrightarrow} A
  \right)
    \,.
$$

This simplicial object can be viewed as a [[resolution]] of $A$, in a sense explained below.

## Definition

Let $\mathbf{E}$ be a [[category]] and let $(T, m: T T \to T, u: 1_{\mathbf{E}} \to T)$ be a [[monad]] on $\mathbf{E}$. We let $\mathbf{E}^T$ denote the [[Eilenberg-Moore category|category of]] $T$-[[algebra over a monad|algebras]], and $U: \mathbf{E}^T \to \mathbf{E}$ the [[forgetful functor]] which is [[monadic functor|monadic]], with [[left adjoint]] $F$.

Recall that the ([[augmented simplicial set|augmented]]) [[simplex category]] $\Delta_a$, viz. the category consisting of [[finite set|finite]] [[ordinals]][^fine1] and order-preserving maps, is the "[[walking structure|walking]] [[monoid]]", i.e., is [[initial object|initial]] among strict [[monoidal categories]] equipped with a [[monoid object]]. The monoidal product on $\Delta_a$ is ordinal addition $[m]+[n] = [m+n]$. If $[n]$ is the $n$-element ordinal, then the [[terminal object]] $[1]$ carries a unique monoid structure and represents the "generic monoid"[^fine2].

[^fine1]: N.B.: including the empty ordinal.

[^fine2]: If $X: \Delta_a^{op} \to \mathbf{C}$ is a simplicial object, then $X([n])$ is what is usually denoted $X_{n-1}$, the object of cells in dimension $n-1$. Note that $X([0]) = X_{-1}$ is the augmented component. The $n$ can be thought of as the number of vertices of a simplex of dimension $n-1$. We choose the index $n$ over the geometric dimension $n-1$ as it is more convenient for our purposes.

Similarly $\Delta_a^{op}$ is the walking [[comonoid]]. Since the [[comonad]] $F U$ on $\mathbf{E}^T$ can be regarded as a [[comonoid]] in the strict monoidal category of [[endofunctors]] $[\mathbf{E}^T, \mathbf{E}^T]$ (with endofunctor composition as monoidal product), there is a [[strong monoidal functor]] (or in fact a unique [[strict monoidal functor]])

$$\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T]$$

that takes the generic monoid $[1]$ to $F U$ and generally $[n]$ to $(F U)^{\circ n}$.

If furthermore $A$ is a $T$-algebra, there is an [[evaluation]] functor

$$[\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T$$

and we have the following definition:

+-- {: .num_defn}
###### Definition
The **bar construction** $Bar_T(A)$ is the simplicial $T$-algebra given by the composite functor

$$\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T.$$

The composite

$$\Delta_a^{op} \stackrel{Bar_T(A)}{\to} \mathbf{E}^T \stackrel{U}{\to} \mathbf{E}$$

will here be called the **bar resolution** of $A$.
=--

In the notation of [[two-sided bar constructions]], the bar construction would be written as $Bar_T(A) = B(F, T, A)$, and the bar resolution as $B(T, T, A)$.

## Décalage and resolutions

### Décalage

To explain the sense in which $U Bar_T(A)$ is an *acyclic resolution* of (the constant simplicial object) $A$, we recall the fundamental [[décalage]] construction. Very simply, putting

$$D = [1] + (-): \Delta_a^{op} \to \Delta_a^{op}$$

the décalage functor on simplicial objects $C^{\Delta_a^{op}}$ (valued in a category $\mathbf{C}$) is the functor

$$P: \mathbf{C}^{\Delta_a^{op}} \stackrel{\mathbf{C}^D}{\to} \mathbf{C}^{\Delta_a^{op}}.$$

Note that $D$ has a comonad structure (inherited from the comonoid structure on $[1]$ in $\Delta_a^{op}$, see also at _[décalage -- comonad structure](decalage#DecalageComonad)_), and therefore $P$ also carries a comonad structure. Notice also that there is a comonad map $D \to [1]\circ !$ (where $[1]: 1 \to \Delta_a^{op}$ is left adjoint to $!: \Delta_a^{op} \to 1$ since $[1]$ is initial in $\Delta_a^{op}$), induced by the evident natural inclusion $[1]+i: [1]+[0] \to [1]+[m]$ in $\Delta_a$. This in turn induces a comonad map $P X \to {|X|}$ where
${|-|}$ is the composite ("discretization"):

$$C^{\Delta_a^{op}} \stackrel{ev_{[1]}}{\to} C \stackrel{diag}{\to} C^{\Delta_a^{op}}.$$

The notation $P$ is chosen because décalage is essentially a kind of [[path space]] construction, i.e., in the case $\mathbf{C} = Set$ it is a [[simplicial sets]] analogue of a topological pullback

$$\array{
P X & \to & X^I & \stackrel{ev_1}{\to} X \\
\downarrow & & \downarrow_\mathrlap{ev_0} & \\
{|X|} & \underset{id}{\to} & X
}$$

where $id: {|X|} \to X$ is the identity inclusion of the underlying set with the discrete topology. $P X$ is essentially a sum of spaces of based paths $(\alpha: (I, 0) \to (X, x_0)$ over all possible choices of basepoint $x_0$, fibered over $X$ by taking $\alpha$ to $\alpha(1)$. Each space of based paths is contractible and therefore $P X$ is acyclic.

The following definition names a nonce expression; this author (Todd Trimble) does not know how this is (or might be) referred to in the literature:

+-- {: .num_defn #acyclic}
###### Definition
An **acyclic structure** on a simplicial object $X: \Delta_a^{op} \to C$ is a $P$-coalgebra structure $X \to P X$.
=--

Here a $P$-coalgebra structure on $X$ is the same as a *right* $D$-coalgebra (or $D$-comodule) structure, given by a simplicial map $h: X \to X \circ D$ satisfying evident equations. In more nuts-and-bolts terms, it consists of a series of maps $h_n: X([n]) \to X([n+1])$ satisfying suitable equations.

The map $h: X \to X D$ may be viewed as a homotopy. Again, turning to the topological analogue for intuition, the corresponding $h: X \to P X$ is a homotopy (or rather, the composite $X \to P X \to X^I$ can be turned into a homotopy $I \times X \to X$). The coalgebra structure $h: X \to P X$ has a retraction given by the counit $\varepsilon: P X \to X$, so $X$ becomes a retract of an acyclic space, hence acyclic itself.

+-- {: .num_remark #rem}
###### Remark
Definition \ref{acyclic} gives an *absolute* notion of acyclicity, in the sense that if $X: \Delta_a^{op} \to \mathbf{C}$ carries an acyclic structure $h: X \to X D$ and $G: \mathbf{C} \to \mathbf{C}'$ is any functor, then $G X$ automatically carries an acyclic structure $G h: F X \to F X D$. (For example, $G X$ becomes acyclic in a standard model category sense under any functor $G: \mathbf{C} \to Set$.)
=--

### Resolutions

Returning now to the bar resolution $U Bar_T(A)$: there is a canonical [[natural isomorphism]] $T \circ U Bar_T \cong U Bar_T \circ D$ obtained as the following 2-cell pasting (where $U Bar_T$ abbreviates the top and bottom horizontal composites)

$$\label{strong}\array{
\Delta_a^{op} & \stackrel{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \stackrel{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}] \\
 _\mathllap{D = [1] + (-)} \downarrow & \cong & _\mathllap{[id, F U]} \downarrow & \cong & \downarrow_\mathrlap{[id, U F] = [id, T]} \\
\Delta_a^{op} & \underset{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \underset{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}],
}$$

whence there is a homotopy

$$h = (U Bar_T \stackrel{u U Bar_T}{\to} T U Bar_T \cong U Bar_T D).$$

+-- {: .num_prop}
###### Proposition
The map $h$ is an acyclic structure, def. \ref{acyclic}, i.e., a right $D$-coalgebra structure.
=--

+-- {: .proof}
###### Proof
We verify the coassociativity condition for the coaction $h: U Bar_T \to U Bar_T D$; the counit condition is checked along similar lines. The comultiplication of the comonad $F U$ is $\delta \coloneqq F u U$, and putting $\eta = u U: U \to U F U$ for a right $F U$-coaction, the coassociativity of $\eta$ follows from a naturality square

$$\array{
U & \stackrel{\eta}{\to} & U F U \\
 _\mathllap{\eta} \downarrow & & \downarrow_\mathrlap{U\delta} \\
U F U & \underset{\eta F U}{\to} & U F U F U.
}$$

Apply $[id_{\mathbf{E}^T}, -]$ to this coassociativity square to get another coassociativity, this time for the comonad $K \coloneqq [id_{\mathbf{E}^T}, F U]$ on $[\mathbf{E}^T, \mathbf{E}^T]$ (with comultiplication denoted $\delta_K$) and coaction $H \coloneqq [id, \eta]: [id, U] \to [id, U] \circ K$. Thus there is an equalizing diagram

$$[id, U] \stackrel{H}{\to} [id, U]K \stackrel{\overset{[id, U]\delta_K}{\to}}{\underset{H K}{\to}} [id, U]K K.$$

Because $Bar_T: \Delta_a^{op} \to [\mathbf{E}^T, \mathbf{E}^T]$ is a strong monoidal functor (see the left isomorphism in (eq:strong)), the squares in

$$\array{
[id, U] Bar_T & \stackrel{H Bar_T}{\to} & [id, U]K Bar_T & \stackrel{\overset{[id, U]\delta_K Bar_T}{\to}}{\underset{H K Bar_T}{\to}} & [id, U]K K Bar_T \\
 & _\mathllap{h}{\searrow} & \downarrow_\mathrlap{\cong} & & \downarrow_\mathrlap{\cong} \\
 & & [id, U] Bar_T D & \stackrel{\overset{[id, U]Bar_T \delta_D}{\to}}{\underset{h D}{\to}} & [id, U]Bar_T D D
}$$

commute serially, with the triangle commuting by definition of $h$. This completes the verification.
=--

By Remark \ref{rem}, it follows that $U Bar_T(A)$, obtained by applying evaluation at a $T$-algebra $A$, carries an acyclic structure as well. In this sense we may say that $U Bar_T(A)$ (which has $A$ as its augmented component in dimension $-1$) is an acyclic resolution of the constant simplicial $T$-algebra at $A$ that carries a $T$-algebra structure.

## Properties

### Universal property

We now state and prove a [[universal property]] of the bar construction $Bar_T(A)$.

+-- {: .num_defn}
###### Definition
Let $(T, m: T T \to T, u: 1 \to T)$ be a monad on a category $\mathbf{E}$. A $T$-**algebra resolution** is a simplicial object $Y: \Delta_a^{op} \to \mathbf{E}^T$ together with an acyclic structure on $U Y: \Delta_a^{op} \to \mathbf{E}$. A morphism between $T$-algebra resolutions is a natural transformation $\phi: Y \to Y'$ such that $U\phi: U Y \to U Y'$ is a $P$-coalgebra map.
=--

Let $AlgRes_T$ be the category of $T$-algebra resolutions. There is a forgetful functor

$$G: AlgRes_T \to \mathbf{E}^T$$

that takes an algebra resolution $Y$ to its augmentation component $Y[0]$.

+-- {: .num_theorem #universal}
###### Theorem
The functor $\hom_{\mathbf{E}^T}(A, G-): AlgRes_T \to Set$ is represented by $Bar_T(A)$; i.e., $Bar_T(-): \mathbf{E}^T \to AlgRes_T$ is [[left adjoint]] to $G$.
=--

The proof is distributed over two lemmas.

+-- {: .num_lemma #unique}
###### Lemma
Given a $T$-algebra resolution $Y$ and a $T$-algebra map $f: A \to Y[0]$, there is at most one $T$-algebra resolution map $\phi: Bar_T(A) \to Y$ such that
$\phi[0] = f$.
=--

+-- {: .proof}
###### Proof
The $P$-coalgebra structure $h: U Bar_T(A) \to U Bar_T(A) \circ D$ is defined on components $U Bar_T(A)[n] = T^n A$ by $h[n] = u T^n(A): T^n(A) \to T^{n+1}(A)$. Thus in order that $U\phi$ be a $P$-coalgebra map, we must have that the diagram

$$\array{
T^n A & \stackrel{u T^n A}{\to} & U F T^n(A) \\
 _\mathllap{U\phi[n]} \downarrow & & \downarrow_\mathrlap{U\phi[n+1]} \\
U Y[n] & \underset{h_Y[n]}{\to} & U Y[n+1]
}$$

commutes. Here $\phi[n]: T^n (A) \to Y[n]$ determines a unique $T$-algebra map $g: F T^n(A) \to Y[n+1]$ such that

$$U(g) \circ u T^n(A) = h_Y[n] \circ U\phi[n]$$

since $F$ is left adjoint to $U$. Thus, starting with $\phi[0] = f$ as given, each algebra map $\phi[n]$ uniquely determines its successor $\phi[n+1] = g$.
=--

+-- {: .num_remark #simp}
###### Remark
The preceding proof does not show that the $\phi[n]$ fit together to form a map $\phi$ of simplicial $T$-algebras (i.e., to respect faces and degeneracies); it merely shows at most one such $T$-algebra resolution map is possible. But once we show that $\phi$ respects faces and degeneracies, the proof of Theorem \ref{universal} will be complete.
=--

+-- {: .num_lemma #exist}
###### Lemma
Given a $T$-algebra resolution $Y$ and an algebra map $f: X \to Y[0]$, there is at least one $T$-algebra resolution map $\phi: Bar_T(X) \to Y$ with $\phi[0] = f$.
=--

+-- {: .proof}
###### Proof
It is enough to produce such a map $\phi: Bar_T(Y[0]) \to Y$ in the case $f = 1_{Y[0]}$, since the case for general $f: X \to Y[0]$ is then given by a composite

$$Bar_T(X) \stackrel{Bar_T(f)}{\to} Bar_T(Y[0]) \stackrel{\phi}{\to} Y.$$

We will do something slightly more general. For any category $\mathbf{C}$, the endofunctor category $[\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]$ has a comonoid object $P = - \circ D$, so that there is an induced strong monoidal functor

$$\Delta_a^{op} \stackrel{\tilde{P}}{\to} [\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]$$

which, upon evaluating at an object $Y$ of $\mathbf{C}^{\Delta_a^{op}}$, gives a functor

$$B(Y, D, D) \coloneqq eval_Y \circ \tilde{P}: \Delta_a^{op} \to \mathbf{C}^{\Delta_a^{op}}$$

with $B(Y, D, D)[n] = Y D^n$, so that $B(Y, D, D)$ is a double simplicial object. Taking $\mathbf{C} = \mathbf{E}^T$ and taking $Y$ to be a $T$-algebra resolution with acyclic structure $h: Y \to Y D$, we will produce a (double) simplicial map

$$\Phi: B(T, T, Y) \to B(Y, D, D)$$

where $\Phi[n]: T^n Y \to Y D^n$ is defined recursively as in the proof of Lemma \ref{unique}, by setting $\Phi[0] = 1_Y$ and taking $\Phi[n+1]: T^{n+1} Y \to Y D^{n+1}$ the unique simplicial $T$-algebra map such that

$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y \\
 _\mathllap{\Phi[n]} \downarrow & & \downarrow_\mathrlap{\Phi[n+1]} \\
Y D^n & \underset{h D^n}{\to} & Y D^{n+1}
}$$

commutes for all $n$. Once we verify the claim that $\Phi$ respects faces and degeneracies, the same will be true for $\phi[n] = \Phi[n][0]: (T^n Y)[0] = T^n(Y[0]) \to (Y D^n)[0] = Y[n]$, whence the proof will be complete by Remark \ref{simp}.

The claim is proved by induction on $n$. Let $\epsilon: D \to 1_{\Delta_a^{op}}$ be the counit and $\delta: D \to D D$ be the comultiplication. We have face maps

$$T^j m T^{n-j-1} Y: T^{n+1} Y \to T^n Y, \qquad Y D^j \epsilon D^{n-j}: Y D^{n+1} \to Y D^n$$

for $j = 0$ to $n$, under the special convention that $m T^{-1}Y: T Y \to Y$ denotes the action $\alpha: T Y \to Y$. We also have degeneracy maps

$$T^j u T^{n-j} Y: T^n Y \to T^{n+1} Y, \qquad Y D^{j-1} \delta D^{n-j}: Y D^n \to Y D^{n+1}$$

for $j = 1$ to $n$. We proceed as follows.

* To check preservation of face maps, we treat separately the cases where $j = 0$ and $j \geq 1$.
  * For $j = 0$, we must check commutativity of the square in
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
 & _\mathllap{id} \searrow & \downarrow_\mathrlap{m T^{n-1} Y} & & \downarrow_\mathrlap{Y\epsilon D^n} \\
 & & T^n Y & \underset{\Phi[n]}{\to} & Y D^n.
}$$
Since all the maps are algebra maps and $u T^n Y: T^n Y \to T^{n+1} Y$ exhibits $T^{n+1} Y$ as the free algebra on $T^n Y$, it suffices to check commutativity around the perimeter. (N.B.: the triangle commutes, even in the case where $n=0$ which we need to start the induction.) By definition of $\Phi[n+1]$, commutativity of the perimeter boils down to commutativity of
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
 & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y\epsilon D^n} \\
 & & Y D^n & \underset{id}{\to} & Y D^n
}$$
where the triangle commutes by one of the acyclic structure equations.
  * For $j \geq 1$, the commutativity of the right square in
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\PhiY[n+1]}{\to} & Y D^{n+1} \\
 _\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & nat. & \downarrow_\mathrlap{T^j m T^{n-j-1} Y} & & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\
T^{n-1} Y & \underset{u T^{n-1} Y}{\to} & T^n Y & \underset{\Phi[n]}{\to} & Y D^n \\
 & _\mathllap{\Phi[n-1]} \searrow & & \nearrow_\mathrlap{h D^{n-1}} & \\
 & & Y D^{n-1} & &
}$$
is again by appeal to a freeness argument where we just need to check commutativity of the perimeter, noting commutativity of the left square by naturality and that of the bottom quadrilateral by the recursive definition of $\Phi[n]$. But the perimeter commutes by examining the diagram
$$\array{
T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\
_\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\
T^{n-1} Y & ind. & Y D^n & nat. & Y D^n \\
 & _\mathllap{\Phi[n-1]} \searrow & \downarrow & \nearrow_\mathrlap{h D^{n-1}} & \\
 & & Y D^{n-1} & &
}$$
(where the middle vertical arrow is $Y D^{j-1} \epsilon D^{n-j}$) using the inductive hypothesis in the bottom left parallelogram.

* To check preservation of degeneracy maps, we treat separately the cases $j=1$ and $j \geq 2$.
  * For $j = 1$, the commutativity of the top right square in
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
 _\mathllap{u T^{n-1} Y} \downarrow & nat. & \downarrow _\mathrlap{T u T^{n-1} Y} & & \downarrow_\mathrlap{Y \delta D^{n-1}} \\
T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\
 & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\
 & & Y D^n & &
}$$
is by appeal to a freeness argument where we just need to check commutativity of the perimeter (the special case $n=1$ being used to start the induction). But this boils down to commutativity of the diagram
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
 _\mathllap{u T^{n-1} Y} \downarrow & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y \delta D^{n-1}} \\
T^n Y & & Y D^{n-1} & & Y D^{n+1} \\
 & _\mathllap{\Phi[n]} \searrow & \downarrow_\mathrlap{h D^{n-1}} & \nearrow_\mathrlap{h D^n} & \\
 & & Y D^n & &
}$$
where the bottom right quadrilateral commutes by one of the acyclic structure equations.
  * For $j \geq 2$, the commutativity of the top right square in
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
 _\mathllap{T^{j-1} u T^{n-j} Y} \downarrow & nat. & \downarrow _\mathrlap{T^j u T^{n-j} Y} & & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\
T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\
 & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\
 & & Y D^n & &
}$$
is once again by appeal to a freeness argument where we just need to check commutativity of the perimeter. Here it boils down to commutativity of
$$\array{
T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\
 _\mathllap{T^{j-1} u T^{n-j} Y} \downarrow  & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\
T^n Y & ind. & Y D^{n-1} & nat. & Y D^{n+1} \\
 & _\mathllap{\Phi[n]} \searrow & \downarrow & \nearrow_\mathrlap{h D^n} & \\
 & & Y D^n & &
}$$
where the middle vertical arrow is $Y D^{j-2} \delta D^{n-j}$.

This completes the proof.
=--

## Special cases

### For modules over an algebra

Let $A$ be a commutative [[associative algebra]]s over some [[ring]] $k$. Write $A Mod$ for the category of [[connective chain complex|connective]] [[chain complex]]es of [[module]]s over $A$.

For $N$ a right module, also $N \otimes_k A$ is canonically a module. This construction extends to a functor

$$
  A \otimes_k (-) :
    A Mod
      \to
    A Mod
  \,.
$$

The [[monoid]]-structure on $A$ makes this a [[monad]] in [[Cat]]: the monad product and unit are given by the product and unit in $A$.

For $N$ a module its right action $\rho :N \otimes A \to N$ makes the module an [[algebra over a monad|algebra over this monad]].

The bar construction $\mathrm{B}(A,N)$ is then the simplicial module

$$
  \cdots
  \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
  N \otimes_k A \otimes_k A \stackrel{\overset{Id \otimes \mu}{\longrightarrow}}{\underset{\rho \otimes Id}{\longrightarrow}} N \otimes_k A
  \,.
$$

Under the [[Moore complex]] functor of the [[Dold-Kan correspondence]] this is identified with a [[chain complex]] whose [[differential]] is given by the alternating sums of the face maps indicated above.

This chain complex is what originally was called the **bar complex** in [[homological algebra]]. Because the first authors denoted its elements using a notation involving vertical bars ([Ginzburg](#Ginzburg))!!

This chain complex provides a resolution that computes the [[Tor]]

$$
  Tor(N, A \times A)
  \,.
$$

This gives the [[Hochschild homology]] of $A$. See there for more details.

### For differential graded (Hopf) algebras

See [[bar and cobar construction]].

### For $E_\infty$-algebras

See ([Fresse](#Fresse)).

##Related entries

* [[simplicial resolution]]; this is essentially the same concept but from a slightly different perspective.

* [[homotopy coherent nerve]]

* [[canonical resolution]]

* [[monadic cohomology]]

## References and Literature

A general discussion of bar construction for monads is at

* {#Trimble} [[Todd Trimble]], _On the Bar Construction_ ([blog](http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html))

Textbook accounts can be found at:

* [[Saunders Mac Lane]], section IV.5 of _Homology_

* [[Charles Weibel]], section 8.6 of _[[An Introduction to Homological Algebra]]_ (1994)

The bar complex of a bimodule is reviewed for instance in

* {#Ginzburg} [[Victor Ginzburg]], _Lectures on noncommutative geometry_ ([arXiv:math/0506603](http://arxiv.org/abs/math.AG/0506603))

around page 16.

The bar complex for [[E-infinity algebra]]s is discussed in

* [[Benoit Fresse]], _The bar complex of an E-infinity algebra_, Advances in Mathematics Volume 223, Issue 6, 1 April 2010, Pages 2049-2096
{#Fresse}

The compositional structure of the bar construction of several monads, as well as its interpretation in terms of [[partial evaluation|partial evaluations]] is studied in

* [[Carmen Constantin]], [[Tobias Fritz]], [[Paolo Perrone]] and [[Brandon Shapiro]], _Partial evaluations and the compositional structure of the bar construction_. ([arXiv](https://arxiv.org/abs/2009.07302))

[[!redirects bar constructions]]

[[!redirects bar complex]]

[[!redirects bar resolution]]
[[!redirects bar resolutions]]


Barak Kol

* [webpage](http://www.phys.huji.ac.il/~barak_kol/)

## Selected writings

On [[(p,q)5-brane webs]]:

* [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))

* [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))

## Related $n$Lab entries

* [[(p,q)5-brane]]

* [[brane intersection]]

* [[orientifold]], [[orientifold plane]]

category: people


Barbara Fantechi

__Barbara Fantechi__ is an algebraic geometer with professorship in Trieste.

* [Website](http://people.sissa.it/~fantechi/)

## Selected writing

On basics of [[algebraic geometry]], following [[Grothendieck]]'s [[FGA]]:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:

  **[[FGA explained]]** (ICTP, Trieste 2003--2005)

  Mathematical Surveys and Monographs __123__

  Amer. Math. Soc. 2005. x+339 pp.

  * [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)

    [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

    [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)

## Related $n$Lab entries

* [[perfect obstruction theory]]

* [[Gromov-Witten invariants]]

category: people


Barcan formula

Under construction

>If it is possible that something is $F$, then something is such that it is possible that it is $F$.

## Related concepts

* [[de dicto and de re]]

[[!redirects Barcan's formula]]


Bargmann-Segal transform

See [[coherent state]].

Bargmann-Segal transform is the integral transform whose kernel is the overlap between the projective measure corresponding to the coherent states and
a measure corresponding to an orthonormal basis comeing from some polarization for $L^2$-sections. The kernel is a special case of a
[[Bergman kernel]] in complex analysis.

Classical case of Heisenberg group:

* V. Bargmann, _On a Hilbert space of analytic functions and an associated integral transform_, Communications on Pure and Applied Mathematics __14__ (1961) 187-214 [MR0157250](http://www.ams.org/mathscinet-getitem?mr=157250) [doi](ttp://dx.doi.org/10.1002/cpa.3160140303)

Vectors in the Hilbert space can be represented in the coherent state representation: $|f\rangle = \int |z\rangle\langle z|f\rangle d\mu$; if $f$ is in $L^2$ then $\langle z|f\rangle$ is a holomorphic function and this passage is called the Bargmann-Segal transform (referring to [Irving Segal](http://en.wikipedia.org/wiki/Irving_Segal)); this way certain Hilbert space of holomorphic function appears, the Bargmann-Fock space.

Further generalization is to Perelomov coherent states.

More recent generalized Segal-Bargmann transform of Hall:

* Brian Charles Hall. _The Segal-Bargmann coherent state transform for Lie groups_. J. Funct. Anal. __122__:103–151, 1994, [doi](http://dx.doi.org/10.1006/jfan.1994.1064); _Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type_, Comm. Math. Phys., __226__:233–268, 2002. [doi](http://dx.doi.org/10.1007/s002200200607)

[[!redirects Segal-Bargmann transform]]
[[!redirects Bargmann transform]]


Barnes G-function

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{:toc}

## Idea

Much like the [[Gamma function]] generalizes the functional equation

$$
  \Gamma(z + 1) \;=\; z \, \Gamma(z)
$$

to non-integer values of $z$, so the *Barnes $G$-function* $G(-)$ corresponds to the functional equation

\[
  \label{TranslationFormula}
  G(z + 1) \;=\; \Gamma(z) \, G(z)
  \,.
\]

Just as for $z =n \in \mathbb{N}$, $\Gamma(n+1) = n!$, so $G(n+2) = n!(n-1)! \cdots 1!$.

## Definition

(...)

## Properties

### Special values

\[
  \label{GOf1}
  G(1) \;=\; 1
\]

$$
  G(1/2)
  \;=\;
  2^{1/24}
    \cdot
  e^{ \tfrac{3}{2} \zeta^'(-1) }
    \cdot
  \pi^{ - 1/4 }
  \,.
$$

([WP here](https://en.wikipedia.org/wiki/Barnes_G-function#Value_at_1/2))

### Stirling-like asymptotic expansion
 {#AsymptoticExpansion}

$$
  ln G(1 + z)
  \;=\;
  z^2
  \left(
    \tfrac{1}{2} ln(z)
    -
    \tfrac{3}{4}
  \right)
  +
  \tfrac{1}{2}
  ln(2 \pi)
  z
  -
  \tfrac{1}{12} ln(z)
  +
  \zeta^'(-1)
  +
  \mathcal{O}(z^{-1})
  \,,
$$

where $\zeta$ denotes the [[Riemann zeta function]].

(e.g. [WP here](https://en.wikipedia.org/wiki/Barnes_G-function#Asymptotic_expansion), [WMW (14)](#WMW))

### Relations to the Gamma-function

A version of the [[Gauss multiplication formula]] for the [[Gamma function]]:

\begin{prop}\label{MultiplicationFormulaForGammaFunctionInTermsOfGFunction}
$$
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \Gamma(j/2)
  \;=\;
  \frac
    {
      G(N/2+ 1)
      \cdot
      G(N/2 + 1/2)
    }
    { G(1/2) }
  \,.
$$
\end{prop}
([Kotěšovec 13, p. 2](#Kotesovec))
\begin{proof}
By repeated use of the translation formula (eq:TranslationFormula)
and using the initial value $G(1) = 1$ (eq:GOf1):

In the case that $N = 2M$ in an [[even number]]:

$$
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \Gamma(j/2)
  \;=\;
\underoverset
    {j = 1}
    {M}
    {\prod}
  \Gamma(j)
\cdot
\underoverset
    {j = 1}
    {M}
    {\prod}
  \Gamma(j - \frac{1}{2})
 \;=\;
G(M+1) \cdot G(M +1/2) /G(1/2)
 \;=\;
G(N/2 + 1) \cdot G(N/2 + 1/2)/G(1/2).
$$

In the case that $N = 2M+1$ is an [[odd number]]:

$$
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \Gamma(j/2)
  \;=\;
\underoverset
    {j = 1}
    {M}
    {\prod}
  \Gamma(j)
\cdot
\underoverset
    {j = 1}
    {M+1}
    {\prod}
  \Gamma(j - \frac{1}{2})
 \;=\;
G(M+1) \cdot G(M +3/2) /G(1/2)
 \;=\;
G(N/2 + 1/2) \cdot G(N/2 + 1)/G(1/2).
$$
\end{proof}

## References

See also:

* Wikipedia, *[Barnes G-function](https://en.wikipedia.org/wiki/Barnes_G-function)*

* {#WMW} WolframMathWorld, *[Barnes G-Function](https://mathworld.wolfram.com/BarnesG-Function.html)*

In the context of [counting](semistandard+Young+tableau#NumberOfSYTWithBoundedNumberOfRows) of [[standard Young tableaux]] of bounded height:

* {#Kotesovec13} Václav Kotěšovec, *Asymptotic of Young tableaux of bounded height*, 2013 ([pdf](http://www.kotesovec.cz/math_articles/kotesovec_young_tableaux_conjecture.pdf), [[KotesovecAsymptoticsOfBoundedYoungTableaux.pdf:file]])

[[!redirects Barnes G-functions]]


Barr embedding theorem

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[[Barr]] proved a theorem about [[embedding of categories|embedding]] [[regular categories]] into [[presheaf category|categories of]] [[small presheaves]], and also a strengthening for [[Barr exact categories]].

## Idea

Barr's embedding theorem has the classical form of many embedding[^note] theorems in mathematics: if a structure $\mathcal{C}$ has certain good properties, then it admits an embedding with certain other good properties into another structure $\mathcal{D}$ which is somehow more explicit than $\mathcal{C}$.

For example,

* by Tychonoff's embedding theorem, if a $T_0$ space $X$ is [[completely regular]], then there exists an embedding of $X$ into a product of [[metric space]]s

* by the [[Whitney embedding theorem]], if an abstract $d$-dimensional real manifold $M$ is smooth, then there exists an embedding, which is an [[embedding of smooth manifolds]], of $M$ into the explicit real manifold that is $\mathbb{R}^{2d}$

* by the [[Freyd-Mitchell embedding theorem]], if a category is [[small]] and [[abelian category|abelian]], then there exist an embedding, which is [[exact functor|exact]], into the category of modules of a (not necessarily commutative) [[ring]].

and,

* by Barr's embedding theorem, if a category is [[small]] and [[regular category|regular]], then there exists an [[embedding]], which is a full, faithful and [[regular functor|regular]], into a category of [[presheaves]] over some [[small category]].

## A proof of Makkai

The proof of (a version of) Barr's theorem given by Makkai in [Makkai1980](#Makkai80) is a nice example of a non-trivial application of [[ultraproduct]]s in category theory.

## References

It has been proved in

* [[Michael Barr|M. Barr]], _Exact categories_, Lecture Notes in Math. __236__, (Springer, Berlin, 1971), 1-120.

and, in a different way, in

* M. Barr, _Representation of categories_, J. Pure Appl. Alg. __41__ (1986) 113-137 (this article has supposedly some fixable errors).

* F. Borceux, _A propos d'un théorème de Barr_, Séminaire de mathématique (nouvelle série) Rapport 28 - Mai 1983, Institute de Mathematique Pure et Appliqee, Univ. Cath. de Louvain.

* M. Makkai, _A theorem on Barr-exact categories, with an infinitary generalization_, Ann. Pure Appl. Logic __47__ (1990), no. 3, 225-268.

[[Michael Barr]]'s full exact embedding theorem for [[Barr exact categories]], proved in (?)

* Michael Barr, _Embedding of categories_, Proc. Amer. Math. Soc. __37__, No. 1 (Jan., 1973), pp. 42-46 ([jstor](http://www.jstor.org/stable/2038702), [pdf](http://www.math.mcgill.ca/barr/papers/embed.pdf))

generalizes the Lubkin-Freyd-Mitchell embedding theorems for abelian categories. The Giraud's theorem for topoi is not much more than a special case of that theorem.

* {#Makkai80} M. Makkai, _On full embeddings I_, Journal of Pure and Applied Algebra __16__, (1980), pp. 183-195

* M. Makkai, _Full continuous embeddings of toposes_, Trans. Amer. Math. Soc. __269__, No. 1 (Jan., 1982), pp. 167-196 [jstor](http://www.jstor.org/stable/1998599)

One can also consider categories enriched over a locally finitely presentable closed symmetric monoidal category $V$. Such a $V$-category $C$ is regular if it is finitely complete, admits the coequalizers of kernel pairs all [[regular epimorphism]]s are universal (i.e. stable under pullbacks) and stable under cotensors with the finite objects. A $V$-functor is regular if it preserves finite limits and regular epimorphisms. The following generalizes the Barr's embedding theorem for regular categories to the regular enriched categories:

* Dimitri Chikhladze, _Barr's embedding theorem for enriched categories_, J. Pure Appl. Alg. __215__, n. 9 (2011) 2148-2153, [arxiv/0903.1173](http://arxiv.org/abs/0903.1173), [doi](http://dx.doi.org/10.1016/j.jpaa.2010.12.004)

Its main result is

> Theorem 10. For a small regular $V$-category $C$ there exists a small category $T$ and a regular fully faithful functor $E : C \longrightarrow [T, V]$.

[[!redirects Barr's embedding theorem]]