ascending chain condition

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## Definition

Given a [[category]] $C$ and an object $B:C$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are [[subobjects]] of $B$ and whose [[morphisms]] are [[monomorphisms]]. A ascending chain of subobjects of $B$ is a [[direct sequence]] in $\mathrm{Mono}(B)$, a [[sequence]] of subobjects $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent sequence]] of monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$.

Given categories $C$ and $D$ with [[forgetful functor]] $F:C \to D$, an object $B$ is said to satisfy the **ascending chain condition** on subobjects of $F(B)$ if for every ascending chain of subobjects $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an [[isomorphism]].

## Examples

There is a [[forgetful functor]] $F:Ring \to BMod$ from the category [[Ring]] of [[rings]] to the category $BMod$ of [[bimodules]], which forgets the multiplicative structure on the bimodule, that the canonical [[left action]] and [[right action]] of each ring $R$ have [[domain]] $R^2$ and are equal to each other and to the multiplicative binary operation of $R$, and that the canonical [[biaction]] of each ring $R$ has domain $R^3$.

Given a ring $R$, a two-sided ideal is a subobject of $F(R)$, a sub-$R$-$R$-bimodule. A [[ring]] $R$ is said to satisfy the ascending chain condition on two-sided ideals if for every ascending chain of two-sided ideals $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an [[isomorphism]].

Similarly, there are forgetful functors $G:Ring \to LeftMod$ from $Ring$ to the category $LeftMod$ of [[left modules]] and $H:Ring \to RightMod$ from $Ring$ to the category $RightMod$ of [[right modules]], and one could similarly define the ascending chain condition on [[left ideals]] and [[right ideals]] for rings.

## See also

* [[descending chain condition]]

* [[Noetherian bimodule]]

* [[Noetherian ring]]

* [[property sup]]

[[!redirects ascending chain conditions]]


Asgar Jamneshan

* [website](https://asgarjam.wixsite.com/home)

## Selected writings

On [[measure theory]] via (secretly) [[Boolean toposes]]:

* [[Asgar Jamneshan]], [[Terence Tao]], _Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration_ ([arXiv:2010.00681](arXiv:2010.00681))

On a generalization of the [[Furstenberg-Zimmer structure theorem]] via Boolean toposes:

* [[Asgar Jamneshan]], _An uncountable Furstenberg-Zimmer structure theory_ ([arXiv:2103.17167](https://arxiv.org/abs/2103.17167)).

category: people


Ash Asudeh

* [webpage](http://www.sas.rochester.edu/lin/sites/asudeh)

### Related entries

* [[monad (in linguistics)]]

category: people


Asher Peres

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

## Selected writings

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

On the [[interpretation of quantum mechanics]]:

* {#Peres97} [[Asher Peres]], *Interpreting the Quantum World*, Stud. History Philos. Modern Physics **29** (1998) 611 [[arXiv:quant-ph/9711003](http://arxiv.org/abs/quant-ph/9711003)]

category: people


Ashkbiz Danehkar

Ashkbiz Danehkar is an astrophysicist studying black holes and galaxies.

* [webpage](https://www.danehkar.net/)

* [arXiv page](https://arxiv.org/a/danehkar_a_1.html)

* [INSPIRE-HEP](https://inspirehep.net/authors/1048604)

* [NASA ADS](https://ui.adsabs.harvard.edu/search/q=orcid%3A0000-0003-4552-5997)

* Scopus ID: [34972723700](https://www.scopus.com/authid/detail.uri?authorId=34972723700)

* ResearcherID: [C-2053-2009](https://www.webofscience.com/wos/author/rid/C-2053-2009)

* ORCID: [0000-0003-4552-5997](https://orcid.org/0000-0003-4552-5997)

## Selected writings

On [[Weyl tensor]]:

* [[Ashkbiz Danehkar]], _Covariant Evolution of Gravitoelectromagnetism_, Universe 8 (2022) 318 ([arXiv:2206.13946](https://arxiv.org/abs/2206.13946), [doi:10.3390/universe8060318](https://doi.org/10.3390/universe8060318))

* [[Ashkbiz Danehkar]], _On the Significance of the Weyl Curvature in a Relativistic Cosmological Model_, Mod.Phys.Lett.A 24 (2009) 3113-3127 ([arXiv:0707.2987](https://arxiv.org/abs/0707.2987), [doi:10.1142/S0217732309032046](https://doi.org/10.1142/S0217732309032046))

* [[Ashkbiz Danehkar]], _Gravitational Fields of the Magnetic-type_, Int.J.Mod.Phys.D 29 (2020) 2043001 ([arXiv:2006.13287](https://arxiv.org/abs/2006.13287), [doi:10.1142/S0218271820430014](https://doi.org/10.1142/S0218271820430014))

On [[electric-magnetic duality ]]:

* [[Ashkbiz Danehkar]], _Electric-magnetic duality in gravity and higher-spin fields_, Front. Phys. 6 (2019) 146 ([doi:10.3389/fphy.2018.00146](https://doi.org/10.3389/fphy.2018.00146))

* [[Ashkbiz Danehkar]],  Hassan Alshal, [[Thomas Curtright]], _Dual fields of massive/massless gravitons in IR/UV completions_, Int.J.Mod.Phys.D 30 (2021) 2142021 ([arXiv:2109.05148](https://arxiv.org/abs/2109.05148), [doi:10.1142/S0218271821420219](https://doi.org/10.1142/S0218271821420219))

* Constantin Bizdadea, Eugen M. Cioroianu, [[Ashkbiz Danehkar]], Marius Iordache, Solange O. Saliu, Silviu C. Sararu, _Consistent interactions of dual linearized gravity in D=5: couplings with a topological BF model_, Eur.Phys.J.C 63 (2009) 491-519 ([arXiv:0908.2169](https://arxiv.org/abs/0908.2169), [doi:10.1140/epjc/s10052-009-1105-0](https://doi.org/10.1140/epjc/s10052-009-1105-0))

On [[black hole]] astrophysics:

* [[Ashkbiz Danehkar]], Michael A. Nowak,  Julia C. Lee,  Gerard A. Kriss, Andrew J. Young,  Martin J. Hardcastle,  Susmita Chakravorty,  Taotao Fang, Joseph Neilsen,  Farid Rahoui, Randall K. Smith, _The Ultra-fast Outflow of the Quasar PG 1211+143 as Viewed by Time-averaged Chandra Grating Spectroscopy_, Astrophys. J. 853 (2018) 165 ([arXiv:1712.07118](https://arxiv.org/abs/1712.07118), [doi:10.3847/1538-4357/aaa427](https://doi.org/10.3847/1538-4357/aaa427))

* Gerard A. Kriss, Julia C. Lee, [[Ashkbiz Danehkar]], Michael A. Nowak, Taotao Fang, Martin J. Hardcastle, Joseph Neilsen, Andrew Young, _Discovery of an Ultraviolet Counterpart to an Ultrafast X-Ray Outflow in the Quasar PG 1211+143_, Astrophys. J. 853 (2018) 166 ([arXiv:1712.08850](https://arxiv.org/abs/1712.08850), [doi:10.3847/1538-4357/aaa42b](https://doi.org/10.3847/1538-4357/aaa42b))

* Rozenn Boissay-Malaquin, [[Ashkbiz Danehkar]], Herman L. Marshall, Michael A. Nowak, _Relativistic Components of the Ultra-fast Outflow in the Quasar PDS 456 from Chandra/HETGS, NuSTAR, and XMM-Newton Observations_, Astrophys. J. 873 (2019) 29 ([arXiv:1901.06641](https://arxiv.org/abs/1901.06641), [doi:10.3847/1538-4357/ab0082](https://doi.org/10.3847/1538-4357/ab0082))

On [[galaxy]] [[astronomy]]:

* [[Ashkbiz Danehkar]], M. S. Oey, William J. Gray, _Catastrophic Cooling in Superwinds. II. Exploring the Parameter Space_, Astrophys. J. 921 (2021) 91 ([arXiv:2106.10854](https://arxiv.org/abs/2106.10854), [doi:10.3847/1538-4357/ac1a76](https://doi.org/10.3847/1538-4357/ac1a76))

* [[Ashkbiz Danehkar]], M. S. Oey, William J. Gray, _Catastrophic Cooling in Superwinds. III. Nonequilibrium Photoionization_, Astrophys. J. 937 (2022) 68 ([arXiv:2208.12030](https://arxiv.org/abs/2208.12030), [doi:10.3847/1538-4357/ac8cec](https://doi.org/10.3847/1538-4357/ac8cec))

* [[Ashkbiz Danehkar]], _Physical and Chemical Properties of Wolf-Rayet Planetary Nebulae_, Astrophys. J. Suppl. 257 (2021) 58 ([arXiv:2106.10762](https://arxiv.org/abs/2106.10762), [doi:10.3847/1538-4365/ac2310](https://doi.org/10.3847/1538-4365/ac2310))

* [[Ashkbiz Danehkar]], _Morpho-kinematic Properties of Wolf-Rayet Planetary Nebulae_, Astrophys. J. Suppl. 260 (2022) 14 ([arXiv:2107.03994](https://arxiv.org/abs/2107.03994), [doi:10.3847/1538-4365/ac5cca](https://doi.org/10.3847/1538-4365/ac5cca))

On [[plasma]] / [[computational physics]]:

* [[Ashkbiz Danehkar]], _Electron beam-plasma interaction and electron-acoustic solitary waves in a plasma with suprathermal electrons_, Plasma Phys. Control. Fusion 60 (2018) 065010 ([arXiv:1804.07299](https://arxiv.org/abs/1804.07299), [doi:10.1088/1361-6587/aabc40](https://doi.org/10.1088/1361-6587/aabc40))

* [[Ashkbiz Danehkar]], _Electrostatic solitary waves in an electron-positron pair plasma with suprathermal electrons_, Phys. Plasmas 24 (2017) 102905 ([arXiv:1711.01141](https://arxiv.org/abs/1711.01141), [doi:10.1063/1.5000873](https://doi.org/10.1063/1.5000873))

category: people


Ashok Das

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

## related $n$Lab entries

* [[gravity]]

* [[rubber-sheet analogy of gravity]]

category: people


Ashoke Sen

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

## Selected writings

On microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:

* [[Ashoke Sen]], _Extremal black holes and elementary string states_,  Mod. Phys. Lett. A10: 2081-2094, 1995 ([arXiv:hep-th/9504147](http://arxiv.org/abs/hep-th/9504147))

* {#Sen07} [[Ashoke Sen]], _Black Hole Entropy Function, Attractors and Precision Counting of Microstates_, Gen. Rel. Grav. 40: 2249-2431, 2008 ([arXiv:0708.1270](http://arxiv.org/abs/0708.1270))

Introducing [[Sen's conjecture]] on [[open string]] [[tachyon condensation]] and the decay of [[D-brane]]/[[anti-D-brane]] pairs in [[superstring theory]] via open [[superstring]] [[tachyon condensation]]:

* {#Sen98} [[Ashoke Sen]], _Tachyon Condensation on the Brane Antibrane System_, JHEP 9808:012,1998 ([arXiv:hep-th/9805170](https://arxiv.org/abs/hep-th/9805170))

* {#Sen99} [[Ashoke Sen]], _Universality of the Tachyon Potential_, JHEP 9912:027, 1999 ([arXiv:hep-th/9911116](http://arxiv.org/abs/hep-th/9911116))

with review in:

* [[Ashoke Sen]], _Tachyon Dynamics in Open String Theory_, Int. J. Mod. Phys. A20:5513-5656, 2005 ([arXiv:hep-th/0410103](https://arxiv.org/abs/hep-th/0410103))

Discussion of [[gravitational waves]] in relation to the [[soft graviton theorem]]:

* Arnab Priya Saha, Biswajit Sahoo, [[Ashoke Sen]], _Proof of the Classical Soft Graviton Theorem in $D=4$_ ([arXiv:1912.06413](https://arxiv.org/abs/1912.06413))

* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

* Sergei Alexandrov, Jan Manschot, [[Boris Pioline]], _D3-instantons, Mock Theta Series and Twistors_, JHEP 1304 (2013)
002 ([arXiv:1207.1109](http://arxiv.org/abs/1207.1109))

On [[Lagrangian densities]] for [[self-dual higher gauge fields]] (generically thought to be [[non-Lagrangian field theories]]) at the cost of an unusual decoupling [[auxiliary field]]:

* {#Sen20} [[Ashoke Sen]], *Self-dual forms: Action, Hamiltonian and Compactification*,  Journal of Physics A: Mathematical and Theoretical, **53** 8 (2020) [[arXiv:1903.12196](https://arxiv.org/abs/1903.12196), [doi:10.1088/1751-8121/ab5423](https://doi.org/10.1088/1751-8121/ab5423)]

On [[D-brane instantons]] and [[moduli stabilization]] in [[type II string theory]]:

* Sergei Alexandrov, Atakan Hilmi Fırat, Manki Kim, [[Ashoke Sen]], [[Bogdan Stefański]], *D-instanton Induced Superpotential*, J. High Energ. Phys. **2022** 90 (2022) [[arXiv:2204.02981](https://arxiv.org/abs/2204.02981), doi:10.1007/JHEP07(2022)090]

and in 0B string theory using methods from [[string field theory]]:

* Joydeep Chakravarty, [[Ashoke Sen]], *Normalization of D instanton amplitudes in two dimensional type 0B string theory* [[arXiv:2207.07138](https://arxiv.org/abs/2207.07138)]

reviewed in

* [[Ashoke Sen]], *D-instanton amplitudes in string theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Sen2023)]

## Related entries

* [[Sen's conjecture]]

* [[electric-magnetic duality]]/[[S-duality]]

* [[M-theory lift of gauge enhancement on D6-branes]]

* [[BFSS matrix model]]

category: people


Ashvin Vishwanath

* [institute page](https://www.physics.harvard.edu/people/facpages/vishwanath)

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

## Selected writings

On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:

* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$

* [[Yi Zhang]], [[Tarun Grover]], [[Ashvin Vishwanath]], *General procedure for determining braiding and statistics of anyons using entanglement interferometry*, Phys. Rev. B **91** (2015) 035127 $[$[arXiv:1412.0677](https://arxiv.org/abs/1412.0677), [doi:10.1103/PhysRevB.91.035127](https://doi.org/10.1103/PhysRevB.91.035127)$]$

On [[topological semi-metals]]:

* [[Ari M. Turner]], [[Ashvin Vishwanath]], Part I of: *Beyond Band Insulators: Topology of Semi-metals and Interacting Phases*, in: *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) 293-324 $[$[arXiv:1301.0330](https://arxiv.org/abs/1301.0330), [ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$

* N. P. Armitage, [[Eugene Mele]], [[Ashvin Vishwanath]], *Weyl and Dirac semimetals in three-dimensional solids*, Rev. Mod. Phys. **90** 015001 (2018) $[$[doi:10.1103/RevModPhys.90.015001](https://doi.org/10.1103/RevModPhys.90.015001)$]$

category: people


Ashwin S. Pande

* [institute page](https://ahduni.edu.in/academics/schools-centres/school-of-arts-and-sciences/faculty/ashwin-pande/)

## Selected writings

On [[topological T-duality]] for non-free torus actions (physically: [[KK-monopoles]]):

* {#Pande06} [[Ashwin S. Pande]], _Topological T-duality and Kaluza-Klein Monopoles_, Adv. Theor. Math. Phys. **12** (2007) 185-215 [[arXiv:math-ph/0612034](https://arxiv.org/abs/math-ph/0612034), [doi:10.4310/ATMP.2008.v12.n1.a3](https://dx.doi.org/10.4310/ATMP.2008.v12.n1.a3)]

On [[topological T-duality]] in relation to [[T-folds]]:

* [[Peter Bouwknegt]], [[Ashwin S. Pande]], *Topological T-duality and T-folds*, Advances in Theoretical and Mathematical Physics **13** 5 (2009) [[arXiv:0810.4374](https://arxiv.org/abs/0810.4374), [doi:10.4310/ATMP.2009.v13.n5.a6](https://dx.doi.org/10.4310/ATMP.2009.v13.n5.a6)]

category: people

[[!redirects Ashwin Pande]]


Asif Equbal

member of [[CQTS]] at NYU Abu Dhabi

* [institute page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/asif-equbal.html)

* [personal page](https://www.asifequbal.com/)

* [GoogleScholar page](https://scholar.google.com.hk/citations?user=q8KQ9gUAAAAJ&hl=en)

## Selected writings

On [[Floquet theory]] in [[nuclear magnetic resonance]]:

* Konstantin L. Ivanov, Kaustubh R. Mote, Matthias Ernst, [[Asif Equbal]], Perunthiruthy K. Madhu, *Floquet theory in magnetic resonance: Formalism and applications*, Progress in Nuclear Magnetic Resonance Spectroscopy, **126**–**127** (2021) 17-58 [[doi:10.1016/j.pnmrs.2021.05.002](https://doi.org/10.1016/j.pnmrs.2021.05.002)]

On Dynamic Nuclear Polarization in [[nuclear magnetic resonance]]:

* [[Asif Equbal]] et al.,  *Role of electron spin dynamics and coupling network in designing dynamic nuclear polarization*, Progress in Nuclear Magnetic Resonance Spectroscopy **126**–**127** (2021) 1-16 [[doi:10.1016/j.pnmrs.2021.05.003](https://doi.org/10.1016/j.pnmrs.2021.05.003)]

On [[nuclear magnetic resonance]] and related spin technologies for the use of [[quantum computation]]:

* [[Asif Equbal]], *Molecular spin qubits for future quantum technology*, talk at [[CQTS]] (Nov 2022) [slides: [[Equbal-CQTS-Nov2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/play/YTjIGL-Bevb1H44UuL-ZimXdxph5cffddWpP3H4ZsuRT2xu3OrnTbC0NZLsKedUGwS68DJ8onVFPAETb.rreoi7Wt6uXFyaYN?continueMode=true&_x_zm_rtaid=e0VPIMlfT9KlVd_wiaOq6A.1669794985784.11f8cd37091ebf6bdd2a878668e26cd6&_x_zm_rhtaid=404)]

On [[nuclear magnetic resonance]] on the [[nitrogen-vacancy center in diamond]] and related systems in [[quantum sensing]]:

* Santiago Bussandri, Daphna Shimon, [[Asif Equbal]], Yuhang Ren, Susumu Takahashi, [[Chandrasekhar Ramanathan]], Songi Han,  *P1 center electron spin clusters are prevalent in type Ib diamond* [[arXiv:2311.05396](https://arxiv.org/abs/2311.05396)]

## Related entries

* [[nuclear magnetic resonance]]

* [[quantum computing]]

category: people


Assaf Libman

* [webpage](http://homepages.abdn.ac.uk/a.libman/pages/home_page/)

## related $n$Lab entries

* [[tower]]

category: people


assembly map

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## Idea

The _analytic assembly map_ is a natural [[morphism]] from $G$-[[equivariant cohomology|equivariant]] [[topological K-theory]] to the [[operator K-theory]] of a corresponding [[crossed product C*-algebra]].

More generally in equivariant [[KK-theory]] this is called the _Kasparov descent map_  and is of the form

$$
  KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)
$$

where on the left we have $G$-equivariant KK-theory and on the right ordinary [[KK-theory]] of [[crossed product C*-algebras]] (which by the discussion there are models for the [[groupoid convolution algebras]] of $G$-[[action groupoids]]).

(recalled as [Blackadar, theorem 20.6.2](#Blackadar))

## Properties

The [[Baum-Connes conjecture]] states that under some conditions the analytic assembly map is in fact an [[isomorphism]]. The [[Novikov conjecture]] makes statements about it being an [[injection]]. The _[[Green-Julg theorem]]_ states that under some (milder) conditions the Kasparov desent map is an [[isomorphism]].

## Related concepts

* [[Novikov conjecture]]

* [[Baum-Connes conjecture]]

## References

The construction goes back to

* [[Gennady Kasparov]], _The index of invariant elliptic operators, K-theory, and Lie group representations_. Dokl. Akad. Nauk. USSR, vol. 268, (1983), 533-537.

An introduction is in

* [[Alain Valette]], _Introduction to the Baum-Connes conjecture_  ([pdf](http://www.univ-orleans.fr/mapmo/membres/chatterji/Valette.pdf))

A textbook account is in

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_
 {#Blackadar}

See also

* Markus Land, _The Analytical Assembly Map and Index Theory_, ([arXiv:1306.5657](http://arxiv.org/abs/1306.5657))
 {#Land13}

[[!redirects assembly maps]]

[[!redirects Baum-Connes assembly map]]
[[!redirects Baum-Connes assembly maps]]

[[!redirects analytic assembly map]]
[[!redirects analytic assembly maps]]


ASSet

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## Definition

The [[category]] $sSet_+$ of **augmented simplicial sets** is the category of [[presheaf|presheaves]] on the _augmented_ version of the [[simplex category]] $\Delta_a$:

$$
  sSet_+ := Set^{\Delta^{op}_a}
  \,.
$$

This is the category whose [[object]]s are [[augmented simplicial set]]s and whose [[morphism]]s are the evident morphisms between these.

## Related concepts

* [[sSet]]

* **asSet**

## Applications

The [[join of simplicial sets]] is most naturally defined via a construction on augmented simplicial sets.

[[!redirects asSet]]

category: category


Assia Mahboubi

* [website](https://people.rennes.inria.fr/Assia.Mahboubi/)

## Publications

* [[Sophie Bernard]], [[Cyril Cohen]], [[Assia Mahboubi]], [[Pierre-Yves Strub]], *Unsolvability of the Quintic Formalized in Dependent Type Theory*, ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France ([hal:hal-03136002](https://hal.inria.fr/hal-03136002))

## Talks

* [[Assia Mahboubi]], *Continuity in dependent type theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 31 March 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Mahboubi-2022-03-31-HoTTEST.pdf))

category: people


assignment > history

< [[assignment]]


assignment operator

# Contents
* table of contents
{: toc}

## Idea

A symbol in [[mathematics]] and [[computer science]] to indicate that a particular variable is being initialized or assigned a value or that a particular symbol is being defined.

One sometimes distinguishes between assignment operators which allow reassignment, with what are known as [[single assignment operators]], which do not allow reassignment.

The assignment operator in purely [[functional programming]] languages like [[Haskell]] amd [[Agda]] is an example of a single assignment operator. As purely functional programming languages can be represented in [[type theory]], and every [[foundations of mathematics]] could also be represented in type theory, the assignment operators used in [[definitions]] in mathematics, such as $\coloneqq$, are single assignment operators; see [[definition]] for more details.

## Related concepts

* [[functional programming]]

* [[equality]]

[[!include mathematical statements --- contents]]

## References

* Wikipedia, _Assignment (computer science)_

[[!redirects assignment]]
[[!redirects assignment operators]]


associahedron

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## Idea

The _associahedra_ or _Stasheff polytopes_ $\{K_n\}$ are [[CW complex]]es that naturally arrange themselves into a topological [[operad]] that resolves the standard associative operad: an [[A-infinity-operad]].

The vertices of $K_n$ correspond to ways in which one can bracket a product of $n$ variables. The edges correspond to rebracketings, the faces relate different sequences of rebracketings that lead to the same result, and so on.

The associahedra were introduced by Jim Stasheff in order to describe [[topological space]]s equipped with a multiplication operation that is associative up to every higher coherent homotopy.

## Definition

Here is the rough idea, copied, for the moment, verbatim from Markl94 [p. 26] (http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=26) (for more details see references below):

For $n \geq 1$ the **associahedron** $K_n$ is an $(n-2)$-dimensional polyhedron whose $i$-dimensional cells are, for $0 \leq i \leq n-2$, indexed by all (meaningful) insertions of $(n-i-2)$ pairs of brackets between $n$ independent indeterminates, with suitably defined incidence maps.

[[simplicial set|Simplicially]] ${ }$ [[subdivision|subdivided]] associahedra (complete with simplicial [[operad|operadic]] structure) can be presented efficiently in terms of an abstract [[bar construction]]. Let $\mathcal{O}: Set/\mathbb{N} \to Set/\mathbb{N}$ be the [[monad]] which takes a [[graded set]] $X$ to the non-permutative [[non-unital operad]] freely generated by $X$, with monad multiplication denoted $m: \mathcal{O}\mathcal{O} \to \mathcal{O}$. Let $t_+$ be the graded set $\{X_n\}_{n \geq 0}$ that is [[empty set|empty]] for $n = 0, 1$ and [[terminal object|terminal]] for $n \geq 2$; this carries a unique non-unital non-permutative operad structure, via a structure map $\alpha: \mathcal{O}t_+ \to t_+$. The bar construction $B(\mathcal{O}, \mathcal{O}, t_+)$ is an (augmented) simplicial graded set (an object in $Set^{\Delta^{op} \times \mathbb{N}}$) whose face maps take the form

$$\ldots \mathcal{O}\mathcal{O}\mathcal{O}t_+
\stackrel{\stackrel{\overset{m\mathcal{O} t_+}{\to}}{\underset{\mathcal{O}m t_+}{\to}}}{\underset{\mathcal{O}\mathcal{O}\alpha}{\to}}
\mathcal{O}\mathcal{O}t_+ \stackrel{\overset{m t_+}{\to}}{\underset{\mathcal{O}\alpha}{\to}} \mathcal{O}t_+ \stackrel{\alpha}{\to} t_+.$$

Intuitively, the (graded set of) $0$-cells $\mathcal{O}t_+$ consists of planar trees where each inner node has two or more incoming edges, with trees graded by number of leaves; the extreme points are binary trees [corresponding to complete binary bracketings of words], whereas other trees are barycenters of higher-dimensional faces of Stasheff polytopes. The construction $B(\mathcal{O}, \mathcal{O}, t_+)$ carries a simplicial (non-permutative non-unital) operad structure, where the [[geometric realization]] of the simplicial set at grade (or [[arity]]) $n$ defines the barycentric subdivision of the Stasheff polytope $K_n$. As the operad structure on $B(\mathcal{O}, \mathcal{O}, t_+)$ is expressed in [[doctrine|finite product logic]] and geometric realization preserves finite products, the (simplicially subdivided) associahedra form in this way the components of a topological operad.

## Loday's realization

[[Jean-Louis Loday]] gave a simple formula for realizing the Stasheff polytopes as a convex hull of integer coordinates in Euclidean space [(Loday 2004)](#Loday04).  Let $Y_n$ denote the set of (rooted planar) binary trees with $n+1$ leaves (and hence $n$ internal vertices).  For any binary tree $t \in Y_n$, enumerate the leaves by left-to-right order, denoted $\ell_1, \ldots, \ell_{n+1}$, and enumerate the internal vertices as $v_1, \ldots, v_n$ where $v_i$ is the closest common ancestor of $\ell_i$ and $\ell_{i+1}$. Define a vector $M(t) \in \mathbb{R}^n$ whose $i$th coordinate is the product $a_i b_i$ of the number $a_i$ of leaves that are left descendants of $v_i$ and the number $b_i$ of leaves that are right descendants of $v_i$.

+-- {: .un_thm}
###### Theorem (Loday)
The convex hull of the points $\{ M(t) \in \mathbb{R}^n \mid t \in Y_n \}$ is a realization of the Stasheff polytope of dimension $n-1$, and is included in the affine hyperplane $\{(x_1, \ldots, x_n): x_1 + \ldots + x_n = \binom{n}{2}\}$.
=--

## Illustrations

* **$K_1$** is the [[empty set]], a degenerate case not usually considered.

* **$K_2$** is simply the shape of a binary operation:
$$ x \otimes y ,$$
which we interpret here as a single [[point]].

* **$K_3$** is the shape of the usual [[associator]] or associative law
$$ (x \otimes y) \otimes z \to x \otimes (y \otimes z) ,$$
consisting of a single [[interval]].

* {#K4} **$K_4$** The fourth associahedron $K_4$ is the [[pentagon identity|pentagon]] which expresses the different ways a product of four elements may be bracketed

\begin{tikzcd}[column sep = -3.1em, row sep = 10em, every label/.append style = {font = \Large}, font=\Large]
    & & (w \otimes x) \otimes (y \otimes z) \arrow[drr, "a_{w, x, y \otimes z}"] & & \\
    ((w \otimes x) \otimes y) \otimes z \arrow[urr, "a_{w \otimes x, y, z}"] \arrow[dr,"a_{w,x,y} \otimes \mathrm{id}_{z}", swap] & & & & w \otimes (x \otimes (y \otimes z)) \\
    & (w \otimes (x \otimes y)) \otimes z \arrow[rr, "a_{w, x \otimes y, z}", swap] & & w \otimes ((x \otimes y) \otimes z) \arrow[ur, "\mathrm{id}_{w} \otimes a_{x,y,z}", swap] &
\end{tikzcd}

One can also think of this as the top-level structure of the 4th [[oriental]]. This controls in particular the _pentagon identity_ in the definition of [[monoidal category]], as discussed there.

* **$K_5$** is the [dual polyhedron](http://en.wikipedia.org/wiki/Dual_polyhedron) to the [triaugmented triangular prism](http://en.wikipedia.org/wiki/Triaugmented_triangular_prism)

+--{: style="text-align:center"}

(image from the [Wikimedia Commons](http://commons.wikimedia.org/wiki/File:Polytope_K3.svg))
=--

* One can rotate and explore Stasheff polyhedra in [this interactive associahedron app](https://ltrujello.github.io/associahedron/).

* Illustrations of some polytopes, including $K_5$, can also be found [here](http://irma.math.unistra.fr/~chapoton/galerie.html).

*  A template which can be cut out and assembled into a $K_5$ can be found in Appendix B of [ChengLauda2004](#ChengLauda2004).

## Relation to other structures

### Relation to orientals

The above list shows that the first few Stasheff polytopes are nothing but the first few [[oriental]]s. This doesn't remain true as $n$ increases.
The orientals are free **strict** [[omega-category|omega-categories]] on [[simplex]]es as parity complexes. This means that certain interchange cells (e.g., Gray tensorators) show up as thin in the oriental description.

The first place this happens is the sixth oriental: where there are three tensorator squares and six pentagons in Stasheff's $K_5$, the corresponding tensorator squares coming from $O(5)$ are collapsed.

It was when [[Todd Trimble]] made this point to [[Ross Street]] that Street began to think about using associahedra to define weak [[n-category|n-categories]].

### Categorified associahedra
 {#Categorification}

There is a [[vertical categorification|categorification]] of associahedra discussed in

* S. Saneblidze, R. Umble, Diagonals on the permutahedra, multiplihedra and associahedra, Homology Homotopy Appl. 6(1) (2004) 363--411.
* {#Forcey12} [[Stefan Forcey]], _Quotients of the multiplihedron as categorified associahedra_, Homology Homotopy Appl. __10__:2 (2008) 227--256 ([Euclid](http://projecteuclid.org/euclid.hha/1251811075))

### Tamari lattice

The associahedron is closely related to a structure known as the _Tamari lattice_, which is especially well-studied in [[combinatorics]].  The Tamari lattice $T_n$ can be defined as the [[poset]] of all parenthesizations of $n+1$ variables with the order generated by rightwards reassociation $(a b)c \le a(b c)$, or equivalently as the poset of all binary trees with $n$ internal nodes (and hence $n+1$ leaves), with the order generated by rightwards tree rotation.  (Note the off-by-one offset from the convention for associahedra: the Tamari lattice $T_n$ corresponds to the associahedron $K_{n+1}$.)  It was originally introduced by Dov Tamari in his thesis "Monoïdes préordonnés et chaînes de Malcev" (Université de Paris, 1951), around a decade before Jim Stasheff's work.[^Stash]

[^Stash]: [[Jim Stasheff]] comments on this in an essay titled "How I 'met' Dov Tamari" [(Tamari Memorial Festschrift 2012)](#TomariFestschrift), writing that the "so-called Stasheff polytope ... more accurately should be called the Tamari or Tamari-Stasheff polytope".

As the name suggests, the Tamari lattice also carries the structure of a [[lattice]].  This property was originally established by Haya Friedman and Tamari (1967), and later simplified by Samuel Huang and Tamari (1972).

## References

The original articles that define associahedra and in which the operad $K$ that gives $A(\infty)$-topological spaces is implicit are

* [[Jim Stasheff]], _Homotopy associativity of H-spaces I_, Trans. Amer. Math. Soc. 108 (1963), 275--312.  ([web](http://www.jstor.org/stable/1993608))

* [[Jim Stasheff]], _Homotopy associativity of H-spaces II_, Trans. Amer. Math. Soc. 108 (1963), 293--312.  ([web](http://www.jstor.org/stable/1993609))

A textbook discussion (slightly modified) is in section 1.6 of the book

* [[Martin Markl]], [[Steven Shnider]], [[Jim Stasheff]], _Operads in Algebra, Topology and Physics_ ([web](http://books.google.de/books?id=fMhZjT9lQo0C&pg=PA56&lpg=PA56&dq=Stasheff+associahedra&source=bl&ots=ZuGXjT4zbp&sig=V-taGG2LHS0msHK-PTxmUXXCvEY&hl=de#PPP1,M1))

Loday's original article on the Stasheff polytope is

* {#Loday04} [[Jean-Louis Loday]], Realization of the Stasheff polytope, _Archiv der Mathematik_ 83 (2004), 267-278. ([doi](https://dx.doi.org/10.1007%2Fs00013-004-1026-y))

Further explanations and references are collected at

* [AMS entry on associahedra](http://www.ams.org/featurecolumn/archive/associahedra.html)

* [[Alexander Postnikov]], _Permutohedra, associahedra and beyond_, [math.CO/0507163](http://arxiv.org/abs/math/0507163) [pdf](http://math.mit.edu/~apost/papers/permutohedron.pdf)

The connection to Tamari lattices as well as other developments are in

* {#TomariFestschrift} Folkert Müller-Hoissen, Jean Marcel Pallo, [[Jim Stasheff]] (editors), _Associahedra, Tamari Lattices, and Related Structures: Tamari Memorial Festschrift_, Birkhäuser, 2012. ([google books](https://books.google.fr/books?id=Y01d6g5UemQC&lpg=PP1&pg=PP1#v=onepage&q&f=false))

For a template of $K_5$, see Appendix B of the following.

* {#ChengLauda2004} [[Eugenia Cheng]], [[Aaron Lauda]], _Higher-dimensional categories: an illustrated guidebook_, 2004, [available here](http://eugeniacheng.com/guidebook/).

category: combinatorics
[[!redirects associahedra]]
[[!redirects Stasheff polytope]]
[[!redirects Tamari polytope]]
[[!redirects Tamari-Stasheff polytope]]


associahedron > K4

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associated bundle

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

#Contents#
* table of contents
{:toc}

## Definition

### Traditional

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

### In geometric homotopy theory
 {#InGeometricHomotopyTheory}

In the context of [[higher topos theory]] there is an elegant and powerful definition and construction of associated bundles. We discuss here some basics and how this recovers the traditional definition. For more see at _[[associated infinity-bundle]]_ and at _[[geometry of physics -- representations and associated bundles]]_.

At _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_ is discussed how smooth [[principal bundles]] for a [[Lie group]] $G$ over a [[smooth manifold]] $X$ are equivalently the [[homotopy fibers]] of morphisms of [[smooth groupoids]] ([[smooth stacks]]) of the form

$$
  X \stackrel{}{\longrightarrow} \mathbf{B}G
  \,.
$$

Now given an [[action]] $\rho$ of $G$ on some [[smooth manifold]] $V$, and regarding this action via its [[action groupoid]] projection $p_\rho \colon V//G \to \mathbf{B}G$ as discussed [above](#ActionsOf1Groups), then we may consider these two morphisms into $\mathbf{B}G$ jointly

$$
  \array{
    && V//G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
$$

and so it is natural to construct their [[homotopy fiber product]].

We now discuss that this is equivalently the [[associated bundle]] which is associated to the principal bundle $P \to X$ via the action $\rho$.

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

For $G$ a [[smooth group]] (e.g. a [[Lie group]]), $X$ a [[smooth manifold]], $P \to X$ a smooth $G$-[[principal bundle]] over $X$ and $\rho$ a smooth [[action]] of $G$ on some [[smooth manifold]] $V$, then the [[associated bundle|associated]] $V$-[[fiber bundle]] $P \times_G V\to X$ is equivalently (regarded as a [[smooth groupoid]]) the [[homotopy pullback]] of the [[action groupoid]]-projection $p_\rho \colon V//G \to \mathbf{B}G$ along a morphism $g \colon X\to\mathbf{B}G$ which [[modulating morphism|modulates]] $P$

$$
  \array{
    P\times_G V &\longrightarrow& V//G
    \\
    \downarrow && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
  \,.
$$

=--

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

By the discussion at _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_, the morphism $g$ of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an [[open cover]] $\{U_i \to X\}$ over wich $P$ trivialize and choosing a trivialization, by the [[zig-zag]]

$$
  \array{
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
$$

where the top morphism is the [[Cech cohomology|Cech cocycle]] of the given local trivialization regarded as a morphism out of the [[Cech groupoid]] of the given cover.

Moreover, by [this proposition](geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG) the morphism $(p_\rho)_\bullet$ is a global fibration of pre-smooth groupoids, hence, by the discussion at _[[geometry of physics -- smooth homotopy types]]_, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of $(p_\rho)_\bullet$ along this $g_\bullet$

$$
  \array{
    C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet
    \\
    \downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}}
    \\
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}&
    (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
  \,.
$$

This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism $(x,i)\to (x,j)$ in the Cech groupoid as well as a morphism $s \stackrel{g}{\to} \rho(s)(g)$ in the action groupoid, such that the group label $g$ of the latter equals the cocycle $g_{i j}(x)$ of the cocycle on the former. Schematically:

$$
  C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet
  =
  \left\{
     ((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g))
  \right\}
  \,.
$$

This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is [[equivalence of groupoids]] equivalent to the [[quotient]] of its space of objects by the [[equivalence relation]] induced by its morphisms:

$$
  \cdots \simeq
  \left(
    \underset{i}{\coprod} U_i \times V
  \right)/_\sim
  \,.
$$

This is a traditional description of the associated bundle in question.

=--

## Examples

* [[adjoint bundle]]

* [[tractor bundle]]

## Related concepts

* [[associated infinity-bundle]]

[[!redirects associated bundles]]

## References

* [[Norman Steenrod]], _The topology of fibre bundles_, Princeton Mathematical Series __14__, 1951. viii+224 pp. [MR39258](http://www.ams.org/mathscinet-getitem?mr=39258); reprinted 1994

* [[Dale Husemöller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.)

* [[Glen Bredon]], Section II.2 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

[[!redirects associated bundles]]

[[!redirects associated fiber bundle]]
[[!redirects associated fiber bundles]]


associated graded object

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

#Contents#
* table of contents
{:toc}

## Definition

For

$$
  \cdots
  \hookrightarrow
   X_{(n)}
  \hookrightarrow X_{(n+1)} \hookrightarrow \cdots \hookrightarrow X
$$

a [[filtered object]] in an [[abelian category]] $\mathcal{C}$, the _associated graded object_ $Gr(X)$ is the [[graded object]] which in degree $n$ is the [[cokernel]] of the $n$th inclusion, fitting into a [[short exact sequence]]

$$
  0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0
  \,,
$$

hence the [[quotient]] of the $n$th layer of $X$ by the next lower one:

$$
  Gr_n(X) := X_{(n)}/X_{(n-1)}
  \,,
$$

## Examples

* For $\mathcal{A}$ an [[abelian category]] and $C_{\bullet, \bullet}$ a [[double complex]] in $\mathcal{A}$, let $X = Tot(C)$ be the corresponding [[total complex]]. This is naturally filtered by either row-degree or by column-degree. The corresponding associated graded complex is what the terms in the _[[spectral sequence of a filtered complex]]_ compute.

* [[associated graded vector space]]

## Related concepts

[[!include filtered objects -- contents]]

## References

Discussion of the universal property of the associated graded construction on [mathoverflow](http://mathoverflow.net/questions/263/what-is-the-universal-property-of-associated-graded)

[[!redirects associated graded]]
[[!redirects associated graded objects]]

[[!redirects associated graded space]]
[[!redirects associated graded spaces]]


associated graded ring

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

#Contents#
* table of contents
{:toc}

## Definition

A [[filtered ring]] is a [[filtered object]] in the [[category]] [[Ring]] of [[rings]].

The **associated graded ring** to a filtered ring is the corresponding _[[associated graded object]]_.

## Properties

A version of [[PBW theorem]] states that if a [[Lie algebra]] $g$ over a [[field]] $k$ is [[flat module|flat]] as a $k$-[[module]] over a commutative ground ring $k\supset \mathbb{Q}$ containing [[rational numbers|rationals]], then the associated graded ring $Gr U(g)$ is [[isomorphism|isomorphic]] to the [[symmetric algebra]] $Sym(g)$ of the underlying $k$-module of $g$.

## Related concepts

[[!include filtered objects -- contents]]

[[!redirects associated graded rings]]
[[!redirects associated graded algebra]]
[[!redirects associated graded algebras]]


associated graded vector space

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

#Contents#
* table of contents
{:toc}

## Definition

An _associated graded vector space_ is an [[associated graded object]] in a [[category]] [[Vect]] of [[vector spaces]].

## Examples

* [[weight systems are associated graded of Vassiliev invariants]]

## Related concepts

[[!include filtered objects -- contents]]

[[!redirects associated graded vector spaces]]


associated infinity-bundle

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

# Contents
* table of contents
{: toc}

## Idea

An **associated $\infty$-bundle** $E \to X$ is a [[fiber bundle]] in an [[(∞,1)-topos]] $\mathbf{H}$ with typical fiber $F \in \mathbf{H}$ that is classified by a [[cocycle]] $X \to \mathbf{B}\underline{Aut}(F)$ with coefficients in the [[delooping]] of the [[internalization|internal]] [[automorphism ∞-group]] of $F$.  We say this is _associated to_ the corresponding  $\underline{Aut}(F)$-[[principal ∞-bundle]].

More generally there should be notions of associated $\infty$-bundles whose fibers are objects in an [[(∞,n)-topos]] over $\mathbf{H}$ for some $n \gt 1$.

## Definition

Let $\mathbf{H}$ be an [[(∞,1)-topos]].

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

For $V,X \in \mathbf{H}$ two objects, say a _$V$-[[fiber ∞-bundle]] over $X$_ is a [[morphism]] $E \to X$ (an object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$) such that there exists an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] $U \to X$ and an [[(∞,1)-pullback]] square

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

=--

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

Let $G \in Grp(\mathbf{H})$ be an [[∞-group]] equipped with an
[[∞-action]] $\rho$ on $V$. Then for $P \to X$ a $G$-[[principal ∞-bundle]] over $X$, the _$\rho$-associated $\infty$-bundle_ is

$$
  P \times_G V \to X
  \,,
$$

where $P \times_G V := (P \times V)//G$ is the homotopy quotient of the diagonal $G$-action.

=--

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

Below in _[Properties](#Properties)_ we see that every $\rho$-associated $\infty$-bundle is a $V$-fiber $\infty$-bundle and that every $V$-fiber $\infty$-bundle arises as associated to an  $\mathbf{Aut}(V)$-[[principal ∞-bundle]]

=--

## Properties
 {#Properties}

### General

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

For $V \in \mathbf{H}$, write $\mathbf{Aut}(V) \in Grp(\mathbf{H})$ for the internal [[automorphism ∞-group]] of $V$. This comes with a canonical action on $V$. Then the operation of sending an $\mathbf{Aut}$-[[principal ∞-bundle]] $P \to X$ to the associated $P \times_G V \to X$ establishes an equivalence

$$
  H^1(X, \mathbf{Aut}(V)) \simeq \{V-fiber\;\infty-bundles\}
  \,.
$$

=--

More specifically, if $\rho$ is an [[∞-action]] of $G$ on some $V \in \mathbf{H}$, then under the [[equivalence of (∞,1)-categories]]

$$
  G Act \simeq \mathbf{H}_{/\mathbf{B}G}
$$

it corresponds to a [[fiber sequence]]

$$
  \array{
    V &\to& V//G
    \\
    && \downarrow^{\mathrlap{\overline{\rho}}}
    \\
    && \mathbf{B}G
  }
$$

in $\mathbf{H}$. This is the **universal $\rho$-associated $V$-bundle** in that for $P \to X$ any $G$-[[principal ∞-bundle]] modulated by $g \colon X \to \mathbf{B}G$ we have a natural equivalence

$$
  P \times_G V \simeq g^* \overline{\rho}
  \,.
$$

This is discussed in ([NSS, section I 4.1](#NSS)).

### Presentation in simplicial presheaves

In ([Wendt](#Wendt)), section 5.5, a [[presentable (infinity,1)-category|presentation]] of the general situation for [[n-localic (infinity,1)-topos|1-localic]] [[(∞,1)-toposes]] is given in terms of the [[model structure on simplicial presheaves]] (as discussed at [[models for ∞-stack (∞,1)-toposes]]) .

Under this presentation we have:

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

The universal $F$-$\infty$-bundle $\mathbf{E} F \to \mathbf{B}Aut(F)$ is presented by the [[bar construction]]

$$
  F \to B(*, Aut(F), F) \to B(*, Aut(F), *)
  \,.
$$

=--

Compare [[universal principal ∞-bundle]].

## Examples

### Fibrations of topological spaces / simplicial sets

For the special case that $\mathbf{H} = $ [[∞Grpd]] and using the [[presentable (∞,1)-category|presentation]] by the [[model structure on topological spaces]]/[[model structure on simplicial sets]] the classification theorem \ref{Classification} reduces to the classical statement of ([Stasheff](#Stasheff), [May](#May)).

### $\infty$-Gerbes

In the case that the fiber $F$ is the [[delooping]] $F = \mathbf{B}G$ of an [[∞-group]] object $G$, the $\underline{Aut}(\mathbf{B}G)$-associated $\infty$-bundles are called **$G$-[[∞-gerbes]]**. See there for more details.

## Related concepts

* [[action]], [[∞-action]]

* [[representation]], [[∞-representation]]

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

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

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

* [[principal ∞-bundle]] / [[∞-gerbe]] / **associated $\infty$-bundle**

* [[vector bundle]]

* [[(∞,1)-vector bundle]]

## References
 {#References}

Early work on associated $\infty$-bundles takes place in the $(\infty,1)$-topos [[∞Grpd]] $\simeq$ [[Top]]. In

* [[Jim Stasheff]], _A classification theorem for fiber spaces_ , Topology 2 (1963) 239-246
 {#Stasheff}

* [[Jim Stasheff]], _H-spaces and classifying spaces: foundations and recent developments_. Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), pp. 247–272. MR0321079 (47 #9612)

a classification of [[fibrations]] of [[CW-complexes]] with given CW-complex fiber in terms of maps into a classifying CW-complex is given.

In

* [[Daniel Gottlieb]], _The total space of universal fibrations._  Pacific J. Math. Volume 46, Number 2 (1973), 415-417.

the total space of the universal $F$-[[fiber ∞-bundle]] in the pointed context is identified with $\mathbf{B}Aut_*(F)$ (the pointed [[automorphism ∞-group]]).

A generalization or more systematic account of the classification theory is then given in

* [[Peter May]], _Classifying Spaces and Fibrations_ Mem. Amer. Math. Soc. 155 (1975) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/Classifying.pdf))
{#May}

This has been reproven in various guises, such as the statement of [[univalence]] in the [[model]] [[sSet]] for [[homotopy type theory]]. See the references at _[[univalence]]_ for more on this.

Generalizations with extra structure on the fibers are discussed in

* Claudio Pacati, Petar Pavesic, Renzo Piccinini, _On the classification of $\mathcal{F}$-fibrations_, Topology and its applications 87 (1998) ([pdf](http://www.fmf.uni-lj.si/~pavesic/RESEARCH/On%20the%20classification%20of%20F-fibrations.pdf))

Consideration of associated $\infty$-bundles / [[fiber sequences]] in general [[n-localic (infinity,1)-topos|1-localic]] [[(∞,1)-toposes]] [[presentable (infinity,1)-category|presented]] by a [[model structure on simplicial presheaves]] (which subsumes the above case for the trivial site) is discussed in

* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ , Journal of  Homotopy and Related Structures 6(1), 2011, pp. 1--38.  ([arXiv](http://arxiv.org/abs/1009.2930)) ([published version](http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm))
{#Wendt}

Related discussion on the behaviour of [[fiber sequences]] under left [[Bousfield localization of model categories]] is in

* [[Matthias Wendt]], _Fibre sequences and localization of simplicial sheaves_ ([pdf](http://home.mathematik.uni-freiburg.de/arithmetische-geometrie/preprints/wendt-flocal.pdf))

Similar considerations and results are in

* Martin Blomgren, [[Wojciech Chacholski]], _On the classification of fibrations_ ([arXiv:1206.4443](http://arxiv.org/abs/1206.4443))
 {#BlomgrenChacholski}

With the advent of [[(∞,1)-topos theory]] all these statements and their generalizations follow from the existence of [[object classifiers]] in an [[(∞,1)-topos]]. For the classical case in [[∞Grpd]] $\simeq$ [[Top]]${}^\circ$ [[sSet]]${}^\circ$ this is discussed in

* _[object classifier in ∞Grpd](http://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid)_,

which reproduces the classical results ([Stasheff](#Stasheff), [May](#May)).

For general [[(∞,1)-toposes]] the classification of associated $\infty$-bundles is discussed in section I 4.1 of

* [[schreiber:Principal ∞-bundles -- theory, presentations and applications]]

Models in [[rational homotopy theory]] of classifying spaces for homotopy types $Aut(F)$ go back to [[Sullivan]]'s remarks on the [[automorphism L-infinity algebra]]. Further developments are reviewed and developed in

* [[Andrey Lazarev]], _Models for classifying spaces and derived deformation theory_ ([arXiv:1209.3866](http://arxiv.org/abs/1209.3866))

[[!redirects associated infinity-bundle]]
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associative 3-form

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## Definition

On the [[Cartesian space]] $\mathbb{R}^7$ the _associative 3-form_ is the [[differential n-form|differential 3-form]] $\omega \in \Omega^3(\mathbb{R}^7)$ which is constant and whose value at the origin on three vectors is

$$
  \omega(u,v,w) \coloneqq \langle u, v \times w\rangle
  \,,
$$

where

* $\langle -,-\rangle$ is the canonical [[bilinear form]] on $\mathbb{R}^7$;

* $(-)\times (-)$ is the [[cross product]] in $\mathbb{R}^7$.

This are the structure constants of the unit [[octonions]].

The Hodge dual of the associative 3-form is sometimes called the _co-associative 4-form_.

## Properties

### Relation to $G_2$

The [[group]] of [[linear map|linear]] [[diffeomorphisms]] of $\mathbb{R}^7$ which preserve this form is the [[exceptional Lie group]] [[G2]].

## Related concepts

[[!include special holonomy table]]

* [[associative submanifold]]

* [[octonions]]

* [[Hitchin functional]]

## References

(...)

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[[!redirects co-associative 4-forms]]


associative dialgebra

__Associative dialgebra__ is an algebraic structure introduced by Loday and Pirashvili to formalize the structure for the universal enveloping of a [[Leibniz algebra]].

* [[Jean-Louis Loday]], T. Pirashvili, _Universal enveloping algebras of Leibniz algebras and (co)homology_, Math. Ann. __296__:1 (1993) 139--158

Associative dialgebras are algebras over a Koszul quadratic operad whose Koszul dual is the operad of [[dendriform dialgebra]]s. In the standard presentation,
associative dialgebras have two algebraic operations.

category: algebra


associative H-space

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## Idea

In [[dependent type theory]], an *associative [[H-space]]* (cf. [BCFR23](#BCFR23)) or *$A_3$-space* is a naive translation of the notion of [[monoid]]. If the underlying type is an [[h-set]] (such as if the type theory is [[extensional type theory|extensional]]), then it yields a correct notion of monoid.  Otherwise, it is only a "partly-coherent" notion of "homotopy monoid": a type-theoretic version of the notion of $A_3$-[[An-space|space]] from homotopy theory.

## Definition

Similar to the definition of [[H-spaces]], there are coherent and non-coherent versions of associative H-spaces:

### Non-coherent associative H-spaces

A **non-coherent associative H-space** or **non-coherent $A_3$-space** consists of

* A type $A$,
* A basepoint $e:A$
* A binary operation $\mu : A \to A \to A$
* A left unitor
$$\lambda:\prod_{x:A} \mu(e,x)=x$$
* A right unitor
$$\rho:\prod_{x:A} \mu(x,e)=x$$
* An asssociator
$$\alpha:\prod_{x:A} \prod_{y:A} \prod_{z:A} \mu(\mu(x, y),z)=\mu(x,\mu(y,z))$$

### Coherent associative H-spaces

A **coherent associative H-space** or **coherent $A_3$-space** is a non-coherent associative H-space $A$ which additionally has the coherence condition

$$\mu_{\lambda \rho}:\lambda(e) = \rho(e)$$

since $\lambda(e)$ and $\rho(e)$ are elements of the identity type $\mu(e, e) = e$.

## Homomorphisms of associative H-spaces

There are also two different notions of homomorphisms between associative H-spaces, depending on whether the H-space is coherent or not.

### Homomorphisms of non-coherent associative H-spaces

A **homomorphism of non-coherent associative H-spaces** between two non-coherent associative H-spaces $A$ and $B$ is a function $\phi:A \to B$ such that

* The basepoint is preserved, i.e. there is a specified identification
$$\phi_e:\phi(e_A) = e_B$$

* The binary operation is preserved, i.e. there is a specified dependent function
$$\phi_\mu:\prod_{x:A} \prod_{y:A} \phi(\mu_A(x, y)) = \mu_B(\phi(x),\phi(y))$$

* The left unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications
$$\phi_\mu(e_A, x):\phi(\mu_A(e_A,x)) = \mu_B(\phi(e_A),\phi(x))$$
$$\mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e):\mu_B(\phi(e_A),\phi(x)) = \mu_B(e_B,\phi(x))$$
$$\lambda_B(\phi(x)):\mu_B(e_B,\phi(x)) = \phi(x)$$
with the image $\mathrm{ap}_{\phi}(\lambda_A(x)):\phi(\mu_A(e_A,x)) = \phi(x)$ of the left unitor of $A$.
$$\phi_\lambda:\prod_{x:A} \phi_\mu(e_A, x) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e) \bullet \lambda_B(\phi(x)) = \mathrm{ap}_{\phi}(\lambda_A(x))$$

* Similarly, the right unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications
$$\phi_\mu(x, e_A):\phi(\mu_A(x,e_A)) = \mu_B(\phi(x),\phi(e_A))$$
$$\mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e):\mu_B(\phi(x),\phi(e_A)) = \mu_B(\phi(x),e_B)$$
$$\rho_B(\phi(x)):\mu_B(\phi(x),e_B) = \phi(x)$$
with the image $\mathrm{ap}_{\phi}(\rho_A(x)):\phi(\mu_A(x,e_A)) = \phi(x)$ of the right unitor of $A$.
$$\phi_\rho:\prod_{x:A} \phi_\mu(x, e_A) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e) \bullet \rho_B(\phi(x)) = \mathrm{ap}_{\phi}(\rho_A(x))$$

* The associator is preserved in an analogous way.

### Homomorphisms of coherent associative H-spaces

A **homomorphism of coherent associative H-spaces** between two coherent associative H-spaces $A$ and $B$ is a homomorphism of non-coherent associative H-spaces $\phi:A \to B$ in which the coherence condition is preserved.

## Examples

* The [[integers]] are an associative H-space under addition, as are the [[natural numbers type|natural numbers]]. More generally, as noted above, any [[h-set]] monoid is an associative H-space, and is coherent.

* Every [[loop space]] type $\Omega_x(X) \equiv (x=_X x)$ is naturally an associative H-space, with [[path]] concatenation as the operation.  In this case, the operation is in fact coherent, and indeed every loop space is a $\infty$-group (although this is difficult to express in type theory).

* The type of endofunctions $A \to A$ has the structure of an associative H-space, with basepoint $id_A$, operation function composition.  Once again the operation is coherent, so this is an $\infty$-monoid (inverses need not exist).

* The type of coherent H-space structures on a central type is a coherent associative H-space.

## Related concepts

* [[higher algebra]]
* [[H-space]]
* [[monoid]]
* [[ring]]
* [[H-category]]
* [[An-space]]
* [[grouplike A3-space]]

## References

* {#BCFR23} [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

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associative magma

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## Definition

A [[magma]] $(S,\cdot)$ is called _associative_ if it satisfies the [[associativity]] condition, saying that for all $x,y,z \in S$ then the [[equation]]

$$
  (x \cdot y) \cdot z
  =
  x \cdot (y \cdot z)
$$

holds.

## Examples

Examples include [[semigroups]]/[[monoids]], [[rings]], [[associative algebras]], etc.

## Related concepts

* [[unital magma]]

* [[commutative magma]]

* [[nonassociative ring]], [[nonassociative algebra]]

* [[alternative algebra]]

[[!redirects associative magmas]]

[[!redirects associative operation]]
[[!redirects associative operations]]


associative n-category

[[!redirects associative n-categories]]

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## Idea

The notion of *associative $n$-categories* (ANCs, [Dorn 2018](#Dorn18), [Reutter & Vicary 2019](#ReutterVicary19)) is a [[semi-strict infinity-category|semi-strict]] [[algebraic definition of higher categories|algebraic model]] for [[higher category theory]] based on the [[geometric shape for higher structures|geometric shape]] of [[globular sets]] with strictly [[associativity|associative]] [[composition]]: All higher [[coherence law|coherence laws]] in associative $n$-categories are strict, except for weak versions of higher [[exchange laws]].

Due to the tight control over [[coherence]]-issues, the theory of (free) associative $n$-categories lends itself to [[formal proof]], for implementation of corresponding [[proof assistants]] see *[[homotopy.io]]* (and its precursor *[[Globular]]*).

Composition operations in associative $n$-categories naturally have stratified-geometric semantics in [[manifold diagram|manifold diagrams]], which makes them an instance of [[geometric n-category|geometrical $n$-categories]]. In particular, the weak coherence laws of an associative $n$-category can be thought of in terms of a notion of homotopy between composites. This is similar to the case of a [[Gray-category]], which is strictly associative and unital, but which has a weak exchange law. In this sense, ANCs can be seen as a generalization of Gray categories.

## Examples

* An associative 0-category is a [[set]]

* An associative 1-category is an [[biased definition|unbiased]] [[1-category]]

* An associative 2-category is an [[biased definition|unbiased]] [[strict 2-category]]

* An associative 3-category is an [[biased definition|unbiased]] [[Gray-category|Gray 3-category]]

A separate notion of 'free' associative n-categories has been developed. These are instances of [[geometric computad|geometric computads]].

## Remarks

* It is conjectured that every [[weak n-category]] is weakly equivalent to an associative $n$-category with strict units. A (proof-wise) potentially more realistic version of this conjecture concerns the case of free associative $n$-categories, and is (as of 2023) a topic of active research.

* A related conjecture is [[Simpson's conjecture]], which states that fully weak higher categories are (in an appropriate sense) equivalent to weakly unital higher categories.

* The underlying idea of this line of modelling higher structures can be traced back to work on [[Gray-category|Gray 3-categories]] and [[surface diagram|surface diagrams]].

## Related concepts

* [[manifold-diagrammatic n-category]]

* [[semistrict n-category]]

* [[homotopy.io]], [[Globular]]

* [[manifold diagram]]

* [[surface diagram]]

* [[globular set]]

* [[Gray-category]]

* [[Simpson's conjecture]]

## References

* [[Christoph Dorn]], _Associative $n$-categories_, talk at _[103rd Peripatetic Seminar on Sheaves and Logic](https://www.math.muni.cz/~loregianf/PSSL103/PSSL103.php)_ ([pdf](https://www.math.muni.cz/~loregianf/PSSL103/slides/Dorn.pdf)).

* {#Dorn18} [[Christoph Dorn]], _Associative $n$-categories_, PhD thesis ([arXiv:1812.10586](https://arxiv.org/abs/1812.10586)).

* {#ReutterVicary19} [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]

* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

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associative operad

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##  Idea

The _associative operad_ $Assoc$ is the [[operad]] whose [[algebras over an operad|algebras]] are [[monoid|monoids]]; i.e. objects equipped with an [[associative]] and [[unital]] binary operation.

One might also consider the non-unital version, whose algebras are objects equipped with a binary operation but not with a [[unit]].

## Definition

### As a $Vect$-operad

The _associative operad_, denoted $Assoc$ or $Ass$, is often taken to be the [[Vect]]-[[operad]] whose [[algebra over an operad|algebras]] are precisely [[associative unital algebras]].

### As a $Set$-operad

As a [[Set]]-enriched [[planar operad]], $Assoc$ is the operad that has precisely one single $n$-ary operation for each $n$.  Accordingly, $Assoc$ in this sense is the [[terminal object]] in the [[category]] of [[planar operads]].

(Here the unique 0-ary operation is the [[unit]]. Hence the non-unital version of $Assoc$ has a single operation in each positive arity and none in arity 0.)

As a [[Set]]-enriched [[symmetric operad]] $Assoc$ has (the set underlying) the [[symmetric group]] $\Sigma_n$ in each degree, with the [[action]] being the action of $\Sigma_n$ on itself by multiplication from one side.

Similarly, as a _planar_ [[dendroidal set]], $Assoc$ is the [[presheaf]] that assigns the singleton to every [[planar tree]] (hence also the [[terminal object]] in the category of dendroidal sets).

But, by the above, as an [[symmetric operad|symmetric]] [[dendroidal set]], $Assoc$ is not the terminal object.

## Properties

### Resolution

The relative [[Boardman-Vogt resolution]] $W([0,1],I_* \to Assoc)$ of $Assoc$ in [[Top]] is [[Jim Stasheff]]'s version of the [[A-∞ operad]] whose [[algebra over an operad|algebras]] are [[A-∞ algebras]].

### Relation to planar operads

A [[planar operad]] may be identified with a [[symmetric operad]] that is equiped with a map to the associative operad. See at _[[planar operad]]_ for details.

## Related concepts

* **associative operad**

* [[commutative operad]]

* [[Lie operad]]

## References

In the context of [[higher algebra]] of [[(infinity,1)-operads]], the associative operad is discussed in section 4.1.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects Ass]]
[[!redirects Assoc]]


associative quasigroup

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## Definitions

A __(left/right/two-sided) [[associative]] [[quasigroup]]__ is a (left/right/two-sided) [[quasigroup]] $(G, \cdot,\backslash,/)$ where $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a$, $b$, and $c$ in $G$.

In every associative quasigroup $G$, $a/a = b\backslash b$ for every $a$ and $b$ in $G$. This is because for every $a$ and $b$ in $G$, $b \cdot a = b \cdot (a/a) \cdot a = b \cdot (b\backslash b) \cdot a$. Left dividing both sides by $b$ and right dividing both sides by $a$ results in $a/a = b\backslash b$. This means in particular that $a/a = a\backslash a$, which means that every associative quasigroup is a [[possibly empty loop]]. Thus, an associative quasigroup is a associative possibly empty loop, or a __possibly empty group__. In particular, there is an additional definition of an associative quasigroup in terms of the left and right divisions alone, without any semigroup operation at all:

A __left associative quasigroup__ is a [[set]] $G$ with a binary operation $(-)\backslash(-):G \times G \to G$ (a [[magma]]) such that:

  * For all $a$ and $b$ in $G$, $a\backslash a=b\backslash b$
  * For all $a$ in $G$, $(a\backslash (a\backslash a))\backslash (a\backslash a)=a$
  * For all $a$, $b$, and $c$ in $G$, $(a\backslash b)\backslash c= b\backslash ((a\backslash (a\backslash a))\backslash c)$.

For any element $a$ in $G$, the element $a\backslash a$ is called a __right identity element__, and the element $a\backslash (a\backslash a)$ is called the __right inverse element__ of $a$. For all elements $a$ and $b$ in $G$, __left multiplication__ of $a$ and $b$ is defined as $(a\backslash (a\backslash a))\backslash b$.

A __right associative quasigroup__ is a [[set]] $G$ with a binary operation $(-)/(-):G \times G \to G$ such that:

  * For all $a$ and $b$ in $G$, $a/a=b/b$
  * For all $a$ in $G$, $(a/a)/((a/a)/a)=a$
  * for all $a$, $b$, and $c$ in $G$, $a/(b/c)=(a/((c/c)/c)/b$

For any element $a$ in a $G$, the element $a/a$ is called a __left identity element__, and the element $(a/a)/a$ is called the __left inverse element__ of $a$. For all elements $a$ and $b$, __right multiplication__ of $a$ and $b$ is defined as $a/((b/b)/b)$.

An __associative quasigroup__ is a possibly empty left and right group as defined above such that the following are true:

* left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$

* left and right inverse elements are equal (i.e. $(a/a)/a = a\backslash (a\backslash a)$) for all $a$ in $G$

* left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$.

This definition [first appeared on the heap article](https://ncatlab.org/nlab/revision/diff/heap/13) and is due to [[Toby Bartels]].

## Pseudo-torsors

Every left associative quasigroup $G$ has a [[pseudo-torsor]] $t_G:G^3 \to G$ defined as $t_G(x,y,z) = x \cdot (y \backslash z)$. Every right associative quasigroup $H$ has a [[pseudo-torsor]] $t_H:H^3 \to H$ defined as $t_H(x,y,z) = (x / y) \cdot z$. This means every associative quasigroup has two pseudo-torsors. If the (left or right) associative quasigroup is inhabited, then those pseudo-torsors are actually [[torsors]] or [[heaps]].

## Category of associative quasigroups

An __associative quasigroup homomorphism__ is a [[semigroup homomorphism]] between associative quasigroups that preserves left and right quotients. Associative quasigroup homomorphisms are the [[morphisms]] in the [[category]] of associative quasigroups $AssocQuasiGrp$.

As the category of associative quasigroups is a concrete category, there is a [[forgetful functor]] $U:AssocQuasiGrp \to Set$. $U$ has a [[left adjoint]], the __free associative quasigroup__ [[free functor|functor]] $F:Set \to AssocQuasiGrp$.

The __empty associative quasigroup__ $0$ whose underlying set is the [[empty set]] is the [[initial object|initial associative quasigroup]], and is [[strict initial object|strictly initial]]. The __trivial associative quasigroup__ $1$ whose underlying set is the [[singleton]] is the [[terminal object|terminal associative quasigroup]].

The __direct product__ $G \times H$ of associative quasigroups $G$ and $H$ is the [[cartesian product]] of sets $U(G) \times U(H)$ with an associative quasigroup structure defined componentwise by

$$
  (g_1, h_1) \cdot_{G \times H} (g_2, h_2) = (g_1 \cdot_G g_2, h_1 \cdot_H h_2)
$$

$$
  (g_1, h_1) /_{G \times H} (g_2, h_2) = (g_1 /_G g_2, h_1 /_H h_2)
$$

$$
  (g_1, h_1) \backslash_{G \times H} (g_2, h_2) = (g_1 \backslash_G g_2, h_1 \backslash_H h_2)
$$

for all $(g_1, h_1), (g_2, h_2) \in U(G) \times U(H)$, with [[product projections]] $p_G: G \times H \to G$ $p_H: G \times H \to H$ where $p_G(g, h) = g$ and $p_H(g,h) = h$ for all $(g,h)\in G \times H$ The direct product of associative quasigroups is thus the [[cartesian product]] in $AssocQuasiGrp$. The [[endofunctor]] $P(G) = G \times 1$ is the [[identity functor]] on $AssocQuasiGrp$ while the endofunctor $P(G) = G \times 0$ is a [[constant functor]] that sends every associative quasigroup to $0$.

An __associative subquasigroup__ of an associative quasigroup $G$ is an associative quasigroup $H$ with an associative quasigroup [[monomorphism]]
$$
  H \hookrightarrow G
  \,.
$$
The empty associative is the initial associative subquasigroup of $G$.

Let $G$ and $H$ be associative quasigroups and let $f:G \to H$ be an associative quasigroup homomorphism. Then given an element $h \in H$, there is an associative subquasigroup
$$
  i: I \hookrightarrow G
  \,.
$$
such that $g \in I$ if and only if $f(g) = h$. $I$ is called the __[[fiber]]__ of $f$ over $h$.

Because $AssocQuasiGrp$ has a terminal object, cartesian products, and fibers, it is a [[finitely complete category]].

Every associative quasigroup $G$ and associative subquasigroup $H \hookrightarrow G$ has a __set of left [[ideal in a semigroup|ideals]]__ $G H$ in $G$ and a __set of right ideals__ $H G$ in $G$.

## Examples

* Every [[group]] is an associative quasigroup.

* The empty associative quasigroup is an associative quasigroup that is not a group.

## Related concepts

* [[quasigroup]]

* [[commutative quasigroup]]

* [[loop (algebra)|loop]]

* [[group]]

* [[n-ary group]]

[[!include oidification - table]]

## References

* [The Group With No Elements](https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html#c059798) at the [n-category café](https://golem.ph.utexas.edu/category/)

[[!redirects associative quasigroups]]

[[!redirects possibly empty group]]
[[!redirects possibly empty groups]]


associative quasigroupoid

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#### Category theory
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#### Algebra
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#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], an associative quasigroupoid should be the oidification of an [[associative quasigroup]].

## Definition

An __associative quasigroupoid__ is a [[magmoid]] $Q$ such that for every [[diagram]]

$$
  a\underset{\quad f \quad}{\to}b\underset{\quad g \quad}{\to}c\underset{\quad h \quad}{\to}d
  \,,
$$

$$
  h \circ (g \circ f) = (h \circ g) \circ f
  \,,
$$

for every [[span]]

$$
  \array{
     && s
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
     \\
     x
     &&&&
     y
  }
$$

in $Q$, there exists morphisms $i:x\to y$ and $j:y \to x$ such that $i \circ f = g$ and $j \circ g = f$, and for every [[cospan]]

$$
  \array{
     && a &&&& b
      \\
      &
      && {}_{f}\searrow
       &
       & \swarrow_g
      &&
     \\
     &&&&
     c
     &&&&
  }
$$

in $Q$, there exists morphisms $d:a\to b$ and $e:b \to a$ such that $g \circ d = f$ and $f \circ e = g$.

## Examples

* Every [[groupoid]] is an associative quasigroupoid.

* A one-object associative quasigroupoid is an [[associative quasigroup]].

* An associative quasigroupoid [[enriched magmoid|enriched]] in [[truth values]] is an [[equivalence relation]].

## Related concepts

[[!include oidification - table]]


associative ring spectrum

#Definition#

An **associative ring spectrum**, or [[A-infinity ring]], is an [[algebra in an (infinity,1)-category|monoid object in]] the [[stable (infinity,1)-category of spectra]] with its [[smash product of spectra]] [[monoidal (infinity,1)-category|monoidal structure]].

Equivalently, this should be the same as an ordinary [[monoid]] with respect to the [[symmetric monoidal smash product of spectra]].

#Remarks#

* A commutative associative ring spectrum is a [[commutative ring spectrum]], or [[E-infinity ring]].

#References#

the $(\infty,1)$-categorical description of associative ring spectra is in section 4.3 of

* [[Jacob Lurie]], [[higher algebra|Noncommutative algebra]]


associative submanifold

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* table of contents
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## Definition

A [[3-manifold|3-dimensional]] [[submanifold]] $\Sigma_3 \hookrightarrow X_7$ of a [[G2-manifold]] $X_7$ is called _associative_ if it is a [[calibrated submanifold]], hence if the canonical 3-form $\phi$ of the $G_2$-manifold (tangent space-wise the [[associative 3-form]]) restricts to the [[volume form]] on $\Sigma_3$.

Accordingly a [[4-manifold|4-dimensional]] [[submanifold]] $\Sigma_4 \hookrightarrow X_7$ is called _coassociative_ if it is a [[calibrated submanifold]] with respect to the coassociative 4-form $\star_X \phi$.

## Application in string theory

In [[M-theory on G2-manifolds]],

* associative 3-manifolds appear as [[supersymmetric cycles]] relevant for [[membrane instantons]] for [[M2-branes]] [[wrapped brane|wrapping them]];

* coassociative 4-manifolds appear as fibers of compactifications dual to [[F-theory on CY4-folds]] ([Gukov-Yau-Zaslow 02](#GukovYauSaslow02)).

## References

* Ian Weiner, _Associative 3-manifolds in $\mathbb{R}^7$_, 2001 ([pdf](https://www.math.hmc.edu/seniorthesis/archives/2001/iweiner/iweiner-2001-thesis.pdf))

* Selman Akbulut, Sema Salur, _Associative submanifolds of a $G_2$-manifold_ ([pdf](http://users.math.msu.edu/users/akbulut/papers/associativeG2.pdf))

* Damien Gayet, _Smooth moduli spaces of associative submanifolds_ ([arXiv:1011.1744](http://arxiv.org/abs/1011.1744))

On [[associative submanifolds]] of the [[7-sphere]]:

* [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](https://arxiv.org/abs/1006.0361))

Discussion of [[wrapped branes]] on associative 3-cycles is in

* [[Katrin Becker]], [[Melanie Becker]], D.R.Morrison, [[Hirosi Ooguri]], Y.Oz, Z.Yin, _Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds_, Nucl.Phys.B480:225-238,1996 ([arXiv:hep-th/9608116](http://arxiv.org/abs/hep-th/9608116))

Further discussion specifically of [[M5-branes]] [[wrapped brane|wrapped]] on associative 3-cycle which are either the [[3-sphere]] or a [[hyperbolic 3-manifold]] is in

* [[Bobby Acharya]], [[Jerome Gauntlett]], Nakwoo Kim, _Fivebranes Wrapped On Associative Three-Cycles_, Phys.Rev. D63 (2001) 106003 ([arXiv:hep-th/0011190](http://arxiv.org/abs/hep-th/0011190))

* {#GukovYauZaslow02} [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Duality and Fibrations on $G_2$ Manifolds_ ([arXiv:hep-th/0203217](http://arxiv.org/abs/hep-th/0203217))

[[!redirects associative submanifolds]]

[[!redirects associative 3-manifold]]
[[!redirects associative 3-manifolds]]

[[!redirects associative 3-cycle]]
[[!redirects associative 3-cycles]]