event

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## Idea

In [[spacetime]] [[physics]] and specifically in [[special relativity|special]]/[[general relativity]]:

* an *event* is a [[point]] in a [[spacetime]].

In [[probability theory]]:

* an *event* is a [[subset]] of the underlying [[measure space]] ([[probability space]]).

## Related concepts

* [[spacelike]], [[timelike]], [[lightlike]]

## References

* Wikipedia, _[Event (relativity)](http://en.wikipedia.org/wiki/Event_%28relativity%29)_

[[!redirects events]]

[[!redirects event (probability theory)]]
[[!redirects event (in probability theory)]]


event horizon

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#Contents#
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## Idea

Fix a [[timelike curve]] $\gamma$ in [[spacetime]], thought of as representing an [[observer]].  If $H$ is a [[hypersurface]] in spacetime, then (at least if spacetime is [[orientation|orientable]] and possibly otherwise), $H$ separates spacetime into two regions, arbitrarily called the inside and outside.  Then $H$ is an __event horizon__ (relative to $\gamma$) if $\gamma$ never intersects the [[future]] of the inside of $H$.  (It follows that $\gamma$ lies entirely outside $H$.)

Although we have given this definition relative to an observer, we can reverse the situation, beginning with a [[lightlike]] hypersurface $H$ and noting that $H$ is a horizon relative to every observer that remains in the [[past]] of $H$ (of which, unlike for a [[spacelike]] hypersurface, there are typically many).  This includes the horizon around a [[black hole]] and the future [[light cone]] of any event.

Theoretically, the observer outside an event horizon will observe [[Bekenstein-Hawking entropy]] at the horizon and [[Hawking radiation]] from it.  In the case of an observer accelerating to remain outside a future light cone, this is the [[Unruh effect]].

## Related concepts

* [[Cauchy horizon]]

* [[naked singularity]]

* [[observable universe]]

* [[generalized second law of thermodynamics]]

* [[near-horizon geometry]]

## References

Resolved image of the direct vicinity of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* {#EHT19} [[Event Horizon Telescope Collaboration]], _First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole_,  The Astrophysical Journal Letters, Volume 875, Number 1, 2019 ([doi:10.3847/2041-8213/ab0ec7](https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7))

Discussion of event horizons of [[black holes]] in terms of [[AdS/CFT]] is in

* Kyriakos Papadodimas, Suvrat Raju, _An Infalling Observer in AdS/CFT_ ([arXiv:1211.6767](http://arxiv.org/abs/1211.6767))

[[!redirects event horizon]]
[[!redirects event horizons]]
[[!redirects horizon]]
[[!redirects horizons]]


Event Horizon Telescope

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#### Experiments
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## References

### General

* _[EHT Home Page](https://eventhorizontelescope.org/)_

See also

* Wikipedia, _[Event Horizon Telescope](https://en.wikipedia.org/wiki/Event_Horizon_Telescope)_

### Publications

Resolved image of the direct vicinity of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* {#EHT19} [[Event Horizon Telescope Collaboration]], _First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole_,  The Astrophysical Journal Letters, Volume 875, Number 1, 2019 ([doi:10.3847/2041-8213/ab0ec7](https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7))

Exposition:

* Shep Doeleman on behalf of the [[EHT Collaboration]], _Focus on the First Event Horizon Telescope Results_, The Astrophysical Journal Letters 2019 ([web](https://iopscience.iop.org/journal/2041-8205/page/Focus_on_EHT))

[[!redirects Event Horizon Telescope Collaboration]]
[[!redirects EHT Collaboration]]

[[!redirects EHT]]


event structure

# Contents
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##Idea##

Event structures were introduced in order to abstract away from the precise 'places' and times at which events occur in distributed systems. The structure focuses on the events and the causal ordering between them.

Event structures in the sense of this article are sometimes also called *prime* event structures. There are other variants, for example where the partial order $\leq$ is replaced by an *enabling relation* $\vdash$. This is usually more expressive, because an event can have *disjunctive causes*: if $a \vdash e$ and $b \vdash e$, then either of $a$ or $b$ suffices for $e$ to occur. In a prime event structure, if $a \leq e$ and $b \leq e$ then *both* $a$ and $b$ must occur before $e$; these are *conjunctive causes*.

##Definition##

\begin{definition}
An *event structure* is a tuple $(E,\leq, \mathrm{Con})$ consisting of a [[poset]] $(E,{\le})$ of _events_, and a nonempty set $\mathrm{Con} \subseteq \mathcal{P}(E)$ of _consistent subsets_, satisfying the following axioms:

*  finite causes: for every event $e$ the set $\{e'\mid e'\le e\}$ is finite;

* if $e \in E$ then $\{ e \} \in \mathrm{Con}$;

* if $X \in \mathrm{Con}$ and $Y \subseteq X$ then $Y \in \mathrm{Con}$;

* if $X \in \mathrm{Con}$, $e \in X$, and $e' \leq e$, then $X \cup \{ e\} \in \mathrm{Con}$.

\end{definition}

A restricted but simpler definition is as follows:

\begin{definition}
An *event structure with binary conflict* is a tuple $(E, \leq, \#)$, where $(E, {\leq})$ is a [[poset]] and $\#$ is an irreflexive binary relation on $E$, the *conflict relation*, satisfying:

* finite causes: for every event $e$ the set $\{e'\mid e'\le e\}$ is finite;

* hereditary conflict: if $e \#  e'$ and $e' \leq e''$ then $e \# e''$.

\end{definition}

Event structures with binary conflict can be characterised as follows:
\begin{proposition}
If $(E, \leq, \#)$ is an event structure with binary conflict, then defining $\mathrm{Con} = \{ X \subseteq E \mid \forall e, e' \in X. \neg (e \# e') \} $
makes $(E, \leq, \Con)$ an event structure.

Conversely, if an event structure $(E, \leq, \mathrm{Con})$ satisfies
\[
\forall X \subseteq E. (\forall e, e' \in X. \neg (e \# e')) \implies X \in \mathrm{Con}
\]
then defining $\# = \{ (e, e') \mid \{ e, e'\} \in \mathrm{Con} \}$  makes $(E, \leq, \#)$ an event structure with binary conflict.
\end{proposition}

That is, event structures with binary conflict correspond to event structures in which pairwise consistency implies mutual consistency.

We write $E$ for $(E, \leq, \Con)$ whenever possible. The possible states of an event structures are called *configurations*.

\begin{definition}

Let $E$ be an event structure. A **configuration** of $E$
 is a subset $x \subseteq E$ which is consistent ($x \in \Con$) and down-closed (if $e\in x$ and $e' \leq e$ then $e' \in x$). The set of _finite_ configurations of $E$ is denoted $\mathscr{C}(E)$.

\end{definition}

##The category of event structures##

\begin{definition}
A (total) **map of event structures** from $(E, \leq, \Con)$ to $(E', \leq', \Con')$ is a function $f : E \to E'$ such that:

* $f$ preserves configurations: if $x\in \mathscr{C}(E)$, then $f(x) \in \mathscr{C}(E')$.

* $f$ is locally injective: if $e, e' \in x \in \mathscr{C}(E)$ and $f(e) = f(e')$, then $e = e'$.

\end{definition}

Intuitively, a map $E \to E'$ expresses that all executions of $E$ can be faithfully simulated within $E'$.

*Partial* maps of event structures are also important in the literature, for example in the event structure model of CCS. The definition is the same when $f$ is a partial function, and the condition $f(e) = f(e')$ means in particular that they are both defined. Winskel and Nielsen explain this definition as follows:

_A morphism $f : E\to E'$ between event structures expresses how behaviour in $E$ determines behaviour in $E'$. The partial function, $f$, expresses how the occurrence of an event in $E$ implies the simultaneous occurrence of an event in $E'$; the fact that $f(e) = e'$ can be understood as expressing that the event $e$ is a ''component'' of the event $e'$ and, in this sense, that the occurrence of $e$ implies the simultaneous occurrence of $e'$. If two distinct events in $E$ have the same image in $E'$ under $f$ then they cannot belong to the same configuration. _

With this definition of morphism, and the obvious notions of identity and composition, we get the category $\mathbf{ES}$ of event structures (and total maps).

##Event structures as presheaves

##References##

* [[G. Winskel]] and M. Nielsen, Models for concurrency. vol. 3, Handbook of Logic in Computer Science, pages 100 - 200, Oxford Univ. Press, 1994. (see also [online technical report](http://www.daimi.au.dk/PB/463/PB-463.pdf)).

* [[G. Winskel]], _Events, causality, and symmetry_, (an earlier version appeared in the BCS conference 'Visions in Computer Science.' September 2008. The final version appears in a special issue of The Computer Journal 2009; doi: 10.1093/comjnl/bxp052; see also  an [online version](http://www.cl.cam.ac.uk/~gw104/CJVision-Revised.pdf)).

* [[Sam Staton]] and [[G. Winskel]], On the expressivity of symmetry in event structures,
[LICS 2010](http://www.cl.cam.ac.uk/~gw104/lics2010final.pdf)

[[!redirects event structure]]
[[!redirects event structures]]


events of interest for the working category theorist

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This page is for announcing events (conferences, workshops, ...) that could be of some interest for the working category theorist.

To add an announcement, hit "edit" at the bottom of this page: In the edit pane that opens, add a bullet item to the [following list](#LatestsEvents) (use inverse chronological order), *and* copy the same item into the comment-box below the edit pane, as that will make the announcement be posted more visibly to the [nForum](https://nforum.ncatlab.org/) (see [this thread](https://nforum.ncatlab.org/discussion/17291/category-theory-events/#Item_1)).

\linebreak

## Latest events
 {#LatestsEvents}

* 2024, July 15-19

  **QPL 2024**

  (annual Quantum Physics and Logic conference)

  home page: [qpl2024.dc.uba.ar](https://qpl2024.dc.uba.ar)

  Buenos Aires (Argentina)

* 2024, July 8th-13th

  **Foundational Methods in Computer Science 2024 (FMCS 2024)**

  homepage: TBA

  Kananaskis (Alberta, Canada) at [Biogeoscience Institute](https://research.ucalgary.ca/biogeoscience-institute)

* 2024, July 1-5th

  **TACL24**

  (annual *conference on Topology, Algebra and Categories in Logic*)

  home page: [iiia.csic.es/tacl2024](https://iiia.csic.es/tacl2024/)

  in Barcelona (Spain)

* 2024, June 24rd-28th

  **Logic Colloqium 2024**

  (annual *conference of the Association of Symbolic Logic*)

  home page: [lc2024.se](https://lc2024.se)

  in Gothenburg (Sweden)

* 2024, June 23rd-29th

  **Category Theory 2024**

  (annual *[[Category Theory conference]]*)

  home page: [www.usc.gal/regaca/ct2024](https://www.usc.gal/regaca/ct2024)

  in Santiago de Compostela (Spain)

* 2023, December 18th-19th

  **ItaCa23**

  (annual meeting of the [[ItaCa|Italian Category Theory Community]])

  home page: [progetto-itaca.github.io/ItaCa-23/](https://progetto-itaca.github.io/ItaCa-23/)

  in Turin (Italy)

* 2023, November 23rd

  **ItaCa Fest**

  (recurrent [[ItaCa]] online seminar)

  home page: [progetto-itaca.github.io/pages/fest23.html#nov23](https://progetto-itaca.github.io/pages/fest23.html#nov23)

* 2023, October 28/29

  **Category Theory Octoberfest**

  website: [richardblute.ca/octoberfest-2023/](https://richardblute.ca/octoberfest-2023/)

* 2023, October 25th

  **ItaCa Fest**

  (recurrent [[ItaCa]] online seminar)

  home page: [progetto-itaca.github.io/pages/fest23.html#oct25](https://progetto-itaca.github.io/pages/fest23.html#oct25)

category: reference

[[!redirects Events of interest for the working category theorist]]
[[!redirects category theory events]]


eventual image

# Eventual image

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## Definition

The **eventual image** of an [[endomorphism]] $f:A\to A$ is the [[intersection]] $\bigcap_{n=0}^{\infty} f^{(n)}(A)$ of the images of its [[iteration|iterates]].  This makes sense in many different [[categories]].

## Properties

On the category of [[finite sets]], the operation assigning to each endomorphism its eventual image is a [[dinatural transformation]].

This example can also be viewed in terms of the [[trace of a category|trace]] of [[FinSet]], defined as $\int^{a: FinSet}\; \hom(a, a)$. Indeed, the value of $f \in \hom(a, a)$ under the canonical map $\hom(a, a) \to \int^a \; \hom(a, a)$ is the same as that of the restriction $f|: Evim(f) \to Evim(f)$ to its eventual image, which is a permutation, and the value may be regarded as the [[conjugacy class]] of that permutation.

## References

* [[Tom Leinster]], [The eventual image](https://golem.ph.utexas.edu/category/2011/12/the_eventual_image.html) (blog post)

* [[Tom Leinster]], [The eventual image, part 2](https://golem.ph.utexas.edu/category/2011/12/the_eventual_image_part_2.html) (blog post)

[[!redirects eventual images]]


eventuality filter

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# Eventuality filters
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## Idea

Eventuality filters are the key translation between [[filters]] and [[nets]] in [[analysis]] and [[topology]].  Specifically:

*  Given a net $n$ (or in particular a [[sequence]]) in a [[set]] $X$, the eventuality filter of $n$ is a [[proper filter]] on $X$;

*  Every proper filter on $X$ is the eventuality filter of some net in $X$;

*  Two nets are equivalent for purposes of [[convergence]] (meaning precisely that they are [[subnets]] of each other in the sense of Årnes & Andenæs) if and only if their eventuality filters are equal.

This last point is not so much a result as the *definition* of the subnet relation (or at least of its [[symmetric relation|symmetrisation]], the relation of equivalence of nets).  One still needs to check that every use of nets in analysis and topology (or other fields) actually respects this notion of equivalence of nets, if one wishes to convert nets to filters.

## Definition

Let $X$ be a [[set]], let $D$ be a [[directed set]], and let $n$ be a [[function]] from $D$ to $X$, so that $n$ is a [[net]] in $X$.  Given a [[subset]] $A$ of $X$, $n$ is __eventually__ in $A$ if, for some $i$ in $D$, for each $j \geq i$ in $D$, $n_j \in A$.  The collection $F_n$ of all those subsets $A$ such that $n$ is eventually in $A$ is a [[proper filter]] on $X$, called the __eventuality filter__ of $n$.

In symbols,
$$ F_n \coloneqq \{ A \subseteq X \;|\; \ess \forall\, j,\; n_j \in A \} ,$$
where $\ess \forall\, j$ is read 'for essentially each $j$' or 'eventually for each $j$' and means $\exists\, i,\; \forall j \geq i$.  In the case where $D$ is the set of [[natural numbers]] directed by $\leq$, so that $n$ is an [[infinite sequence]] ($\omega$-sequence) in $X$, then $\ess \forall\, j$ may be read as 'for all but finitely many $j$'.

## Converse

Given a [[filter]] $F$ on $X$, let $D$ be the [[binary relation]] $\in_X$ restricted to $F$, viewed as a subset of the [[cartesian product]] $X \times F$, [[preorder|ordered]] so that $(y, A) \leq (z, B)$ means simply that $B \subseteq A$.  Let $n\colon D \to X$ map $(y, A)$ simply to $y$.  Then $D$ is [[direction|directed]] (so that $n$ is a net in $X$) iff $F$ is [[proper filter|proper]]; and in that case, the eventuality filter of the net $n$ is $F$.

It is possible to define $D$ as ${{\in_X}|_F} \times \mathbb{N}$ (instead of as ${\in_X}|_F$ as done above) so that the direction on $D$ is a [[partial order]], if one really wants to.  (Then $(y, A, i) \leq (z, B, j)$ means that $B \subseteq A$, $i \leq j$, and $y = z$ if $i = j$.)

[[!redirects eventuality filter]]
[[!redirects eventuality filters]]

[[!redirects eventuality]]
[[!redirects eventualities]]
[[!redirects eventual]]
[[!redirects eventually]]


Evgeny Buchbinder

## Selected writings

On [[Horava-Witten theory]], [[heterotic string theory]]:

* {#BuchbinderOvrut04} [[Evgeny Buchbinder]], [[Burt Ovrut]], _Vacuum Stability in Heterotic M-Theory_, Phys.Rev. D69 (2004) 086010 ([arXiv:hep-th/0310112](http://arxiv.org/abs/hep-th/0310112))

On [[heterotic string theory|heterotic]] [[string phenomenology]] using [[heterotic line bundle models]]:

* [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], _The Moduli Space of Heterotic Line Bundle Models: a Case Study for the Tetra-Quadric_, 	JHEP03 (2014) 025 ([arXiv:1311.1941](https://arxiv.org/abs/1311.1941))

On computation of [[Yukawa couplings]] in [[heterotic string theory]]:

* Stefan Blesneag, [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], [[Eran Palti]], _Matter Field Kähler Metric in Heterotic String Theory from Localisation_ ([arXiv:1801.09645](https://arxiv.org/abs/1801.09645))

On [[ekpyrotic cosmology]]:

* [[Evgeny Buchbinder]], [[Justin Khoury]], [[Burt Ovrut]], _New Ekpyrotic Cosmology_, Phys. Rev. D76:123503, 2007 ([arXiv:hep-th/0702154](https://arxiv.org/abs/hep-th/0702154))

## Related $n$Lab entries

* [[Yukawa coupling]]

* [[moduli stabilization]]

* [[ekpyrotic cosmology]]

category: people


Evgeny Lifshitz

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

## related $n$Lab entries

* _[[Course of Theoretical Physics]]_

category: people


Evgeny Likhtman

* encyclopedia entry: **[[Likhtman_Evgeny_Pinkhasovich.pdf:file]]** (by [[Mikhail Shifman]])

## Selected writings

Introducing the [[super Poincaré Lie algebra]] ("[[supersymmetry]]"):

* {#GolfandLikhtman72} [[Yuri Golfand]], [[Evgeny Likhtman]], _On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: [[Victor Ginzburg]] et al. (eds.) _I. E. Tamm Memorial Volume Problems of Theoretical Physics_, (Nauka, Moscow 1972), page 37,

  translated and reprinted in: [[Mikhail Shifman]] (ed.), _[[The Many Faces of the Superworld]]_ pp. 44-53,  World Scientific (2000) ([doi:10.1142/9789812793850_0006](https://doi.org/10.1142/9789812793850_0006))

* {#Likhtman01} [[Evgeny Likhtman]], _Around SuSy 1970_, talk at "30 years of supersymmetry", Nucl.Phys.Proc.Suppl. 101 (2001) 5-14 ([arxiv:hep-ph/0101209](https://arxiv.org/abs/hep-ph/0101209))

category: people


evolution equation

#Contens#
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#Idea
An evolution equation is an equation of the form
$$
   \partial_t f = L f
$$
where $\partial_t$ represents the partial derivative with respect to time, $L$ is a differential operator and $f$ is usually assumed to be an element of some [[topological vector space]]. Examples are the [[Schrödinger equation]] and the [[Fokker-Planck equation]]. These equations describe the time evolution of a physical system, hence the name.

[[!redirects evolution equations]]


evolutionary derivative

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## Idea

The _evolutionary derivative_ or "[[Fréchet derivative]] of a [[tuple]] of differential functions" ([Olver 93, def. 5.24](#Olver93))) is the [[derivative]] of a [[section]] of some [[vector bundle]] $V$ depending on [[jets]] of a "[[field bundle]]" $E$ (def. \ref{FieldDependentSections} below) along the prolongation of an [[evolutionary vector field]] on $E$. Equivalently this is a jet-dependent [[differential operator]] on the [[vertical tangent bundle]] of $E$ and as such is usefully related to the [[Euler-Lagrange derivative]] on $E$ (example \ref{DifferentialOperatorDerivativeOfLagrangianFunction} and prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives} below).

## Definition

In the following _[[fiber bundles]]_ are considered in [[differential geometry]] and in particular _[[vector bundle]]_ means _[[smooth vector bundle]]_.

+-- {: .num_defn #FieldDependentSections}
###### Definition
**([[field (physics)|field]]-dependent [[sections]])**

For

$$
  E \overset{fb}{\longrightarrow} \Sigma
$$

a [[fiber bundle]], regarded as a [[field bundle]], and for

$$
  E' \overset{fb'}{\longrightarrow} \Sigma
$$

any other [[fiber bundle]] over the same base space ([[spacetime]]), we write

$$
  \Gamma_{J^\infty_\Sigma(E)}(E')
  \;\coloneqq\;
  \Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' )
  \;=\;
  Hom_\Sigma(J^\infty_\Sigma(E), E')
  \;\simeq\;
  DiffOp(E,E')
$$

for the [[space of sections]] of the [[pullback of bundles]] of $E'$ to the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$  along $jb$.

$$
  \Gamma_{J^\infty_\Sigma(E)}(E')
  \;=\;
  \left\{
    \array{
       && E'
       \\
       & {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'}
       \\
       J^\infty_\Sigma(E)
       &\underset{jb}{\longrightarrow}&
       \Sigma
    }
    \phantom{A}\,\,
  \right\}
  \,.
$$

(Equivalently this is the space of [[differential operators]] from sections of $E$ to sections of $E'$. )

=--

In ([Olver 93, section 5.1, p. 288](#Olver93)) the field dependent sections of def. \ref{FieldDependentSections}, considered in [[local coordinates]], are referred to as [[tuples]] of _differential functions_.

+-- {: .num_example #EvolutionaryVectorFieldsAsFieldDependentSections}
###### Example
**([[source forms]] and [[evolutionary vector fields]] are field-dependent sections)**

For $E \overset{fb}{\to} \Sigma$ a [[field bundle]], write $T_\Sigma E$ for its [[vertical tangent bundle]] and $T_\Sigma^\ast E$ for its [[dual vector bundle]], the [[vertical cotangent bundle]].

Then the field-dependent sections of these bundles according to def. \ref{FieldDependentSections} are identified as follows:

* the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of [[evolutionary vector fields]] $v$ as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$:

  $$
    \left(
      \array{
        && T_\Sigma E
        \\
        & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}}
        \\
        J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma
      }
    \right)
    \;\in\;
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
  $$

* $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of [[source forms]] $E$ as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:

  $$
    \left(
      \array{
        && T^\ast_\Sigma E
        \\
        & {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}}
        \\
        J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \overset{fb}{\longrightarrow}& \Sigma
      }
    \right)
    \;\in\;
    \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
  $$

This makes manifest the duality pairing between [[source forms]] and [[evolutionary vector fields]]

$$
  \array{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
    \otimes
    \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
    &\longrightarrow&
    C^\infty(J^\infty_\Sigma(E))
  }
$$

which in local coordinates is given by

$$
  (v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a)
  \mapsto
  v^a \omega_a
$$

for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ [[smooth functions]] on the [[jet bundle]].

=--

+-- {: .num_defn #FieldDependentDifferentialOperatorDerivative}
###### Definition
**([[evolutionary derivative of field-dependent section]])**

Let

$$
  E \overset{fb}{\to} \Sigma
$$

be a [[fiber bundle]] regarded as a [[field bundle]] and let

$$
  V \overset{vb}{\to} \Sigma
$$

be a [[vector bundle]]. Then for

$$
  P \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$

a field-dependent section of $E$ accoring to def. \ref{FieldDependentSections}, its _evolutionary derivative_ is the morphism

$$
  \array{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
      & \overset{  \mathrm{D}P }{\longrightarrow} &
    \Gamma_{J^\infty_\Sigma(E)}(V)
    \\
    v &\mapsto& \hat v(P)
  }
$$

which, under the identification of example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}, sense an [[evolutionary vector field]] $v$ to the [[derivative]] of $P$ along the prolongation [[tangent vector field]] $\hat v $ of $v$.

In the case that $E$ and $V$ are [[trivial vector bundles]] over [[Minkowski spacetime]] with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu),  (\rho^b))$, respectively, then this is given by

$$
  ((\mathrm{D}P)(v))^b
   \;=\;
  \left(
     v^a \frac{\partial P^b}{\partial \phi^a}
     +
     \frac{d v^a}{d x^\mu}
     \frac{\partial P^b}{\partial \phi^a_{,\mu}}
     +
      \frac{d^2 v^a}{d x^\mu d x^\nu}
      \frac{\partial  P^b}{\partial \phi^a_{,\mu \nu}}
      +
     \cdots
  \right)
$$

This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent [[differential operator]] from the [[vertical tangent bundle]] $T_\Sigma E$ to $V$, namely a morphism of the form

$$
  \mathrm{D}_P
  \;\colon\;
  J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E
  \longrightarrow
  V
$$

in that

$$
  \label{FrechetDerivativeAsDifferentialOperatorEquality}
  \mathrm{D}_P(-,v)
  =
  \mathrm{D}P(v)
  =
  \hat v (P)
  \,.
$$

=--

([Olver 93, def. 5.24](#Olver93))

## Examples

+-- {: .num_example #DifferentialOperatorDerivativeOfLagrangianFunction}
###### Example
**([[evolutionary derivative]] of [[Lagrangian function]])**

Over a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] $\Sigma$, let $\mathbf{L} = L dvol \in \Omega^{p,0}_\Sigma(E)$ be a [[Lagrangian density]], with coefficient function regarded as a field-dependent section (def. \ref{FieldDependentSections}) of the [[trivial bundle|trivial]] [[real line bundle]]:

$$
  L \;\in \; \Gamma_{J^\infty_\Sigma}(E)(\Sigma \times \mathbb{R})
  \,,
$$

Then the [[formally adjoint differential operator]]

$$
  (\mathrm{D}_L)^\ast
    \;\colon\;
  J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast
  \longrightarrow
  T_\Sigma^\ast E
$$

of its [[evolutionary derivative]], def. \ref{FieldDependentDifferentialOperatorDerivative}, regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section

$$
  1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)
$$

is the [[Euler-Lagrange derivative]]

$$
  \delta_{EL}\mathbf{L}
  \;=\;
  \left(\mathrm{D}_{L}\right)^\ast(1)
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast)
   \simeq
   \Omega^{p+1,1}_\Sigma(E)_{source}
$$

via the identification from example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}.

=--

([Olver 93, above (5.80)](#Olver93))

## Properties

+-- {: .num_prop #EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}
###### Proposition
**([[Euler-Lagrange derivative]] is [[derivation]] via [[evolutionary derivatives]])**

Let $V \overset{vb}{\to} \Sigma$ be a [[vector bundle]] and write $V^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]].

For field-dependent sections (def. \ref{FieldDependentSections})

$$
  \alpha \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$

and

$$
  \beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast)
$$

we have that the [[Euler-Lagrange derivative]] of their canonical pairing to a [[smooth function]] on the [[jet bundle]] is the sum of the derivative of either one via the [[formally adjoint differential operator]] of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) of the other:

$$
  \delta_{EL}( \alpha \cdot \beta^\ast )
  \;=\;
  (\mathrm{D}_\alpha)^\ast(\beta^\ast)
  +
  (\mathrm{D}_{\beta^\ast})^\ast(\alpha)
$$

=--

([Olver 93 (5.80)](#Olver93))

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

It is sufficient to check this in [[local coordinates]]. By the [[product law]] for [[differentiation]] we have

$$
  \begin{aligned}
    \frac{
      \delta_{EL} \left(\alpha \cdot \beta^\ast \right)
    }
    {
      \delta \phi^a
    }
    & =
    \frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a}
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}}
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}}
    \right)
    -
    \cdots
    \\
    & =
    \phantom{+}
    \frac{\partial \alpha }{\partial \phi^a}
    \cdot \beta^\ast
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \alpha }{\partial \phi^a_{,\mu}}
       \cdot
       \beta^\ast
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \alpha  }{\partial \phi^a_{,\mu \nu}}
       \cdot
       \beta^\ast
    \right)
    -
    \cdots
    \\
    & \phantom{=}
    +
    \frac{\partial \beta^\ast }{\partial \phi^a}
    \cdot \alpha
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}}
       \cdot
       \alpha
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \beta^\ast  }{\partial \phi^a_{,\mu \nu}}
       \cdot
       \alpha
    \right)
    -
    \cdots
    \\
    & =
    (\mathrm{D}_\alpha)^\ast(\beta^\ast)
    +
    (\mathrm{D}_{\beta^\ast})^\ast(\alpha)
  \end{aligned}
$$

=--

+-- {: .num_prop #EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}
###### Proposition
**([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] over [[Minkowski spacetime]] and regard the [[Euler-Lagrange derivative]]

$$
  \delta_{EL}\mathbf{L}
  \;=\;
  \delta_{EL}L \wedge dvol_\Sigma
$$

as a field-dependent section of the [[vertical cotangent bundle]]

$$
  \delta_{EL}L
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
$$

as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative})  is [[formally self-adjoint differential operator|formally self-adjoint]]:

$$
  (D_{\delta_{EL}L})^\ast
  \;=\;
  D_{\delta_{EL}L}
$$

=--

([Olver 93, theorem 5.92](#Olver93))

+-- {: .num_prop #EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}
###### Proposition
**([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] over [[Minkowski spacetime]] and regard the [[Euler-Lagrange derivative]]

$$
  \delta_{EL}\mathbf{L}
  \;=\;
  \delta_{EL}L \wedge dvol_\Sigma
$$

as a field-dependent section of the [[vertical cotangent bundle]]

$$
  \delta_{EL}L
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
$$

as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative})  is [[formally self-adjoint differential operator|formally self-adjoint]]:

$$
  (D_{\delta_{EL}L})^\ast
  \;=\;
  D_{\delta_{EL}L}
$$

=--

([Olver 93, theorem 5.92](evolutionary+derivative#Olver93)) The following proof is due to [[Igor Khavkine]].

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

By definition of the [[Euler-Lagrange form]] we have

$$
  \frac{\delta_{EL} L }{\delta \phi^a}
  \delta \phi^a
  \,
  \wedge
  dvol_\Sigma
  \;=\;
  \delta L \,\wedge dvol_\Sigma \;+\; d(...)
  \,.
$$

Applying the [[variational derivative]] $\delta$ to both sides of this equation yields

$$
  \left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right)
  \wedge
  \delta \phi^a
  \,
  \wedge
  dvol_\Sigma
  \;=\;
  \underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma
  \;+\;
  d(...)
  \,.
$$

It follows that for $v,w$ any two [[evolutionary vector fields]] the contraction of their prolongations $\hat v$ and $\hat w$ into the [[variational differential form|differential 2-form]] on the left is

$$
  \left(
    \delta \frac{\delta_{EL} L }{\delta \phi^a}
    \wedge
    \delta \phi^a
  \right)(v,w)
  =
    w^a
    (\mathrm{D}_{\delta_{EL}})_a(v)
    -
    v^b(\mathrm{D}_{\delta_{EL}})_b(w)
  \,,
$$

by inspection of the definition of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) and their contraction into the form on the right is

$$
  \iota_{\hat v}
  \iota_{\hat w}
  d(...)
  \;=\;
  d(...)
$$

by the fact (prop. \ref{EvolutionaryVectorFieldProlongation}) that contraction with prolongations of evolutionary vector fields coommutes with the [[total spacetime derivative]].

Hence the last two equations combined give

$$
  w^a
  (\mathrm{D}_{\delta_{EL}})_a(v)
  -
  v^b(\mathrm{D}_{\delta_{EL}})_b(w)
  \;=\;
  d(...)
  \,.
$$

This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be [[formally self-adjoint differential operator]].

=--

## References

* {#Olver93} [[Peter Olver]], _Applications of Lie groups to Differential equations_, Graduate Texts in Mathematics, Springer 1993

* {#Barnich10} [[Glenn Barnich]], equation(3) of _A note on gauge systems from the point of view of Lie algebroids_, in P. Kielanowski, V. Buchstaber, A. Odzijewicz,
M.. Schlichenmaier, T Voronov,  (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) ([arXiv:1010.0899](https://arxiv.org/abs/1010.0899), [doi:/10.1063/1.3527427]( https://doi.org/10.1063/1.3527427))

* [[Igor Khavkine]], starting with p. 45 of _Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory_ ([arXiv:1211.1914](https://arxiv.org/abs/1211.1914))

[[!redirects evolutionary derivatives]]

[[!redirects evolutionary derivative of field-dependent section]]
[[!redirects evoolutionary derivatives of field-dependent sections]]


evolutionary vector field

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

# Contents
* table of contents
{: toc}

## Idea

In [[variational calculus]] an _evolutionary vector_ field is a particular type of [[vertical vector field]] on the infinite-order [[jet bundle]] $J^\infty_\Sigma(E)$ of a [[fiber bundle]] $E \overset{fb}{\to} \Sigma$, which may be interpreted as the generator of a transformation of the [[space of sections]] $\Gamma_\Sigma(E)$ of $E$.

Evolutionary vector fields describe [[infinitesimal]] [[symmetries]] of [[Lagrangian field theories]] in the formulation of the [[variational bicomplex]], and as such form an ingredient of [[Noether's theorem]].

## Definition

There are (at least) two different definitions, def. \ref{EvolutionaryVectorFieldPerceivedOnJetBundle} and def. \ref{AsGeneralizedVectorFieldOnE} below, which are equivalent (prop. \ref{EquivalenceOfDefinitions} below).

### As a vector field on the jet bundle

+-- {: .num_defn #EvolutionaryVectorFieldPerceivedOnJetBundle}
###### Definition
**(as a vector field on the [[jet bundle]])**

An **evolutionary vector field** is a [[vector field]] $v$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ which is [[vertical vector field|vertical]] with respect to the [[projection]] $\pi_\infty: J^\infty_\Sigma E \to \Sigma$ such that the following equivalent conditions hold:

1. The [[Lie derivative]] $\mathcal{L}_v$ satisfies

   1. $\mathcal{L}_v d = d \mathcal{L}_v$,

   1. $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$,

   where $d$ is the [[horizontal derivative]] on differential forms on $J^\infty E$, while $\delta$ is the [[vertical derivative]].

1. $v$ preserves the [[contact ideal]], i.e., for which $\mathcal{L}_v\theta$ is a [[contact form]] whenever $\theta$ is a [[contact form]].

We denote the subspace of evolutionary vector fields as $\mathfrak{X}_{ev}(J^\infty_\Sigma(E)) \subset \mathfrak{X}_V(J^\infty_\Sigma(E))$.

=--

Note that every evolutionary vector field is uniquely defined by its action on the [[pullbacks of functions]] on $E$ to $J^\infty_\Sigma(E)$. To formalize this, one may use the following notion.

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

Let $v \in \mathfrak{X}_{ev}(J^\infty E)$ be an evolutionary vector field. Its **characteristic** is the map $\pi_{\infty,0*} \circ v: J^\infty E \to V E$.

=--

### As a generalized vector field on the fiber bundle

Alternatively, a generalized vector field on the [[fiber bundle]] $E$ may be seen as a vector field on $E$ whose coefficients are functions on the [[jet bundle]] $J^\infty E$. More formally, it is a particular map from $J^\infty E$ to $T E$. Here we are interested only in a particular class of vertical vector fields.

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

An **evolutionary vector field** is a map $w \colon J^\infty_\Sigma(E) \to V_\Sigma E$ from the [[jet bundle]] $J^\infty E$ to the [[vertical tangent bundle]] $V_\Sigma E$, such that

$$
  \nu \circ w = \pi_{\infty,0}
  \phantom{AAAAAAA}
  \array{
    J^\infty_\Sigma(E)
    && \overset{w}{\longrightarrow} &&
    V_\Sigma E
    \\
    & {}_{\mathllap{\pi_{\infty,0}}}\searrow && \swarrow_{\mathllap{\nu}}
    \\
    && E
  }
  \,,
$$

where $\nu: V_\Sigma E \to E$ is the bundle map of $V_\Sigma E$ and $\pi_{\infty,0}: J^\infty_\Sigma(E) \to E$ is the target map of the jet bundle.

=--

Every evolutionary vector field has a unique prolongation to a vector field on $J^\infty_\Sigma(E)$.

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

Let $w$ be an evolutionary vector field in the sense of def. \ref{AsGeneralizedVectorFieldOnE}. Its **prolongation** to $J^\infty_\Sigma E$ is the unique vertical vector field $pr w$ on $J^\infty_\Sigma E$ such that

1. $w$ and $pr w$ agree on functions on $E$

1. $pr w$ preserves the contact ideal, i.e., $\mathcal{L}_v\theta$ is a [[contact form]] whenever $\theta$ is a [[contact form]].

=--

### Equivalence of the definitions

+-- {: .num_prop #EquivalenceOfDefinitions}
###### Proposition
**(equivalence of the two definitions)**

If $v$ is an evolutionary vector field in the sense of def. \ref{EvolutionaryVectorFieldPerceivedOnJetBundle}, then its characteristic $\pi_{\infty,0*} \circ v$ (def. \ref{Characteristic}) is an evolutionary vector field in the sense of def. \ref{AsGeneralizedVectorFieldOnE}.

Conversely, if $w$ is an evolutionary vector field in the sense of def. \ref{AsGeneralizedVectorFieldOnE}, then its prolongation $pr w$ (def. \ref{Prolongation}) is an evolutionary vector field in the sense of def. \ref{EvolutionaryVectorFieldPerceivedOnJetBundle}.

Furthermore, $pr (\pi_{\infty,0*} \circ v) = v$ and $\pi_{\infty,0*} \circ (pr w) = w$, so that this establishes a [[bijection]] between both definitions.

=--

## Related concepts

* [[source form]]

* [[evolutionary derivative]]

* [[point symmetry]]

* [[conserved current]]

* [[Noether's theorem]]

## References

See also

* {#AndersonTorre96} [[Ian Anderson]], [[Charles Torre]], section 2A of _Classification of Local Generalized Symmetries for  the Vacuum Einstein Equations_, Communications in Mathematical Physics 176, 479-539 (1996)

* [[Joseph Krasil'shchik]], Alexander Verbovetsky, section 1.4 of _Geometry of jet spaces and integrable systems_, J.Geom.Phys.61:1633-1674, 2011 ([arXiv:1002.0077](https://arxiv.org/abs/1002.0077))

[[!redirects evolutionary vector fields]]


ex falso quodlibet

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

* table of contents
{: toc}

**Ex falso quodlibet** is Latin for "from falsehood, anything". It is also called the **principle of explosion**.

In [[logic]] it refers to the principle that when a [[contradiction]] can be derived in a logical system, then any [[proposition]] follows.

In [[type theory]] it is the [[elimination rule]] of the [[empty type]] (see [there](empty+type#RecursionPrinciple)).

*Ex falso quodlibet* holds in many systems of logic, such as [[classical logic]] and [[intuitionistic logic]], but it fails to hold in [[paraconsistent logic]], which was devised to allow controlled inconsistency.

Variants of the principle's name include *ex falso sequitur quodlibet*, "from falsehood, anything follows", and *ex contradictione (sequitur) quodlibet*, "from contradiction, anything (follows)").

## Related concepts

* [[empty type]]

* [[excluded middle]]

* [[law of double negation]]

## References

* Wikipedia, *[Principle of explosion](https://en.wikipedia.org/wiki/Principle_of_explosion)*

* Pedro Francisco, Valencia Vizcaíno, *Relations between ex falso, tertium non datur, and double negation elimination* [[arXiv:1304.0272](https://arxiv.org/abs/1304.0272)]

[[!redirects principle of explosion]]


ex-space > history


exact (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Exact $(\infty,1)$-categories_ are the analog in [[(∞,1)-category theory]]  of [[exact categories]] in [[category theory]].

## Definition

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

Let $\mathcal{C}$ be an [[(∞,1)-category]]. This is called an _exact $(\infty,1)$-category_ if

1. $\mathcal{C}$ has a [[terminal object in an (∞,1)-category|terminal object]] and [[(∞,1)-fiber products]];

1. [[groupoid object in an (infinity,1)-category|groupoid objects]] in $\mathcal{C}$ are effective:

1. realization of groupoid objects is [[universal colimit|universal]].

=--

## Other notions of "exact"

There is another meaning for "exact (∞,1)-category" for which there is a [[Quillen Q-construction]] for exact (∞,1)-categories which allows to compute its [[algebraic K-theory]].

## Related concepts

* [[exact category]]

* [[finitely complete (infinity,1)-category]]

* [[regular (infinity,1)-category]], [[coherent (infinity,1)-category]], [[(infinity,1)-pretopos]]

* [[Quillen Q-construction]]

* [[theorem of the heart]]

## References

On exact $\infty$-categories

* [[Clark Barwick]], _On the Q construction for exact quasicategories_ ([arXiv:1301.4725](http://arxiv.org/abs/1301.4725))

and the [[theorem of the heart]]:

* [[Clark Barwick]], *On exact infinity-categories and the Theorem of the Heart*, Compositio Mathematica **151** (2015) 2160-2186 [[arXiv:1212.5232](https://arxiv.org/abs/1212.5232), [doi:10.1112/S0010437X15007447](https://doi.org/10.1112/S0010437X15007447)]

[[!redirects exact (infinit,1)-categories]]

[[!redirects exact (∞,1)-category]]
[[!redirects exact (∞,1)-categories]]


exact 2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _exact 2-catory_ is the analog in [[2-category theory]] of the notion of _[[exact category]]_ in [[category theory]].

## Definition

A [[regular category|regular 1-category]] is [[exact category|exact]] when every [[congruence]] has a kernel.  The definition for 2-categories is analogous.

+--{: .num_defn}
###### Definition
Let $-1\lt n\le 2$ be directed (see [[(n,r)-category]]).  A [[2-category]] is **$n$-exact** if it is [[regular 2-category|regular]] and every [[(n,r)-congruence]] is a kernel.

=--

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

Recall (from [[regular 2-category]]) that a 1-category is [[regular category|regular]] as a 1-category iff it is regular as a homwise-discrete 2-category.  However, a regular 1-category is [[exact category|exact]] as a 1-category precisely when it is _1-exact_ as a 2-category.  Since general 2-congruences in a 1-category are internal categories, they will obviously not all be kernels.  This is why we add a prefix to "exact:" the best notion for 2-categories, which we call "2-exact," is not a conservative extension of the established meaning of "exact" for 1-categories.

Likewise, it is unreasonable to expect a (2,1)-category to be any more than (2,1)-exact, or a (1,2)-category to be any more than (1,2)-exact.

=--

## Examples

* [[Cat]] is 2-exact.  Likewise, [[Gpd]] is (2,1)-exact, [[Pos]] is (1,2)-exact, and of course [[Set]] is 1-exact.

* Every regular (0,1)-category (that is, every [[meet-semilattice]]) is (0,1)-exact, and in fact even 2-exact, since there are no nontrivial congruences of any sort in a poset.

* If $K$ is 2-exact, then by the [[2-congruence|classification of congruences]], $gpd(K)$ is (2,1)-exact, $pos(K)$ is (1,2)-exact, $disc(K)$ is 1-exact, and $Sub(1)$ is (0,1)-exact.

* If $K$ is $n$-exact, then so is $K^{co}$, by the remarks about opposite [[2-congruences]].  For 1-categories and (2,1)-categories, of course, this is contentless, but for 2-categories and (1,2)-categories it is contentful.

## Properties

### Exact completion

See at _[[2-congruence]]_ the section _[Exactness](http://ncatlab.org/nlab/show/2-congruence#Exactness)_.

## References

The definition is originally due to

* [[Michael Shulman]], _[[michaelshulman:exact 2-category]]_

based on

* [[Ross Street]], _[[StreetCBS]]_

[[!redirects exact 2-categories]]

[[!redirects 2-exact 2-category]]
[[!redirects (2,1)-exact 2-category]]
[[!redirects 1-exact 2-category]]

[[!redirects (2,1)-exact (2,1)-category]]

[[!redirects 2-exact 2-categories]]
[[!redirects (2,1)-exact 2-categories]]
[[!redirects 1-exact 2-categories]]

[[!redirects (2,1)-exact (2,1)-categories]]


exact category

> The term **exact category** has several different meanings.  This page is about exact categories *in the sense of [Barr 1971](#Barr71)*, also called "Barr-exact categories" or "effective regular categories."  This is distinct from the notion of *[[Quillen exact category]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Regular and Exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

An **exact category** (in the sense of [Barr 1971](#Barr71)) is a [[regular category]] in which every [[congruence]] is a [[kernel pair]] (that is, every [[internalization|internal]] [[equivalence relation]] is _[[effective epimorphism|effective]]_).  These are also called **effective regular categories**.

\begin{remark}
See *[[familial regularity and exactness]]* for a generalization of exactness and its relationship to [[extensive category|extensivity]].
\end{remark}

## Properties

\begin{remark}
If $R\hookrightarrow X\times X$ is a congruence which is the kernel pair of $f:X \to Y$, then if $f = m \circ p$ is the [[image factorization]] of $f$, one can show that $p$ is a [[coequalizer]] of $R$.  Therefore, congruences have [[quotient object|quotient]]s in an exact category.

However, not _every_ parallel pair of morphisms need have a coequalizer, and there are also regular categories having all coequalizers which are not exact.
\end{remark}

\begin{remark}
The [[codomain fibration]] of an exact category is a [[stack]] ([[2-sheaf]]) for its [[regular category|regular topology]].  However, being exact is not a necessary condition for this to hold in a regular category; all that is required is that if $R\rightrightarrows A$ is a kernel pair, then so is $f^*R \rightrightarrows B$ for any $f\colon B\to A$.
\end{remark}

## Examples ##

* Any [[topos]] is an exact category.

* More generally, so is every [[pretopos]], by definition.

* Any [[regular category]] with [[van Kampen colimit|van Kampen]] [[quotients]] of [[congruences]] is exact. See at [[van Kampen colimit]] for a proof.

* Any category which is [[monad|monadic]] over a power of [[Set]] is exact. A proof may be found [here](/nlab/show/colimits+in+categories+of+algebras#exact).

* Any [[abelian category]] is exact. In fact an abelian category is precisely an exact [[additive category]].

* If $C$ is exact and $T$ is a [[Lawvere theory]], then the category $Mod(T, C)$ of $T$-models in $C$ is also exact. See Theorem 5.11 of Barr's [Exact Categories](#Barr).

* Any [[slice]] or [[co-slice]] of an exact category is also exact. (Source: [Borceux and Bourn](http://www.springer.com/us/book/9781402019616), Appendix A.)

* One can construct, for any regular category $C$, a "free" exact category $C_{ex/reg}$ on $C$ by adjoining formal quotient objects for congruences.  One way to define $C_{ex/reg}$ is as the (locally discrete) [[2-category]] whose objects are congruences in $C$ and whose morphisms are [[anafunctor|anafunctors]].  If $C$ is already exact, then $C_{ex/reg}$ is equivalent to $C$.  See [[regular and exact completions]].

* Similarly, one can construct the "free" exact category $C_{ex/lex}$ on any category $C$ with finite limits, or even with [[weak limit|weak finite limits]].  The exact categories of the form $C_{ex/lex}$ for a category $C$ with weak finite limits are exactly those which have [[projective object|enough (regular) projectives]]; in this case the projective objects are the retracts of objects of $C$ (Carboni-Vitale 1998).  See [[regular and exact completions]].

## Related concepts

* [[exact 2-category]]

* [[Puppe exact category]]

* [[Barr embedding theorem]]

* [[regular category]]

* [[exact functor]]

## References

* {#BarrGrilletvonOsdool} [[Michael Barr]],  [[Pierre Grillet]], [[Donovan van Osdol]], *Exact Categories and Categories of Sheaves*,  Lec. Notes in Math. **236**, Springer (1971) [[doi:10.1007/BFb0058579](https://link.springer.com/book/10.1007/BFb0058579)]

* {#Barr71}  [[Michael Barr]], _Exact categories_, in: *Exact categories and categories of sheaves*, Springer Lec. Notes in Math. __236__ (1971) 1-120 [[doi:10.1007/BFb0058580](http://dx.doi.org/10.1007/BFb0058580), [pdf](https://www.math.mcgill.ca/barr/papers/exact.pdf), [[Barr-ExactCategories.pdf:file]]]

* [[Aurelio Carboni]], [[Enrico Vitale]], _Regular and exact completions_, _JPAA_ 125, 1998.

* [[Francis Borceux]], [[Dominique Bourn]], *[[Borceux-Bourn|Mal'cev, protomodular, homological and semi-abelian categories]]*

[[!redirects exact categories]]

[[!redirects Barr-exact category]]
[[!redirects Barr-exact categories]]

[[!redirects Barr exact category]]
[[!redirects Barr exact categories]]

[[!redirects effective regular category]]
[[!redirects effective regular categories]]

[[!redirects Barr-exact]]


exact couple

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

#Contents#
* table of contents
{:toc}

## Idea

_Exact couples_ are a way to encode data that makes a [[spectral sequence]], specially adapted to the case that the underlying filtering along which the spectral sequence proceeds is induced from a tower of [[homotopy fibers]], such as a [[Postnikov tower]] or [[Adams tower]] (see also at _[[Adams spectral sequence]]_).

## Definition

### Exact couples

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

Given an [[abelian category]] $\mathcal{C}$, an **[[exact couple]]** in $\mathcal{C}$ is a cyclic [[long exact sequence]] of three [[morphisms]] among two [[objects]] of the form

$$
  \cdots \stackrel{k}{\longrightarrow} E \overset{j}{\to} D \overset{\varphi}{\longrightarrow} D \overset{k}{\to} E \overset{j}{\longrightarrow} \cdots
  \,.
$$

=--

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

This being cyclic, it is usually depicted as a triangle

$$
  \array{
    D && \stackrel{\varphi}{\longrightarrow} && D
    \\
    & {}_{\mathllap{j}}\nwarrow && \swarrow_{\mathrlap{k}}
    \\
    &&
    E
  }
$$

=--

The archetypical example from which this and the following definition draw their meaning is example \ref{ExactCoupleOfATower} below.

### Spectral sequences from exact couples
 {#SpectralSequencesFromExactCouples}

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

A _cohomology [[spectral sequence]]_ $\{E_r^{p,q}, d_r\}$ is

1. a sequence $\{E_r^{\bullet,\bullet}\}$ $r \in \mathbb{Z}$, $r \geq 2$ of [[bigraded object|bigraded]] [[abelian groups]];

1. a sequence of [[differentials]] $\{d_r \colon E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}$

such that

* $H_{r+1}^{\bullet,\bullet}$ is the [[cochain cohomology]] of $d_r$:, i.e. $E_{r+1}^{\bullet, \bullet} = H(E_r^{\bullet,\bullet},d_r)$.

Given a $\mathbb{Z}$-[[graded abelian group]]_ $C^\bullet$ equipped with a decreasing [[filtration]]

$$
  C^\bullet \supset \cdots \supset F^s C^\bullet \supset F^{s+1} C^\bullet \supset \cdots \supset 0$$

such that

$$
  C^\bullet = \underset{s}{\cup} F^s C^\bullet \;\;\;\; and \;\;\;\; 0 = \underset{s}{\cap} F^s C^\bullet
$$

then the spectral sequence is said to _converge_ to  $C^\bullet$, denoted,

$$
  E_2^{\bullet,\bullet} \Rightarrow C^\bullet
$$

if

1.  in each bidegree $(s,t)$ the sequence $\{E_r^{s,t}\}_r$ eventually becomes constant on a group

    $E_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t}$;

1. $E_\infty^{\bullet,\bullet}$ is the [[associated graded]] of the filtered $C^\bullet$ in that

   $E_\infty^{s,t} \simeq F^s C^{s+t} / F^{s+1}C^{s+t}$.

The converging spectral sequence is called _multiplicative_ if

1. $\{E_2^{\bullet,\bullet}\}$ is equipped with the structure of a [[bigraded object]] [[associative algebra|algebra]];

1. $F^\bullet C^\bullet$ is equipped with the structure of a filtered [[graded algebra]] ($F^p C^k \cdot F^q C^l \subset F^{p+q} C^{k+l}$);

such that

1. each $d_{r}$ is a [[derivation]] with respect to the (induced) algebra structure on ${E_r^{\bullet,\bullet}}$, graded of degree 1 with respect to total degree;

1. the multiplication on $E_\infty^{\bullet,\bullet}$ is compatible with that on $C^\bullet$.

=--

+-- {: .num_defn #ExactCoupleAndDerivedExactCouple}
###### Definition
**(derived exact couples)**

An _[[exact couple]]_ is three [[homomorphisms]] of [[abelian groups]] of the form

$$
  \array{
    D && \stackrel{g}{\longrightarrow} && D
    \\
    & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h}}
    \\
    && E
  }
$$

such that the [[image]] of one is the [[kernel]] of the next.

$$
  im(h) = ker(f)\,,\;\;\; im(f) = ker(g)\,, \;\;\; im(g) = ker(h)
  \,.
$$

Given an exact couple, then its _derived exact couple_ is

$$
  \array{
    im(g) && \stackrel{g}{\longrightarrow} && im(g)
    \\
    & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h \circ g^{-1}}}
    \\
    && H(E, h \circ f)
  }
  \,.
$$

=--

Here and in the following we write $g^{-1}$ etc. for the operation of choosing a [[preimage]] under a given function $g$. In each case it is left implicit that the given expression is independent of which choice is made.

+-- {: .num_prop #CohomologicalSpectralSequenceOfAnExactCouple}
###### Proposition
**(cohomological spectral sequence of an exact couple)**

Given an exact couple, def. \ref{ExactCoupleAndDerivedExactCouple},

$$
  \array{
    D_1 && \stackrel{g_1}{\longrightarrow} && D_1
    \\
    & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}}
    \\
    && E_1
  }
$$

its derived exact couple

$$
  \array{
    D_2 && \stackrel{g_2}{\longrightarrow} && D_2
    \\
    & {}_{\mathllap{f_2}}\nwarrow && \swarrow_{\mathrlap{h_2}}
    \\
    && E_2
  }
$$

is itself an exact couple. Accordingly there is induced a sequence of exact couples

$$
  \array{
    D_r && \stackrel{g_r}{\longrightarrow} && D_r
    \\
    & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}}
    \\
    && E_r
  }
  \,.
$$

If the abelian groups $D$ and $E$ are equipped with [[bigraded object|bigrading]] such that

$$
  deg(f) = (0,0)\,,\;\;\;\; deg(g) = (-1,1)\,,\;\;\; deg(h) = (1,0)
$$

then $\{E_r^{\bullet,\bullet}, d_r\}$ with

$$
  \begin{aligned}
    d_r & \coloneqq h_r \circ f_r
     \\
     & = h \circ g^{-r+1} \circ f
  \end{aligned}
$$

is a cohomological spectral sequence, def. \ref{CohomologySpectralSequence}.

(As before in prop. \ref{CohomologicalSpectralSequenceOfAnExactCouple}, the notation $g^{-n}$ with $n \in \mathbb{N}$ denotes the function given by choosing, on representatives, a [[preimage]] under $g^n = \underset{n\;times}{\underbrace{g \circ \cdots \circ g \circ g}}$, with the implicit claim that all possible choices represent the same equivalence class.)

If for every bidegree $(s,t)$ there exists $R_{s,t} \gg 1$ such that for all $r \geq R_{s,t}$

1. $g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1}$;

1. $g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}$

then this spectral sequence converges to the [[inverse limit]] group

$$
  G^\bullet \coloneqq \underset{}{\lim}
  \left(
    \cdots \stackrel{g}{\to} D^{s,\bullet-s} \stackrel{g}{\longrightarrow} D^{s-1, \bullet - s + 1}
   \stackrel{g}{\to}
   \cdots
  \right)
$$

filtered by

$$
  F^p G^\bullet \coloneqq ker(G^\bullet \to D^{p-1, \bullet - p+1})
  \,.
$$

=--

(e.g. [Kochmann 96, lemma 2.6.2](#Kochmann96))

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

We check the claimed form of the $E_\infty$-page:

Since $ker(h) = im(g)$ in the exact couple, the kernel

$$
  ker(d_{r-1}) \coloneqq ker(h \circ g^{-r+2} \circ f)
$$

consists of those elements $x$ such that $g^{-r+2} (f(x)) = g(y)$, for some $y$, hence

$$
  ker(d_{r-1})^{s,t} \simeq f^{-1}(g^{r-1}(D^{s+r-1,t-r+1}))
  \,.
$$

By assumption there is for each $(s,t)$ an $R_{s,t}$ such that for all $r \geq R_{s,t}$ then $ker(d_{r-1})^{s,t}$ is independent of $r$.

Moreover, $im(d_{r-1})$ consists of the image under $h$ of those $x \in D^{s-1,t}$ such that $g^{r-2}(x)$ is in the image of $f$, hence (since $im(f) = ker(g)$ by exactness of the exact couple) such that $g^{r-2}(x)$ is in the kernel of $g$, hence such that $x$ is in the kernel of $g^{r-1}$. If $r \gt R$ then by assumption $g^{r-1}|_{D^{s-1,t}} = 0$ and so then $im(d_{r-1}) = im(h)$.

(Beware this subtlety: while $g^{R_{s,t}}|_{D^{s-1,t}}$ vanishes by the convergence assumption, the expression $g^{R_{s,t}}|_{D^{s+r-1,t-r+1}}$ need not vanish yet. Only the higher power $g^{R_{s,t}+ R_{s+1,t+2}+2}|_{D^{s+r-1,t-r+1}}$ is again guaranteed to vanish. )

{#InfinityPage} It follows that

$$
  \begin{aligned}
    E_\infty^{p,n-p}
    & =
    ker(d_R)/im(d_R)
    \\
    &
    \simeq
    f^{-1}(im(g^{R-1}))/im(h)
    \\
    &
    \simeq
    f^{-1}(im(g^{R-1}))/ker(f)
    \\
    &
    \underoverset{\simeq}{f}{\longrightarrow}
    im(g^{R-1}) \cap im(f)
    \\
    &
    \simeq
    im(g^{R-1}) \cap ker(g)
  \end{aligned}
$$

where in last two steps we used once more the exactness of the exact couple.

{#InfinityPageIsSubgroupOfImageOfFirstPageUnderf}(Notice that the above equation means in particular that the $E_\infty$-page is a sub-group of the image of the $E_1$-page under $f$.)

The last group above is that of elements $x \in G^n$ which map to zero in $D^{p-1,n-p+1}$ and where two such are identified if they agree in $D^{p,n-p}$, hence indeed

$$
  E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n
  \,.
$$

=--

## Examples

### Exact couple of a tower of (co)-fibrations
 {#ExactCoupleOfATowerOfFibrations}

...[[spectral sequence of a tower of fibrations]]...

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

A [[filtered spectrum]] is a [[spectrum]] $X$ equipped with a sequence $X_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Spectra$  of spectra of the form

$$
  \cdots
   \longrightarrow
  X_3
    \stackrel{f_2}{\longrightarrow}
  X_2
    \stackrel{f_1}{\longrightarrow}
  X_1 \stackrel{f_0}{\longrightarrow} X_0 = X
  \,.
$$

=--

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

More generally a [[filtered object in an (infinity,1)-category|filtering]] on an object $X$ in (stable or not) [[homotopy theory]] is a $\mathbb{Z}$-graded sequence $X_\bullet $ such that $X$ is the [[homotopy colimit]]  $X\simeq \underset{\longrightarrow}{\lim} X_\bullet$. But for the present purpose we stick with the simpler special case of def. \ref{FilteredSpectrum}.

=--

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

There is _no_ condition on the [[morphisms]] in def. \ref{FilteredSpectrum}. In particular, they are _not_ required to be [[n-monomorphisms]] or [[n-epimorphisms]] for any $n$.

On the other hand, while they are also not explicitly required to have a presentation by [[cofibrations]] or [[fibrations]], this follows automatically: by the existence of [[model structures for spectra]], every filtering on a spectrum is equivalent to one in which all morphisms are represented by [[cofibrations]] or by [[fibrations]].

This means that we may think of a filtration on a spectrum $X$ in the sense of def. \ref{FilteredSpectrum} as equivalently being a [[tower of fibrations]] over $X$.

=--

The following remark \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum} unravels the structure encoded in a filtration on a spectrum, and motivates the concepts of [[exact couples]] and their [[spectral sequences]] from these.

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

Given a [[filtered spectrum]] as in def. \ref{FilteredSpectrum},
write $A_k$ for the [[homotopy cofiber]] of its $k$th stage, such as to obtain the diagram

$$
  \array{
   \cdots
     &\stackrel{}{\longrightarrow}&
   X_3
     &\stackrel{f_2}{\longrightarrow}&
   X_2
     &\stackrel{f_2}{\longrightarrow} &
   X_1
     &\stackrel{f_1}{\longrightarrow}&
   X
   \\
   && \downarrow && \downarrow && \downarrow && \downarrow
   \\
   && A_3 && A_2 && A_1 && A_0
  }
$$

where each stage

$$
 \array{
   X_{k+1} &\stackrel{f_k}{\longrightarrow}& X_k
   \\
   && \downarrow^{\mathrlap{cofib(f_k)}}
   \\
   && A_k
 }
$$

is a [[homotopy fiber sequence]].

To break this down into invariants, apply the [[stable homotopy groups]]-[[functor]]. This yields a diagram of $\mathbb{Z}$-[[graded abelian groups]] of the form

$$
  \array{
   \cdots
     &\stackrel{}{\longrightarrow}&
   \pi_\bullet(X_3)
     &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}&
   \pi_\bullet(X_2)
     &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} &
   \pi_\bullet(X_1)
     &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}&
   \pi_\bullet(X_0)
   \\
   && \downarrow && \downarrow && \downarrow && \downarrow
   \\
   && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0)
  }
  \,.
$$

Here each hook at stage $k$ extends to a [[long exact sequence of homotopy groups]] via [[connecting homomorphisms]] $\delta_\bullet^k$

$$
  \cdots
    \to
  \pi_{\bullet+1}(A_k)
    \stackrel{\delta_{\bullet+1}^k}{\longrightarrow}
  \pi_\bullet(X_{k+1})
    \stackrel{\pi_\bullet(f_k)}{\longrightarrow}
  \pi_\bullet(X_k)
    \stackrel{}{\longrightarrow}
  \pi_\bullet(A_k)
   \stackrel{\delta_\bullet^k}{\longrightarrow}
  \pi_{\bullet-1}(X_{k+1})
  \to
  \cdots
  \,.
$$

If we understand the [[connecting homomorphism]]

$$
  \delta_k \colon \pi_\bullet(A_k) \longrightarrow \pi_\bullet(X_{k+1})
$$

as a morphism of degree -1, then all this information fits into one diagram of the form

$$
  \array{
   \cdots
     &\stackrel{}{\longrightarrow}&
   \pi_\bullet(X_3)
     &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}&
   \pi_\bullet(X_2)
     &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} &
   \pi_\bullet(X_1)
     &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}&
   \pi_\bullet(X_0)
   \\
   &&
   \downarrow &{}_{\mathllap{\delta_2}}\nwarrow &
   \downarrow &{}_{\mathllap{\delta_1}}\nwarrow &
   \downarrow &{}_{\mathllap{\delta_0}}\nwarrow
   & \downarrow
   \\
   && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0)
  }
  \,,
$$

where each triangle is a rolled-up incarnation of a [[long exact sequence of homotopy groups]] (and in particular is _not_ a commuting diagram!).

If we furthermore consider the [[bigraded object|bigraded]] [[abelian groups]] $\pi_\bullet(X_\bullet)$ and $\pi_\bullet(A_\bullet)$, then this information may further be rolled-up to a single diagram of the form

$$
  \array{
     \pi_\bullet(X_\bullet)
       & \stackrel{\pi_\bullet(f_\bullet)}{\longrightarrow} &
     \pi_\bullet(X_\bullet)
     \\
       & {}_{\mathllap{\delta}}\nwarrow
       & \downarrow^{\mathrlap{\pi_\bullet(cofib(f_\bullet))}}
     \\
     &&
     \pi_\bullet(A_\bullet)
  }
$$

where the morphisms $\pi_\bullet(f_\bullet)$, $\pi_\bullet(cofib(f_\bullet))$ and $\delta$ have bi-degree $(0,-1)$, $(0,0)$ and $(-1,1)$, respectively.

Here it is convenient to shift the bigrading, equivalently, by setting

$$
  \mathcal{D}^{s,t} \coloneqq \pi_{t-s}(X_s)
$$

$$
  \mathcal{E}^{s,t} \coloneqq \pi_{t-s}(A_s)
  \,,
$$

because then $t$ counts the cycles of going around the triangles:

$$
  \cdots
   \to
  \mathcal{D}^{s+1,t+1}
    \stackrel{\pi_{t-s}(f_s)}{\longrightarrow}
  \mathcal{D}^{s,t}
    \stackrel{\pi_{t-s}(cofib(f_s))}{\longrightarrow}
  \mathcal{E}^{s,t}
    \stackrel{\delta_s}{\longrightarrow}
  \mathcal{D}^{s+1,t}
    \to
  \cdots
$$

Data of this form is called an _[[exact couple]]_, def. \ref{ExactCouple} below.

=--

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

An _unrolled [[exact couple]]_ (of Adams-type) is a diagram of [[abelian groups]] of the form

$$
  \array{
     \cdots
       &\stackrel{}{\longrightarrow}&
     \mathcal{D}^{3,\bullet}
       &\stackrel{i_2}{\longrightarrow}&
     \mathcal{D}^{2,\bullet}
       &\stackrel{i_1}{\longrightarrow} &
     \mathcal{D}^{1,\bullet}
       &\stackrel{i_0}{\longrightarrow}&
     \mathcal{D}^{0,\bullet}
     \\
     &&
     \downarrow^{\mathrlap{}}  &{}_{\mathllap{k_2}}\nwarrow &
     {}^{\mathllap{j_2}}\downarrow &{}_{\mathllap{k_1}}\nwarrow &
     {}^{\mathllap{j_1}}\downarrow &{}_{\mathllap{k_0}}\nwarrow
     & {}_{\mathllap{j_0}}\downarrow
     \\
     && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet}
     && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet}
   }
$$

such that each triangle is a rolled-up [[long exact sequence]] of [[abelian groups]] of the form

$$
  \cdots
   \to
  \mathcal{D}^{s+1,t+1}
    \stackrel{i_s}{\longrightarrow}
  \mathcal{D}^{s,t}
    \stackrel{j_s}{\longrightarrow}
  \mathcal{E}^{s,t}
    \stackrel{k_s}{\longrightarrow}
  \mathcal{D}^{s+1,t}
    \to
  \cdots
  \,.
$$

=--

The collection of this "un-rolled" data into a single diagram of [[abelian groups]] is called the corresponding _[[exact couple]]_.

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

An _[[exact couple]]_ is a [[diagram]] (non-commuting) of [[abelian groups]] of the form

$$
  \array{
    \mathcal{D}
    &\stackrel{i}{\longrightarrow}&
    \mathcal{D}
    \\
    & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}}
    \\
    && \mathcal{E}
  }
  \,,
$$

such that this is [[exact sequence]] exact in each position, hence such that the [[kernel]] of every [[morphism]] is the [[image]] of the preceding one.

=--

The concept of exact couple so far just collects the sequences of long exact sequences given by a filtration. Next we turn to extracting information from this sequence of sequences.

+-- {: .num_remark #Observingd1}
###### Remark

The sequence of long exact sequences in remark \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum} is inter-locking, in that every $\pi_{t-s}(X_s)$ appears _twice_:

$$
  \array{
    && & \searrow && \nearrow
    \\
    && && \pi_{t-s-1}(X_{s+1})
    \\
    && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow
    && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}}
    && && && \nearrow
    \\
    && \pi_{t-s}(A_s) && \underset{def: \;\;d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1})
    && \stackrel{def: \; d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2})
    \\
    & \nearrow && && && {}_{\mathllap{\delta_{t-s-1}^{s+1}}}\searrow
    && \nearrow_{\mathrlap{\pi_{t-s-2}(cofib(f_{s+2}))}}
    \\
    && && && && \pi_{t-s-2}(X_{s+2})
    \\
    && && && & \nearrow && \searrow
  }
$$

This gives rise to the horizontal composites $d_1^{s,t}$, as show above, and by the fact that the diagonal sequences are long exact, these are differentials: $d_1^2 = 0$, hence give a [[chain complex]]:

$$
  \array{
    \cdots & \stackrel{}{\longrightarrow}
    && \pi_{t-s}(A_s) && \overset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1})
    && \stackrel{d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2})
    &&\longrightarrow & \cdots
  }
  \,.
$$

We read off from the interlocking long exact sequences what these differentials _mean_: an element $c \in \pi_{t-s}(A_s)$ lifts to an element $\hat c \in \pi_{t-s-1}(X_{s+2})$ precisely if $d_1 c = 0$:

$$
  \array{
    &\hat c \in & \pi_{t-s-1}(X_{s+2})
    \\
    && & \searrow^{\mathrlap{\pi_{t-s-1}(f_{s+1})}}
    \\
    && && \pi_{t-s-1}(X_{s+1})
    \\
    && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow
    && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}}
    \\
    & c \in  & \pi_{t-s}(A_s) && \underset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1})
  }
$$

This means that the [[cochain cohomology]] of the complex $(\pi_{\bullet}(A_\bullet), d_1)$ produces elements of $\pi_\bullet(X_\bullet)$ and hence of $\pi_\bullet(X)$.

In order to organize this observation, notice that in terms of the exact couple of remark \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum}, the differential

$$
  d_1^{s,t}  \;\coloneqq \; \pi_{t-s-1}(cofib(f_{s+1})) \circ \delta_{t-s}^s
$$

is a component of the composite

$$
  d \coloneqq j \circ k
  \,.
$$

=--

Some terminology:

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

Given an exact couple, def. \ref{ExactCouple},

$$
  \array{
    \mathcal{D}^{\bullet,\bullet}
    &\stackrel{i}{\longrightarrow}&
    \mathcal{D}^{\bullet,\bullet}
    \\
    & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}}
    \\
    && \mathcal{E}^{\bullet,\bullet}
  }
$$

its _page_ is the [[chain complex]]

$$
  (E^{\bullet,\bullet}, d \coloneqq j \circ  k)
  \,.
$$

=--

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

Given an exact couple, def. \ref{ExactCouple}, then the induced _derived exact couple_ is the diagram

$$
  \array{
    \widetilde {\mathcal{D}}
    &\stackrel{\tilde i}{\longrightarrow}&
    \widetilde {\mathcal{D}}
    \\
    & {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}}
    \\
    && \widetilde{\mathcal{E}}
  }
$$

with

1. $\tilde{\mathcal{E}} \coloneqq ker(d)/im(d)$;

1. $\tilde {\mathcal{D}} \coloneqq im(i)$;

1. $\tilde i \coloneqq i|_{im(i)}$;

1. $\tilde j \coloneqq j \circ (im(i))^{-1}$;

1. $\tilde k \coloneqq k|_{ker(d)}$.

=--

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

A derived exact couple, def. \ref{DerivedExactCouple},
is again an exact couple, def. \ref{ExactCouple}.

=--

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

Given an exact couple, def. \ref{ExactCouple},
then the induced [[spectral sequence]], def. \ref{SpectralSequence}, is the sequence of pages, def. \ref{PageOfAnExactCouple}, of the induced sequence of derived exact couples, def. \ref{DerivedExactCouple}, prop. \ref{DerivedExactCoupleIsExactCouple}.

=--

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

Consider a [[filtered spectrum]], def. \ref{FilteredSpectrum},

$$
  \array{
   \cdots
     &\stackrel{}{\longrightarrow}&
   X_3
     &\stackrel{f_2}{\longrightarrow}&
   X_2
     &\stackrel{f_2}{\longrightarrow} &
   X_1
     &\stackrel{f_1}{\longrightarrow}&
   X
   \\
   && \downarrow && \downarrow && \downarrow && \downarrow
   \\
   && A_3 && A_2 && A_1 && A_0
  }
$$

and its induced [[exact couple]] of [[stable homotopy groups]], from remark \ref{UnrolledExactCoupleOfAFiltrationOnASpectrum}

$$
  \array{
    \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D}
    \\
    &{}_{\mathllap{k}}\nwarrow& \downarrow^{\mathrlap{j}}
    \\
    && \mathcal{E}
  }
  \;\;\;\;\;\,\;\;\;\;\;\;
  \array{
    \mathcal{D} &\stackrel{(-1,-1)}{\longrightarrow}& \mathcal{D}
    \\
    &{}_{\mathllap{(1,0)}}\nwarrow& \downarrow^{\mathrlap{(0,0)}}
    \\
    && \mathcal{E}
  }
$$

with bigrading as shown on the right.

<div style="float:right;margin:0 10px 10px 0;">
<img src="http://ncatlab.org/nlab/files/adamstypedifferentials.jpg" width="360" >
</div>

As we pass to derived exact couples, by def. \ref{DerivedExactCouple},
the bidegree of $i$ and $k$ is preserved, but that of $j$ increases by $(1,1)$ in each step, since

$$
  deg(\tilde j) = deg( j \circ im(i)^{-1}) = deg(j) + (1,1)
  \,.
$$

Therefore the induced [[spectral sequence]] has differentials of the form

$$
  d_r \;\colon\; \mathcal{E}_r^{s,t} \longrightarrow \mathcal{E}_r^{s+r, t+r-1}
  \,.
$$

=--

## References

The original article is

*  [[W. S. Massey]], _Exact Couples in Algebraic Topology (Parts I and II)_, Annals of Mathematics, Second Series, Vol. 56, No. 2 (Sep., 1952), pp. 363-396 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/massey6.pdf))

also

* [[Beno Eckmann]], [[Peter Hilton]], _Exact couples in an abelian category_, Journal of Algebra Volume 3, Issue 1, January 1966, Pages 38-87 ([jorunal](http://www.sciencedirect.com/science/article/pii/0021869366900196))

A class of examples leading to what later came to be known as the [[Atiyah-Hirzebruch spectral sequence]] is discussed in section XV.7 of

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press (1956)

Textbook accounts include

* [[Frank Adams]], part III, section 7 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochmann96} [[Stanley Kochmann]], section 2.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#McCleary01} [[John McCleary]], section 2.2 (from p. 37 on) in _[A user's guide to spectral sequences](https://pages.vassar.edu/mccleary/books/users-guide-to-spectral-sequences/)_, Cambridge University Press, 2001

* [[Charles Weibel]], section 5.9 _[[An Introduction to Homological Algebra]]_

Another review with an eye towards application to the [[Adams spectral sequence]] is in

* [[Doug Ravenel]], chapter 2, section 1 of _[[Complex cobordism and stable homotopy groups of spheres]]_

[[!redirects exact couples]]

[[!redirects spectral sequence of an exact couple]]
[[!redirects spectral sequences of exact couples]]


exact form

A [[differential form]] $\omega \in \Omega^n(X)$ is _exact_ if there is a differential form $\lambda \in \Omega^{n-1}(X)$ such that $d_{dR}\lambda = \omega$.


exact functor

> This entry is about the concept of "exact functors" in plain [[category theory]]. For the different concept of that name in [[triangulated category]] theory see at _[[triangulated functor]]_.

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

#Contents#
* table of contents
{: toc}

## Idea

A left/right _exact functor_ is a [[functor]] that [[preserved limit|preserves]] [[finite limits]]/finite [[colimits]].

The term originates in [[homological algebra]], see remark \ref{OriginOfTerminology} below, where a central role is played by [[exact sequences]] (originally of [[modules]], more generally in any [[abelian category]]) and the fact that various [[functors]] preserve or destroy exactness of sequences to some extent gave vital information on those functors.

In this context, one says that an **exact functor** is one that preserves exact sequences.  However, many functors are only "exact on one side or the other".  For instance, for all modules $M$ and [[short exact sequences]] $0 \to A \to B \to C \to 0$  of modules (over some ring $R$), the sequence

$$0 \to Mod_R(M, A) \to Mod_R(M,B) \to Mod_R(M,C)$$

is exact -- but note that there is no 0 on the right hand.  Thus $F(-) = Mod_R(M,-)$ converts an exact sequence into a _left_ exact sequence; such a functor is called a **left exact functor**.  Dually, one has **right exact functors**.

It is easy to see that an [[additive functor]] between [[additive categories]] is left exact in this sense if and only if it preserves [[finite limits]].

Since merely preserving left exact sequences does not require a functor to be additive, in a non-additive context one defines a **left exact functor** to be one which preserves finite limits, and dually. Below we give the [general definition](#Definition) and then discuss the relation to the concept in homological algebra in the section [Properties - On abelian categories](#OnAbelianCategories).

## Definition
 {#Definition}

A [[functor]] between [[finitely complete categories]] is called __left exact__ (or [[flat functor|flat]]) if it preserves [[finite limits]].  Dually, a functor between finitely cocomplete categories is called __right exact__ if it preserves [[finite colimits]].  A functor is called __exact__ if it is both left and right exact.

Specifically, [[Ab]]-[[enriched functor]]s between [[abelian categories]] are exact if they preserve [[exact sequence]]s.

## Properties

### Characterisations

#### General

+-- {: .num_prop}
###### Proposition
* A [[functor]] $F : C \to D$ between finitely cocomplete categories is right exact if and only if for all [[objects]] $d \in D$ the [[comma category]] $F/d$ is [[filtered category|filtered]].

* A [[functor]] $F : C \to D$ between finitely complete categories is left exact if and only if for all [[objects]] $d \in D$ the [[opposite category|opposite]] [[comma category]] $(d/F)^{op}$ is [[filtered category|filtered]].
=--

In other language, this says that a functor between finitely complete categories is left exact if and only if it is (representably) [[flat functor|flat]].  Conversely, one can show that a representably flat functor preserves all finite limits that exist in its domain.

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

A functor between categories with finite limits preserves [[finite limit]]s if and only if:

* it preserves [[terminal objects]], binary [[product]]s, and [[equalizer]]s; or

* it preserves [[terminal object]]s and binary [[pullback]]s.

=--

Since these conditions frequently come up individually, it may be worthwhile listing them separately:

* $F: C \to D$ **preserves terminal objects** if $F(t_C)$ is terminal in $D$ whenever $t_C$ is terminal in $C$;

* $F: C \to D$ **preserves binary products** if the pair of maps
$$F(c) \stackrel{F(\pi_1)}{\leftarrow} F(c \times d) \stackrel{F(\pi_2)}{\to} F(d)$$
exhibits $F(c \times d)$ as a product of $F(c)$ and $F(d)$, where $\pi_1: c \times d \to c$ and $\pi_2: c \times d \to d$ are the product projections in $C$;

* $F: C \to D$ **preserves equalizers** if the map
$$F(i): F(e) \to F(c)$$
is the equalizer of $F(f), F(g): F(c) \stackrel{\to}{\to} F(d)$, whenever $i: e \to c$ is the equalizer of $f, g: c \stackrel{\to}{\to} d$ in $C$.

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

Some author use the term "left exact" when $C$ does not have all [[finite limits]], defining it to mean a [[flat functor]].

'Left exact' is sometimes abbreviated **lex**.  In particular, [[Lex]] is the 2-category of categories with [[finite limits]] and lex functors.  See also [[continuous functor]].   Similarly, but more rarely, 'right exact' is sometimes abbreviated as **rex**.

Left exact functors correspond to [[pro-representable functors]], provided some smallness conditions are satisfied.

=--

#### Between categories of modules

Right exact functors between [[categories of modules]] are characterized by the _[[Eilenberg-Watts theorem]]_. See there for more details.

### On abelian categories / in homological algebra
 {#OnAbelianCategories}

In the context of [[homological algebra]], the notion of left/right exact functors is considered specifically in [[abelian categories]]. In this context the above formulation is equivalently formulated in terms of the behaviour of the functor on [[short exact sequences]]. We now discuss this case.

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

A functor $F : C \to D$ between [[abelian categories]] is left exact if and only if it preserves [[direct sums]] and [[kernels]].

A functor $F : C \to D$ between [[abelian categories]] is right exact if and only if it preserves [[direct sums]] and [[cokernels]].

=--

+-- {: .num_cor #ExactFunctorPreservesExactSequences}
###### Corollary

In particular for $0 \to A \to B \to C \to 0$ an [[exact sequence]] in the abelian category $C$, we have that

* if $F$ is left exact then

  $$
    0 \to F(A) \to F(B) \to F(C)
  $$

  is an exact sequence in $D$;

* if $F$ is right exact then

  $$
    F(A) \to F(B) \to F(C) \to 0
  $$

  is an exact sequence in $D$;

* if $F$ is exact then

  $$
    0 \to F(A) \to F(B) \to F(C) \to 0
  $$

  is an exact sequence in $D$.

Also: if $F$ is exact then it preserves [[chain homology]].

=--

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

We discuss the first case. The second is formally dual. The third combines the two cases.

For the first case notice that $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ being an [[exact sequence]] is equivalent to $i$ being a [[monomorphism]] and $p$ being an [[epimorphism]], hence to $0 \to A$ being the [[kernel]] of $i$, $i$ being the kernel of $p$ and $C \to 0$ being the [[cokernel]] of $p$. Since the functor $F$ is assumed to preserve this kernel-property, but not the cokernel property, it follows that $F(0) \to F(A)$ is the kernel of $F(A) \stackrel{F(i)}{\to} F(B)$, but not more than that. This means that

$$
  0 \to F(A) \to F(B) \to F(C)
$$

is an exact sequence, as claimed.

=--

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

The properties of corollary \ref{ExactFunctorPreservesExactSequences} explain the "left"/"right"-terminology: a _left_ exact functor preserves exactness of sequences to the left of a morphism (only), while a _right_ exact functor preserves exactness to the right.

=--

## Related concepts

* [[LexCat]]

* [[derived functor]]

* [[exact (∞,1)-functor]]

* [[satellite]]

* [[exact category]]

## References

Early use of _left exact_ and _exact_:

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

*  [[A. Grothendieck]], 1959, _Technique de descente et théorèmes d'existence en géométrie algèbrique. II. Le théorème d'existence en théorie formelle des modules_, in Séminaire Bourbaki, Vol. 5 , Exp. No. 195, 369 – 390,
Soc. Math. France [Numdam](http://www.numdam.org/numdam-bin/item?id=SB_1958-1960__5__369_0), Paris.

General discussion

* [[Francis Borceux]], Section 6.1 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 3.3 in: _[[Categories and Sheaves]]_

See also:

* [[Michael Barr]], *Right exact functors*, J. Pure Appl. Algebra **4** (1974) 1–8 [doi:10.1016/0022-4049(74)90025-5, [pdf](https://www.math.mcgill.ca/barr/papers/right.exact.pdf), [[Barr-ExactFunctors.pdf:file]]]

A detailed discussion of how the property of a functor being exact is related to the property of it preserving homology in generalized situations:

* [[Michael Barr]], _Preserving homology_ , Theory and Applications of Categories,  Vol. 16, 2006, No. 7, pp 132-143. ([TAC](http://www.tac.mta.ca/tac/volumes/16/7/16-07abs.html))

Discussion of left exactness (or [[flat functor]]) in the context of [[(∞,1)-category theory]] is in

* [[Jacob Lurie]], def. 5.3.2.1 in _[[Higher Topos Theory]]_

[[!redirects exact functors]]

[[!redirects left exact]]
[[!redirects left exact functor]]
[[!redirects left exact functors]]

[[!redirects lex functor]]
[[!redirects lex functors]]
[[!redirects lex]]

[[!redirects right exact functor]]
[[!redirects right exact functors]]

[[!redirects rex functor]]
[[!redirects rex functors]]
[[!redirects rex]]

[[!redirects finitely continuous]]
[[!redirects finitely continuous functor]]
[[!redirects finitely continuous functors]]
[[!redirects finitely cocontinuous]]
[[!redirects finitely cocontinuous functor]]
[[!redirects finitely cocontinuous functors]]


exact real computer arithmetic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

_Exact real computer arithmetic_ refers to treatment of [[real number]] [[arithmetic]] on [[computation|computers]] to finite (necessarily) but arbitrary precision. This is in contrast with what is called [[floating-point arithmetic]] which uses just one fixed finite approximation of the real numbers by [[natural numbers]].

Exact real computer arithmetic essentially implements what in mathematical [[computability theory]] is known as the [[Type Two Theory of Effectivity|type-II theory]] (in contrast to the "type-I" theory of [[partial recursive functions]] acting just on [[natural numbers]]). The formal mathematical definition of [[computable function (analysis)]] is the core topic of [[constructive analysis]]/[[exact analysis]].

## Related concepts

* [[topological domain theory]]

## References

For more see the references at *[[real number]]* and at *[[constructive analysis]]*.

* [[Chee-Keng Yap]], [[Thomas Dubé]], *The exact computation paradigm*,  in: *Computing in Euclidean Geometry*, Lecture Notes Series on Computing, World Scientific (1995) 452-492  [[doi:10.1142/9789812831699_0011](https://doi.org/10.1142/9789812831699_0011), [[YapDube-ExactComputationParadigm.pdf:file]]]

* [[Jean Vuillemin]], *Exact real computer arithmetic with continued fractions*, in *LFP '88: Proceedings of the 1988 ACM conference on LISP and functional programming* (1988) 14–27 [[doi:10.1145/62678.62681](https://doi.org/10.1145/62678.62681)]

* Peter Potts, Abbas Edalat, *Exact real computer arithmetic* (1997) [[pdf](http://www.doc.ic.ac.uk/research/technicalreports/1997/DTR97-9.pdf), [[PottsEdalat-ExactRealComputerArithmetic.pdf:file]]]

* Dave Plume (supervised by [[Martín Escardó]], [[Alex Simpson]]): *A Calculator for Exact Real Number Computation*, University of Edinburgh  (1998) [[web](https://www.dcs.ed.ac.uk/home/mhe/plume/)]

A collection of further references is listed at

* [Exact computation](https://web.archive.org/web/20150512035451/http://wwwhomes.doc.ic.ac.uk/~ae/exact-computation/)

Discussion relating to [[computability theory]], [[Type Two Theory of Effectivity]] and [[constructive analysis]]/[[computable analysis]] includes

* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]

* The Haskell Wiki, _[Exact real arithmetic](http://www.haskell.org/haskellwiki/Exact_real_arithmetic)_

Efficient exact representations of [[pi]] and related [[irrational numbers]] ([[Bailey-Borwein-Plouffe algorithm]])

* [[David Bailey]], [[Peter Borwein]], [[Simon Plouffe]]: *On the Rapid Computation of Various Polylogarithmic Constants*, Math. Comp. **66** (1997) 903-913 [[doi:1997-66-218/S0025-5718-97-00856-9](https://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-00856-9), [pdf](https://www.davidhbailey.com/dhbpapers/digits.pdf)]

further discussed in:

* [[Jerzy Karczmarczuk]], *Infinite precision real fractions, and lazy carry propagation* or: *The Most Unreliable Technique in the World to Compute $\pi$*, A Braga School (1998) [[pdf](https://karczmarczuk.users.greyc.fr/arpap/lazypi.pdf), [[Karczmarczuk-Pi.pdf:file]]]

* [[Simon Plouffe]], *On the computation of the $n^{th}$ decimal digit of various transcendental numbers* [[arXiv:0912.0303](https://arxiv.org/abs/0912.0303)]

* MathWorld, *[BBP-Type formula](https://mathworld.wolfram.com/BBP-TypeFormula.html)*

Implementation of [[Cauchy real numbers]] (in [[Errett Bishop|Bishop]]-style [[constructive analysis]]) in [[Agda]]:

* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [pdf, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

Coding of something like the [[HoTT book real numbers]] in [[Coq]]:

* [[Gaëtan Gilbert]], *Formalising Real Numbers in Homotopy Type Theory*, in *CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs*  (2017) 112–124 [[arXiv:1610.05072](https://arxiv.org/abs/1610.05072), [doi:10.1145/3018610.3018614](https://doi.org/10.1145/3018610.3018614)]

[[!redirects exact real computer arithmetic]]
[[!redirects exact real arithmetic]]
[[!redirects exact arithmetic]]

[[!redirects Bailey-Borwein-Plouffe algorithm]]
[[!redirects BBP algorithm]]


exact sequence

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

#Contents#
* table of contents
{:toc}

## Idea

An **exact sequence** may be defined in a [[semi-abelian category]], and more generally in a [[homological category]]. It is a [[sequential diagram]] in which the [[image]] of each [[morphism]] is equal to the [[kernel]] of the next morphism.

## Definition
 {#Definition}

### Definition in additive categories
Let $\mathcal{A}$ be an [[additive category]] (often assumed to be an [[abelian category]], for instance $\mathcal{A} = R$[[Mod]] for $R$ some [[ring]]).

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

An **exact sequence** in $\mathcal{A}$ is a [[chain complex]] $C_\bullet$ in $\mathcal{A}$ with vanishing [[chain homology]] in each degree:

$$
  \forall n \in \mathbb{N} . H_n(C) = 0
  \,.
$$

=--

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

A **short exact sequence** is an exact sequence, def. \ref{ExactSequence} of the form

$$
  \cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots
  \,.
$$

One usually writes this just "$0 \to A \to B \to C \to 0$" or even just "$A \to B \to C$".

=--

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

A general exact sequence is sometimes called a **long exact sequence**, to distinguish from the special case of a short exact sequence.

=--

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

Explicitly, a sequence of morphisms

$$
  0 \to A \stackrel{i}\to B \stackrel{p}\to C  \to 0
$$

is short exact, def. \ref{ShortExactSequence}, precisely if

1. $i$ is a [[monomorphism]],

1. $p$ is an [[epimorphism]],

1. and the [[image]] of $i$ equals the [[kernel]] of $p$  (equivalently, the [[coimage]] of $p$ equals the [[cokernel]] of $i$).

=--

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

The third condition is the definition of exactness at $B$. So we need to show that the first two conditions are equivalent to exactness at $A$ and at $C$.

This is easy to see by looking at elements when $\mathcal{A} \simeq R$[[Mod]], for some ring $R$ (and the general case can be reduced to this one using one of the [embedding theorems](abelian%20category#EmbeddingTheorems)):

The sequence being exact at

$$
  0 \to A \to B
$$

means, since the [[image]] of $0 \to A$ is just the element $0 \in A$, that the [[kernel]] of $A \to B$ consists of just this element. But since $A \to B$ is a [[group homomorphism]], this means equivalently that $A \to B$ is an [[injection]].

Dually, the sequence being exact at

$$
  B \to C \to 0
$$

means, since the [[kernel]] of $C \to 0$ is all of $C$, that also the [[image]] of $B \to C$ is all of $C$, hence equivalently that $B \to C$ is a [[surjection]].

=--

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

A **[[split exact sequence]]** is a short exact sequence as above in which $i$ is a [[split monomorphism]], or (equivalently) in which $p$ is a [[split epimorphism]].

=--

In this case, $B$ may be decomposed as the [[biproduct]] $A \oplus C$ (with $i$ and $p$ the usual biproduct inclusion and projection); this sense in which $B$ is 'split' into $A$ and $C$ is the origin of the general terms 'split (mono/epi)morphism'.

### Definition in pointed sets
It is also helpful to consider a similar notion in the case of a [[pointed set]].

+-- {: .num_defn}
###### Definition
In the category $Set_*$ of [[pointed sets]], a sequence
$$
\array{
  (A, a) & \overset{f}{\to} & (B, b) & \overset{g}{\to} & (C, c)
}
$$
is said to be **exact** at $(B, b)$ if $im f = g^{-1}(c)$.

For [[concrete category|concrete]] pointed categories (ie. a category $\mathcal{C}$ with a faithful functor $F: \mathcal{C} \to Set_*$), a sequence is exact if the image under $F$ is exact.
=--

In the case of (abelian) categories like $Ab$ and $R-Mod$, the two notions of exactness coincide if we pick the point of each group/module to be $0$. Such a general notion is useful in cases such as the [[long exact sequence of homotopy groups]] where the homotopy "groups" for small $n$ are just pointed sets without a group structure.

## Properties

### Computing terms in an exact sequence

A typical use of a long exact sequence, notably of the [[homology long exact sequence]], is that it allows to determine some of its entries in terms of others.

The characterization of short exact sequences in prop. \ref{CharacterizationByMonoEpi} is one example for this: whenever in a long exact sequence one entry vanishes as in $\cdot \to 0 \to C_n \to \cdot$ or $\cdot \to C_n \to 0 \to \cdots$, it follows that the _next_ morphism out of or into the vanishing entry is a [[monomorphism]] or [[epimorphism]], respectively.

In particular:

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

If part of an exact sequence looks like

$$
  \cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots
  \,,
$$

then $\partial_n$ is an [[isomorphism]] and hence

$$
  C_{n+1} \simeq C_n
  \,.
$$

=--

### Exactness and quasi-isomorphisms
 {#ExactnessAndQuasiIsomorphisms}

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

A chain complex $C_\bullet$ is exact (is a long exact sequence), precisely if the unique [[chain map]] from the [[initial object]], the 0-complex

  $$
    0 \to C_\bullet
  $$

is a [[quasi-isomorphism]].

=--

### Short exact sequences and quotients
 {#SESAndQuotients}

The following are some basic lemmas that show how given a short exact sequence one obtains new short exact sequences from forming [[quotients]]/[[cokernels]] (see [Wise](#Wise)).

Let $\mathcal{A}$ be an [[abelian category]].

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

For

$$
  A \to B \to C \to 0
$$

an exact sequence in $\mathcal{A}$ and for $X \to B$ any morphism in $\mathcal{A}$, also

$$
  A \to B/X \to C/X \to 0
$$

is a short exact sequence.

=--

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

We have an exact sequence of complexes of length 2

$$
  \array{
    0 &\to& X &\stackrel{id}{\to}& X &\to& 0
    \\
    \downarrow & & \downarrow & & \downarrow & & \downarrow
    \\
    A &\to& B &\to& C &\to& 0
  }
$$

and the exact sequence to be demonstrated is degreewise the [[cokernel]] of this sequence. So the statement reduces to the fact that forming cokernels is a [[right exact functor]].

=--

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

For

$$
  0 \to A \to B \to C
$$

an exact sequence and $X \to A$ any morphism, also

$$
  0 \to A/X \to B/X  \to C
$$

is exact.

=--

## Examples

### Specific examples
 {#SpecificExamples}

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

Let $\mathcal{A} = \mathbb{Z}$[[Mod]] $\simeq$ [[Ab]]. For $n \in \mathbb{N}$ with $n \geq 1$ let $\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}$ be the [[linear map]]/[[homomorphism]] of [[abelian groups]] which acts by the ordinary multiplication of [[integers]] by $n$. This is clearly an [[injection]]. The [[cokernel]] of this morphism is the projection to the [[quotient group]], which is the [[cyclic group]] $\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}$. Hence we have a short exact sequence

$$
  0 \to \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}
  \to \mathbb{Z}_n
  \,.
$$

=--

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

The [[connecting homomorphism]] of the [[long exact sequence in homology]] induced from short exact sequences of the form in example \ref{MultiplicationAndCyclicGroup} is called a _[[Bockstein homomorphism]]_.

=--

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

* [[exponential exact sequence]]

* [[Kummer sequence]]

* [[Artin-Schreier sequence]]

* [[Kummer-Artin-Schreier-Witt exact sequence]]

=--

### Classes of examples

* [[short exact sequence of vector bundles]]

* [[long exact sequence of homotopy groups]]

* [[long exact sequence in homology]]/[[long exact sequence in homology|in cohomology]]

  * [[long exact sequence in chain homology]]

  * [[long exact sequence in generalized homology]]

* [[Serre long exact sequence]]

## Related concepts

* [[splicing of short exact sequences]]

* [[exact functor]]

* [[exact sequence of Hopf algebras]]

## References

A standard introduction is for instance in section 1.1 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

The quotient lemmas from [above](#SESAndQuotients) are discussed in

* [[Jonathan Wise]], _The Snake Lemma_ ([pdf](http://math.stanford.edu/~jonathan/papers/snake.pdf))
 {#Wise}

in the context of the [[salamander lemma]] and the [[snake lemma]].

[[!redirects exact sequences]]

[[!redirects long exact sequence]]
[[!redirects short exact sequence]]

[[!redirects long exact sequences]]
[[!redirects short exact sequences]]