FGA

This page compiles pointers for the collection of seminar notes

* [[Alexander Grothendieck]],

  **Fondements de la Géométrie Algébrique**

  Extraits du Séminaire Bourbaki 1957-1962

  * original: [pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/FGA.pdf), [[FGA.pdf:file]]

    English translation by [[Tim Hosgood]]: [web version](https://thosgood.com/fga).

    [Wikipedia entry](https://en.wikipedia.org/wiki/Fondements_de_la_G%C3%A9ometrie_Alg%C3%A9brique)

on basics of [[algebraic geometry]] and including a fair bit of basics of [[category theory]] -- a precursor to *[[Éléments de géométrie algébrique]]* ([[EGA]]),

consisting of the following parts:

* (...)

* [[Alexander Grothendieck]],

  *Technique de descente et théorèmes d’existence engéométrie algébrique. I:*

  *Généralités. Descente par morphismes fidèlement plats*,

  Séminaire N. Bourbaki, 1960, exp. no190, p. 299-327

  ([numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0), [original pdf](http://www.numdam.org/item/SB_1958-1960__5__299_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-I.pdf), [web version](https://thosgood.com/fga/book/FGA-3-I))

* [[Alexander Grothendieck]],

  *Technique de descente et théorèmes d'existence en géométrie algébriques. II:*

  *Le théorème d'existence en théorie formelle des modules*,

  Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195

  ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/SB_1958-1960__5__369_0), [original pdf](http://www.numdam.org/item/SB_1958-1960__5__369_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-II.pdf), [web version](https://thosgood.com/fga/book/FGA-3-II)).

* [[Alexander Grothendieck]],

  *Techniques de construction et théorèmes d'existence en géométrie algébrique III:*

  *préschémas quotients*,

  Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212

  ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [original pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-III.pdf), [web version](https://thosgood.com/fga/book/FGA-3-III)).

* (...)

Related references:

* [[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)

category: reference

[[!redirects Fondements de la Géométrie Algébrique]]


FGA explained

One of the series of Grothendieck's works is [[FGA]] (see entry [[EGA]] for overall description of EGA, FGA and [[SGA]]). A summer school in Trieste 2003, has tried to summarize some of the main historical breakthroughs of FGA in modern exposition. The proceedings of that school are an updated version of freely available materials on the ICTP web.

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

  **Fundamental algebraic geometry -- Grothendieck's FGA explained**

  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)

## Contents

See the [conference page](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable) for scans of the lecture notes.

On [[stacks]] in the guise of [[fibered categories]] satisfying [[descent]]:

* [[Angelo Vistoli]], *Grothendieck topologies, fibered categories and descent theory* (1--104) [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406); [math.AG/0412512](http://arxiv.org/abs/math/0412512).

On [[Hilbert schemes]]:

* [[Nitin Nitsure]], *Construction of Hilbert and [[Quot schemes]]* (105--137)
[MR2223407](http://www.ams.org/mathscinet-getitem?mr=2223407); [math/0504590](https://arxiv.org/abs/math/0504590)

* [[Barbara Fantechi]], [[Lothar Göttsche]], *Local properties and Hilbert schemes of points* (139--178) MR2223408; (draft [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s7t5&ifd=13843&down=1&type=lecture_notes))

On [[Picard schemes]]:

* [[Steven Kleiman|Steven L. Kleiman]], *The Picard scheme* (235--321) MR2223410 (draft [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s6t3&ifd=15022&down=1&type=lecture_notes))

See also:

* [[Luc Illusie]], *[[Grothendieck's existence theorem]] in formal geometry* (179--233) MR2223409; (draft version [pdf](https://indico.ictp.it/event/a0255/session/28/contribution/17/material/0/0.pdf))

category: reference

[[!redirects Fundamental algebraic geometry -- Grothendieck's FGA explained]]


FI-representation

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In variation of how a [[symmetric group representation]] (or "$Sym(n)$-module") may equivalently be thought of as a [[functor]]

* from the [[delooping groupoid]] $\mathbf{B}Sym(n)$

  or equivalently from the [[core]] [[groupoid]] of the [[full subcategory]] of [[FinSet|$FinSet$]] on the [[finite sets]] of [[cardinality]] $n$

* to [[Vect]] (or to $R$[[Mod]] for some [[ring]] $R$)

so an *FI-representation* or *FI-modules* is a [[functor]] to [[Vect]] (or $R$[[Mod]]) from the larger category [[FinSetInj|$FinSet_{inj}$]] of *all* [[finite sets]] with [[morphism]] all the [[injective maps]] between these.

So an FI-representation is, in particular, one [[symmetric group representation|$Sym(n)$-representation]] in the sense of ordinary [[representation theory]], for all $n \in \mathbb{N}$, but in addition equipped with a system of [[intertwiners]] between these.

## Properties

### Relation to representation stability

One reason that FI-representation draw attention is that they exhibit the phenomenon called *[[representation stability]]*, and in fact the FI-representation theory thereby serves to "explain" the occurence of various stability phenomena seen the the study of [[moduli spaces]], notably in the study of ordered [[configuration spaces of points]].

## References

Original articles on the notion of FI-modules:

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]

Further discussion in relation to [[representation stability]]:

* [[Trevor Hyde]], *$FI$-Modules and Representation Stability* (2016) [[pdf](https://math.uchicago.edu/~tghyde/Hyde%20--%20FI-modules%20and%20representation%20stability.pdf), [[Hyde-FIModules.pdf:file]]]

* [[Nir Gadish]], *Categories of FI type: a unified approach to generalizing representation stability and character polynomials*, Journal of Algebra **480** (2017) 450-486 [[arXiv:1608.02664](https://arxiv.org/abs/1608.02664), [doi:10.1016/j.jalgebra.2017.03.010](https://doi.org/10.1016/j.jalgebra.2017.03.010)]

* [[Joe Moeller]], *Extensions of representation stable categories* [[arXiv:2209.03879](https://arxiv.org/abs/2209.03879)]

Discussion of [[FI-representations]] in the generality of [[infinity-representations|$\infty$-representations]] in [[stable infinity-categories|stable $\infty$-categories]] and their analysis via [[Goodwillie calculus]]:

* [[Kaya Arro]], *FI-Calculus and Representation Stability* (2023) [[arXiv:2306.13597](https://arxiv.org/abs/2306.13597)]

[[!redirects FI-representations]]

[[!redirects FI-module]]
[[!redirects FI-modules]]


fiber

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###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _fiber_ of a [[morphism]] or [[bundle]] $f : E \to B$ over a [[point]] of $B$ is the collection of [[generalized element|elements]] of $E$ that are mapped by $f$ to this point.

## Definition

Fibers make sense in any category with a terminal object $*$ and pullbacks.  For $f : A \to B$ a [[morphism]] in such a [[category]] and $B$ equipped with the [[structure]] of a [[pointed object]] $pt : * \to B$, the **fiber** of $f$ is the [[fiber product]] of $f$ with $pt$, hence the [[pullback]]

$$
  \array{
    A \times_B * &\to& *
    \\
    \downarrow && \downarrow_pt
    \\
    A &\stackrel{f}{\to}& B
  }
$$

## Examples

### Bundles

Given a [[bundle]] $p: E \to B$ and a [[global element]] $x: 1 \to B$, the __fibre__ or __fiber__ $E_x$ of $(E,p)$ over $x$ is the [[pullback]]
$$ \array {
   E_x        & \to             & E \\
   \downarrow &                 & \downarrow_p \\
   1          & \stackrel{x}\to & B
} $$
if it exists.

In a [[fiber bundle]], all fibres are [[isomorphism|isomorphic]] to some standard fibre $F$ in a coherent way.

### Kernels

In an [[additive category]] fibers over the [[zero object]] are called _[[kernels]]_.

### Fibers of a sheaf of modules

The fiber of a sheaf $\mathcal{E}$ of $\mathcal{O}$-modules over a [[locally ringed space]] $(X,\mathcal{O})$ at a point $x \in X$ is defined as the vector space $\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x)$ over the [[residue field]] $k(x)$. If $\mathcal{E}$ is [[quasicoherent sheaf|quasicoherent]], the associated vector bundle of the fiber is the pullback of the associated vector bundle of $\mathcal{E}$:
$$ \array {
   V(\mathcal{E}(x)) = \mathrm{Spec} \mathrm{Sym} \mathcal{E}(x)
     & \to
     & \underline{Spec}_X \mathrm{Sym} \mathcal{E} = V(\mathcal{E}) \\
   \downarrow & & \downarrow \\
   \mathrm{Spec} k(x) & \to & X
} $$

## Related concepts

* [[essential fiber]]

* [[homotopy fiber]], [[fiber sequence]]

* [[cofiber]], [[homotopy cofiber]]

* [[kernel]], [[cokernel]]

* [[fiber type]]

[[!redirects fibre]]
[[!redirects fibres]]
[[!redirects fibers]]


fiber bundle

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

#Contents#
* table of contents
{:toc}

## Idea

A _fibre bundle_ or _fiber bundle_ is a [[bundle]] in which every [[fibre]] is [[isomorphism|isomorphic]], in some coherent way, to a _standard fibre_ or _typical fiber_. Usually one also requires that it be [[locally trivial]], hence locally of the form of a [[Cartesian product]] with that typical fiber.

## Definitions

In the most general sense, a __bundle__ over an object $B$ in a [[category]] $C$ is a [[morphism]] $p: E \to B$ in $C$.

In appropriate contexts, a __fibre bundle__ over $B$ with standard fibre $F$ may be defined as a bundle over $B$ such that, given any [[global element]] $x: 1 \to B$, the [[pullback]] of $E$ along $x$ is isomorphic to $F$.
Certainly this definition is appropriate whenever $C$ has a [[terminal object]] $1$ which is a [[separator]], as in a [[well-pointed category]]; even then, however, one often wants the more restrictive notion below.

One often writes a typical fibre bundle in shorthand as $F \to E \to B$ or
$$ \array {
   F & \rightarrow & E \\
     &             & \downarrow \\
     &             & B
} $$
even though there is not a single morphism $F \to E$ but instead one for each global element $x$ (and none at all if $B$ has no global elements!).

If $C$ is a [[site]], then a __[[locally trivial|locally]] [[trivial fibre bundle]]__ over $B$ with typical fibre $F$ is a bundle over $B$ with a [[cover]] $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, the pullback $E_\alpha$ of $E$ along $j_\alpha$ is isomorphic in the [[slice category]] $C/{U_\alpha}$ to the [[trivial bundle]] $U_\alpha \times F$ (a _[[local trivialization]]_).

One can also drop $F$ and define a slightly more general notion of __[[locally trivial]] bundle__ over $B$ as a bundle over $B$ with a cover $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, there is a fibre $F_\alpha$ and an isomorphism in $C/{U_\alpha}$ between the pullback $E_\alpha$ and the trivial bundle $U_\alpha \times F_\alpha$.  Every [[locally trivial]] fibre bundle is obviously a locally trivial bundle; the converse holds if $B$ is [[connected object|connected]].

+-- {: #TransitionMorphisms}
Now suppose that $E$ is a fibre bundle over $B$ with [[typical fibre]] $F$, [[local trivialization|locally trivialised]] over an ([[open cover|open]]) [[cover]] $(U_\alpha)_\alpha$ via [[isomorphisms]] $(t_\alpha \colon E_\alpha \simeq U_\alpha\times F)_\alpha$.  Given an index $\alpha$ and an index $\beta$, let $U_{\alpha,\beta}$ be the [[fibred product]] (pullback) of $U_\alpha$ and $U_\beta$. Note that we can pull back the commutative triangle formed by $t_\alpha$ along $j_\alpha^*j_\beta$ to get a new isomorphism $t_{\alpha,\beta} \colon E_{\alpha,\beta} \simeq U_{\alpha,\beta} \times F$:

\begin{centre}
\begin{tikzcd}
	{U_\alpha\times F} &&& {U_{\alpha,\beta}\times F} \\
	& \textcolor{rgb,255:red,100;green,100;blue,100}{U_\alpha} & \textcolor{rgb,255:red,100;green,100;blue,100}{U_{\alpha,\beta}} \\
	{E_\alpha} &&& {E_{\alpha,\beta}}
	\arrow["{j_\alpha^*j_\beta}", color={rgb,255:red,100;green,100;blue,100}, from=2-3, to=2-2]
	\arrow[from=3-1, to=2-2]
	\arrow[from=1-1, to=2-2]
	\arrow["\sim"{sloped}, "{t_\alpha}"', from=3-1, to=1-1]
	\arrow[from=1-4, to=2-3]
	\arrow[from=3-4, to=2-3]
	\arrow[from=1-4, to=1-1]
	\arrow["{t_{\alpha,\beta}}", "\sim"{sloped,swap}, from=3-4, to=1-4]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-90}, draw=none, from=1-4, to=2-2]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-4, to=2-2]
	\arrow[from=3-4, to=3-1]
\end{tikzcd}
\end{centre}

We can do the same on the other side, leading to the following situation:

\begin{centre}
\begin{tikzcd}
	&&& {U_{\alpha,\beta}\times F} \\
	&&& {E_{\alpha,\beta}} \\
	&&& \textcolor{rgb,255:red,100;green,100;blue,100}{U_{\alpha,\beta}} \\
	{U_\alpha\times F} & {E_\alpha} & \textcolor{rgb,255:red,100;green,100;blue,100}{U_\alpha} && \textcolor{rgb,255:red,100;green,100;blue,100}{U_\beta} & {E_\beta} & {U_\beta\times F} \\
	&&& \textcolor{rgb,255:red,100;green,100;blue,100}{B} \\
	&&& E \\
	&&& {B \times F}
	\arrow["{j_{\alpha}}", color={rgb,255:red,100;green,100;blue,100}, from=4-3, to=5-4]
	\arrow["{j_{\beta}}"', color={rgb,255:red,100;green,100;blue,100}, from=4-5, to=5-4]
	\arrow["{j_\alpha^*j_\beta}", color={rgb,255:red,100;green,100;blue,100}, from=3-4, to=4-3]
	\arrow["{j_\beta^*j_\alpha}"', color={rgb,255:red,100;green,100;blue,100}, from=3-4, to=4-5]
	\arrow[from=2-4, to=3-4]
	\arrow[from=4-6, to=4-5]
	\arrow[from=6-4, to=5-4]
	\arrow[from=4-2, to=4-3]
	\arrow[from=4-2, to=6-4]
	\arrow[from=4-6, to=6-4]
	\arrow[from=2-4, to=4-2]
	\arrow[from=2-4, to=4-6]
	\arrow[from=4-1, to=7-4]
	\arrow[from=4-7, to=7-4]
	\arrow[from=1-4, to=4-1]
	\arrow[from=1-4, to=4-7]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-90}, draw=none, from=2-4, to=4-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-4, to=4-5]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=4-2, to=5-4]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-90}, draw=none, from=4-6, to=5-4]
	\arrow["{t_\alpha}"', "\sim"{sloped,swap}, from=4-2, to=4-1]
	\arrow["{t_\beta}", "\sim"{sloped,swap}, from=4-6, to=4-7]
	\arrow["{t_{\beta,\alpha}}"'{pos=0.4}, bend right=30, "\sim"{sloped}, from=2-4, to=1-4]
	\arrow["{t_{\alpha,\beta}}"{pos=0.4}, bend left=30, "\sim"{sloped}', from=2-4, to=1-4]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=-45}, color={rgb,255:red,100;green,100;blue,100}, draw=none, from=3-4, to=5-4]
\end{tikzcd}
\end{centre}

Then we have an [[automorphism]] $g_{\alpha,\beta}$ of $U_{\alpha,\beta} \times F$ in $C/{U_{\alpha,\beta}}$ given by $g_{\alpha,\beta} = t_{\beta,\alpha} \circ t_{\alpha,\beta}^{-1}$. The $g_{\alpha,\beta}$ are the __[[transition morphisms]]__ of the locally trivial fibre bundle $E$.

If $E$ in fact admits a *global* [[trivial fiber bundle|trivialisation]] $t \colon E \simeq B \times F$, then one can see that the two isomorphisms $t_{\alpha,\beta}$ and $t_{\beta,\alpha}$ are equal to the pullback of (the commutative triangle formed by) $t$ along $j_{\alpha,\beta} : U_{\alpha,\beta} \to B$, so that the [[transition morphisms]] $g_{\alpha,\beta}$ are all [[identity morphism|identities]].
=--

Often one considers special kinds of bundles, by requiring structure on the standard fibre $F$ and/or conditions on the [[transition morphisms]] $g_{\alpha,\beta}$.  For example:

## Special cases

### $G$-bundle

If $G$ is a [[group object]] in $C$ that [[action|acts]] on $F$, then a __$G$-bundle__ (or bundle with __structure group__ $G$) over $B$ with standard fibre $F$ is a locally trivial fibre bundle over $B$ with standard fibre $F$ together with morphisms $U_{\alpha,\beta} \to G$ that, relative to the action of $G$ on $F$, give the transition maps $g_{\alpha,\beta}$.  (The morphism $U_{\alpha,\beta} \to G$ is also written $g_{\alpha,\beta}$, conflating action with application.)

### $G$-principal bundles

More specifically, a (right or left) __[[principal bundle|principal]] $G$-bundle__ over $B$ is a $G$-bundle over $B$ with standard fibre $G$, associated with the action of $G$ on itself by (right or left) multiplication.

### Structure-preserving bundles

If $F$ is an object of a [[concrete category]] over $C$, then we can consider locally trivial fibre bundles with standard fibre $F$ such that the transition morphisms are structure-preserving morphisms.  If the [[automorphism group]] $Aut(F)$ can be internalised in $C$, then this the same as an $Aut(F)$-bundle, but the concept makes sense in any case.

### Vector bundles

As a fairly specific example, if $F$ is a [[topological vector space]] (and $C$ is a category with structure to support this, such as [[Top]] or [[Diff]]), then a __[[vector bundle]]__ over $B$ with standard fibre $F$ is a $GL(F)$-bundle over $B$ with standard fibre $F$, where $GL(F)$ is the [[general linear group]] with its defining action on $F$.

### Associated bundles

Given a right [[principal bundle|principal]] $G$-bundle $\pi: P\to X$ and a left $G$-space $F$, all in a sufficiently strong category $C$ (such as [[Top]]), one can form the [[quotient object]] $P\times_G F = (P\times F)/{\sim}$, where $P \times F$ is a [[product]] and $\sim$ is the smallest [[congruence]] such that (using [[generalized element]]s) $(p g,f)\sim (p,g f)$; there is a canonical projection $P\times_G F\to X$ where the class of $(p,f)$ is mapped to $\pi(p)\in X$, hence making $P\times_G F\to X$ into a fibre bundle with typical fiber $F$, and the transition functions belonging to the action of $G$ on $F$. We say that  $P\times_G F\to X$ is the __[[associated bundle]]__ to $P\to X$ with fiber $F$.

## Generalizations

### In higher category theory

In [[higher category theory]] the notion of fiber bundle generalizes. See

* [[associated ∞-bundle]].

### In commutative algebra

Under the interpretation of [modules as generalized vector bundles](http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction), locally trivial fiber bundles correspond to _[[locally free modules]]_. See there for more.

### In noncommutative geometry

In [[noncommutative geometry]] both principal and associated bundles have analogues. The principal bundles over noncommutative spaces typically have structure group replaced by a [[Hopf algebra]]; the most well-known class whose base is described by a single algebra are [[Hopf--Galois extensions]]; the global sections of the associated bundle are formed using cotensor product. Transition functions can be to some extent emulated using noncommutative localizations, which yield nonaffine generalizations of Hopf--Galois extensions. Another generalization is when Hopf--Galois extensions in the sense of comodule algebras are replaced by entwining structures with analogous Galois condition.

## Properties

### Relation to fibrations
 {#RelationToFibrations}

\begin{prop}

Every [[locally trivial]] topological fiber bundle projection is a [[Serre fibration]], and a [[Hurewicz fibration]] if it is also a [[numerable bundle]].

\end{prop}

\begin{proof}
  It is clear that the [[projection]] out of a [[Cartesian product]] is a [[Serre fibration]]. With this, the statement follows from [[local triviality]] and the local recognition of Serre fibrations ([this Prop](Serre+fibration#MapIsSerreFibrationIfLocallySo)) and of [[Hurewicz fibrations]] ([this Prop.](Hurewicz+fibration#MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo)), respectively.
\end{proof}

\begin{remark}
  Beware that the converse statement is far from being true.
\end{remark}

## Related concepts

* [[bundle]]

* [[Ehresmann's theorem]]

* **fiber bundle** / [[fiber ∞-bundle]]

  * [[associated bundle]]

  * [[vector bundle]]

  * [[sphere fiber bundle]], [[spherical fibration]]

  * [[empty bundle]]

  * [[equivariant bundle]]

  * [[fiber bundles in physics]]

  * [[Leray-Hirsch theorem]]

* [[principal bundle]] / [[torsor]] / **associated bundle**

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]

* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]

## References

Fiber bundles (then spelled fibre-bundles) were originally defined in:

* [[Hassler Whitney]], *On the theory of sphere-bundles*, Proceedings of the National Academy of Sciences **26** 2 (1940) 148-153 [[doi:10.1073/pnas.26.2.148](https://doi.org/10.1073/pnas.26.2.148)]

  > (with focus on [[sphere bundles]])

Other sources:

* [[Norman Steenrod]], Section I.7 of _The topology of fibre bundles_, Princeton Mathematical Series **14**, Princeton Univ. Press (1951) [[jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5)]

* [[Alexander Grothendieck]], *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]

  > (in relation to [[nonabelian cohomology|nonabelian]] [[Čech cohomology]])

* [[Jean Frenkel]], *Cohomologie non abélienne et espaces fibrés*, Bulletin de la Société Mathématique de France, **85** (1957) 135-220 [[numdam:BSMF_1957__85__135_0](http://www.numdam.org/item/BSMF_1957__85__135_0)]

* [[Glen Bredon]], Section II.1 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* [[Dale Husemöller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/husemoller))

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,
Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

* [[Ralph Cohen]], _The Topology of Fiber Bundles_, Stanford University (2017) ([pdf](http://math.stanford.edu/~ralph/fiber.pdf), [OMN:201707.110706](https://www.ams.org/open-math-notes/omn-view-listing?listingId=110706))

* [[Dai Tamaki]], *Fiber Bundles and Homotopy*, World Scientific 2021 ([doi:10.1142/12308](https://doi.org/10.1142/12308))

  > (in the context of [[homotopy theory]])

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 9 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Gerd Rudolph]], [[Matthias Schmidt]], Chapter 1 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

Discussion of fiber bundles [[internalization|internal]] to [[finitely complete categories]]:

* [[Anders Kock]], *Fibre bundles in general categories*, Journal of Pure and Applied Algebra **56** 3 (1989) 233-245 [doi:10.1016/0022-4049(89)90059-5]

* [[Anders Kock]], *Generalized fibre bundles*, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics **1348** (2006) 194-207 [[doi:10.1007/BFb0081359](https://doi.org/10.1007/BFb0081359)]

[[!redirects fiber bundles]]

[[!redirects fibre bundle]]
[[!redirects fibre bundles]]

[[!redirects standard fiber]]
[[!redirects standard fibre]]

[[!redirects typical fiber]]
[[!redirects typical fibre]]

[[!redirects locally trivial fiber bundle]]
[[!redirects locally trivial fiber bundles]]

[[!redirects locally trivial fibre bundle]]
[[!redirects locally trivial fibre bundles]]

[[!redirects locally trivial bundle]]
[[!redirects locally trivial bundles]]

[[!redirects G-bundle]]
[[!redirects G-bundles]]

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[[!redirects bundle with structure groups]]
[[!redirects bundles with structure groups]]

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fiber bundles in physics

> This entry is supposed to be a survey of and motivation for the role of [[fiber bundles]] in [[physics]].

***

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Fiber Bundles in Physics#
* table of contents
{:toc}

## Idea
 {#Idea}

\begin{imagefromfile}
    "file_name": "WuYang-PhysicsBundlesDictionary.jpg",
    "float": "right",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Wu and Yang 1975](#WuYang75)"
\end{imagefromfile}

All of [[physics]] has two aspects: a local or even [[infinitesimal object|infinitesimal]] aspect, and a global aspect. Much of traditional lore deals just with the local and infinitesimal aspects -- the _[[perturbative field theory|perturbative]]_ aspects -- and [[fiber bundles]] play little role there. But they are the all-important structure that govern the global -- the _[[non-perturbative field theory|non-perturbative]]_ -- aspects. [[bundles|Bundles]] are the _global_ structure of [[physical fields]] and they are irrelevant only for the crude local and perturbative description of reality.

##  Basic idea of the definition of fiber bundles
 {#BasicIdeaOfDefinition}

One way to think of fiber bundles is that they are the data to _globally twist functions_ (on [[spacetime]], say) where "global twist" is much in the sense of "global [[anomaly]]" and the like, namely an effect visible on topologically nontrivial spaces when moving around non-contractible cycles. The concept of _[[monodromy]]_ -- which may be more familiar to physicists -- is closely related: monodromy is something exhibited by a [[connection on a bundle]] and specifically by a [[flat connection|flat bundle]]. For a [[discrete group|discrete]] structure group ([[gauge group]]) _every_ bundle is flat, and in this case non-trivial bundles and non-trivial [[monodromy]] come down to essentially the same thing (see also at _[[local system]]_).

More explicitly, suppose $X$ denotes [[spacetime]] and $F$ denotes some space that one wants to map into. For instance $F$ might be the [[complex numbers]] and a [[free field|free]] [[scalar field]] would be a [[function]]  $X\to F$. For the following it is useful to talk about functions a bit more indirectly: observe that the [[projection]] $pr_2 \colon F \times X \to X$ from the [[product]] of $F$ with $X$ down to $X$ is such that a [[section]] of this map, namely a $\Phi \colon X \to X\times F$ such that

$$
  pr_2 \circ \Phi = id_X
  \phantom{AAAAA}
  \array{
     && F \times X
     \\
     &
     {}^{\mathllap{\Phi}}\nearrow
     & \downarrow\mathrlap{pr_2}
     \\
     X &=& X
  }
$$

is just the same data as a function $X \to F$. We think of the projection $X \times F \overset{pr_2}{\to} X$ as encoding the fact that there is one copy of $F$ associated with each point of $X$, and think of a function with values in $F$ as something that, of course, takes values in $F$ over each point of $X$. One says that this projection, suggestively shown vertically,

$$
  \array{
    F \times X
    \\
    \downarrow\mathrlap{pr_2}
    \\
    X
  }
$$

is the _[[trivial bundle|trivial]]_ $F$-[[fiber bundle]] over $X$.

If $F$ is equipped with the [[structure]] of a [[vector space]] then one calls this a _[[trivial vector bundle|trivial]] [[vector bundle]]_, etc.

The point being that more generally we may add a global "twist" to the $F$-valued functions by making the space $F$ vary to some degree as we move along $X$. For a [[fiber bundle]] one requires that it doesn't change _much_: in fact the word "fiber" in "[[fiber bundle]]" refers to the fact that _all_ fibers (over all points of $X$) are _[[equivalence|equivalent]]_. But the point is that any $F$ may be equivalent to _itself_ in more than one way (it may have "[[automorphisms]]"), and this allows non-trivial global structure even though all fibers look alike.

In this sense, a general $F$-[[fiber bundle]] on some $X$ is defined to be a [[space]] $P$ equipped with a map

$$
  \array{
     P
     \\
     \downarrow \mathrlap{fb}
     \\
     X
  }
$$

to the base space $X$ (e.g. to spacetime), such that _locally_ it looks like the trivial $F$-fiber bundle, up to equivalence.

To say this more technically: $P \overset{fb}{\to} X$ is called an $F$-[[fiber bundle]] if there exists a [[cover]] ([[open cover]]) of $X$ by patches (e.g. [[coordinate charts]]!) $U_i \to X$ for some index [[set]] $I$, such that for each patch $U_i$ (with $i \in I$) there exists a fiberwise [[equivalence]] between the restriction $P|_{U_i}$ of $P$ to $U_i$, and the trivial $F$-fiber bundle $F\times U_i \to U_i$ over the patch $U_i$.

To say this again in terms of sections: this means that a [[section]] $\Phi$ of $P \overset{fb}{\to} X$

$$
  fb \circ \Phi = id_X
  \phantom{AAAAA}
  \array{
     && P
     \\
     &
     {}^{\mathllap{\Phi}}\nearrow
     & \downarrow\mathrlap{fb}
     \\
     X &=& X
  }
$$

is locally on each ([[coordinate chart|coordinate]]) patch $U_i$ simply an $F$-valued function, but when we change patches (change coordinates) then there may be a non-trivial identifications (notably: [[gauge transformations]]) that relates the values of the function on one patch to that on another patch, where they overlap.

Even if this may seem a bit roundabout on first sight, this is actually something at the very heart of modern physics, in that it embodies the two central principles of modern physics, namely

1. the _principle of locality_;

1. the _gauge principle_.

The first roughly says that every global phenomenon in physics must come from local data. In the above discussion this means that any "globally $F$-valued thing on spacetime $X$" must come from just $F$-valued functions on local (coordinate) charts $U_i \hookrightarrow X$ of spacetime. BUT -- and this is key now --, second, the _gauge principle_ says that we may never strictly identify any two phenomena in physics (neither locally nor globally) but we must always ask instead for [[gauge transformations]] connecting two maybe seemingly different phenomena. Hence combining the gauge principle with the locality principle means that if an $F$-valued something on spacetime is locally given by plain $F$-valued functions, then it should be globally given by gluing these $F$-valued functions together not by _identification_ but by [[gauge equivalence]]. The result may be a structure that has global twists, and the nature of these global twists is precisely what an $F$-fiber bundle embodies.

## Examples of fiber bundles in physics

For instance the [[gauge fields]] in [[Yang-Mills theory]], hence in [[electromagnetism]], in [[QED]] and in [[QCD]], hence in the [[standard model of particle physics|standard model of the known universe]], are not really just the local [[differential 1-forms]] "$A_\mu^a$" known from so many textbooks, but are _globally_ really [[connections on principal bundles]] (or their [[associated bundles]]) and this is all-important once one passes to [[non-perturbative field theory|non-perturbative]] [[Yang-Mills theory]], hence to the full story, instead of its infinitesimal or local approximation.

Notably what is called a _[[Yang-Mills instanton]]_ in general and the _[[QCD instanton]]_ in particular is nothing but the underlying nontrivial class of the principal bundle underlying the Yang-Mills [[gauge field]]. Specifically, what physicists call the _[[instanton number]]_ for [[special unitary group|SU(2)]]-[[gauge field theory]] in 4-dimensions is precisely what mathematically is called the [[Chern class|second Chern-class]], a "[[characteristic class]]" of these gauge bundles.

> _YM Instanton = class of principal bundle underlying the non-perturbative gauge field_

To appreciate the utmost relevance of this, observe that the non-perturbative vacuum of the [[experiment|observable world]] is a "sea of instantons" with about one [[Yang-Mills instanton|YM instanton]] per [[femtometer]] to the 4th. See for instance the first sections of ([Schaefer-Shuryak 98](instanton+in+QCD#SchaeferShuryak98))
for a review of this fact. So the very substance of the physical world, the very vacuum that we inhabit, is all controled by non-trivial fiber bundles and is inexplicable without these.

Also [[monopole]] solutions in physics are mathematically nontrivial [[principal bundles]]. For instance the [[Dirac monopole]] (that appears in [[Dirac charge quantization]]) or the [[Yang monopole]].

Similarly [[fiber bundles]] control all other [[topology|topologically]] non-trivial aspects of [[physics]]. For instance most [[quantum anomalies]] are the statement that what looks like an [[action functional|action]] _[[function]]_ to feed into the [[path integral]], is globally really the [[section]] of a non-trivial bundle -- notably a [[Pfaffian line bundle]] resulting from the [[fermionic path integrals]]. Moreover _all_ [[classical anomalies]] are statements of nontrivializability of certain fiber bundles.

Indeed, as the discussion there shows, [[quantization]] as such, if done [[non-perturbative field theory|non-perturbatively]], is all about lifting [[differential form]] data to [[line bundle]] data, this is called the _[[prequantum line bundle]]_ which exists over any globally quantizable [[phase spaces]] and controls all of its [[quantum theory]]. It reflects itself in many [[central extensions]] that govern [[quantum physics]], such as the [[Heisenberg group]] central extension of the Hamiltonian translation and generally and crucially the [[quantomorphism group]] central extension of the [[Hamiltonian diffeomorphisms]] of [[phase space]]. All these [[central extensions]] are non-trivial fiber bundles, and the "quantum" in "quantization" to a large extent a reference to the discrete (quantized) [[characteristic classes]] of these bundles. One can indeed understand quantization as such as the lift of infinitesimal classical differential form data to global bundle data. This is described in detail at _[quantization -- Motivation from classical mechanics and Lie theory](quantization#MotivationFromClassicalMechanicsAndLieTheory)_.

But actually the role of fiber bundles reaches a good bit deeper still. Quantization is just a certain [[extension]] step in the general story, but already [[classical field theory]] cannot be understood globally without a notion of bundle. Notably the very formalization of what a _[[field (physics)|classical field]]_ really is says: a [[section]] of a _[[field bundle]]_. For instance the global nature of [[spinors]], hence _[[spin structures]]_ and their subtle effect on [[fermion]] physics are all encoded by the corresponding [[spinor bundles]].

Two aspects of bundles in physics come together in the theory of [[gauge fields]] and combine to produce [[fiber infinity-bundle|higher fiber bundles]]: namely we saw above that a [[gauge field]] is itself already a bundle (with a [[connection on a bundle|connection]]), and hence the bundle of which a gauge field is a section has to be a "second-order bundle". This is called _[[gerbe]]_ or _[[principal 2-bundle|2-bundle]]_: the only way to realize the [[Yang-Mills field]] both locally and globally accurately is to consider it as a section of a bundle whose typical fiber is $\mathbf{B}G$, the universal [[moduli stack]] of $G$-[[principal bundles]]. For more on this see on the nLab at _[The traditional idea of field bundles and its problems](field%20%28physics%29#IdeaOfFieldBundlesAndItsProblems)_.

All of this becomes even more pronounced as one digs deeper into _[[local quantum field theory]]_, with locality formalized as in the [[cobordism hypothesis|cobordism theorem]] that classifies local [[topological field theories]]. Then already the [[local Lagrangians]] and [[local action functionals]] themselves are [[principal infinity-connection|higher connections]] on higher bundles over the [[moduli infinity-stack|higher moduli stack]] of fields. For instance the fully local formulation of [[Chern-Simons theory]] exhibits the Chern-Simons [[action functional]]  --- with all its global [[gauge invariance]] correctly realized -- as a [[universal Chern-Simons circle 3-bundle]]. This is such that by [[transgression]] to lower [[codimension]] it reproduces all the global gauge structure of this field theory, such as in codimension 2 the _[[WZW gerbe]]_ (itself a fiber 2-bundle: the [[background gauge field|background]] [[B-field]] of the [[WZW model]]!), in codimension 1 the [[prequantum line bundle]] on the [[moduli space of connections]] whose sections in turn yield the [[Hitchin connection|Hitchin bundle]] of [[conformal blocks]] on the [[moduli space of Riemann surfaces]].

And so on and so forth. In short: all global structure in [[field theory]] is controled by [[fiber bundles]], and all the more the more the field theory is [[quantum field theory|quantum]] and [[gauge field theory|gauge]]. The only reason why this can be ignored to some extent is because field theory is a complex subject and maybe the majority of discussions about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The [[QCD]] [[vacuum]] that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic at its very heart.

## Related concepts

* [[Aharonov-Bohm effect]]

* [[geometry of physics]]

* [[twisted smooth cohomology in string theory]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[motivation for higher differential geometry]]

* [[applications of (higher) category theory]]

* [[motivation for cohesion]]

* [[Hilbert's sixth problem]]

* [[motives in physics]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

## References

> See also the references at _[[Dirac charge quantization]]_ at *[[gauge potential]]*.

The understanding of [[electromagnetic potentials]] as [[connection on a bundle|connections]] on [[fiber bundles]] originates with:

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

Further discussion and review:

* {#EguchiGilkeyHanson80} [[Tohru Eguchi]], [[Peter Gilkey]], [[Andrew Hanson]], _Gravitation, gauge theories and differential geometry_, Physics Reports **66** 6 (1980) 213-393 [doi:10.1016/0370-1573(80)90130-1]

* [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam, A. Stern, _Gauge Theories and Fibre Bundles - Applications to Particle Dynamics_, Springer "Lecture Notes in Physics", 188 (1983) ([arXiv:1702.08910](https://arxiv.org/abs/1702.08910), [doi:10.1007/3-540-12724-0](https://link.springer.com/book/10.1007/3-540-12724-0))

* P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, _Classical Topology and Quantum States_ (World Scientific, Singapore, 1991).

* L. Mangiarotti, [[Gennadi Sardanashvily]], _Connections in Classical and Quantum Field Theory_, World Scientific, 2000 ([doi:10.1142/2524](https://doi.org/10.1142/2524))

* Luciano Boi, _Geometrical and topological foundations of theoretical physics: From gauge theories to string program_, 2003 ([pdf](http://www.emis.de/journals/HOA/IJMMS/2004/33-361777.pdf))

* [[Mikio Nakahara]], _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,  Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

* [[Adam Marsh]], _Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results_ ([arXiv:1607.03089](https://arxiv.org/abs/1607.03089)), chapter 10 in: [[Adam Marsh]], _Mathematics for Physics: An Illustrated Handbook_, World Scientific 2018 ([doi:10.1142/10816](https://doi.org/10.1142/10816), [book webpage](https://www.mathphysicsbook.com/))

* {#RudolphSchmidt17} [[Gerd Rudolph]], [[Matthias Schmidt]], _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

Specifically in [[solid state physics]] ([[valence bundles]] and their [[K-theory classification of topological phases of matter]]):

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

category: motivation

[[!redirects fiber bundles and physics]]


fiber functor

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#### Category theory
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#### Algebra
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[[!include higher algebra - contents]]
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#### Duality
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[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[forgetful functor]] from a [[category]] of [[actions]]/[[representations]] to the underlying [[sets]]/[[spaces]] is often called a _[[fiber functor]]_, notably in the context of [[Tannaka duality]] and [[Galois theory]].

The archetypical example which gives rise to the term is the following. If one has the  [[category]] $Et(X)$ of [[covering spaces]]  of a (nice enough) [[topological space]] $X$, then after picking any point $x \in X$ the operation of forming the [[fibre]] over that point gives a [[functor]] $fib_x \colon Et(X)\to Set$ to the category [[Set]] of [[sets]]. The [[natural automorphisms]] of this functor form the (algebraic) [[Chevalley fundamental group|fundamental group]], $\pi_1(X)$. The main theorem of the [[Galois theory|Galois-Poincaré theory]] of [[covering spaces]] can be viewed as stating that this sets up an [[equivalence of categories]] between that category of covering spaces and the category of $\pi_1$[[permutation representation|-sets]]. This equivalence is compatible with the chosen fibre functor and the further [[forgetful functor]] from $\pi_1-Sets$ to $Sets$. Extracting from this situation, that forgetful functor is thought of as being a _fibre functor_ as well.  Any category of [[G-sets]], for $G$ a [[group]], gives a [[monoidal category]], and the forgetful functor is a [[monoidal functor]]; of course, the category of [[G-sets]] corresponds to the category of [[permutation representations]] of $G$, and generalising this basic example leads to the following idea.

The [[forgetful functor|forgetful]] [[strict monoidal functor]] from a [[monoidal category]] to some standard monoidal category, usually the category [[Vect]] of [[vector spaces]] over a field is called the __fiber functor__ in some contexts, especially in [[Tannaka reconstruction]] in which the symmetry object is reconstructed from the (object of) endomorphisms of the fiber functor. In mixed Tannaka duality, a single fiber functor does not suffice for reconstruction, but rather a family of fiber functors to different bases.

Historically, the notion was used extensively, starting in the 1960s by [[Grothendieck]] and his collaborators. The terminology is from the [[Grothendieck Galois theory]]: namely Grothendieck reconstructs the ([[profinite group|profinite]]) [[fundamental group]] in algebraic geometry from a fiber functor: the fundamental group acts on a covering by deck transformations and by monodromy transformation for bundles over the covering, algebraic analogues of such a picture can thus be used to define a fundamental group, not by using some idea of loops which are often hard to define in abstract setups, but by a form of Tannakian reconstruction. Grothendieck also introduced the idea of using many "base points" that is, many fibre functors, thus giving an abstract analogue of the  [[fundamental groupoid]] of a space.

## Warning

Please do not confuse the terminology with the case of a functor which is a [[Grothendieck fibration]] (i.e. a fibered category); nor with a fiber ("preimage" of a sort) of a functor. These are related ideas but are best kept separate.

## Properties

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## See also

* [[Tannakian category]]

## References

* [[P. Cartier]], _A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry_, Bull. Amer. Math. Soc. __38__ (2001), 389-408, [pdf](http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf)

* [[P. Deligne]], _[[Catégories Tannakiennes]]_

For [[fusion 2-categories]]:

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]]. *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories*. (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

[[!redirects fiber functors]]
[[!redirects fibre functor]]
[[!redirects fibre functors]]


fiber infinity-bundle

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#Contents#
* table of contents
{:toc}

## Definition

Let $\mathbf{H}$ be an ambient [[(∞,1)-topos]]. Let $V, X$ be two [[objects]] of $\mathbf{H}$. Then a _$V$-[[fiber bundle]]_ over $X$ in $\mathbf{H}$ is a morphism $E \to X$ such that there is an [[effective epimorphism]] $U \to X$ and an [[(∞,1)-pullback]] square of the form

$$
  \array{
    U \times V &\to& E
    \\
    \downarrow && \downarrow
    \\
    U &\to& X
  }
  \,.
$$

Externally this is a _$V$-fiber $\infty$-bundle_.

See at _[[associated ∞-bundle]]_ for more.

## Example

A fiber $\infty$-bundle whose typical fiber $V$ is a pointed [[connected object in an (∞,1)-topos|connected object]], hence a [[delooping]] $\mathbf{B}G$ of an [[∞-group]] $G$

$$
  V \simeq \mathbf{B}G
$$

is a _$G$-[[∞-gerbe]]_.

## Properties
 {#Properties}

Every $V$-fiber $\infty$-bundle is the [[associated ∞-bundle]] to an [[automorphism ∞-group]]-[[principal ∞-bundle]].

For let $Type$ be the [[object classifier]]. Then any bundle $E \to X$ is classified by a morphism

$$
  X \longrightarrow Type
$$

On the other hand, since the pullback to the bundle on some $U$ is trivializable, that bundle over $U$ is classsified by a map that factors through the point which is the name of the fiber $V$

$$
  U \longrightarrow \ast \stackrel{\vdash V}{\longrightarrow} Type
  \,.
$$

The [[1-image]]-of this point inclusion is the [[delooping]] of the [[automorphism ∞-group]] of $V$ :

$$
  U \longrightarrow \ast \longrightarrow \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Type
  \,.
$$

Therefore the fact that $E$ is trivialized over $U$ means that there the classifying maps fit into a [[commuting diagram]] of the form

$$
  \array{
     U  &\longrightarrow& \mathbf{B}\mathbf{Aut}(V)
     \\
     \downarrow && \downarrow
     \\
     X &\longrightarrow& Type
  }
$$

By assumption the left morphism is a [[1-epimorphism]] and by the above construction the right morphism is a [[1-monomorphism]]. Therefore by the [[(n-connected, n-truncated) factorization system]] this diagram has an essentially unique lift

$$
  \array{
     U  &\longrightarrow& \mathbf{B}\mathbf{Aut}(V)
     \\
     \downarrow &\nearrow& \downarrow
     \\
     X &\longrightarrow& Type
  }
$$

This diagonal lift classifies an $\mathbf{Aut}(V)$-[[principal ∞-bundle]] and the commutativity of the bottom right triangle exhibits the original bundle $E \to X$ as the [[associated ∞-bundle]] to that.

## Related concepts

* [[associated ∞-bundle]], [[principal ∞-bundle]]

* [[∞-gerbe]]

* [[local coefficient ∞-bundle]]

## References

See the references at _[[associated ∞-bundle]]_.

The explicit general definition appears as def. 4.1 in part I of

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal infinity-bundles]]_

[[!redirects fiber ∞-bundle]]
[[!redirects fiber ∞-bundles]]


fiber integration

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+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

_Fiber integration_ or _push-forward_ is a process that sends [[generalized cohomology]] classes on a [[bundle]] $E \to B$ of [[manifolds]] to cohomology classes on the base $B$ of the bundle, by _evaluating them on each fiber_ in some sense.

This sense is such that if the cohomology in question is [[de Rham cohomology]] then fiber integration is ordinary [[integration]] of [[differential form]]s over the fibers. Generally, the fiber integration over a bundle of $k$-dimensional fibers reduces the degree of the cohomology class by $k$.

Composing pullback of cohomology classes with fiber integration yields the notion of [[transgression]].

## Definition

### Fiberwise integration of ordinary differential forms

Consider a smooth [[submersion]]
$$f\colon T\to B$$
of [[smooth manifolds]],
whose fibers have dimension $d$.
The fiberwise integration of [[differential forms]]
is a map
$$f_*\colon\Omega^n(T)\to\Omega^{n-d}(B)$$
defined as follows.
Given $\omega\in\Omega^n(T)$,
the differential $(n-d)$-form $f_*(\omega)$ is constructed as follows.

The value of $f_*(\omega)$ at some
collection of $(n-d)$ tangent vector fields $v_1$, …, $v_{n-d}$
is computed as follows.

First, lift each $v_i$ to a section of the quotient bundle
$T(T)/T(f)$, where $T(f)$ is the [[relative tangent bundle]].
Lift these sections to sections $u_i$ of $T(T)$ in some arbitrary way.
Substitute the resulting vector fields on $T$ into the differential
$n$-form $\omega$, obtaining a differential $d$-form $\psi$.
Pull back $\psi$ to each of the fibers of the map $f\colon T\to B$,
obtaining a $d$-form on each of these fibers,
which does not depend on the choices of liftings $u_i$.
Now integrate the resulting $d$-form over each of these fibers.
This gives a number for each $b\in B$, which depends smoothly on $b$.
This is the value of $f_*(\omega)$ on $v_1$, …, $v_{n-d}$.

See Greub, Halperin, and Vanstone, Volume I, Section VII.7.12.

### In generalized cohomology via Pontryagin-Thom collapse maps

#### Along maps of manifolds

Here is the rough outline of the construction via [[Pontryagin-Thom collapse maps]].

The basic strategy is this:

1. start with a map $E \to B$

1. make $B$ bigger without changing its homotopy type such that the map from $E$ becomes an [[embedding]];

1. choose an [[orientation in generalized cohomology|orientation]] structure that makes the cohomology of $E$ equivalent to that of $Th(E)$ (the [[Thom isomorphism]]);

1. compose the Thom isomorphism with the pullback along $B \to Th(E)$ to get an "Umkehr" map from cohomology of $E$ to cohomology of $B$.

Now in detail.

Let $p : E \to B$ be a [[bundle]] of smooth compact [[manifolds]] with typical [[fiber]] $F$.

By the [[Whitney embedding theorem]] one can choose an embedding $e:E \hookrightarrow \mathbb{R}^n$ for some $n \in \mathbb{N}$. From this one obtains an embedding

$$
  (p,e) : E \hookrightarrow B \times \mathbb{R}^n
  \,.
$$

Let $N_{(p,e)} (E)$ be the [[normal bundle]] of $E$ relative to this embedding. It is a rank $n- dim F$ bundle over the image of $E$ in $B \times \mathbb{R}^n$.

Fix a [[tubular neighbourhood]] of $E$ in $B \times \mathbb{R}^n$ and identify it with the total space of $N_{(p,e)}$. Then collapsing the whole [[complement]] $B \times \mathbb{R}^n \setminus N_{(p,e)}(E)$ to a point gives the [[Thom space]] of $N_{(p,e)}(E)$, and the quotient map

$$
  B \times \mathbb{R}^n
  \to
  B \times \mathbb{R}^n / (B \times \mathbb{R}^n - N_{(p,e)}(E))
  \simeq
  Th(N_{(p,e)}(E))
$$

factors through the [[one-point compactification]] $(B \times \mathbb{R}^n)^*$
of $B \times \mathbb{R}^n$. Since $(B \times \mathbb{R}^n)^*\cong \Sigma^n B_+$ (see [here](one-point+compactification#MonoidalFunctoriality)),
the $n$-fold [[suspension]] of $B_+$ (or, equivalently, the [[smash product]] of $B$ with the $n$-sphere: $\Sigma^n B_+= S^n \wedge B_+$), we obtain a factorization
$$
  B \times \mathbb{R}^n \to \Sigma^n B_+ \stackrel{\tau}{\to} Th(N_{(p,e)}(E))
  \,,
$$
where $\tau$ is called the [[Pontrjagin-Thom collapse map]].

Explicitly, as sets we have $\Sigma^n B_+ \simeq B \times \mathbb{R}^n \cup \{\infty\}$ and $Th(N_{(e,p)}(E)) = N_{(e,p)} \cup \{\infty\}$, and for $U \subset \Sigma^n B_+$ a tubular neighbourhood of $E$ and $\phi : U \to N_{(e,p)}(E)$ an isomorphism, the map

$$
  \tau : \Sigma^n B_+ \stackrel{}{\to} Th(N_{(p,e)}(E))
$$

is defined by

$$
  \tau : x \mapsto
  \left\{
    \array{
      \phi(x) & | & x \in U
      \\
      \infty & | & otherwise
    }
  \right.
  \,.
$$

Now let $H$ be some [[multiplicative cohomology theory]], and assume that the Thom space $Th(N_{(p,e)}(E))$ has an $H$-[[orientation in generalized cohomology|orientation]], so that we have a  [[Thom isomorphism]]. Then combined with the [[suspension isomorphism]] the pullback along $\tau$ produces a morphism

$$
  \int_F
    \;
    \colon
    \;
     H^\bullet(E)
     \longrightarrow
     H^{\bullet - dim F}(B)
$$

of [[cohomology rings]]:

$$
  \array{
   H^\bullet(E)
   \\
   \big\downarrow
   {}^{
      \mathrlap{
        \simeq_{Thom\;iso}
      }
   }
   \\
   \tilde H^{
      \bullet + n - dim F
   }
   \big(
     Th(N_{(p,e)}(E))
   \big)
   \\
   \big\downarrow
   \mathrlap{
     \tau^\ast_{
       Pontrjagin-Thom\;collapse
     }
   }
   \\
   \tilde H^{\bullet + n - dim F}
   \big(
     \Sigma^n B_+
   \big)
   \\
   \big\downarrow
   {}^\mathrlap{
     \simeq_{suspension\;iso}
   }
   \\
   H^{\bullet - dim F}(B)
   }
   \,.
$$

This operation is independent of the choices involved. It is the **fiber integration** of $H$-cohomology along $p : E \to B$.

#### Along representable morphisms of stacks

The above definition generalizes to one of push-forward
in generalized cohomology on [[stacks]] over [[SmthMfd]]
along [[representable morphisms of stacks]].

(...)

### In generalized cohomology by Umkehr maps via abstract duality

We discuss now a general abstract reformulation
in terms of [[duality]] in [[stable homotopy theory]] and [[higher algebra]] of the above traditional constructions.

#### Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse

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

Write

$$
  D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod
$$

for the [[Spanier-Whitehead duality]] map which sends a [[topological space]] first to its [[suspension spectrum]] and then that to its [[dual object]] in the [[(∞,1)-category of spectra]].

=--

([ABG 11, def 10.3](#ABG11)).

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

For $X$ a [[compact manifold]], let $X \to \mathbb{R}^n$ be an [[embedding]] and write $S^n \to X^{\nu_n}$ for the classical [[Pontryagin-Thom collapse map]] for this situation, and write

$$
  \mathbb{S} \to X^{-T X}
$$

for the corresponding [[looping]] map from the [[sphere spectrum]] to the [[Thom spectrum]] of the negative [[tangent bundle]] of $X$. Then [[Atiyah duality]] produces an [[equivalence]]

$$
  X^{- T X} \simeq D X
$$

which identifies the [[Thom spectrum]] with the [[dual object]] of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a [[commuting diagram]]

$$
  \array{
    && X^{- T X}
    \\
    & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbb{S}
    &\underset{D(X \to \ast)}{\to}&
    D X
  }
$$

identifying the classical [[Pontryagin-Thom collapse map]] with the abstract [[dual morphism]] construction of prop. \ref{SpanierDualityOperation}.

More generally, for $W \hookrightarrow X$ an [[embedding]] of [[manifolds]], then [[Atiyah duality]] identifies the [[Pontryagin-Thom collapse maps]]

$$
  \mathbb{S} \to X^{-T X} \to W^{- T W}
$$

with the abstract [[dual morphisms]]

$$
  \mathbb{S} \to D X \to D W
  \,.
$$

=--

([ABG 11, prop. 10.5](#ABG11)).

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

Given now $E \in CRing_\infty$ an [[E-∞ ring]], then the [[dual morphism]] $\mathbb{S} \to D X$ induces under [[smash product]] a similar Pontryagin-Thom collapse map, but now not in [[sphere spectrum]]-[[(∞,1)-modules]] but in $E$-[[(∞,1)-modules]].

$$
  E \to D X \otimes_{\mathbb{S}} E
  \,.
$$

The image of this under the $E$-[[generalized cohomology theory|cohomology]] functor produces

$$
  [D X \otimes_{\mathbb{S}} E, E] \to E
  \,.
$$

If now one has a [[Thom isomorphism]] ($E$-[[orientation in generalized cohomology|orientation]]) $ [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]$ that identifies the cohomology of the dual object with the original cohomology, then together with produces the __Umkehr map__

$$
  [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E
$$

that pushes the $E$-cohomology of $X$ to the $E$-cohomology of the point. Analogously if instead of the terminal map $X \to \ast$ we start with a more general map $X \to Y$.

More generally a [[Thom isomorphism]] may not exists, but $[D X \otimes_{\mathbb{S}} E, E]$ may still be equivalent to a [[twisted cohomology]]-variant $[X,E]_{\chi}$ of $[X,E]$, namely to $[\Gamma_X(\chi),E]$, where $\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod$ is an ([[flat (∞,1)-bundle|flat]]) $E$-[[(∞,1)-module bundle]] on $X$ and and $\Gamma \simeq \underset{\to}{\lim}$ is the [[(∞,1)-colimit]] (the [[generalized Thom spectrum]] construction). In this case the above yields a _[[twisted Umkehr map]]_.

=--

([ABG 10, 9.1](#ABG10))

#### In linear homotopy-type theory

We may formulate the above still a bit more abstractly in
[[linear homotopy-type theory]] (following _[[schreiber:Homotopy-type semantics for quantization]]_: see at [[indexed monoidal infinity-category]] the section on _[Fundamental classes](indexed+monoidal+infinity-category#FundamentalClasses)_ and following.

\linebreak

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

(...)

### In generalized differential cohomology

See

* [[fiber integration in differential cohomology]]

  * [[fiber integration in ordinary differential cohomology]]

  * [[fiber integration in differential K-theory]]

### In KK-theory
 {#InKKTheory}

We discuss fiber integration/push-forward/[[Gysin maps]] in
[[operator K-theory]], hence in [[KK-theory]] ([Connes-Skandalis 85](#ConnesSkandalis84), [BMRS 07, section 3](#BMRS07)).
For more see at _[[fiber integration in K-theory]]._

The following discusses KK-pushforward

1. _[Along an embedding](#KKPushForwardAlongEmbedding)_

1. _[Along a submersion](#KKPushforwardAlongSubmersion)_

1. _[Along a fibration of closed spin^c manifolds](#AlongAFibrationOfClosedSpinCManifolds)_

1. _[Along a general K-oriented map](#KKPushforwardAlongGeneralMap)_

1. _[In twisted K-theory](#KKPushforwardAlongGeneralMap)_

The construction goes back to ([Connes 82](#Connes82)), where it is given over smooth manifolds. Then ([Connes-Skandalis 84](#ConnesSkandalis84), [Hilsum-Skandalis 87](#HilsumSkandalis87)) generalize this to maps between [[foliations]] by KK-elements betwen the [[groupoid convolution algebras]] of the coresponding [[holonomy groupoids]] and  ([Rouse-Wang 10](#RouseWang10)) further generalize to the case where a [[circle 2-bundle]] twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary [[noncommutative topology|noncommutative spaces]]/[[C*-algebras]] is in ([BMRS 07](#BMRS07)).

#### Along an embedding
 {#KKPushForwardAlongEmbedding}

([Connes-Skandalis 84, above prop. 2.8](ConnesSkandalis84))

Let $h \colon X \hookrightarrow Y$ be an [[embedding]] of [[compact topological space|compact]] [[smooth manifolds]].

The push-forward constructed from this is supposed to be an element in [[KK-theory]]

$$
  h! \colon KK_d(C(X), C(Y))
$$

in terms of which the push-forward on [[operator K-theory]] is induced by postcomposition:

$$
  h_!
    \;\colon\;
  K^\bullet(X)
   \simeq
  KK_\bullet(\mathbb{C}, X)
  \stackrel{h!\circ (-)}{\to}
  KK_{\bullet+d}(\mathbb{C},Y)
  \simeq
 KK^{\bullet+d}(Y)
 \,,
$$

where $d = dim(X) - dim(Y)$.

Now, if we could "thicken" $X$ a bit, namely to a [[tubular neighbourhood]]

$$
  h  \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y
$$

of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element

$$
  j! \in KK(C_0(U), C(Y))
$$

induced directly from the [[C*-algebra]] homomorphism $C_0(U) \to C(Y)$ from the [[algebra of functions]] [[vanishing at infinity]] of $U$ to functions on $Y$, given by extending these functions by 0 to functions on $Y$.
Or rather, it will be that element composed with the assumed KK-equivalence

$$
  \psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
$$

The bulk of the technical work in constructing the push-forward is in constructing this equivalence. ([BMRS 07, example 3.3](#BMRS07))

In order for it to exist at all, assume that the [[normal bundle]]

$$
  N_Y X \coloneqq h^\ast(T Y)/ T X
$$

has a [[spin^c structure]]. Write $S(N_Y X)$ for the associated [[spinor bundle]].

Then there is an invertible element in [[KK-theory]]

$$
  \iota^X! \in KK_n(C(X), C_0(N_Y X))
$$

hence a KK-equivalence $\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X)$, where $C_0(-)$ denotes the [[algebra of functions]] [[vanishing at infinity]].

This is defined as follows. Consider the pullback $\pi_n^\ast S(N_Y X) \to N_Y X$ of this spinor to the normal bundle itself along the projection $\pi_N \colon N_Y X \to X$.  Then...

Moreover, a choice of a [[Riemannian metric]] on $X$ allows to find a [[diffeomorphism]] between the [[tubular neighbourhood]] $U_{h(X)}$ of $h(X)$ and a neighbourhood of the zero-section of of the normal bundle

$$
  \Phi \colon U_{h(X)} \hookrightarrow N_Y X
  \,.
$$

This induces a KK-equivalence

$$
  [\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
$$

Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in [[KK-theory]] with

$$
  h! \colon
  C(X)
    \underoverset{\simeq_{KK}}{i^X!}{\to}
  C_0(N_Y X)
    \underoverset{\simeq_{KK}}{\Phi}{\to}
  C_0(U)
    \stackrel{j!}{\to}
  C(Y)
  \,.
$$

#### Along a proper submersion
 {#KKPushforwardAlongSubmersion}

([Connes-Skandalis 84, above prop. 2.9](ConnesSkandalis84))

For $\pi \colon X \to Z$ a [[K-orientation|K-oriented]] [[proper map|proper]] [[submersion]] of compact smooth manifolds, the push-forward map along it is reduced to the [above](#KKPushForwardAlongEmbedding) case of an embedding by

1. using that by the [[Whitney embedding theorem]] every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding

   $$
     h \colon X \to Z \times \mathbb{R}^{2 q}
   $$

1. using that there is a KK-equivalence

   $$
     \iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q})
     \,.
   $$

The resulting push-forward is then given by postcomposition in [[KK-theory]] with

$$
  \pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q)
   \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to}
  C(Z)
  \,.
$$

([BMRS 07, example 3.4](#BMRS07))

#### Along a smooth fibration of closed $Spin^c$-manifolds
 {#AlongAFibrationOfClosedSpinCManifolds}

Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose [[fibers]] $X/Z$ are

* [[closed manifold|closed]] [[smooth manifolds|smooth]] [[spin^c structure|spin^c]] [[manifolds]] of even [[dimension]]

the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ is given by the [[Fredholm module|Fredholm]]-[[Hilbert module]] obatined from the fiberwise [[spin^c Dirac operator]] acting on the fiberwise [[spinors]]. ([Connes-Skandalis 84, proof of lemma 4.7](ConnesSkandalis84), [BMRS 07, example 3.9](BMRS07)).

In detail, write

$$
  T(X/Z) \hookrightarrow T X
$$

for the sub-bundle of the total [[tangent bundle]] on the [[vertical vectors]] and choose a [[Riemannian metric]] $g^{X/Z}$ on this bundle (hence a collection of Riemannian metric on the fibers $X/Z$ smoothly varying along $Z$). Write $S_{X/Z}$ for the corresponding [[spinor bundle]].

A choice of horizontal complenet $T X \simeq T^H X \oplus T(X/Z)$ induces an [[affine connection]] $\nabla^{X/Z}$. This combined with the [[symbol map]]/Clifford multiplication of $T^\ast (X/Z)$ on $S_{X/Z}$ induces a fiberwise [[spin^c Dirac operator]], acting in each fiber on the [[Hilbert space]] $L^2(X/Z, S_{X/Z})$.

This yields a [[Fredholm module|Fredholm]]-[[Hilbert bimodule]]

$$
  (D_{X/Z}, L^2(X/Z, S_{X/Z}))
$$

which defines an element in [[KK-theory]]

$$
  \pi ! \in KK(C_0(X), C_0(Z))
  \,.
$$

Postcompositon with this is the push-forward map in K/KK-theory, equivalently the [[index]] map of the collection of Dirac operators.

#### Along a general K-oriented map
 {##KKPushforwardAlongGeneralMap}

([Connes-Skandalis 84, def. 2.1](ConnesSkandalis84))

Now for $f \colon X \to Y$ an arbitray [[K-orientation|K-oriented]] smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its [[graph map]], followed by projection to $Y$:

$$
  f \;\colon\;
  X \stackrel{graph(f)}{\to}
  X \times Y
  \stackrel{p_Y}{\to}
  Y
  \,.
$$

Hence push-forward along such a general map is postcomposition in [[KK-theory]] with

$$
  f! \coloneqq p_Y !\circ graph(f)!
  \,.
$$

([BMRS 07, example 3.5](#BMRS07))

#### In twisted K-theory
 {#KKPushforwardAlongGeneralMap}

We discuss push forward in K-theory more generally by [[Poincaré duality C*-algebras]] hence [[dual objects]] in [[KK-theory]].

Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

$$
  C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
$$

with notation as in [this definition](Poincaré+duality+algebra#CStarAlgebraOf2BundleOnManifold). By [this proposition](Poincaré+duality+algebra#DualOfCompactManifoldWithTwist) the [[dual morphism]] is of the form

$$
  C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{W_3(\tau_X)}{\chi}}(X)
  \,.
$$

If we assume that $X$ has a [[spin^c structure]] then this is

$$
  C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{1}{\chi}}(X)
  \,.
$$

Postcomposition with this map in [[KK-theory]] now yields a map from the $\frac{W_3(\tau_Q)}{i^\ast \chi}$-[[twisted K-theory]] of $Q$ to the $\chi^{-1}$-[[twisted K-theory]] of $X$:

$$
  i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi}
  \,.
$$

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element

$$
  [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)
$$

is called (the K-class of) a _[[Chan-Paton gauge field]]_ on the D-brane satisfying the _[[Freed-Witten-Kapustin anomaly cancellation]]_ mechanism. (The orginal _[[Freed-Witten anomaly cancellation]]_ assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin^c structure]] on $Q$.) Finally its [[fiber integration|push-forward]]

$$
  [i_! \xi] \in K_{\bullet- \chi}(X)
$$

is called the corresponding _[[D-brane charge]]_.

### Of cohesive differential form data

In [[differential cohomology]] realized in [[cohesive homotopy type theory]] there is a canonical fiber integration map for the curvature coefficients of a given diffential cohomology theory. See at _[integration of differential forms -- In cohesive homotopy-type theory](integration+of+differential+forms#InCohesiveHomotopyTypeTheory)_.

## Examples

### To the point

When $B$ is a point, one obtains integration aginst the [[fundamental class]] of $E$,
$$
\int_E:H^\bullet(E)\to H^{\bullet-dim E}(*)
$$
taking values in the coefficients of the given cohomology theory. Note that in this case $\Sigma^n B_+=S^n$, and this hints to a relationship between the Thom-Pontryagin construction and [[Spanier-Whitehead duality]]. And indeed [[Atiyah duality]] gives a homotopy equivalence between the [[Thom spectrum]] of the stable normal bundle of $E$ and the Spanier-Whitehead dual of $E$.
...

## Related concepts

* [[Grothendieck-Riemann-Roch theorem]]

* [[transfer context]], [[sheaf with transfer]], [[Becker-Gottlieb transfer]]

* [[geometric quantization by push-forward]]

* [[twisted Umkehr map]]

  * [[fiber integration in K-theory]]

[[!include generalized fiber integration synonyms - table]]

## References

### General

* {#Adams74} [[Frank Adams]], pages 25-27 in _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974

* {#Kochmann96} [[Stanley Kochmann]], section 4.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Fiber integration of differential forms is discussed in section VII of volume I of

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

A quick summary can be found from [slide 14](http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf#page=14) on in

* [[Robin Koytcheff]], _A homotopy-theoretic view of Bott-Taubes integrals_ ([pdf slides](http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf))

More details are in

* [[Ralph Cohen]], [[John Klein]], _Umkehr Maps_ ([arXiv:0711.0540](http://arxiv.org/abs/0711.0540))

### In noncommutative topology and KK-theory
 {#ReferencesInKKTheory}

Push-forward in [[twisted K-theory]] is discussed in

* [[Alan Carey]], [[Bai-Ling Wang]], _Thom isomorphism and Push-forward map in twisted K-theory_ ([arXiv:math/0507414](http://arxiv.org/abs/math/0507414))
 {#CareyWang05}

and section 10 of ([ABG, 10](#ABG10))

Discussion of fiber integration [[Gysin maps]]/Umkehr maps in [[noncommutative topology]]/[[KK-theory]] as [above](#InKKTheory) is in the following references.

The definition of the element $f! \in KK(C(X), C(Y))$ for a $K$-oriented map $f \colon X \to Y$ between smooth manifolds goes back to section 11 in

* {#Connes82} [[Alain Connes]], _A survey of foliations and operator algebras_, Proceedings of the A.M.S., 38, 521-628 (1982) ([pdf](http://www.alainconnes.org/docs/foliationsfine.pdf))

The functoriality of this construction is demonstrated in section 2 of the following article, which moreover generalizes the construction to maps between [[foliations]] hence to KK-elements between [[groupoid convolution algebras]] of [[holonomy groupoids]]:

* {#ConnesSkandalis84} [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.4218))

More on this is in

* {#HilsumSkandalis87} [[Michel Hilsum]], [[Georges Skandalis]], _Morphismes K-orienté d'espace de feuille et fonctoralité en théorie de Kasparov_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 20 no. 3 (1987), p. 325-390 ([numdam](http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0))

(the article that introuced [[Hilsum-Skandalis morphisms]]).

This is further generalized to [[circle 2-bundle]]-twisted convolution algebras of foliations in

* {#RouseWang10} Paulo Carrillo Rouse, [[Bai-Ling Wang]], _Twisted longitudinal index theorem for foliations and wrong way functoriality_ ([arXiv:1005.3842](http://arxiv.org/abs/1005.3842))

Dicussion for general [[C*-algebras]] is in section 3 of

* {#BMRS07} Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))

and specifically including also [[twisted K-theory]] again (and the relation to [[D-brane charge]]) in section 7 of

* {#BrodzkiMathaiRosenbergSzabo06} Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_, Commun. Math. Phys. 277:643-706,2008 ([arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020))

### Abstract formulation

The abstract formulation in [[stable homotopy theory]] via [[(infinity,1)-module bundles]] is sketched in section 9 of

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

and in section 10 of

* {#ABG11} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_ ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203))

This is reviewed and used also in

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of boundary prequantum field theory]]_, MSc thesis, Utrecht 2013

Formulation of this in [[linear homotopy-type theory]] is discussed in

* [[Urs Schreiber]], _[[schreiber:Homotopy-type semantics for quantization]]_

[[!redirects umkehr map]]
[[!redirects umkehr maps]]
[[!redirects Umkehr map]]
[[!redirects Umkehr maps]]

[[!redirects fiber integration in generalized cohomology]]
[[!redirects integration in generalized cohomology]]

[[!redirects push-forward in generalized cohomology]]
[[!redirects pushforward in generalized cohomology]]

[[!redirects fiber integral]]
[[!redirects fiber integrals]]

[[!redirects fiberwise integral]]
[[!redirects fiberwise integrals]]


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#Contents#
* table of contents
{:toc}

## Idea

The refinement of the notion of [[fiber integration]] in [[generalized (Eilenberg-Steenrod) cohomology]] to [[differential cohomology]].

General discussion of fiber integration for [Hopkins-Singer coefficients](differential%20cohomology%20diagram#HopkinsSingerCoefficients) is in ([HopkinsSinger, section 4.10](#HopkinsSinger)) and for integration over [[circle]] fibers in ([Bunke-Nikolaus-Völkl 13, section 4.5](#BunkeNikolausVoelkl13)).

In ([Bunke-Schick 09](#BunkeSchick09)) and precursors, existence of fiber integration in differential cohomology over circle fibers was formalized as a separate [[axiom]] ([Bunke-Schick 09, def. 1.3](#BunkeSchick09)).

## Examples

* [[integration of differential forms]]

* [[fiber integration in ordinary differential cohomology]]

* [[fiber integration in differential K-theory]]

## Related concepts

* [[higher holonomy]]

## References

* {#GomiTerashima00} [[Kiyonori Gomi]], [[Yuji Terashima]], _A Fiber Integration Formula for the Smooth Deligne Cohomology_, International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf), [doi:10.1155/S1073792800000386](https://doi.org/10.1155/S1073792800000386))

* {#HopkinsSinger} [[Mike Hopkins]], [[Isadore Singer]], section 4.10 in  _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

* {#BunkeSchick09} [[Ulrich Bunke]], [[Thomas Schick]], _Uniqueness of smooth extensions of generalized cohomology theories_, J. Topol. 3 (2010), no. 1, 110-156 ([arXiv:0901.4423](http://arxiv.org/abs/0901.4423))

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

[[!redirects integration in differential cohomology]]


fiber integration in differential K-theory

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#Contents#
* table of contents
{:toc}

## Idea

The special case of [[fiber integration in differential cohomology]] for the case of [[differential K-theory]].

## Related concepts

* [[index theory]]

* [[fiber integration in K-theory]]

* [[geometric quantization by push-forward]]

## References

* [[Dan Freed]], [[John Lott]], _An index theorem in differential K-theory_, Geometry and Topology 14 (2010) ([pdf](http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf))

[[!redirects integration in differential K-theory]]


fiber integration in K-theory

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#### Integration theory
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[[!include integration theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The special case of [[fiber integration in generalized cohomology]]/[[twisted Umkehr maps]] for [[KU]]-cohomology.

## Models
 {#Models}

There are various different models for describing and constructing fiber integration in K-theory.

1. [In terms of bundles of Fredholm operators](#InTermsOfBundlesOfFredholmOperators)

1. [In operator KK-theory](#InOperatorKKTheory)

### In terms of bundles of Fredholm operators
 {#InTermsOfBundlesOfFredholmOperators}

We discuss here fiber integration in the model of [[twisted K-theory]] [by bundles of spaces of Fredholm operators](twisted%20K-theory#ByBundlesOfFredholmOperators). Related literature includes ([Carey-Wang 05](CareyWang05)).

1. _[Along a fibration of closed spin^c manifolds](#AlongAFibrationOfClosedSpinCManifolds)_

#### Along a fibration of closed $Spin^c$-manifolds

Let $f\colon Y \longrightarrow X$ be a [[fiber bundle]] of [[compact manifolds|compact]] [[smooth manifolds]] carrying fiberwise a [[spin^c structure]].

For $x\in X$ write

* $Y_x$ for the [[fiber]] over $x$;

* $P_x \to Y_x$ for the fiberwise [[spin^c]]-[[principal bundle]];

* $Cl(Y)_x \coloneqq P_x \underset{Spin^c}{\times} Cl_n$ for the fiberwise [[Clifford bundle]];

* $S_x$ for the [[spin^c]] [[spin bundle]] on $Y_x$;

* $D_x\colon \Gamma(S_x)\to \Gamma(S_x)$ for the [[Spin^c Dirac operator]] on the [[space of sections]] of [[spinors]];

* $\tilde D_x \coloneqq \frac{D_x}{\sqrt{1+ D_x^\ast D_x}} \colon L^2(S_x)\to L^2(S_x)$ for the [[bounded linear operator|bounded]] version;

Here $D_x$ depends smoothly on $x$ while $\tilde D_x$ still depends continuously on $x$. Equip $CL(Y)_y$ with the $Cl_n$-[[action]] given on elements $v \in \mathbb{R}^n \hookrightarrow Cl_n$ joint right Clifford product by $v$ and left Clifford product by the volume element

$$
  vol \cdot (-) \cdot v
  \,.
$$

This is such that $D_x$ [[graded commutator|graded-commutes]] with this $Cl_n$-action. Hence the assignment $x \mapsto \tilde D_x$ defines a map

$$
  \tilde D_{(-)} \colon X \longrightarrow Fred^{(n)}
$$

from the base to the space [[Fredholm operators]] [[graded-commutator|graded-commuting]] with $Cl_n$, as defined [here](twisted%20K-theory#ByBundlesOfFredholmOperators) at _[[twisted K-theory]]_.

More generally, let then $V \to Y$ be a [[vector bundle]] representing a class in $K^0(Y)$. With a choice of [[connection]] this twists the above constrction to yield $V$-twisted Dirac operator $\tilde D^V_x$ and hence a map

$$
  \tilde D_{(-)}^V \colon X \longrightarrow Fred^{(n)}
  \,.
$$

This represents the push-forward class in $K^{dim(Y_x)}(X)$, and this construction gives a map

$$
  \int f
  \colon
  K^0(Y) \longrightarrow K^{dim(Y)-dim()X}(X)
  \,.
$$

For $dim(Y_x)$ even and hence ignoring the compatibility with the $Cl_n$-action, this is discussed in ([Carey-Wang 05](CareyWang05)).

### In operator KK-theory
 {#InOperatorKKTheory}

We discuss fiber integration /push-forward/[[Umkehr maps]]/[[Gysin maps]] in
[[operator K-theory]], hence in [[KK-theory]] ([Connes-Skandalis 85](#ConnesSkandalis84), [BMRS 07, section 3](#BMRS07)).

The following discusses KK-pushforward

1. _[Along an embedding](#KKPushForwardAlongEmbedding)_

1. _[Along a submersion](#KKPushforwardAlongSubmersion)_

1. _[Along a fibration of closed spin^c manifolds](#AlongAFibrationOfClosedSpinCManifolds)_

1. _[Along a general K-oriented map](#KKPushforwardAlongGeneralMap)_

1. _[In twisted K-theory](#KKPushforwardAlongGeneralMap)_

The construction goes back to ([Connes 82](#Connes82)), where it is given over smooth manifolds. Then ([Connes-Skandalis 84](#ConnesSkandalis84), [Hilsum-Skandalis 87](#HilsumSkandalis87)) generalize this to maps between [[foliations]] by KK-elements betwen the [[groupoid convolution algebras]] of the coresponding [[holonomy groupoids]] and  ([Rouse-Wang 10](#RouseWang10)) further generalize to the case where a [[circle 2-bundle]] twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary [[noncommutative topology|noncommutative spaces]]/[[C*-algebras]] is in ([BMRS 07](#BMRS07)).

#### Along an embedding
 {#KKPushForwardAlongEmbedding}

([Connes-Skandalis 84, above prop. 2.8](ConnesSkandalis84))

Let $h \colon X \hookrightarrow Y$ be an [[embedding]] of [[compact topological space|compact]] [[smooth manifolds]].

The push-forward constructed from this is supposed to be an element in [[KK-theory]]

$$
  h! \colon KK_d(C(X), C(Y))
$$

in terms of which the push-forward on [[operator K-theory]] is induced by postcomposition:

$$
  h_!
    \;\colon\;
  K^\bullet(X)
   \simeq
  KK_\bullet(\mathbb{C}, X)
  \stackrel{h!\circ (-)}{\to}
  KK_{\bullet+d}(\mathbb{C},Y)
  \simeq
 KK^{\bullet+d}(Y)
 \,,
$$

where $d = dim(X) - dim(Y)$.

Now, if we could "thicken" $X$ a bit, namely to a [[tubular neighbourhood]]

$$
  h  \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y
$$

of $h(X)$ in $Y$ without changing the K-theory of $X$, then the element in question will just be the KK-element

$$
  j! \in KK(C_0(U), C(Y))
$$

induced directly from the [[C*-algebra]] homomorphism $C_0(U) \to C(Y)$ from the [[algebra of functions]] [[vanishing at infinity]] of $U$ to functions on $Y$, given by extending these functions by 0 to functions on $Y$.
Or rather, it will be that element composed with the assumed KK-equivalence

$$
  \psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
$$

The bulk of the technical work in constructing the push-forward is in constructing this equivalence. ([BMRS 07, example 3.3](#BMRS07))

In order for it to exist at all, assume that the [[normal bundle]]

$$
  N_Y X \coloneqq h^\ast(T Y)/ T X
$$

has a [[spin^c structure]]. Write $S(N_Y X)$ for the associated [[spinor bundle]].

Then there is an invertible element in [[KK-theory]]

$$
  \iota^X! \in KK_n(C(X), C_0(N_Y X))
$$

hence a KK-equivalence $\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X)$, where $C_0(-)$ denotes the [[algebra of functions]] [[vanishing at infinity]].

This is defined as follows. Consider the pullback $\pi_n^\ast S(N_Y X) \to N_Y X$ of this spinor to the normal bundle itself along the projection $\pi_N \colon N_Y X \to X$.  Then...

Moreover, a choice of a [[Riemannian metric]] on $X$ allows to find a [[diffeomorphism]] between the [[tubular neighbourhood]] $U_{h(X)}$ of $h(X)$ and a neighbourhood of the zero-section of of the normal bundle

$$
  \Phi \colon U_{h(X)} \hookrightarrow N_Y X
  \,.
$$

This induces a KK-equivalence

$$
  [\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U)
  \,.
$$

Therefore the push-forward in operator K-theory along $f \colon X \hookrightarrow Y$ is given by postcomposing in [[KK-theory]] with

$$
  h! \colon
  C(X)
    \underoverset{\simeq_{KK}}{i^X!}{\to}
  C_0(N_Y X)
    \underoverset{\simeq_{KK}}{\Phi}{\to}
  C_0(U)
    \stackrel{j!}{\to}
  C(Y)
  \,.
$$

#### Along a proper submersion
 {#KKPushforwardAlongSubmersion}

([Connes-Skandalis 84, above prop. 2.9](ConnesSkandalis84))

For $\pi \colon X \to Z$ a [[K-orientation|K-oriented]] [[proper map|proper]] [[submersion]] of compact smooth manifolds, the push-forward map along it is reduced to the [above](#KKPushForwardAlongEmbedding) case of an embedding by

1. using that by the [[Whitney embedding theorem]] every compact $X$ may be embedded into some $\mathbb{R}^{2q}$ such as to yield an embedding

   $$
     h \colon X \to Z \times \mathbb{R}^{2 q}
   $$

1. using that there is a KK-equivalence

   $$
     \iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q})
     \,.
   $$

The resulting push-forward is then given by postcomposition in [[KK-theory]] with

$$
  \pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q)
   \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to}
  C(Z)
  \,.
$$

([BMRS 07, example 3.4](#BMRS07))

#### Along a smooth fibration of closed $Spin^c$-manifolds
 {#AlongAFibrationOfClosedSpinCManifolds}

Specifically, for $\pi \colon X \to Z$ a smooth fibration over a closed smooth manifold whose [[fibers]] $X/Z$ are

* [[closed manifold|closed]] [[smooth manifolds|smooth]] [[spin^c structure|spin^c]] [[manifolds]] of even [[dimension]]

the push-forward element $\pi! \in KK(C_0(X), C_0(Z))$ is given by the [[Fredholm module|Fredholm]]-[[Hilbert module]] obatined from the fiberwise [[spin^c Dirac operator]] acting on the fiberwise [[spinors]]. ([Connes-Skandalis 84, proof of lemma 4.7](ConnesSkandalis84), [BMRS 07, example 3.9](BMRS07)).

In detail, write

$$
  T(X/Z) \hookrightarrow T X
$$

for the sub-bundle of the total [[tangent bundle]] on the [[vertical vectors]] and choose a [[Riemannian metric]] $g^{X/Z}$ on this bundle (hence a collection of Riemannian metric on the fibers $X/Z$ smoothly varying along $Z$). Write $S_{X/Z}$ for the corresponding [[spinor bundle]].

A choice of horizontal complenet $T X \simeq T^H X \oplus T(X/Z)$ induces an [[affine connection]] $\nabla^{X/Z}$. This combined with the [[symbol map]]/Clifford multiplication of $T^\ast (X/Z)$ on $S_{X/Z}$ induces a fiberwise [[spin^c Dirac operator]], acting in each fiber on the [[Hilbert space]] $L^2(X/Z, S_{X/Z})$.

This yields a [[Fredholm module|Fredholm]]-[[Hilbert bimodule]]

$$
  (D_{X/Z}, L^2(X/Z, S_{X/Z}))
$$

which defines an element in [[KK-theory]]

$$
  \pi ! \in KK(C_0(X), C_0(Z))
  \,.
$$

Postcompositon with this is the push-forward map in K/KK-theory, equivalently the [[index]] map of the collection of Dirac operators.

#### Along a general K-oriented map
 {##KKPushforwardAlongGeneralMap}

([Connes-Skandalis 84, def. 2.1](ConnesSkandalis84))

Now for $f \colon X \to Y$ an arbitray [[K-orientation|K-oriented]] smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its [[graph map]], followed by projection to $Y$:

$$
  f \;\colon\;
  X \stackrel{graph(f)}{\to}
  X \times Y
  \stackrel{p_Y}{\to}
  Y
  \,.
$$

Hence push-forward along such a general map is postcomposition in [[KK-theory]] with

$$
  f! \coloneqq p_Y !\circ graph(f)!
  \,.
$$

([BMRS 07, example 3.5](#BMRS07))

#### In twisted K-theory
 {#KKPushforwardAlongGeneralMap}

We discuss push forward in K-theory more generally by [[Poincaré duality C*-algebras]] hence [[dual objects]] in [[KK-theory]].

Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

$$
  C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
$$

with notation as in [this definition](Poincaré+duality+algebra#CStarAlgebraOf2BundleOnManifold). By [this proposition](Poincaré+duality+algebra#DualOfCompactManifoldWithTwist) the [[dual morphism]] is of the form

$$
  C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{W_3(\tau_X)}{\chi}}(X)
  \,.
$$

If we assume that $X$ has a [[spin^c structure]] then this is

$$
  C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{1}{\chi}}(X)
  \,.
$$

Postcomposition with this map in [[KK-theory]] now yields a map from the $\frac{W_3(\tau_Q)}{i^\ast \chi}$-[[twisted K-theory]] of $Q$ to the $\chi^{-1}$-[[twisted K-theory]] of $X$:

$$
  i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi}
  \,.
$$

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element

$$
  [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q)
$$

is called (the K-class of) a _[[Chan-Paton gauge field]]_ on the D-brane satisfying the _[[Freed-Witten-Kapustin anomaly cancellation]]_ mechanism. (The orginal _[[Freed-Witten anomaly cancellation]]_ assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin^c structure]] on $Q$.) Finally its [[fiber integration|push-forward]]

$$
  [i_! \xi] \in K_{\bullet- \chi}(X)
$$

is called the corresponding _[[D-brane charge]]_.

## Related concepts

* [[geometric quantization by push-forward]]

* [[geometric quantization with KU-coefficients]]

## References

Discussion in [[KK-theory]]

* {#Connes82} [[Alain Connes]], _A survey of foliations and operator algebras_, Proceedings of the A.M.S., 38, 521-628 (1982) ([pdf](http://www.alainconnes.org/docs/foliationsfine.pdf))

* {#ConnesSkandalis84} [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.4218))

* {#HilsumSkandalis87} [[Michel Hilsum]], [[Georges Skandalis]], _Morphismes K-orienté d'espace de feuille et fonctoralité en théorie de Kasparov_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 20 no. 3 (1987), p. 325-390 ([numdam](http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0))

* {#BMRS07} Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))

* {#RouseWang10} Paulo Carrillo Rouse, [[Bai-Ling Wang]], _Twisted longitudinal index theorem for foliations and wrong way functoriality_ ([arXiv:1005.3842](http://arxiv.org/abs/1005.3842))

Discussion for integration [[twisted K-theory]] over [[manifolds]]:

* {#CareyWang05} [[Alan Carey]], [[Bai-Ling Wang]], _Thom isomorphism and Push-forward map in twisted K-theory_ ([arXiv:0507414](http://arxiv.org/abs/math/0507414))

Discussion for integration of [[twisted K-theory]] along [[representable morphisms]] of  [[local quotient stacks]]:

* {#FHT} [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], _[[Loop Groups and Twisted K-Theory]]_ I (2011) ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906)),

Review in the context of [[geometric quantization with KU-coefficients]] is in

* {#Nuiten} [[Joost Nuiten]], section 4 of _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, 2013

[[!redirects fiber integration in KU-theory]]
[[!redirects push-forward in K-theory]]