ambidextrous adjunction

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
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## Definition

An [[adjoint triple]] $F \dashv G \dashv H$ is called an _ambidextrous adjunction_ (or sometimes _ambiadjunction_ or _ambijunction_, for short) if the [[left adjoint]] $F$ and the [[right adjoint]] $H$ of $G$ are [[equivalence|equivalent]] $F \simeq H$, or
more precisely: equipped with a specified equivalence.

In fact, often $F$ is *identified* with $H$, which is the situation of a *strongly adjoint pair* $(F \dashv G \dashv F)$ originally considered by [Morita 1965](#Morita65). Some authors refer to this situation by saying that $G$ is a *[[Frobenius functor]]* (ie. a functor which has a [[left adjoint]] that is also a [[right adjoint]]).

Sometimes $F$ is said to be *biadjoint* to $G$ (not to be confused with [[biadjoint]] in the sense of *[[biadjunction]]*). Functor $G$ which has a left and right adjoint which are equivalent is said to be [[Frobenius functor]].

In the special case that $G$ is a [[fully faithful functor]] with an ambidextrous adjoint one also speaks of an *[[essential localization]]* (cf. *[[bireflective subcategory]]*).

## Properties

### Frobenius algebra structure

The [[monad]] induced by an ambidextrous adjunction is a [[Frobenius monoid]] object in [[endofunctors]]. (e.g. [Lauda 05, theorem 17](#Lauda05)), hence a [[Frobenius monad]].

### Fiberwise characterization of ambidextrous Kan extension
 {#FiberwiseCharacterization}

Let $\mathcal{D} \in Cat_\infty$ be an [[(∞,1)-category]] with small [[(∞,1)-colimits]]. For $f \;\colon\; X \longrightarrow Y$ a morphism of [[∞-groupoids]], write

$$
  f^\ast \;\colon\; [Y,\mathcal{D}] \longrightarrow [X,\mathcal{D}]
$$

for the induced pullback of [[(∞,1)-functor (∞,1)-categories]] (which one may think of as the categories of $\mathcal{D}$-valued [[local systems]] over $X$ and $Y$, respectively). The [[left adjoint]] and [[right adjoint]] (if it exists) of this are left and right [[(∞,1)-Kan extension]].

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

Say that a morphism $f$ is $\mathcal{D}$-ambidextrous if $(f_! \dashv f^\ast)$  is an ambidextrous adjunction $(f_! \simeq f_\ast)$ and in addtion all pullbacks of $f$ satisfy some property (...).

Say that an [[∞-groupoid]] $A \in Grpd_\infty$ is _$\mathcal{D}$-ambidextrous_ if its [[terminal object in an (∞,1)-category|terminal]] map is.

=--

([Hopkins-Lurie 14, def. 4.1.11](#{#HopkinsLurie14}))

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###### Proposition

A morphism $f \colon X \to Y$ between [[∞-groupoids]], is $\mathcal{D}$-ambidextrous, def. \ref{AmbidextrousKanExtension}, precisely if each [[homotopy fiber]] $X_y$ of $f$ is.

=--

([Hopkins-Lurie 14, prop. 4.3.5](#HopkinsLurie14))

## Examples

+-- {: .num_example}
###### Example
**(coincident limits and colimits)**

Let $\mathcal{C}$ be a [[small category]] and $\mathcal{D}$ any category and consider the functor $const \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}]$ that sends objects to constant [[presheaves]] with this value. Then the [[right adjoint]] of this functor is, if it exists, the [[limit]] construction, and the [[left adjoint]] is, if it exists, the [[colimit]] construction. (See also at _[[Kan extension]]_.) Therefore if both exist as an ambidextrous adjunction, then this means that limits in $\mathcal{D}$ over [[diagrams]] of shape $\mathcal{C}$ coincide with the [[colimits]] over these diagrams. If $\mathcal{C}$ is a [[finite set]], then this situation is traditionally referred to as _[[biproducts]]_. Generally therefore this is sometimes called _[[bilimits]]_ (but see the discussion of the terminology there).

In ([Hopkins-Lurie 14, section 4.3](#HopkinsLuire14)) such $\mathcal{C}$ is called _$\mathcal{D}$-ambidextrous_ (or rather, they consider $\mathcal{C}$ an [[∞-groupoid]] and hence call it a _$\mathcal{D}$-ambidextrous space_). Concrete examples of this include those discussed at _[[K(n)-local stable homotopy theory]]_.

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###### Example
**(Yoga of six functors)**

A [[Wirthmüller context]] in the presence of an un-twisted [[Wirthmüller isomorphism]] is an ambidextrous adjunction.

###### Example

Every [[self-adjoint functor]] forms an ambidextrous adjunction.

=--

## Related concepts

* [[Wirthmüller context]]

* [[infinitesimal cohesion]]

## References

Ambidextrous adjunctions were maybe first considered under the name *strongly adjoint pairs* (of functors), in:

* {#Morita65} [[Kiiti Morita]], *Adjoint pairs of functors and Frobenius extensions*, Science Reports of the Tokyo Kyoiku Daigaku, Section A **9** 202/208 (1965) 40-71 [[jstor:43698658](https://www.jstor.org/stable/43698658)]

The terminology "[[Frobenius functors]]" for "strongly adjoint pairs" is due to

* {#CaenepeelMilitaruZhu97}  [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], *Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties*, Trans. Amer. Math. Soc. **349** (1997) 4311-4342 [[1997-349-11/S0002-9947-97-02004-7](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7), [pdf](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7/S0002-9947-97-02004-7.pdf)]

* F. Castaño Iglesias, [[José Gómez Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

The case of [[bireflective subcategories]]:

* [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

On the [[Frobenius monads]] induced by ambidextrous adjuntions:

* {#Street04} [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Phys. **45** 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]

* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]

See also:

* {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], *[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]* (2014)

with some review in:

* [[Peter Haine]], *Ambidexterity* (2018) [[pdf](https://math.berkeley.edu/~phaine/files/Ambidexterity_4.pdf), [[Haine-Ambidexterity.pdf:file]]]

On the issue of equipping an ambidextrous adjunction $F \dashv G \dashv H$  with a specific equivalence between $F$ and $H$:

* [[Qiaochu Yuan]], [MO:377104](https://mathoverflow.net/a/377104)

Connection to [[Hopf adjunction]]s

* Harshit Yadav, _Frobenius monoidal functors from (co)Hopf adjunctions_, [arXiv:2209.15606](https://arxiv.org/abs/2209.15606)

[[!redirects ambidextrous adjunctions]]

[[!redirects ambidextrous adjoint]]
[[!redirects ambidextrous adjoints]]

[[!redirects ambidextrous space]]
[[!redirects ambidextrous spaces]]

[[!redirects strongly adjoint pair]]
[[!redirects strongly adjoint pairs]]

[[!redirects biadjoint pair]]

[[!redirects ambiadjunction]]
[[!redirects ambiadjunctions]]
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[[!redirects ambijunctions]]


ambient category

An ambient category $C$ is just a category from which [[internalization|internal structures]] take their objects and structure morphisms.

If $C$ has extra structure, like being ([[semi-abelian category|semi]]-)[[abelian category|abelian]] or a [[site]], then one can do extra things; this is made precise through the concept of [[doctrine]].

More generally an ambient category could be seen as a 'universe of discourse', as when $C$ is a [[topos]]; see [[foundations]].


ambimorphic object > history

< [[ambimorphic object]]
[[!redirects ambimorphic object -- history]]


Ambrus Kaposi

* [website](https://akaposi.github.io/)

## Selected writings

* [[Thorsten Altenkirch]], [[Ambrus Kaposi]], *Type Theory in Type Theory using Quotient Inductive Types*, POPL17 (2017) [[pdf](http://www.cs.nott.ac.uk/~psztxa/publ/tt-in-tt.pdf)]

On [[higher inductive-inductive types]]:

* {#KaposiKovacs} [[Ambrus Kaposi]], [[András Kovács]], *A Syntax for Higher Inductive-Inductive Types*, at *3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)* (2018) 20:1-20:18 [[pdf](http://drops.dagstuhl.de/opus/volltexte/2018/9190/), [doi:10.4230/LIPIcs.FSCD.2018.20](https://doi.org/10.4230/LIPIcs.FSCD.2018.20)]

* [[Ambrus Kaposi]], [[András Kovács]], *Signatures and Induction Principles for Higher Inductive-Inductive Types*, Logical Methods in Computer Science **16** 1 (2020) lmcs:6100 [[arXiv:1902.00297](https://arxiv.org/abs/1902.00297), doi:10.23638/LMCS-16(1:10)2020]

Exposition:

* [[Ambrus Kaposi]], *Quotient inductive-inductive types and higher friends*, [[Homotopy Type Theory Electronic Seminar Talks]], 22 October 2020 ([video](https://www.youtube.com/watch?v=a9_KjX1WM84), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kaposi-2020-10-22-HoTTEST.pdf))

category: people


amenable category

An **amenable category** is an [[additive category]] in which all [[idempotent]]s split.


amenable topological groupoid

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###Context###
#### Higher geometry
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#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

([AD-Renault 00](#AD-Renault00))
Recalled for instance in ([Sims-Williams 12, p. 4](#SimsWilliams)).

([ANS 05, def. 1.3](#ANS05))

## Properties

### Groupoid convolution algebra

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###### Proposition

The [[groupoid convolution algebra]] of an amenable topological groupoid is in the [[bootstrap category]].

=--

([Tu 99, prop. 10.7](#Tu99)), recalled as ([Uuye 11, example 3.6](#Uuye11)).

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

For an amenable [[Lie groupoid]] $\mathcal{G}$, the full [[groupoid convolution algebra]] and the reduced one are [[natural isomorphism|naturally isomorphic]].

=--

This is due to ([AD-Renault 00](#AD-Renault00)), recalled for instance as ([ANS 05, prop. 1.9](#ANS05))

## References

* C. Anantharaman-Delaroche, [[Jean Renault]], _Amenable Groupoids_ Monogr. Enseign. Math., vol. 36 Enseignement Math., Geneva (2000) ([webpage, including extensive review](http://www.unige.ch/math/EnsMath/EM_MONO/m36.html))
 {#AD-Renault00}

* Aidan Sims, Dana P. Williams, _Amenability for Fell bundles over groupoids_ ([arXiv:1201.0792](http://arxiv.org/abs/1201.0792))
 {#SimsWilliams12}

* Johannes Aastrup, Ryszard Nest, Elmar Schrohe, _A Continuous Field of C*-algebras and the Tangent Groupoid for Manifolds with Boundary_ ([arXiv:math/0507317](http://arxiv.org/abs/math/0507317))
 {#ANS05}

* [[Jean-Louis Tu]], _La conjecture de Baum-Connes pour les feuilletages moyennables_, K-Theory 17 (1999), no. 3, 215–264. MR 1703305 (2000g:19004) ([Portico](http://www.portico.org/Portico/article/access/viewHtml.por?journalId=ISSN_09203036&issueId=ISSN_09203036v17i3&articleId=pgg1zfpwkq7&fileType=html:Header&fileValid=true), subscription needed)
 {#Tu99}

* [[Otgonbayar Uuye]], _A note on the Künneth theorem for nonnuclear C*-algebras_ ([arXiv:1111.7228](http://arxiv.org/abs/1111.7228))
 {#Uuye11}

[[!redirects amenable topological groupoids]]

[[!redirects amenable topological group]]
[[!redirects amenable topological groups]]

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[[!redirects amenable Lie groupoids]]


Amer Iqbal

* [Wikipedia entry](https://en.wikipedia.org/wiki/Amer_Iqbal)

## Selected writings

On [[string junctions]] in relation to [[Lie algebras]] and [[Lie algebra representations]]:

* Oliver DeWolfe, Tamas Hauer, [[Amer Iqbal]], [[Barton Zwiebach]], _Uncovering the Symmetries on $[p,q]$ 7-branes: Beyond the Kodaira Classification_, Adv. Theor. Math. Phys. 3 (1999) 1785-1833 ([arXiv:hep-th/9812028](https://arxiv.org/abs/hep-th/9812028))

On a [[mysterious duality]]:

* {#INV01} [[Amer Iqbal]], [[Andrew Neitzke]], [[Cumrun Vafa]], _A mysterious duality_, ([arXiv:hep-th/0111068](https://arxiv.org/abs/hep-th/0111068))

On [[M-strings]]/[[self-dual strings]]:

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

and the [[M-string elliptic genus]]:

* [[Stefan Hohenegger]], [[Amer Iqbal]],  _M-strings, Elliptic Genera and $\mathcal{N}=4$ String Amplitudes_, Fortschritte der PhysikVolume 62, Issue 3 ([arXiv:1310.1325](http://arxiv.org/abs/1310.1325))

* [[Stefan Hohenegger]], [[Amer Iqbal]], Soo-Jong Rey, _M String, Monopole String and Modular Forms_, Phys. Rev. D 92, 066005 (2015) ([arXiv:1503.06983](https://arxiv.org/abs/1503.06983))

On [[E-strings]]:

* [[Amer Iqbal]], _A note on E-strings_, Adv. Theor. Math. Phys. 7 (2003) 1-23 ([arXiv:hep-th/0206064](https://arxiv.org/abs/hep-th/0206064))

## Related $n$Lab entries

* [[mysterious duality]]

category: people


Amihay Hanany

Amihay Hanany is professor for theoretical [[physics]] at Imperial College London.

* [webpage](http://www.imperial.ac.uk/people/a.hanany)

## Selected writings

On [[small instantons]] as [[NS5-branes]]/[[M5-branes]] in [[E8]]-[[heterotic string theory]]/[[heterotic M-theory]]:

* {#GanorHanany96} [[Ori Ganor]], [[Amihay Hanany]], _Small $E_8$ Instantons and Tensionless Non-critical Strings_, Nucl. Phys. B474 (1996) 122-140 ([arXiv:hep-th/9602120](https://arxiv.org/abs/hep-th/9602120))

* [[Amihay Hanany]], Noppadol Mekareeya, _The Small $E_8$ Instanton and the Kraft Procesi Transition_, JHEP07 (2018) 098 ([arXiv:1801.01129](https://arxiv.org/abs/1801.01129))

On [[geometric engineering of quantum field theories]] involving [[NS5-branes]] and introducing the [[Hanany-Witten effect]] and other effects of [[NS5-brane]]/[[D-brane]] [[brane intersection|intersections]]:

* {#HanayWitten97} [[Amihay Hanany]], [[Edward Witten]], *Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics*, Nucl. Phys.B **492** (1997) 152-190 [[arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230), doi:10.1016/S0550-3213(97)80030-2]

On [[D-branes]] [[brane intersection|intersecting]] [[NS5-branes]]:

* {#BrodieHanany97} [[John Brodie]], [[Amihay Hanany]], _Type IIA Superstrings, Chiral Symmetry, and N=1 4D Gauge Theory Dualities_, Nucl.Phys. B506 (1997) 157-182 ([arXiv:hep-th/9704043](https://arxiv.org/abs/hep-th/9704043))

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

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

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

and their intersection with [[orientifolds]]:

* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))

Discussion of a [[BFSS matrix model|BFSS-like]] [[matrix model]] for [[MK6-branes]]:

* [[Amihay Hanany]], Gilad Lifschytz, _M(atrix) Theory on $T^6$ and a m(atrix) Theory Description of KK Monopoles_, Nucl. Phys. B519:195-213, 1998 ([arXiv:hep-th/9708037](https://arxiv.org/abs/hep-th/9708037))

On [[toric duality]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))

* [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228))

On [[discrete torsion]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))

On [[vortex strings]]:

* [[Amihay Hanany]], [[David Tong]], _Vortices, Instantons and Branes_, JHEP 0307 (2003) 037 ([arXiv:hep-th/0306150](https://arxiv.org/abs/hep-th/0306150))

* [[Amihay Hanany]], [[David Tong]], _Vortex Strings and Four-Dimensional Gauge Dynamics_, JHEP 0404 (2004) 066 ([arXiv:hep-th/0403158](https://arxiv.org/abs/hep-th/0403158))

* {#Tong09} [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060))

* [[David Tong]], _Vortices, Strings, and Vortex Strings_ ([[TongVortexStrings.pdf:file]])

On [[heterotic M-theory on ADE-orbifolds]]:

* Santiago Cabrera, [[Amihay Hanany]], [[Marcus Sperling]], _Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories_, J. High Energ. Phys. (2019) 2019: 71 ([arXiv:1904.12293](https://arxiv.org/abs/1904.12293))

On [[D=3 N=4 super Yang-Mills theory]]:

* [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657))

## Related entries

* [[super Yang-Mills theory]]

* [[Seiberg duality]]

* [[AdS-CFT]]

category: people


Amir Dembo

* [webpage](http://statweb.stanford.edu/~adembo/)

## related $n$Lab entries

* [[probability theory]]

category: people


Amir-Kian Kashani-Poor

* [webpage](www.phys.ens.fr/~kashani/index.html)

## related $n$Lab entries

* [[topological recursion]]

* [[topological string]]

* [[matrix model]]

category: people


Amirhossein Tajdini

* [InSpire page](https://inspirehep.net/authors/1889144)

* [ResearchGate page](https://www.researchgate.net/scientific-contributions/Amirhossein-Tajdini-2116760582)

## Selected writings

Review of the claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020)  \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\]

Review:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]

category: people


Amit Giveon

* [webpage](http://old.phys.huji.ac.il/~hep/amit.giveon.html)

## Selected writings

On [[geometric engineering of QFT]] in [[intersecting D-brane models]] subject to the [[s-rule]]:

* [[Amit Giveon]], [[David Kutasov]], _Brane Dynamics and Gauge Theory_, Rev. Mod. Phys. 71:983-1084, 1999 ([arXiv:hep-th/9802067](https://arxiv.org/abs/hep-th/9802067))

On [[NS5-brane]]-[[brane intersections|intersections]] with [[D-branes]]:

* {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], *D-Branes in the Background of NS Fivebranes*, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)]

category: people


Amit Patel

* [institute page](https://www.math.colostate.edu/~akp/)

* [personal page](http://akpatel.org/)

## Selected writings

Introducing [[well groups]]:

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

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

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

category: people


Amit Sever

* [Institute page](https://en-exact-sciences.tau.ac.il/profile/asever)

## Selected writings

On [[Wilson line]]-[[quantum observables]] and [[bosonization]] in [[Chern-Simons theory|Chern-Simons]]/matter theory (such as the [[ABJM model]]):

* [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, talk at *[[Strings 2022]]* [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)]

category: people


Amit Sharma

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=214520)

## Selected writings

On [[compact closed category|compact closure]] in [[homotopical algebra]] and relating to the [Barrat-Priddy theorem](stable+cohomotopy#AsAlgebraicKTheoryOverTheFieldWithOneElement):

* [[Amit Sharma]], *Compact closed categories and Γ-categories (with an appendix by [[André Joyal]])*, Theory and Applications of Categories **37** 37 (2021) 1222-1261 [[arXiv:2010.09216](https://arxiv.org/abs/2010.09216), [tac:37-37](http://www.tac.mta.ca/tac/volumes/37/37/37-37abs.html)]

On [[Picard groupoids]]:

* [[Amit Sharma]], *Picard groupoids and $\Gamma$-categories*, [[Cahiers]] **LXIV** 3 (2023) [[arXiv:2002.05811](https://arxiv.org/abs/2002.05811), [pdf](http://cahierstgdc.com/wp-content/uploads/2023/07/SHARMA_LXIV-3.pdf)]

category: people


Amitai Regev

* [Wikipedia entry](https://en.wikipedia.org/wiki/Amitai_Regev)

## Selected writings

Expressing the number of [[standard Young tableaux]] with at most 3 rows through [[Catalan numbers]] and [[Motzkin numbers]]:

* [[Amitai Regev]], *Asymptotic values for degrees associated with strips of young diagrams*, Advances in Mathematics Volume 41, Issue 2, August 1981, Pages 115-136 (doi:10.1016/0001-8708(81)90012-8)

category: people


Amitsur complex

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[commutative ring]] $R$ and an $R$-[[associative algebra]] $A$, hence a [[ring]] [[homomorphism]] $R \longrightarrow A$, the _Amitsur complex_ is the [[Moore complex]] of the dual [[Cech nerve]] of $Spec(A) \to Spec(R)$, hence the [[chain complex]] of the form

$$
  R \to A \to A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots
$$

with [[differentials]] given by the alternating sum of the coface-maps.

(See also at _[[Sweedler coring]]_, at _[[commutative Hopf algebroid]]_ and at _[[Adams spectral sequence]]_ for the same or similar constructions.)

## Properties

### Descent theorem

+-- {: .num_theorem }
###### Theorem
**(descent theorem)**

If $A \to B$ is [[faithfully flat]] then its Amitsur complex is [[exact sequence|exact]].

=--

This is due to ([[Grothendieck]], [[FGA]]1)

The following reproduces the proof in low degree from [Milne, prop. 6.8](#Milne)

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

We show that

$$
  0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B
$$

is an [[exact sequence]] if $f \colon A \longrightarrow B$ is [[faithfully flat]].

First observe that the statement follows if $A \to B$ admits a [[section]] $s \colon B \to A$. Because then we can define a map

$$
  k \colon B \otimes_A B \longrightarrow B
$$

$$
  k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2))
  \,.
$$

This is such that applied to a coboundary it yields

$$
  k(1 \otimes b - b \otimes 1) = f(s(b)) - b
$$

and hence it exhibits every cocycle $b$ as a coboundary $b = f(s(b))$.

So the statement is true for the special morphism

$$
  B \to B \otimes_A B
$$

$$
  b \mapsto b \otimes 1
$$

because that has a section given by the multiplication map.

But now observe that the morphism $B \to B \otimes_A B$ is the [[tensor product]] of the morphism $f$ with $B$ over $A$. That $A \to B$ is [[faithfully flat]] by assumption, hence that it exhibits $B$ as a [[faithfully flat module]] over $A$ means by definition that the Amitsur complex for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact.

=--

### As a bar construction

For $\phi \colon B \longrightarrow A$ a [[homomorphism]]
of suitable [[monoids]], there is the corresponding pull-push [[adjunction]]
([[extension of scalars]] $\dashv$ [[restriction of scalars]]) on [[categories of modules]]

$$
  ((- )\otimes_B A \dashv \phi^\ast )
  \;\colon\;
   Mod_A
    \stackrel{\overset{(-)\otimes_B A}{\leftarrow}}{\underset{\phi^\ast}{\longrightarrow}}
   Mod_B
  \,.
$$

The [[bar construction]] of the corresponding [[monad]] -- the [[higher monadic descent|higher]] [[monadic descent]] objects -- is the corresponding Amitsur complex.

(e.g. [Hess 10, section 6](#Hess10))

## Related concepts

* [[Adams-Novikov spectral sequence]]

## References

The Amitsur complex was introduced in

* [[Shimshon Amitsur]], _Simple algebras and cohomology groups of arbitrary fields_, Transactions of the American Mathematical Society
Vol. 90, No. 1 (Jan., 1959), pp. 73-112 ([JSTOR](http://www.jstor.org/stable/1993268))

His results were simplified in

* Alex Rosenberg and Daniel Zelinsky, _On Amitsur's complex_,  Transactions of the American Mathematical Society Vol. 97, No. 2 (Nov., 1960), pp. 327-356

The statement of proof the descent theorem for the Amitsur complex is due to

* [[Alexander Grothendieck]], [[FGA]]1

A review of the proof in low degree is in

* {#Milne} [[James Milne]], prop. 6.8 of _[[Lectures on Étale Cohomology]]_

Discussion from the point of view of [[Sweedler corings]] and a full proof of the descent theorem is in

*  [[Tomasz Brzezinski]], [[Robert Wisbauer]], section 29 of _Corings and Comodules_, Cambridge University Press, London Math. Soc. LN 309 (2003), ([errata pdf](http://www.math.uni-duesseldorf.de/~wisbauer/corinerr.pdf))

Disucssion from the point of view of [[higher monadic descent]] is in

* {#Hess10} [[Kathryn Hess]], section 6 of _A general framework for homotopic descent and codescent_ ([arXiv:1001.1556](http://arxiv.org/abs/1001.1556))

Discussion of [[formal completion]] of [[(infinity,1)-modules]] in terms of [[totalization]] of Amitsur complexes is in

* {#Carlsson07} [[Gunnar Carlsson]], _Derived completions in stable homotopy theory_ ([arXiv:0707.2585](http://arxiv.org/abs/0707.2585))

[[!redirects Amitsur complexes]]

[[!redirects descent theorem]]
[[!redirects descent theorems]]


amnestic functor

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

# Amnestic functors
* table of contents
{: toc}

## Idea

A [[strict functor]] is _amnestic_ if its [[domain]] has no more duplication of [[isomorphic]] [[objects]] than its [[codomain]].

As ‘amnestic’ is basically a fancy synonym of ‘forgetful’, the idea is to identify a property that one would like in a [[forgetful functor]].

For example, the notion of amnestic functors formalizes the sense in which the [[concrete category]] $Met_cont$ of [[metric spaces]] and [[continuous maps]] is often better thought of as the category $Met Top$ of [[metrizable topological spaces]] (and [[continuous maps]]), by identifying a property that the [[forgetful functor]]  $Met Top \to Set$ has but $Met_cont \to Set$ does not.

There is a corresponding notion of _amnesticization_ of a functor (called the _amnestic modification_ in _[[The Joy of Cats]]_), which replaces the domain with an [[equivalent category]], relative to which the functor becomes amnestic.  Applying this to $Met_cont \to Set$ produces a category [[isomorphic category|isomorphic]] to $Met Top$.  Although not needed in this case, we need the [[axiom of choice]] (AC) in general to prove that every functor has an amnesticization.  Without AC, we can use amnestic [[anafunctors]] to make everything work out, although much of the convenience is lost.

Amnesticity is really a property of [[strict functors]] (or anafunctors) between [[strict category|strict]] [[groupoids]].  Groupoids, because the non-isic [[morphisms]] play no role in the definition; only the categories\' [[cores]] matter, and a functor is amnestic iff its core is amnestic.  Strict, because the definition requires us to state (in two places) that some isomorphic objects are equal; weakening the definition to follow the [[principle of equivalence]] leads to a trivial property that every functor satisfies.  (That is, up to [[equivalence of categories|equivalence]], every functor is amnestic, which is because every functor is equivalent to its amnesticization.)

## Definitions

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

Let $C$ and $D$ be two [[strict categories]], and let $U$ be a [[strict functor]] from $D$ to $C$.  We say that $U$ is __amnestic__ if its [[groupoid core]] reflects [[identity morphisms]].

Explicitly, $U$ is amnestic iff, for every [[isomorphism]] $f \colon a \to B$ in $D$ that $U$ takes to an [[identity morphism]] $U(f) = id_{U(a)} = id_{U(b)}$, then already $f$ itself is an [[identity morphism]].

In other words, a functor is amnestic if its strict [[fibers]] are [[gaunt category|gaunt]].

=--

Observe that this is similar to a [[conservative functor]], which reflects isomorphisms rather than identities.

If we follow the [[principle of equivalence]] and refuse to state [[equalities]] between objects, then we must modify the hypothesis to say that $U(a)$ and $U(b)$ are *[[isomorphic]]* in $C$ (say via $g\colon U(a) \to U b$) and $U(f)$ is the identity *relative to* this isomorphism (so $U(f) = \id_{U(b)} \circ g \circ \id_{U(a)}$; since we can simply let $g$ be $U(f)$, this is trivial (beyond the initial isomorphism $f\colon a \to b$).  Similarly, we must modify the conclusion to say that $a$ and $b$ are *isomorphic* (say via $h\colon a \to b$) and $f$ is the identity *relative to* this isomorphism (so $f = \id_b \circ h \circ \id_a$); since we can simply let $h$ be $f$, this is also trivial.  Thus up to [[equivalence of categories|equivalence]], this property is trivial; on the other hand, it is preserved by [[isomorphism of categories|isomorphism]].

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

Now let $C$ and $D$ be two [[strict categories]], and let $U$ be an [[anafunctor]] from $D$ to $C$.  We again say that $U$ is __amnestic__ if its [[core]] reflects [[identity morphisms]].  Explicitly, now $U$ is amnestic iff, whenever $a$ and $b$ are [[objects]] of $D$, $f$ is an [[isomorphism]] in $D$ from $a$ to $b$, $\alpha$ is a specification of $a$ for the anafunctor $U$, $\beta$ is a specification of $b$ for the anafunctor $U$, $U_\alpha(a)$ and $U_\beta(b)$ are equal objects in $C$, and $U_{\alpha,\beta}(f)$ is the [[identity morphism]] on this object in $C$, then $a$ and $b$ are equal objects in $D$, and $f$ is the identity morphism on this object in $D$.
=--

We might also demand that $\alpha = \beta$; this is automatic if $U$ is saturated.

## Properties

* An amnestic [[full and faithful functor]] is automatically an [[isocofibration]], i.e. injective on objects: if $U D' = U D$, then there is some _isomorphism_ $f : D' \to D$ in $\mathcal{D}$ such that $U f = id_{U D}$, but then we must have $f = id_{U D'} = id_{U D}$, so $D' = D$.

* An amnestic [[isofibration]] has the following lifting property: for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a _unique_ isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$. Indeed, if $\tilde{f}' : \tilde{C}' \to D$ were any other isomorphism such that $U \tilde{f}' = f$, then $U (\tilde{f}^{-1} \circ \tilde{f}') = id_C$, so we must have $\tilde{f} = \tilde{f}'$. Amnestic isofibrations are occasionally called **discrete isofibrations**, but this term may be misleading, because they are not [[isofibrations]] with discrete fibres.

* If the [[composite]] $U \circ K$ is an amnestic functor, then $K$ is also amnestic.

## Examples

* Most famous [[forgetful functors]] are amnestic, such as $Grp \to Set$, $Top \to Set$, $Top Grp \to Grp$, and $Top Grp \to Top$.  Even $Met \to Set$ is amnestic, since the morphisms in [[Met]] are [[short maps]].  However, $Met_cont \to Set$, where the morphisms are [[continuous maps]], is *not* amnestic, as is shown by any set with two different but topologically equivalent metrics (such as $\mathbb{R}^2$ with the $l^1$ and $l^\infty$ metrics).

* The forgetful functor from a groupoid of [[structured sets]] is amnestic.  The examples above may all be defined by starting from such a groupoid and specifying which functions preserve the structure (and so are morphisms in the category of structured sets).  So long as this produces no additional isomorphisms, the forgetful functor will be amnestic.  But in the case of $Met_cont$, any non-isometric homeomorphism will be an isomorphism that was not in the original groupoid (consisting only of surjective isometries), and so this forgetful functor is not amnestic.

* Any strictly [[monadic functor]] is amnestic. Conversely, any monadic functor that is also an amnestic isofibration is necessarily strictly monadic.

## Related concepts

* [[conservative functor]]
* [[concrete category]]
* [[structure]]
* [[stuff, structure, property]]

## References

* J. Adámek, H. Herrlich and G.E. Strecker: [[The Joy of Cats]], Chapter I, Definition 3.27.

* G. Preuß: Theory of Topological Structures: An Approach to Categorical Topology. D. Reidel Publishing Company. Mathematics and Its Applications, Dordrecht, Holland 1988.  p. 178, footnote 31

[[!redirects amnestic functor]]
[[!redirects amnestic functors]]
[[!redirects amnestic anafunctor]]
[[!redirects amnestic anafunctors]]

[[!redirects amnesticization]]
[[!redirects amnesticizations]]
[[!redirects amnesticisation]]
[[!redirects amnesticisations]]
[[!redirects amnestic modification]]
[[!redirects amnestic modifications]]

[[!redirects amnestic isofibration]]
[[!redirects amnestic isofibrations]]
[[!redirects discrete isofibration]]
[[!redirects discrete isofibrations]]


Amnon Neeman

# Amnon Neeman

* [web site](http://wwwmaths.anu.edu.au/~neeman/)

## Selected writing

On [[triangulated categories]]:

* {#Neeman01} [[Amnon Neeman]], *Triangulated Categories*, Annals of Mathematics Studies **213**, Princeton University Press (2001) [[ISBN:9780691086866](https://press.princeton.edu/titles/7102.html), [pdf](http://www.mi-ras.ru/~akuznet/homalg/Neeman%20Triangulated%20categories.pdf)]

The [[theorem of the heart]]:

* [[Amnon Neeman]], *K-Theory for Triangulated Categories III(A): The Theorem of the Heart*, Asian Journal of Mathematics **2** 3 (1998) 495-589 [[arXiv:math/9901091](https://arxiv.org/abs/math/9901091), [doi:10.4310/AJM.1998.v2.n3.a4](https://doi.org/10.4310/AJM.1998.v2.n3.a4)]

* [[Amnon Neeman]], *K-theory for triangulated categories III(B): The theorem of the heart*, Asian Journal of Mathematics **3** 3 (1999) 557–608 [[doi:10.4310/AJM.1999.v3.n3.a2](https://doi.org/10.4310/AJM.1999.v3.n3.a2), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0003/AJM-1999-0003-0003-a002.pdf)]

* [[Amnon Neeman]], *K-Theory for Triangulated Categories $3\frac{3}{4}$: A Direct Proof of the Theorem of the Heart*, K-Theory **22** 1-2 (2001) 1-144 [[doi:10.1023/A:1011172502978](http://dx.doi.org/10.1023/A:1011172502978)]

category: people


Amnon Yekutieli

Here are abstracts of a few papers of mine that connect to higher category theory, with links to the full texts. I will be glad to discuss this material.

### 1. Twisted Deformation Quantization of Algebraic Varieties ###

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.

[full paper - eprint](http://arxiv.org/abs/0905.0488),
[survey](http://arxiv.org/abs/0801.3233)
[lecture notes](http://www.math.bgu.ac.il/%7Eamyekut/lectures/twisted-defs/notes.pdf)

### 2. Nonabelian Multiplicative Integration on Surfaces ###

We construct a 2-dimensional twisted nonabelian multiplicative
integral. This is done in the context of a Lie crossed module
(an object composed of two Lie groups interacting), and a
pointed manifold. The integrand is a connection-curvature pair,
that consists of a Lie algebra valued 1-form and a Lie algebra
valued 2-form, satisfying a certain differential equation.
The geometric cycle of the integration is a kite in the pointed
manifold. A kite is made up of a 2-dimensional simplex in the
manifold, together with a path connecting this simplex to the
base point of the manifold. The multiplicative integral is an
element of the second Lie group in the crossed module.

We prove several properties of the multiplicative integral.
Among them is the 2-dimensional nonabelian Stokes Theorem,
which is a generalization of Schlesinger's Theorem. Our main result
is the 3-dimensional nonabelian StokesTheorem. This is a totally
new result.

The methods we used are: the CBH Theorem for the nonabelian
exponential map; piecewise smooth geometry of polyhedra; and
some basic algebraic topology.

The motivation for this work comes from twisted deformation
quantization. In the paper (no. 1 above) we encountered a problem of
gluing nonabelian gerbes, where the input was certain data in
differential graded algebras. The 2-dimensional multiplicative
integral gives rise, in that situation, to a nonabelian
2-cochain; and the 3-dimensional Stokes Theorem shows that
this cochain is a twisted 2-cocycle.
(This was superseded by a simpler approach; see no. 3 below.)

[eprint](http://arxiv.org/abs/1007.1250)
[lecture notes](http://www.math.bgu.ac.il/~amyekut/lectures/multi-integ/notes.pdf)
[book](http://www.worldscientific.com/worldscibooks/10.1142/9537)

### 3. Combinatorial Descent Data for Gerbes ###

We consider descent data in cosimplicial crossed groupoids.
This is a combinatorial abstraction of the descent data for gerbes in
algebraic geometry. The main result is this: a weak equivalence between
cosimplicial crossed groupoids induces a bijection on gauge equivalence
classes of descent data. This result is used to construct the twisted quantization in paper no. 1 above (replacing the earlier approach with surface integration).

[eprint](http://arxiv.org/abs/1109.1919)
[lecture notes](http://www.math.bgu.ac.il/~amyekut/lectures/higher-descent/notes.pdf)

**Contact:** email to <amyekut@math.bgu.ac.il>

* [webpage](http://www.math.bgu.ac.il/~amyekut/)

[[!redirects Amnon Yekutieli]]
[[!redirects amnon yekutieli]]
[[!redirects A. Yekutieli]]
[[!redirects Amnon+Yekutieli]]

category: people


ample sheaf

## Definition

### General

(...)

### In complex geometry

In [[complex analytic geometry]] a [[line bundle]] is ample precisely if its [[Chern class]] is represented by a [[Kähler form]].

See also at _[[positive line bundle]]_.

## References

* Wikipedia, _[Ample line bundle](http://en.wikipedia.org/wiki/Ample_line_bundle)_

[[!redirects ample sheaves]]

[[!redirects ample line bundle]]
[[!redirects ample line bundles]]

[[!redirects ample vector bundle]]
[[!redirects ample vector bundles]]

[[!redirects ample module]]
[[!redirects ample modules]]


amplimorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _amplimorphism_ is a way to present [[bimodules]] in terms of linear maps.

For $A$ an [[associative algebra]] (not necessarily unital) over some [[ring]] $R$ (possibly with extra structure, in applications often a [[C*-algebra]]) then an _amplimorphism_ from $A$ to $A$ is an algebra homomorphism of the form

$$
  \alpha \;\colon\; A \longrightarrow A \otimes_R Mat_n(R) \simeq Mat_n(A)
$$

for $n \in \mathbb{N}$ and $Mat_n(R)$ the ring of [[matrices]] with [[coefficients]] in $R$ under [[matrix multiplication]].

This map induces the $R$-[[bimodules]] $N_\alpha \subset A \otimes R^n$ on elements $N_\alpha = \{\psi \;|\;  \alpha(1)\psi = \psi\}$ with left $A$-action given by $\alpha$ and right $A$ action given by componentwise multiplication with $A$ from the right.

(If $R = \mathbb{C}$ and $A$ is a [[C*-algebra]] then this is canonically equipped with the structure of a [[Hilbert bimodule]]).

## References

At least in the context of [[AQFT]] amplimorphisms were  introduced

* K. Szlachányi, K. Vecsernyes, _Quantum symmetry and braid group statistics in $G$-spin models_, Commun. Math. Phys. __156__:1 (1993), 127-168, [euclid](http://projecteuclid.org/euclid.cmp/1104253519) [doi](http://dx.doi.org/10.1007/BF02096735)

The concept is recalled for instance in

* Ezio Vasselli, page 6 of _The $C^\ast$-algebra of a vector bundle and fields of Cuntz algebras_, Journal of Functional Analysis 222(2) (2005), 491-502, [arXiv:math/0404166](http://arxiv.org/abs/math/0404166)

* Fernando Lledó, Ezio Vasselli, section 3.3 of _Realization of minimal $C^\ast$-dynamical systems in terms of Cuntz-Pimsner algebras_ ([arXiv:math.OA/0702775](http://arxiv.org/abs/math.OA/0702775))

[[!redirects amplimorphisms]]


amplitude

+-- {: .rightHandSide}
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### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _amplitude_ of a [[wave]] is a measure for its maximal change with [[space]] and [[time]].

## Definition

For $n \in \mathbb{N}$ and $\mathbb{R}^n$ the corresponding [[Cartesian space]], a [[plane wave]] is a function $\mathbb{R}^n \to \mathbb{C}$ of the form

$$
  \vec x \mapsto  A e^{-2 \pi i \vec k \cdot \vec x}
  \,.
$$

Here $a \in \mathbb{C}$ is the _amplitude_ of the [[plane wave]] (while $\vec k$ s its [[wave vector]]).

## Related concepts

* [[probability amplitude]]

* [[absolute value]]

[[!redirects amplitudes]]


amplituhedron

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

#Contents#
* table of contents
{:toc}

## Idea

The [[scattering amplitudes]] in [[Yang-Mills theory]] are traditionally expressed as sums of typically very many [[Feynman diagrams]], which in turn are [[integrals]] over certain [[algebraic functions]]. At least in  highly [[supersymmetry|supersymmetric]] versions of [[Yang-Mills theory]] such as [[N=4 D=4 super Yang-Mills theory]] this can be expressed more efficiently (see [Dixon 13](#Dixon13) for a review of the general method and see at _[[string theory results applied elsewhere]]_) as fewer [[integrals]] of some integrand over some [[domain]] of certain convex [[polyhedra]] inside a [[positive Grassmanian]] (which is an infinite dimensional space related to the study of the phenomenon of [[total positivity]]). Therefore this has been called the _amplituhedron_ ([Arkani-Hamed & Trnka13](#ArkaniHameTrnka13)).

In slightly more detail, the scattering amplitudes in [[N=4 D=4 super Yang-Mills theory]] can be given as functions of  $4n$ real variables, the [[momenta]] of $n$ scattering [[particles]], and at loop number $\ell$ they are given by $4 \ell$-fold integrals of [[algebraic functions]] of $4n+4 \ell$ real variables. (When $\ell=0$, then ([ABCGPT 12, sections 16.4](#ABCGPT12)) claims that all the [[Galois conjugates]] of the [[algebraic function]] occur symmetrically; so that one indeed has [[rational functions]].)

This formulation is typically a drastic improvement of computational complexity for fixed $k$ ([[helicity]]) and fixed $\ell$ (loop number) but variable $n$. The [[Feynman diagram]] description is exponential in $n$, while the complexity of the amplituhedron description grows much more mildly.

In particular for $k = 0$ and $k = 1$ the resulting amplitude is just 0 and the amplituhedron description gives this immediately, but for $k = 1$ there are still many nontrivial Feynman integrals. That the sum of all these indeed vanish was maybe known earlier, though, due to a result by Parke and Taylor.

For $k = 2$ again there is the "[[Parke-Taylor formula]]" efficiently expressing the amplitudes [[MHV amplitudes]].

See also at [[motives in physics]]

## References

Review:

* {#Dixon13} [[Lance Dixon]], _Calculating Amplitudes_, December 2013 ([web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/))

* [[Henriette Elvang]], [[Yu-tin Huang]], *Scattering Amplitudes*, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)]

* Livia Ferro, Tomasz Lukowski, _Amplituhedra, and Beyond_, Topical Review invited by Journal of Physics A: Mathematical and Theoretical ([arXiv:2007.04342 ](https://arxiv.org/abs/2007.04342))

For scattering amplitudes via the "amplituhedron" the integrand is discussed in

* {#ABCGPT12} [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, [[Alexander Goncharov|Alexander B. Goncharov]], [[Alexander Postnikov]], Jaroslav Trnka, _Scattering amplitudes and the positive Grassmannian_, [arxiv/1212.5605](http://arxiv.org/abs/1212.5605)

and the integration domain in

* {#ArkaniHameTrnka13} [[Nima Arkani-Hamed]], Jaroslav Trnka, _The Amplituhedron_, [arxiv/1312.2007](http://arxiv.org/abs/1312.2007)

Simple aspects of four particle scattering are treated in

* [[Nima Arkani-Hamed]], Jaroslav Trnka, _Into the Amplituhedron_, [arxiv/1312.7878](http://arxiv.org/abs/1312.7878)
 {#ArkaniHameTrnka13b}.

Earlier lecture notes and announcements include

* Jaroslav Trnka, _The amplituhedron_, [pdf](http://www.staff.science.uu.nl/~tonge105/igst13/Trnka.pdf)

* Nima Arkani-Hamed, _Grassmannian polytopes and scattering amplitudes_, lecture at Perimeter Institute, [video](http://pirsa.org/11080047)

Informed online discussion includes

* Logan Maingi on MathOverflow [here](http://mathoverflow.net/questions/142841/the-amplituhedron-minus-the-physics/143421#143421)

* David Speyer [here](http://www.scottaaronson.com/blog/?p=1537#comment-88216)

Journalistic coverage includes

* Natalie Wolchover, _A Jewel at the Heart of Quantum Physics_, Quanta Magazine, Sep 17, 2013 (hosted at Simons Foundation) [html](https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics)

See also

* Wikipedia: _[Amplituhedron](http://en.wikipedia.org/wiki/Amplituhedron)_
* Thomas Lam, _Amplituhedron cells and Stanley symmetric functions_,
[arxiv/1408.5531](http://arxiv.org/abs/1408.5531)


Amélia Liao

* [Website](https://amelia.how/)

## Talks

On [[univalent categories]] in [[homotopy type theory]]:

* [[Amélia Liao]], *Univalent Category Theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 6 October 2022, ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Liao-2022-10-06-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=asY6dXkR2yg))

On [[displayed categories]]:

*  [[Amélia Liao]], [*Displayed Categories as Building Blocks*](https://www.fmf.uni-lj.si/en/news/event/698/amelia-liao-displayed-categories-as-building-blocks/), [Seminar for foundations of mathematics and theoretical computer science](https://www.fmf.uni-lj.si/en/research/seminar-for-foundations-of-mathematics/), 22 September 2022, [Faculty of Mathematics and Physics](https://www.fmf.uni-lj.si/en/), University of Ljubljana

## Related entries

* [[1lab]]

category: people

[[!redirects Amelia Liao]]


An Descheemaeker

* [Institute page I](https://www.kuleuven.be/wieiswie/nl/person/00011206)

* [Institute page II (publications)](https://kuleuven.academia.edu/AnDescheemaeker)

## Selected writing

On relative [[completions of groups]]:

* [[Carles Casacuberta]], [[An Descheemaeker]], *Relative group completions*, Journal of Algebra, **285** 2 (2005) 451-469 [[doi:10.1016/j.jalgebra.2004.10.023](https://doi.org/10.1016/j.jalgebra.2004.10.023)]

category: people


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

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

## Purpose of this page

The purpose of this page is to examine how much of the theory of [[Hilbert space]]s can be done in an "elementary" fashion.  More specifically, to develop as much as possible of the standard theory without using the fact that Hilbert spaces are special normed [[vector space]]s, or special [[metric space]]s.

The reason for trying this is in the spirit of [[centipede mathematics]].  One can view the theory of [[locally convex space|locally convex topological vector spaces]] as the result of pulling off most of the legs of Hilbert spaces, though perhaps one should go one step further and say that Hilbert spaces themselves are the result of pulling the "finite dimensional" leg off the Euclidean centipede.  Indeed, this is almost the standard treatment of LCTVSs except that the usual starting point is [[normed vector space|normed vector spaces]] rather than Hilbert spaces.  In that view, Hilbert spaces are special normed vector spaces rather than normed vector spaces being Hilbert spaces without a leg or two.

Although the intention is that the treatment be elementary, we shall remark on the relationship to the standard theory and thus the commentary will not necessarily be elementary.  For example, concepts that are traditionally defined by using the metric space structure of a Hilbert space will need recasting and we shall need to reassure the reader that the new definition is equivalent to the original.

We shall work over $\mathbb{C}$ throughout.

## Basic Definitions

We start with the basic definition of an inner product space.

+-- {: .num_defn #ipsp}
###### Definition
An **inner product space** is a vector space, say $V$, equipped with a function $|V| \times |V| \to \mathbb{C}$, written $\langle u, v \rangle$, satisfying:

1. $\langle v, u\rangle = \overline{\langle u, v\rangle}$,
2. $\langle v + \lambda w, u \rangle = \langle v, u \rangle + \lambda \langle w, u\rangle$,
3. $\langle v, v\rangle \in [0,\infty)$ with $\langle v, v\rangle = 0$ if and only if $v = 0$.
=--

Ideally, we want to deal solely with Hilbert spaces but first we need to figure out how to deal with completeness without recourse to metric space theory.  We do this by using orthonormal families.

+-- {: .num_defn #ofam}
###### Definition
Let $(V,\langle -, - \rangle)$ be an inner product space.  An **orthogonal family** in $V$ is a subset $B \subseteq V$ with the property that $\langle b, b'\rangle = 0$ whenever $b,b' \in B$ are distinct.

The family is said to be **orthonormal** if, in addition, $\langle b, b\rangle = 1$ for all $b \in B$.

Two vectors, say $u$ and $v$, are said to be orthogonal if the family $\{u,v\}$ is an orthogonal family.
=--

Using orthogonal families, we can express the notion of completeness as follows.

+-- {: .num_defn #hilb}
###### Definition
A **Hilbert space** is an inner product space, $(H, \langle -, -\rangle)$ in which the following property holds.  Let $(b_n)$ be an orthonormal sequence and $(\lambda_n)$ a sequence of positive real numbers such that $\sum \lambda_n^2$ is bounded.  Then there is some $v \in H$ such that for all $u \in H$,
$$
\sum \lambda_n \langle b_n, u \rangle = \langle v, u \rangle.
$$
=--

+-- {: .un_remark}
###### Remark
We need to justify this notion of completeness.  One direction is simple: the sequence of partial sums of the series $\sum \lambda_n b_n$ is Cauchy and so if the space is complete, it has a limit and this limit satisfies the criterion.

The other direction takes a little more effort.  The quickest (but dirtiest) route is simply to observe that if that condition is satisfied then whenever we have an isometry from $\ell^0$ (with its standard inner product) into our space then it extends to $\ell^2$.  A slightly more concrete route is as follows.  Start with a Cauchy sequence in $H$, say $(x_n)$, and then apply the [[Gram–Schmidt process]].  This results in an orthonormal sequence, say $(b_n)$.  For each $k$, we define $\lambda_k$ as the limit (in $\mathbb{C}$) of $(\langle x_n, b_k\rangle)$.  We can make $\lambda_k$ a positive real number by taking the phase factor in to $b_k$.  Let $(s_n)$ be the sequence of partial sums of the series $\sum \lambda_k b_k$.  Then $(s_n)$ is Cauchy and the interpolation of $(s_n)$ and $(x_n)$ is also Cauchy.  By assumption, $(s_n)$ has a weak limit.  As it is Cauchy, the existence of a weak limit is enough to show that it has a strong limit.  Thus $(x_n)$ also converges and so $H$ is complete.

The tenor of the new definition and its equivalence to the standard one is a theme that will run throughout this page.  In our elementary treatment, weak definitions are to be preferred to the standard strong ones.  Their equivalence exposes some of the deep results of Hilbert space theory.
=--

As this is intended as an elementary treatment, it is likely that at some point we will want to assume that our Hilbert space is "small", by which (of course) we mean "[[separable space|separable]]".  Fortunately, it is not hard to formulate separability without recourse to metric spaces.

+-- {: .num_defn #sep}
###### Definition
An inner product space is **separable** if it contains a sequence, say $(x_n)$ with the property that $\langle x, x_n\rangle = 0$ for all $n$ implies that $x = 0$.
=--

## Cauchy--Schwarz

The Cauchy--Schwarz inequality is one of the basic results of Hilbert space theory.  As we wish to avoid any mention of a norm, we state it as follows.

+-- {: .num_proposition #csineq}
###### Proposition
Let $H$ be a Hilbert space, $u,v \in H$.  Then
$$
|\langle u, v \rangle|^2 \le \langle u, u \rangle \langle v, v \rangle
$$
with equality if and only if $u$ and $v$ are collinear.
=--

+-- {: .query}
I removed the square roots here, since there doesn\'t seem to be much point to them if you\'re not going to connect with a metric.  (On the other hand, surely even an elementary treatment of Hilbert spaces may deign to mention the geometric concept of distance?  It\'s one thing to not assume a knowledge of metric spaces, but it\'s another thing to refuse to even mention norms.)  ---Toby

As you can (probably) tell, I'm making this up as I go along!  I've not decided yet whether to allow distances or not, so at the moment I'm avoiding them if possible.  Part of my goal is to avoid overwhelming the "reader" with notation just for the sake of it.  So I may well implicitly use the norm as $\langle v, v\rangle$ but without introducing the $\|v\|$ notation.  But although I expect I'll be the main contributor here, I don't particularly want to be so I'm more than happy for others to weigh in with their opinions on what an "elementary treatment" would look like.  If I disagree then it'll force me to think carefully _why_ I disagree and that can only improve things.

Pondering a little more, I think that I'm avoiding distance not because I don't think it belongs in an elementary treatment - as you imply, what could be more simple to understand than distance? - but because it's a way of keeping metric spaces at bay: once I start talking about distances then it'll be easy to talk about metrics and the like.

I think you're right about the square roots, by the way.  ---Andrew

Since accidentally using metric space theory is always a danger, I understand not wanting to mention distances and norms right away.  On the other hand, I do think that any introductory treatment ought to mention them at some point, and even to prove that they satisfy the triangle inequality and Cauchy completeness.  But those should be theorems specifically about the concept of distance in a Hilbert space, not anything fundamental to the development of the general theory of Hilbert spaces.  So sure, keep them out for now.  ---Toby
=--

+-- {: .proof}
###### Proof
The "if and only if" part gives us the key to the most direct proof.  It will simplify matters a little if we assume that $u$ and $v$ are non-zero, the case where one of them is zero being easy to establish.

Now if $u$ and $v$ are collinear then there is some $\lambda \in \mathbb{C}$ such that $u = \lambda v$.  There is only one possibility for $\lambda$ which can be found by taking inner products with $v$: if $\lambda$ exists such that $u = \lambda v$ then $\lambda = \langle u, v\rangle/\langle v, v\rangle$.  So we consider the question: is $\langle v, v\rangle u = \langle u, v\rangle v$?  Or, equivalently, is $\langle v, v\rangle u - \langle u, v\rangle v = 0$?

We have a test for when a vector is non-zero using the inner product.  One of the axioms for an inner product says that a vector $w$ is zero if and only if $\langle w, w\rangle = 0$.  Moreover, we also know that in any case $\langle w, w\rangle \ge 0$.  Thus we have

\[
\label{csineq}
\left\langle \langle v, v\rangle u - \langle u, v\rangle v, \langle v, v\rangle u - \langle u, v\rangle v \right\rangle \ge 0
\]

with equality if and only if $u$ and $v$ are collinear.

Before expanding out the left hand side of this, we note that

$$
\left\langle \langle v, v\rangle u - \langle u, v\rangle v , v \right\rangle = 0
$$

whence \eqref{csineq} simplifies to

$$
\left\langle \langle v, v\rangle u - \langle u, v\rangle v, u \right\rangle \ge 0
$$

with equality if and only if $u$ and $v$ are collinear.  Expanding this out yields

$$
\langle v,v\rangle \langle u,u\rangle - \langle u, v \rangle \langle v, u\rangle \ge 0
$$

with equality if and only if $u$ and $v$ are collinear.  Rearranging and square-rooting produces the traditional statement of the Cauchy--Schwarz inequality.
=--

## Subspaces and Complements


An Essay on the Foundations of Geometry

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

* [[Bertrand Russell]],

  _An Essay on the Foundations of Geometry_,

  Cambridge 1897,

  [online version](http://bertrandrussellsocietylibrary.org/br-efg/br-efg.html)

In his early philosophical work Russell concerned himself with German [[idealism]], which was at that time propounded in Britain by [[J.M. Ellis McTaggart]], while he studied as well the mathematical innovations of the 19th century. The latter questioned the [[Kant|Kantian]] view of [[space]] -- especially in [[geometry]] where a developing plethora of new concepts of space, e.g. [[metric space]] or [[Riemannian manifold]], were hardly compatible with the Kantian Transcendental Aesthetic in which there exists a perception ["Anschauung"] of a space such that every space occurs only as a subspace of this space ([[Critique of Pure Reason]] [A24](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_024.png)/B39):

> ... when we talk of divers spaces, we mean only parts of one and the same space.

In the essay under consideration Russell proposes an alternative version of Kant's argument that continues the idea of a space given _a priori_, but only as a "form of externality" and not a concrete perception of a space. About this space three properties can be deduced _a priori_ (constant [[curvature]], [[finite number|finite]] [[dimension|dimensionality]], and existence of straight [[lines]]). At the end of his essay Russell recapitulates:

> ...

> In the second chapter, we endeavoured, by a criticism of some geometrical philosophies, to prepare the ground for a constructive theory of Geometry. We saw that Kant, in applying the argument of the Transcendental Aesthetic to space, had gone too far, since its logical scope extended only to some form of externality in general. We saw that Riemann, Helmholtz and Erdmann, misled by the quantitative bias, overlooked the qualitative substratum required by all judgments of quantity, and thus mistook the direction in which the necessary axioms of Geometry are to found. We also rejected Helmholtz's view that Geometry depends on Physics, because we found that Physics must assume a knowledge of Geometry before it can become possible. ...

> Proceeding, in the third chapter, to a constructive theory of Geometry, we saw that projective Geometry, which has no reference to quantity, is necessarily true for any form of externality. Its three axioms---homogeneity, dimension, and the straight line---were all deduced from the conception of a form of externality, and, since some such form is necessary to experience, were all declared _a priori_. In metrical Geometry, on the contrary, we found an empirical element arising out of the alternatives of Euclidean and non-Euclidean space. ...

> In the present chapter, we completed our proof of the apriority of the projective and equivalent metrical axioms by showing the necessity, for experience, of some form of externality, given by sensation or intuition, and not merely inferred from other data. Without this, we said, a knowledge of diverse but interrelated things, the corner-stone of all experience, would be impossible. ...

## Related entries

* [[space form]]

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

## References

* [[Kant]], _[[Critique of Pure Reason]]_, second edition (referenced to as "B") 1787

category: reference


An Exercise in Kantization

## Motivation

Once, [[Johan Alm]] wrote some very neat notes:

* [Quantization as a Kan Extension](http://ncatlab.org/nlab/files/Kantization09May27.pdf)

A discussion is available at the [$n$Café](https://golem.ph.utexas.edu/category/2009/05/alm_on_quantization_as_a_kan_e.html). The paper proves an aspect of the "[$n$-Café Quantum Conjecture](http://golem.ph.utexas.edu/category/2007/06/the_ncaf_quantum_conjecture.html)", suggested by [[Urs Schreiber]]. There is some old discussion at previous versions of the page [[Quantization as a Kan Extension]], and some on-page discussion that is now stored [at the nForum](https://nforum.ncatlab.org/discussion/12529/an-exercise-in-kantization/?Focus=90941#Comment_90941).

## Further work

Discussion of quantization in terms of Kan extension and [[pushforward in generalized cohomology]] is now at _[[motivic quantization]]_, with notes being at _[[schreiber:master thesis Nuiten]]_ and _[[schreiber:Quantization via Linear homotopy types]]_ .

category: reference


An Huang

* [webpage](http://people.brandeis.edu/~anhuang/)

* [institute page](http://www.brandeis.edu/facultyguide/person.html?emplid=e225bcb771bc4cefeed959ca4b802ac0055303e5)

## selected writings

* {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013))

  (on [[p-adic string theory]] and the [[Riemann zeta function]])

## related $n$Lab entries

* [[p-adic string theory]]

* [[Riemann zeta function]]

category: people


An Introduction to Homological Algebra

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

This entry provides a hyperlinked index for the textbook

* [[Charles Weibel]],

  **An Introduction to Homological Algebra**

  Cambridge University Press (1994)

  [doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136)

  [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)

which gives a first exposition to central concepts in _[[homological algebra]]_.

For a more comprehensive account of the theory see also chapters 8 and 12-18 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

and see the $n$Lab lecture notes

* _[[schreiber:Introduction to Homological Algebra]]_ ([[IntroductionToHomologicalAlgebra-170509.pdf:file]])

#Contents#
* table of contents
{:toc}

## 1 Chain complexes

### 1.1 Complexes of $R$-modules

* [[abelian group]], [[commutative ring]], [[module]]

* [[exact sequence]]

**Definition 1.1.1** [[chain complex]]

* [[chain map]], [[chain homotopy]]

* [[category of chain complexes]]

**Exercise 1.1.2** [homology is functorial](chain+map#OnHomology)

**Exercise 1.1.3** [exact sequences of chain complexes are split](split+exact+sequence#OfVectorSpaces)

**Exercise 1.1.4** [[internal hom of chain complexes]]

**Definition 1.1.2** [[quasi-isomorphism]]

[[cochain complex]], [[bounded chain complex]]

**Exercise 1.1.5** [exactness and weak nullity](exact+sequence#ExactnessAndQuasiIsomorphisms)

**Application 1.1.3** [[chain on a simplicial set]], [[simplicial homology]]

**Exercise 1.17** [simplicial homology of the tetrahedron](simplicial+homology#OfTetrahedon)

**Application 1.1.4** [[singular homology]]

### 1.2 Operations on chain complexes

* [[additive and abelian categories]]

* [[Ab-enriched category]]

  * [[pre-additive category]]

  * [[additive category]],

    * [[additive functor]]

**Exercise 1.2.1** [homology respects direct product](chain+homology+and+cohomology#ChainHomologyRespectsDirectProduct)

**Definition 1.2.1** [[kernel]], [[cokernel]]

* [[quotient]]

**Exercise 1.2.2** [in an abelian category kernels/cokernels are the monos/epis](Mod#RModIsAbelian)

**Exercise 1.2.3** [(co)kernels of chain maps are degreewise (co)kernels](category+of+chain+complexes#KernelsOfChainComplexes)

**Definition 1.2.2** [[abelian category]], [[abelian subcategory]]

**Theorem 1.2.3** [a category of chain complexes is itself abelian](category+of+chain+complexes#IsAbelian)

**Exercise 1.2.4** [exact sequence of chain complexes is degreewise exact](category+of+chain+complexes#ShortExactSequencesDegreewise)

$R$[[Mod]]

**Example 1.2.4** [[double complex]]

**Sing trick 1.2.5** [double complex with commuting/anti-commuting differentials](double+complex#EquivalenceOfTheTwoDefinitions)

**Total complex 1.2.6** [[total complex]]

**Exercise 1.2.5** [total complex of a bounded degreewise exact double complex is itself exact](total+complex#TotOfBoundedDegreewiseExactIsExact)

**Example 1.2.4** [[double complex]]

**Truncations 1.2.7** [[truncation of a chain complex]]

**Translation 1.2.8** [[suspension of a chain complex]]

**Exercise 1.2.8** [[mapping cone]]

### 1.3 Long exact sequences

**Theorem 1.3.1** [[connecting homomorphism]], [[long exact sequences in homology]]

**Exercise 1.3.1** [[3x3 lemma]],

**Snake lemma 1.3.2** [[snake lemma]]

**Exercise 1.3.3** [[5 lemma]]

**Remark 1.3.5** [[exact triangle]]

### 1.4 Chain homotopies

* [[homotopy theory]]

**Definition 1.4.1** [[split exact sequence]]

**Exercise 1.4.1** [splitness of exact sequences of free modules](split+exact+sequence#OfVectorSpaces)

**Definition 1.4.3** [[null homotopy]]

**Exercise 1.4.3** [split exact means identity is null homotopic](split+exact+sequence#RelationToChainHomotopy)

**Definition 1.4.4** [[chain homotopy]]

**Lemma 1.4.5** [chain homotopy respects homology](chain%20map#OnHomology)

**Exercise 1.4.5** [[homotopy category of chain complexes]]

### 1.5 Mapping cones and cyclinders

**1.5.1** [[mapping cone]]

**1.5.5** [[mapping cylinder]]

**1.5.8** [[fiber sequence]]

### 1.6 More on abelian categories

**Theorem 1.6.1** [[Freyd-Mitchell embedding theorem]]

**Functor categories 1.6.4** [[functor category]]

[[presheaf]]

**Definition 1.6.5** [[abelian sheaf]]

**Definition 1.6.6** left/right [[exact functor]]

**Yoneda embedding 1.6.10** [[Yoneda embedding]]

**Yoneda lemma 1.6.11** [[Yoneda lemma]]

[proof of the Freyd-Mitchell embedding theorem](http://ncatlab.org/nlab/show/Freyd-Mitchell+embedding+theorem#Proof)

## 2 Derived functors

[[derived functor in homological algebra]]

### 2.1 $\delta$-Functor

**Definition 2.1.1** [[delta-functor]]

### 2.2 Projective resolutions

[[projective module]] ([[cofibrant object]] in the [[model structure on chain complexes]])

**Definition 2.2.4**  [[projective resolution]] ([[cofibrant replacement]])

**Horseshoe lemma 2.2.8** [[horseshoe lemma]]

### 2.3 Injective resolutions

[[injective module]] ([[fibrant object]] in the other [[model structure on chain complexes]])

**Baer's criterion 2.3.1** [[Baer's criterion]]

**Definition 2.3.5** [[injective resolution]] ([[fibrant replacement]])

**Definition 2.3.9** [[adjoint functor]]

### 2.4 Left derived functors

[[left derived functor]]

### 2.5 Right derived functors

[[right derived functor]]

**Application 2.5.4** [[global section functor]], [[abelian sheaf cohomology]]

### 2.6 Adjoint functors and left/right exactness

[[adjoint functor]]

**Definition 2.6.4** [[Tor]]

**Application 2.6.5** [[sheafification]]

**Application 2.6.6** [[direct image]], [[inverse image]]

**Application 2.6.7** [[colimit]]

**Variation 2.6.9** [[limit]]

**Definition 2.6.13** [[filtered category]], [[filtered colimit]]

### 2.7 Balancing $Tor$ and $Ext$

**Tensor product of complexes 2.7.1** [[tensor product of chain complexes]]

**Lemma 2.7.3** [[acyclic assembly lemma]]

## 3 Tor and Ext

[[Tor]] and [[Ext]]

### 3.1 $Tor$ for abelian groups

**Proposition 3.1.2-3.1.3** [relation to torsion subgroups](Tor#RelationToTorsionGroups)

### 3.2 $Tor$ and flatness

**Definition 3.2.1** [[flat module]]

**Definition 3.2.3** [[Pontrjagin duality]]

**Flat resolution lemma 3.2.8** [[flat resolution lemma]]

**Corollary 3.2.13** [Localization for Tor](Tor#Localization)

### 3.3 $Ext$ for nice rings

**Corollary 3.3.11** [Localization for Ext](Ext#Localization)

### 3.4 $Ext$ and extensions

[[extension]]

[[group extension]]

**Vista 3.4.6** [[Yoneda extension group]]

### 3.5 Derived functors of the inverse limit

[[tower]]

[[additive and abelian categories|(AB4)-category]]

[[directed limit]]

**Definition 3.5.1** [[lim^1]]

**Definition 3.5.6** [[Mittag-Leffler condition]]

**Exercise 3.5.5** [[pullback]]

### 3.6 Universal coefficient theorem

**Theorem 2.6.1** [[Künneth formula]]

**Universal cofficient theorem for homology 3.6.2** [universal coefficient theorem in homology](universal+coefficient+theorem#InHomology)

**Theorem 3.6.3** [[Künneth formula for complexes]]

**Application 3.6.4** [universal coefficient theorem in topology](universal+coefficient+theorem#InTopology)

**Universal coefficient theorem in cohomology 3.6.5** [universal coefficient theorem in cohomology](universal+coefficient+theorem#InCohomology)

[[Eilenberg-Zilber theorem]]

**Exercise 3.6.2** [[hereditary ring]]

## 4 Homological dimension

### 4.1 Dimensions

* [[dimension]]

* [[global dimension theorem]]

* [[homological dimension]]

### 4.2 Rings of Small Dimension

### 4.3 Change of Rings Theorem

### 4.4 Local rings

* [[local ring]]

### 4.5 Koszul Complexes

* [[Koszul complex]]

### 4.6 Local Cohomology

## 5 Spectral sequences

### 5.1 Introduction

### 5.2 Terminology

* [[spectral sequence]]

### 5.3 Leray-Serre Spectral Sequence

* [[Leray-Serre spectral sequence]]

### 5.4 Spectral sequence of a filtration

* [[spectral sequence of a filtration]]

### 5.5 Convergence

### 5.6 Spectral sequence of a double complex

* [[spectral sequence of a double complex]]

### 5.7 Hypercohomology

* [[hypercohomology]]

### 5.8 Grothendieck spectral sequence

* [[Grothendieck spectral sequence]]

### 5.9 Exact couples

* [[exact couple]]

* [[Bockstein spectral sequence]]

## 6 Group homology and cohomology

* [[group cohomology]]

## 7 Lie algebra homology and cohomology

* [[Lie algebra homology]], [[Lie algebra cohomology]]

## 8 Simplicial methods in homological algebra

### 8.1 Simplicial object

* [[simplicial object]]

### 8.2 Operations on simplicial objects

### 8.3 Simplicial homotopy groups

* [[simplicial homotopy group]]

### 8.4 The Dold-Kan correspondence

* [[Dold-Kan correspondence]]

### 8.5 The Eilenberg-Zilber theorem

* [[Eilenberg-Zilber theorem]]

### 8.6 Canonical resolutions

* [[monad]](=triple) [[comonad]]

  [[bar construction]],

  [[classifying space]]

### 8.7 Cotriple homology

* [[bar construction]]

* [[monadic descent]]

### 8.8 Andre-Quillen Homology and Cohomology

* [[Kähler differential]]

* [[cotangent complex]]

## 9 Hochschild and cyclic homology

### 9.1 Hochschild Homology and Cohomology of Algebras

* [[Hochschild cohomology]]

### 9.2 Derivations, Differentials and Separable Algebra

* [[derivation]]

* [[differential]]

* [[separable algebra]]

### 9.3 $H^2$, Extensions, Smooth Algebras

* [[smooth algebra]]

### 9.4 Hochschild products

* [[Hochschild-Kostant-Rosenberg theorem]]

### 9.5 Morita Invariance

### 9.6 Cyclic Homology

* [[cyclic cohomology]]

### 9.7 Group Rings

### 9.8 Mixed Complexes

### 9.9 Graded Algebras

### 9.10 Lie Algebras of Matrices

## 10 The derived category

* [[derived category]]

## A Category Theory Language

* [[category theory]]

### A.1 Categories

* [[category]]

### A.2 Functors

* [[functor]]

### A.3 Natural transformations

* [[natural transformation]]

### A.4 Abelian categories

* [[abelian category]]

### A.5 Limits and colimits

* [[limit]], [[colimit]]

### A.6 Adjoint functors

* [[adjoint functor]]

category: reference

[[!redirects An introduction to homological algebra]]


Ana Canas da Silva

* [webpage](https://people.math.ethz.ch/~acannas/)

## related $n$Lab entries

* [[geometric quantization]]

* [[geometric quantization by push-forward]]

category: people


Ana Cannas da Silva

* [webpage](http://www.math.ethz.ch/~acannas/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ana_Cannas_da_Silva)

## Selected writings

On [[geometric quantization]]:

*  [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], *Quantization of Presymplectic Manifolds and Circle Actions*, Transactions of the AMS **352** 2 (2000) 525-552  [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)]

On [[symplectic geometry]]:

* [[Ana Cannas da Silva]], *Lectures on Symplectic Geometry*, Lecture Notes in Mathematics **1764**, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7)]

## Related entries

* [[Kähler manifold]]

* [[tubular neighbourhood]]

* [[tubular neighbourhood theorem]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

category: people

[[!redirects Ana Cannas]]


Ana-Maria Raclariu

* [website](https://perimeterinstitute.ca/people/ana-maria-raclariu)

## Selected publications

On [[celestial holography]]:

* [[Nima Arkani-Hamed]], Monica Pate, [[Ana-Maria Raclariu]], [[Andrew Strominger]], *Celestial Amplitudes from UV to IR*, JHEP 08 (2021) 062 ([arXiv:2012.04208](https://arxiv.org/abs/2012.04208))

* [[Ana-Maria Raclariu]], *Lectures on Celestial Holography* ([arXiv:2107.02075](https://arxiv.org/abs/2107.02075))

category:people


anabelian geometry

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#### Geometry
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#Contents#
* table of contents
{:toc}

## Idea

In _anabelian geometry_ one studies how much information about a [[space]] $X$ (specifically: an [[algebraic variety]]) is contained already in its first [[étale homotopy group]] $\pi^{et}_1(X,x)$ (specifically: the [[algebraic fundamental group]]).

The term "anabelian" is supposed to be alluding to the fact that "the _less_ [[abelian group|abelian]] $\pi^{et}_1(X,x)$ is, the more information it carries about $X$." Precisely: an [[anabelian group]] is a non-[[trivial group]] for which every [[finite index subgroup]] has trivial [[center]].

Accordingly, an algebraic variety whose [[isomorphism class]] is entirely determined by $\pi^{et}_1(X,x)$ is called an **anabelian variety**.

An early conjecture motivating the theory (in [Grothendieck 84](#Grothendieck)) was that all [[hyperbolic curves]] over [[number fields]] are anabelian varieties. This was eventually proven by various authors in various cases.  In ([Uchida](#Uchida)) and ([Neukirch](#Neukirch)) it was shown that an isomorphism between [[Galois groups]] of [[number fields]] implies the existence of an isomorphism between those number fields. For algebraic curves over [[finite fields]], over [[number fields]] and over [[p-adic field]] the statement was eventually completed by ([Mochizuki 96](#Mochizuki96)).

Grothendieck also conjectured the existence of higher-dimensional anabelian varieties, but these are still very mysterious.

## Étale homotopy types

* [[Alexander Schmidt]], [[Jakob Stix]], _Anabelian geometry with étale homotopy types_, ([pdf](https://www.mathi.uni-heidelberg.de/~schmidt/papers/anab-hotype.pdf)).

##Related entries

* [[algebraic fundamental group]] also called the 'geometric fundamental
group' by Grothendieck.

* [[child's drawing]]/ Dessins d'enfant.

* [[inter-universal Teichmüller theory]]

* [[profinite completion]]

* [[Grothendieck-Teichmüller group]]

* [[section conjecture]]

* [[arithmetic geometry]]

## References

The notion of anabelian geometry originates in

* [[Alexandre Grothendieck]], letter to Faltings (June 27, 1983) ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf)),

written in response to [[Faltings]]' work on the [[Mordell Conjecture]], and the note of the [[Long March]]:

* [[Alexandre Grothendieck]], La Longue Marche, Notes ([link](https://agrothendieck.github.io/galoispoincaregrothendieck/galois.pdf)).

There is some discussion of the area in

* {#Grothendieck} [[Alexander Grothendieck]], _[[Esquisse d'un programme]]_
 (1984).

A relation with the theory of [[motive]]s is in

* [[Alexander Schmidt]], _Motivic aspects of Anabelian geometry_, Advanced studies in pure mathematics 63, 2012. Galois-Teichmüller theory and Arithmetic geometry. pp 503 - 517 [pdf](https://www.mathi.uni-heidelberg.de/~schmidt/papers/63117.pdf)

Surveys:

* {#Tschinkel14} [[Yuri Tschinkel]], _Introduction to anabelian geometry_, talk at _[Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html)_, Oxford 2014 ([slides pdf](http://www.cims.nyu.edu/~tschinke/.talks/oxford14/oxford14.pdf))

* [[Leila Schneps]], page 60 (2) of _Grothendieck's "Long march through Galois theory"_ ([pdf](http://www.math.jussieu.fr/~leila/SchnepsLM.pdf))

A comprehensive introduction is in

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Introduction to birational anabelian geometry* [[pdf](http://www.math.nyu.edu/~tschinke/papers/yuri/10msri/msri7.pdf), [arXiv:1011.0883](https://arxiv.org/abs/1011.0883)]

Examples are discussed in

* Yasutaka Ihara, Hiroaki Nakamura, _Some illustrative examples for anabelian geometry in high dimensions_, in [[Leila Schneps]], P. Lochak (eds) _Geometric Galois Actions I_, London Math. Soc. Lect. Note Series 242 ([pdf](http://www.math.okayama-u.ac.jp/~h-naka/zoo/lion/INanabel.pdf))

The classification of anabelian varieties for [[number fields]] was shown in

* J. Neukirch, _Kennzeichnung der $p$-adischen und der endlichen algebraischen Zahlkörper_, Invent. Math. 6 (1969), p. 296–314.
 {#Neukirch}

* J. Neukirch, _Über die absoluten Galoisgruppen algebraischer Zahlkörper_, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris (1977)

* {#Uchida} K. Uchida. _Isomorphisms of Galois groups_, J. Math. Soc. Japan 28 (1976), no. 4, 617–620.

* K. Uchida, _Isomorphisms of Galois groups of algebraic function fields_, Ann. Math. (2) 106 (1977), no. 3, p. 589–598.

and for [[algebraic curves]] in

* [[Shinichi Mochizuki]], _The profinite Grothendieck conjecture for hyperbolic curves over number fields_, J. Math. Sci. Univ. Tokyo 3 (1996), 571–627.
 {#Mochizuki96}

* [[Shinichi Mochizuki]], _The absolute anabelian geometry of canonical curves_ (2002) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Canonical%20Liftings.pdf))

See also

* Frans Oort, [Lecture notes](http://www.staff.science.uu.nl/~oort0109/IC.AnabelianWorkshop.ps). Informal notes (not for publication) made available for the Lorentz Center workshop 'Anabelian number theory and geometry', December 3-5, 2001

* [[Nikolai Durov|N. V. Durov]], _Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties)_, preprint POMI 13/2012 (in Russian) [abstract](http://www.pdmi.ras.ru/preprint/2012/12-13.html), [pdf.gz](ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2012/13-12_rus.pdf.gz)

* [[Taylor Duypuy]], _[Anabelian geometry](https://www.youtube.com/playlist?list=PLJmfLfPx1OedXBno8vlpGd89_3wsdfCEc)_

* {#Morrow17} [[Jackson Morrow]], _Kummer classes and Anabelian geometry_, notes from Super [QVNTS](http://www.dms.umontreal.ca/%7Eqvnts/): [Kummer Classes and Anabelian Geometry](http://www.uvm.edu/%7Etdupuy/anabelian.html) 2017 ([pdf](https://www.dropbox.com/s/kiszgkrrsjrqu4p/VermontNotes_Final.pdf?dl=0))

[[!redirects anabelian variety]]
[[!redirects anabelian varieties]]


anabelian group

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Idea

An _anabelian group_ is a group that is "far from being an [[abelian group]]" in a precise sense:

## Definition

It is a non-[[trivial group]] for which every [[finite index subgroup]] has trivial [[center]].

## Related concepts

* [[anabelian geometry]].

[[!redirects anabelian groups]]


anabelioid

\tableofcontents

\section{Introduction}

An anabelioid is a category intended to play the role of a 'generalised geometric object' in algebraic/arithmetic geometry. Its definition is simple: a finite product of [[Galois category|Galois categories]], or in other words of [[classifying topos|classifying topoi]] of [[profinite group|profinite groups]]. The significance comes from the fact that in [[anabelian geometry]], an [[algebraic variety]] is essentially determined by its [[étale fundamental group|algebraic fundamental group]], which arises from a Galois category associated to the algebraic variety. The idea, due to [[Shinichi Mochizuki]], is that one can develop the geometry of these Galois categories themselves, and products of Galois categories in general; thus, develop a form of categorical algebraic geometry.

To quote from Remark 1.1.4.1 of [Mochizuki2004](#Mochizuki2004):

> >_The introduction of anabelioids allows us to work with both "algebro-geometric anabelioids" (i.e., anabelioids arising from (anabelian) varieties) and "abstract anabelioids" (i.e., those which do not
necessarily arise from an (anabelian) variety) as geometric objects on
an equal footing.
_
>
>The reason that it is important to deal with "geometric objects" as opposed to groups, is that:
>
> > _We wish to study what happens as one **varies the basepoint** of one of these geometric objects._

\section{Details}

The following definitions follow [Mochizuki2004](#Mochizuki2004).

\begin{defn} A _connected anabelioid_ is exactly a [[Galois category]]. \end{defn}

\begin{defn} An _anabelioid_ is a category equivalent to a [[finite product]] of connected anabelioids, that is, to a finite product of Galois categories. \end{defn}

\begin{rmk} An anabelioid is also known as a _multi-Galois category_. \end{rmk}

\section{Associated notions}

* [[finite étale morphism of anabelioids]]

[[!redirects multi-Galois category]]

\section{References}

* {#Mochizuki2004} _The geometry of anabelioids_, [[Shinichi Mochizuki]], 2004, Publ. Res. Inst. Math. Sci., 40, No. 3, 819-881. [paper](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Anabelioids.pdf) [Zentralblatt review](https://zbmath.org/?q=an%3A1113.14021)


anabicategory

An **anabicategory** is a particular notion of _weak [[2-category]]_ appropriate in the absence of the [[axiom of choice]] (including in many [[internal category|internal]] contexts).  It is derived from the notion of [[bicategory]] by replacing the composition functors $\circ: B(y,z) \times B(x,y) \to B(x,z)$ with [[anafunctor]]s (and therefore the [[associator]]s and [[unitor]]s with [[ananatural transformation|ananatural transformations]]).

+-- {.query}
[[Zoran Škoda]]: I understand that there is a version of $Cat$ using anafunctors, but it is not clear to me
what are the axioms for a variant of 2-category which this example belongs to. I do not understand what you mean by replacing hom-functor with anafunctor in that definition: I mean which definition of bicategory is phrased in terms of hom-functors; standard definition talks associators and so on...Please be more explicit.

[[Mike Shulman|Mike]]: Is this version clearer?

_Toby_:  Or look at the complete definition in the reference that I just added.
=--

If [[Cat]] is defined as consisting of ([[small category|small]]) [[category|categories]], anafunctors, and [[ananatural transformation|ananatural transformations]] (as is most appropriate in the absence of choice), then $Cat$ is more naturally an anabicategory rather than any stricter notion.

+--{: .query}
[[Mike Shulman|Mike]]: Is that only because of the non-canonicity of pullbacks in $Set$?  If our version of $Set$ has canonical chosen pullbacks, do we get an ordinary bicategory?

_Toby_:  H\'m ... as I recall, you get an ordinary bicategory using canonical pullbacks and general anafunctors, but if you move to *saturated* anafunctors, then you still only get an anabicategory.  I should check this and then rewrite (here and at [[Cat]]) to say it correctly.  (Of course, if you really use anafunctors, then you should only *want* an anabicategory.)
=--

# Explicit definition #

An anabicategory $\mathcal{C}$ consists of a set $\mathcal{C}_0$ of objects, a set $\mathcal{C}_1$ of 1-cells, which are relations $\mathcal{C}_0 \times \mathcal{C}_0$, and a set $\mathcal{C}_2$ of 2-cells, which are relations $\mathcal{C}_1 \times \mathcal{C}_1$. (Note that only $\mathcal{C}_2$ needs a defined equality relation, so the other two sets only need to be [[preset]]s.) On these sets are defined several relations; for each relation $R$ if we have $R(x, y)$ $y$ is said to be a _value_ of $R$ at $x$. Some pairs $x, y$ with $R(x, y)$ are also said to be _specified values_; generally if $R$ has some value at $x$, at least one such value will be specified. We omit the word "value" when it is clear from context.

1. There are _source_ and _target_ relations $\mathcal{C}_1 \times \mathcal{C}_0$ and $\mathcal{C}_2 \times \mathcal{C}_1$, and every 1- or 2- cell $f$ has some specified value for its source and target.
1. There are _2-source_ and _2-target_ relations $\mathcal{C}_2 \times \mathcal{C}_0$, and every 2- cell $f$ has some specified value for its 2-source and 2-target.
1. There are _identity_ relations $Id:\mathcal{C}_0 \times \mathcal{C}_1$ and $\mathcal{C}_1 \times \mathcal{C}_2$, and every object or 1-cell has some specified value for its identity.
1. There is a _composition_ relation $\circ:(\mathcal{C}_1 \times \mathcal{C}_1) \times \mathcal{C}_1$; whenever $f_1$ and $f_2$ are such that some specified target of $f_1$ is also a specified source of $f_2$, there is some specified value of the composite of $f_1$ and $f_2$.
1. There is a _1-composition_ relation $\circ_1:(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$; whenever $f_1$ and $f_2$ are such that some specified 2-target at $f_1$ is also a specified 2-source at $f_2$, there is some specified value of the 1-composite of $f_1$ and $f_2$.
1. There is a _2-composition_ relation $\circ_2:(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$; whenever $f_1$ and $f_2$ are such that some specified target at $f_1$ is also a specified source at $f_2$, there is some specified value of the 2-composite of $f_1$ and $f_2$.

The relations satisfy the following conditions:

1. For any 2-cell $f$, and any values $x$ and $y$ of its source and target (respectively), the set of targets of $x$, of targets of $y$, and of 2-targets of $f$ are all equal (as subsets of $\mathcal{C}_1$), and likewise for sources/2-sources. If $x$ or $y$ is a specified value of either relation, each specified value of its source is a specified value of the 2-source, and likewise for targets/2-targets.
1. Any source of an identity $x$ is also a target of $x$, and vice versa. (However, we do not require the sets of specified values to match.) For any specified identity $y$ of $x$, $x$ should be a specified source and target of $y$.
1. If $h$ is composition of $f \circ g$, the set of targets of $f$ is equal to the set of targets of $h$, and the set of sources of $g$ is equal to the set of sources of $h$. If $h$ is a specified composition, any specified target of $f$ is a specified target of $h$, and any specified source of $g$ is a specified source of $h$. (In general the converses may fail.) Moreover, if the composition $f \circ g$ has any value, the set of targets of $g$ is equal to the set of sources of $f$.
1. If $h$ is a value of the 2-composition $f \circ_2 g$, the set of targets of $f$ is equal to the set of targets of $h$, and the set of sources of $g$ is equal to the set of sources of $h$. If $h$ is a specified value of the 2-composition, any specified target of $f$ is a specified target of $h$, and any specified source of $g$ is a specified source of $h$. Moreover, if the 2-composition $f \circ_2 g$ has any value, the set of targets of $g$ is equal to the set of sources of $f$.
1. If $h$ is a value of the 1-composition $f \circ_1 g$, each source of $h$ is a composition of some source of $f$ and some source of $g$, and each such composition is a source of $h$. For any specified sources of $f$ and $g$, each specified value of their composition is a specified source of $h$. The same results hold with source replaced by target. Moreover, if the 1-composition $f \circ_1 g$ has any value, the set of 2-targets of $g$ is equal to the set of 2-sources of $f$.

There are also conditions defining equivalence relations on $\mathcal{C}_0$ and $\mathcal{C}_1$. (The equivalence relation for $\mathcal{C}_2$ is simply equality.) All the relations defined above are required to be invariant under the appropriate equivalence relations (note, however, that their specified values do not need to be invariant). So in particular, the identity relation $\mathcal{C}_1\times \mathcal{C}_2$ and the 1- and 2- composition relations $(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$ are functional: any two values of the identity on some 1-cell $x$, or any two values of one of the compositions on some 2-cells $f$ and $g$, must be equal (as elements of $\mathcal{C}_2$.)

* A 2-cell $f$ is said to be an _isomorphism_ (between any of its sources and any of its targets) if there are some 2-cell $g$ and 1-cell $x$ such that the 2-composition of $f$ and $g$ is the identity on $x$. We say that $g$ is an _inverse_ of $f$.
* Two 1-cells $f$ and $g$ are _isomorphic_ if there is a 2-cell isomorphism $h$ with $f$ as a source and $g$ as a target (or equivalently, vice versa). This is the chosen equivalence relation for $\mathcal{C}_1$.
* A 1-cell $f$ is said to be an _equivalence_ (between any of its sources and any of its targets) if there are some 1-cell $g$ and object $x$ such that some composition of $f$ and $g$ in either direction is isomorphic to an identity on $x$.
* Two objects $x$ and $y$ are _equivalent_ if there is a 1-cell equivalence $g$ with $x$ as a source and $y$ as a target (or equivalently, vice versa). This is the chosen equivalence relation for $\mathcal{C}_0$.

Finally there are coherence relations:

1. Let $f$ be an identity 2-cell (in other words, a value of the identity relation for some 1-cell), and $g$ be any 2-cell. Then, if there is a 2-composition $f \circ_2 g$ or $g \circ_2 f$, any such value is equal to $g$. It follows (by considering their 2-composition) that identities on isomorphic 1-cells are equal, and that when two 1-cells are isomorphic, this fact will be witnessed by the identity on either object.
1. Let $f$ and $g$ be identity 2-cells on, say, $x$ and $y$ respectively, and suppose that $f \circ_1 g$ exists. Then it is an identity 2-cell on any value of $x \circ y$. Equivalently: the 1-composition of isomorphisms is an isomorphism.
1. Let $f$ be an identity 1-cell, and let $g$ be any 1-cell. Then, if there is a composition $f \circ g$ or $g \circ f$, any such value is isomorphic to $g$. It follows (by considering their composition) that identities on equivalent objects are isomorphic, and that when two objects are equivalent, this fact will be witnessed by the identity on either object.
1. Let $f$, $g$, and $h$ be 2-cells. If $F = f \circ_2 g$, and $G = g \circ_2 h$, then $F \circ_2 h = f \circ_2 G$. (The latter values exist because the set of targets of $G$ and $g$ are equal, as are the set of sources of $g$ and $F$.)
1. Let $f$, $g$, and $h$ be 2-cells. If $F = f \circ_1 g$, and $G = g \circ_1 h$, then $F \circ_1 h = f \circ_1 G$. (The latter values exist because the set of 2-targets of $G$ and $g$ are equal, as are the set of 2-sources of $g$ and $F$.) It follows that composition of 1-cells is also associative, up to isomorphism.
1. Let $f_1$, $f_2$, $g_1$, and $g_2$ be 2-cells. If $F = f_1 \circ_1 f_2$, $G = g_1 \circ_1 g_2$, $H_1 = f_1 \circ_2 g_1$, and $H_2 = f_2 \circ_2 g_2$, then $F \circ_1 G = H_1 \circ_2 H_2$. (It follows from the preceding relations that these compositions exist.)

So every anabicategory is "ana[[skeletal]]" (every equivalence is an identity), and "[[strict category|anastrict]]:" its associators and unitors are also identities.

Every [[bicategory]] can be made into an anabicategory; we simply make the source, target, identity, and composition maps into relations with the original functions as specified values, then define the full values by isomorphism and equivalence. This will satisfy the coherence relations because the original bicategory satisfied the bicategory coherence relations.

On the other hand, we can also assume that every value of the defining relations is specified; then the anabicategory is said to be _saturated_. Any such anabicategory is clearly [[anafunctors|anaequivalent]] to a strict 2-category (just map each object or 1-cell to its equivalence class, and vice versa). For there to be a weak equivalence between the anabicategory and the 2-category, however, we would need the [[Axiom of Choice]] (to pick out a single element of each equivalence class to serve as its image).

# Reference #

*  [[Michael Makkai]]; Avoiding the axiom of choice in general category theory; section 3 (which is part 4 [here](http://www.math.mcgill.ca/makkai/anafun/)).