adiabatic quantum computation

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### Context
#### Computation
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[[!include constructivism - contents]]
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
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## Idea

By *adiabatic quantum computation* one means models of [[quantum computation]] on [[parameterized quantum systems]] where the [[quantum gates]] are [[unitary transformations]] on a [[topological phase of matter|gapped]] (and possibly [[topological order|topologically ordered]]) [[ground state]] which are induced, via the [[quantum adiabatic theorem]], by sufficiently slow movement of external [[dependent linear type theory|parameters]].

Often the term *adiabatic quantum computation* is used by default for [[optimization theory|optimization]] problems ("[[quantum annealing]]", see the references [below](#QuantumAnnealingReferences)).

On the other hand, the possibly most prominent example of adiabatic quantum computation is often not advertized as such (but see [CLBFN 2015](#CLBFN15)), namely [[topological quantum computation]] by adiabatic [[braid group|braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects) (whose positions is the external parameter, varying in a [[configuration space of points]]). This is made explicit in [Freedman, Kitaev, Larsen & Wang 2003, pp. 6](#FreedmanKitaevLarsenWang03); [Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6)](#NayakSimonSternFreedman08); and [Cheng, Galitski & Das Sarma 2011, p. 1](#ChengGalitskiDasSarma11); see also [Arovas, Schrieffer, Wilczek & Zee 1985, p. 1](#ArovasSchriefferWilczekZee85) and [Stanescu 2020, p. 321](#Stanescu20); [Barlas & Prodan 2020](#BarlasProdan20).

The following graphics shows this with labelling indicative of [momentum-space anyons](braid+group+statistics#ReferencesAnyonicBraidingInMomentumSpace):

\begin{imagefromfile}
    "file_name": "AdiabaticBraidingOfBandNodes-220604.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

> (graphics from [[schreiber:Topological Quantum Computation in TED-K|SS22]])

## Related concepts

* [[quantum adiabatic theorem]]

* [[Berry phase]]

* [[quantum computation]]

  * [[measurement-based quantum computation]]

  * [[topological quantum computation]]

## References
 {#References}

### General
 {#ReferencesGeneral}

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Michael Sipser, *Quantum Computation by Adiabatic Evolution* [[arXiv:quant-ph/0001106](https://arxiv.org/abs/quant-ph/0001106)]

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Joshua Lapan, Andrew Lundgren, Daniel Preda, *A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem*, Science  **292** 5516 (2001) 472-475 [[doi:10.1126/science.1057726](https://doi.org/10.1126/science.1057726)]

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

Review:

* [[Andrew Childs]],  *Overview of adiabatic quantum computation*, talk at *[CIFAR](https://cifar.ca/research-programs/quantum-information-science/) Workshop on Quantum Information Processing* (2013) [[pdf](https://www.cs.umd.edu/~amchilds/talks/cifar13-tutorial.pdf), [[Childs-AdiabaticQuantumComputation.pdf:file]]]

On robustness of adiabatic quantum computation (such as against [[decoherence]]):

* {#ChildsFarhiPreskill02} [[Andrew Childs]], [[Edward Farhi]], [[John Preskill]], *Robustness of adiabatic quantum computation*, Phys.Rev. A **65** (2002) 012322 [[arXiv:quant-ph/0108048](https://arxiv.org/abs/quant-ph/0108048), [doi:10.1103/PhysRevA.65.012322](https://doi.org/10.1103/PhysRevA.65.012322)]

### In optimization -- quantum annealing
 {#QuantumAnnealingReferences}

Review with focus on optimization problems ([[quantum annealing]]):

* Catherine C. McGeoch, *Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice* Synthesis Lectures on Quantum Computing (2014) [[doi:10.2200/S00585ED1V01Y201407QMC008)](https://doi.org/10.2200/S00585ED1V01Y201407QMC008)]

* Tameem Albash, Daniel A. Lidar,  *Adiabatic Quantum Computing*, Rev. Mod. Phys. **90** (2018) 015002 [[arXiv:1611.04471](https://arxiv.org/abs/1611.04471), [doi:10.1103/RevModPhys.90.015002](https://doi.org/10.1103/RevModPhys.90.015002)]

* Salvador E. Venegas-Andraca, William Cruz-Santos, Catherine McGeoch, Marco Lanzagorta, *A cross-disciplinary introduction to quantum annealing-based algorithms*, Contemporary Physics **59** 02 (2018) 174-196 [[arXiv:1803.03372](https://arxiv.org/abs/1803.03372), [doi:10.1080/00107514.2018.1450720](https://doi.org/10.1080/00107514.2018.1450720)]

* Erica K. Grant and Travis S. Humble, *Adiabatic Quantum Computing and Quantum Annealing*, Oxford research Encyclopedia (2020) [[doi:10.1093/acrefore/9780190871994.013.32](https://doi.org/10.1093/acrefore/9780190871994.013.32)]

* Atanu Rajak, Sei Suzuki, Amit Dutta, Bikas K. Chakrabarti, *Quantum Annealing: An Overview*, Philos Trans A Math Phys Eng Sci  **381** 2241 (2023) 20210417 [[arXiv:2207.01827](https://arxiv.org/abs/2207.01827), [doi:10.1098/rsta.2021.0417](https://doi.org/10.1098/rsta.2021.0417)]

See also:

* Wikipedia, *[Adiabatic quantum computation](https://en.wikipedia.org/wiki/Adiabatic_quantum_computation)*

* Kristen L. Pudenz, Tameem Albash, Daniel A. Lidar: *Error-corrected quantum annealing with hundreds of qubits*, Nature Communications **5** 3243 (2014) [[doi:10.1038/ncomms4243](https://doi.org/10.1038/ncomms4243)]

  > (with [[quantum error correction]])

* Minjae Jo, Michael Hanks, M. S. Kim, *Divide-and-conquer embedding for QUBO quantum annealing* [[arXiv:2211.02184](https://arxiv.org/abs/2211.02184)]

* Yusuke Kimura, Hidetoshi Nishimori, *Rigorous convergence condition for quantum annealing*, J. Phys. A: Math. Theor. **55** (2022) 435302 [[arXiv:2207.12096](https://arxiv.org/abs/2207.12096), [doi:10.1088/1751-8121/ac9dce](https://doi.org/10.1088/1751-8121/ac9dce)]

A more high-brow mathematical desription via "tangle machines":

* Avishy Y. Carmi, [[Daniel Moskovich]], §5 of: *Tangle Machines*, Proc. R. Soc. A **471** (2015) 20150111 [[arXiv:1404.2862](https://arxiv.org/abs/1404.2862), [doi:10.1098/rspa.2015.0111](https://doi.org/10.1098/rspa.2015.0111)]

On adiabatic quantum computation combined with [[parameterized quantum system|parameterized]] [[quantum circuits]]:

* Ioannis Kolotouros, Ioannis Petrongonas, Miloš Prokop, Petros Wallden, *Adiabatic quantum computing with parameterized quantum circuits* [[arXiv:2206.04373](https://arxiv.org/abs/2206.04373)]

### Geometric phase gates, holonomic quantum computation

References which consider quantum gates operating by (nonabelian) geometric Berry phases due to adiabatic parameter movement (*holonomic quantum computation*):

* [[Paolo Zanardi]], [[Mario Rasetti]], *Holonomic Quantum Computation*, Phys. Lett. A **264** (1999) 94-99 [doi:10.1016/S0375-9601(99)00803-8, [arXiv:quant-ph/9904011](https://arxiv.org/abs/quant-ph/9904011)]

* [[Jiannis Pachos]], [[Paolo Zanardi]], [[Mario Rasetti]], *Non-Abelian Berry connections for quantum computation*, Phys. Rev. A **61** (2000) 010305 [[arXiv:quant-ph/9907103](https://arxiv.org/abs/quant-ph/9907103), [doi:10.1103/PhysRevA.61.010305](https://doi.org/10.1103/PhysRevA.61.010305)]

* Jonathan A. Jones, Vlatko Vedral, Artur Ekert, Giuseppe Castagnoli, *Geometric quantum computation using nuclear magnetic resonance*, Nature **403** (2000) 869–871 [[doi:10.1038/35002528](https://doi.org/10.1038/35002528)]

* Giuseppe Falci, Rosario Fazio, G. Massimo Palma, Jens Siewert & Vlatko Vedral, *Detection of geometric phases in superconducting nanocircuits*, Nature **407** 355–358 (2000) [[doi:10.1038/35030052](https://doi.org/10.1038/35030052)]

* [[Jiannis Pachos]], [[Paolo Zanardi]], *Quantum Holonomies for Quantum Computing*, Int. J. Mod. Phys. B **15** (2001) 1257-1286 [[arXiv:quant-ph/0007110](https://arxiv.org/abs/quant-ph/0007110), [doi:10.1142/S0217979201004836](https://doi.org/10.1142/S0217979201004836)]

* L. M. Duan, J. I. Cirac, P. Zoller, *Geometric Manipulation of Trapped Ions for Quantum Computation*, Science **292** (2001) 1695 [[arXiv:quant-ph/0111086](https://arxiv.org/abs/quant-ph/0111086), [doi:10.1126/science.1058835](https://doi.org/10.1126/science.1058835)]

* Jiang Zhang, Thi Ha Kyaw, Stefan Filipp, Leong-Chuan Kwek, Erik Sjöqvist, Dianmin Tong, *Geometric and holonomic quantum computation* [[arXiv:2110.03602](https://arxiv.org/abs/2110.03602)]

* Daniel Turyansky et al., *Inertial geometric quantum logic gates* [[arXiv:2303.13674](https://arxiv.org/abs/2303.13674)]

* Logan W. Cooke et al., *Demonstration of Floquet engineered non-Abelian geometric phase for holonomic quantum computing* [[arXiv:2307.12957](https://arxiv.org/abs/2307.12957)]

### In topological quantum computation

References which make explicit that [[topological quantum computation]] by [[braiding]] of [[anyon]] [[worldlines]] is a form of adiabatic quantum computation:

* {#ArovasSchriefferWilczekZee85} [[Daniel P. Arovas]], [[Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](anyon#Wilczek90)) [doi:10.1016/0550-3213(85)90252-4]

* [Childs, Farhi & Preskill (2002), p. 2](#ChildsFarhiPreskill02)

* {#FreedmanKitaevLarsenWang03} [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], pp. 6 of *Topological quantum computation*,  Bull. Amer. Math. Soc. __40__ (2003) 31-38 [[arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf)]

* {#BonesteelHormoziZikosSimon05} [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], p. 1 of: *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) [[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)]

* {#NayakSimonSternFreedman08} [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], §II.A.2 (p. 6) of: _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. **80** 1083 (2008) [[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)]

* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 [[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)]

* [[Gustavo Rigolin]], [[Gerardo Ortiz]], p. 1 of: *The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy*, Phys. Rev. A **85** 062111 (2012) [[arXiv:1111.5333](https://arxiv.org/abs/1111.5333), [doi:10.1103/PhysRevA.85.062111](https://doi.org/10.1103/PhysRevA.85.062111)]

* [[Jiannis K. Pachos]], *Introduction to Topological Quantum Computation*, Cambridge University Press (2012) [[doi:10.1017/CBO9780511792908]( https://doi.org/10.1017/CBO9780511792908)]

  > [p. 50]: "topological quantum computation resembles an adiabatic quantum computation with constant energy gap, where the quasiparticle coordinates provide the control parameters of the Hamiltonian."

  > [p. 52]: "Holonomic quantum computation resembles the adiabatic scheme [...] topological quantum computation can be considered as holonomic computation where the employed adiabatic evolutions have topological characteristics."

* {#CLBFN15} Chris Cesare, Andrew J. Landahl, Dave Bacon, Steven T. Flammia, Alice Neels, *Adiabatic topological quantum computing*, Phys. Rev. A **92** (2015) 012336  [[arXiv:1406.2690](https://arxiv.org/abs/1406.2690), [doi:10.1103/PhysRevA.92.012336](https://doi.org/10.1103/PhysRevA.92.012336)]

* {#LahtinenPachos17} [[Ville Lahtinen]], [[Jiannis K. Pachos]], p. 1 of: _A Short Introduction to Topological Quantum Computation_,  SciPost Phys. **3** 021 (2017) [[arXiv:1705.04103](https://arxiv.org/abs/1705.04103), [doi:10.21468/SciPostPhys.3.3.021](https://scipost.org/SciPostPhys.3.3.021)]

* E. Macaluso, T. Comparin, L. Mazza, and I. Carusotto, *Fusion Channels of Non-Abelian Anyons from Angular-Momentum and Density-Profile Measurements*, Phys. Rev. Lett. **123** 266801 (2019) [[arXiv:1903.03011](https://arxiv.org/abs/1903.03011), [doi:10.1103/PhysRevLett.123.266801](https://doi.org/10.1103/PhysRevLett.123.266801)]

* {#Stanescu20} [[Tudor D. Stanescu]], p. 321 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

* {#BarlasProdan20} Yafis Barlas, Emil Prodan, *Topological braiding of non-Abelian mid-gap defects in classical meta-materials*, Phys. Rev. Lett. **124** (2020) 146801 [[arXiv:1903.00463](https://arxiv.org/abs/1903.00463), [doi:10.1103/PhysRevLett.124.146801](https://doi.org/10.1103/PhysRevLett.124.146801)]

[[!redirects quantum annealing]]

[[!redirects holonomic quantum computation]]


adiabatic switching

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## Idea

### Adiabatic switching

In [[perturbative quantum field theory]], the term _adiabatic switching_ refers to considering a smooth transition between vanishing and non-vanishing [[interaction]] [[coupling constant|coupling]]: the interaction is slowly, hence (borrowing a term from [[thermodynamics]]) "[[adiabatic|adiabatically]]", switched on or off. This is mostly a mathematical device, not meant to directly reflect a physical situation of changing coupling, but it does serve to construct physical quantities. This is closely related to the role of [[operator-valued distributions]] which are quantities that give well defined [[linear operators]] (hence [[quantum observables]]) only when evaluated on any [[bump function]].

Originally adiabatic switching was considered ([Lippmann-Schwinger 50](#LippmannSchwinger50)) only in the [[time]]-direction (for a fixed choice of time on [[Minkowski spacetime]]) by multiplying the [[interaction]] term of the [[Lagrangian density]]/[[Hamiltonian]] by the [[exponential]] $\exp(- \epsilon {\Vert t \Vert})$ (for $\epsilon \in (0,\infty)$ a [[positive number|positive]] [[real number]] and for ${\Vert t\Vert}$ the [[absolute value]] of the time [[coordinate]]). Review is for instance in ([Strocchi 13, section 6.3](#Strocchi13)).

Using this, the _Gell-Mann and Low formula_ ([Gell-Mann & Low 51](#GellMannLow51), see [Molinari 06](#Molinari06)) expresses the [[eigenstates]] $\vert \psi \rangle$ of an interacting [[Hamiltonian]] $H = H_{free} + H_{int}$ in terms of the eigenstates $\vert \Psi_{free} \rangle$ of the free Hamiltonian by the "adiabatic limit"

$$
  \vert \Psi^{\pm}_{int} \rangle
  \;\propto\;
  \underset{\epsilon \to 0}{\lim}
  S_\epsilon(0, \pm \infty)
  \vert \Psi_{free} \rangle
$$

(if the [[limit of a sequence|limit]] exists) where $S_\epsilon$ denotes the [[S-matrix]] of the adiabatically switched Hamiltonian $H_\epsilon \coloneqq H_{free} + e^{- \epsilon {\Vert t\Vert}}H_{int}$.

More generally, one may consider adiabatic switching taking place not just in time, but in all of [[spacetime]]. This the basis of [[causal perturbation theory]] and [[locally covariant perturbative quantum field theory]]:

In the construction of [[perturbative quantum field theory]] via the method of [[causal perturbation theory]] the [[interaction]] terms $L_{int}$ used in the mathematical construction of the [[S-matrix]] are multiplied with a "[[coupling constant]]" $g$ which is in fact taken to be a [[smooth function]] of [[compact support]] on [[spacetime]], hence a [[bump function]]:

$$
  L_g = L_{free} + g L_{int}
  \,.
$$

This means that the the [[interaction]] as modeled by the [[S-matrix]]

$$
  S_g \coloneqq T \exp( \tfrac{i}{\hbar} \int_{X} g :L_{int}(x): )
$$

is non-trivial only on a [[compact topological space|compact]] [[subspace]] of [[spacetime]], towards its boundary it smoothly drops to zero. Hence outside this region the interaction is "switched off".

Since the actual interactions in [[physics]] are of course _not_ "switched off" anywhere, the use of an adiabatic switching is just an intermediate mathematical step.
Originally in ([Epstein-Glaser 73](#EpsteinGlaser73)) the idea was that after having constructed the [[S-matrix]] for any adiabatic switching $g$, the [[limit of a sequence|limit]] ("adiabatic limit") $g \to 1$ had to be taken to remove the switching in the end. Failure of this limit to exist is interpreted as "[[infrared divergency]]" of the [[perturbative quantum field theory]] (since the divergency comes from large scales, hence long [[wavelength]]).

But as observed in ([Il'in-Slavnov 78](#IlinSlavnov78)) and rediscovered in ([Brunetti-Fredenhagen 00](#BrunettiFredenhagen00)), an adiabatic switching map that is unity on a globally hyperbolic sub-spacetime $O \subset X$ is sufficient to compute the perturbative [[interacting field algebra]], hence the [[algebra of quantum observables]] $A(O)$ on that subspace, and the collection of all of these as $O$ ranges forms a [[causally local net of observables]] which fully captures the [[quantum field theory]] in the sense of the [[Haag-Kastler axioms]] ([this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet)). This perspective is now known as _[[locally covariant algebraic quantum field theory]]_.

### Adiabatic limit

The [[limit of a sequence|limit]] of the [[perturbative S-matrix]] as the [[adiabatic switching]] is removed (if it exists) is called the _adiabatic limit_ or _strong adiabatic limit_.

If one just asks that the corresponding limit exists for the [[n-point functions]] one speaks of a _weak adiabatic limit_.

Even with the adiabatically switched S-matrix elements (not taking a limit) the [[local net of quantum observables]] is well defined ([this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet)), this is hence a [[functor]]

$$
  \mathcal{O} \mapsto \mathcal{A}(\mathcal{O})
$$

that assigns [[algebras of observables]] to [[causally closed subsets]] of [[spacetime]]. The [[colimit]] algebra

$$
  \mathcal{A} \coloneqq \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim} \mathcal{A}(\mathcal{O})
$$

over this [[functor]] (in the sense of [[category theory]]) always exists. This is also called the _algebraic adiabatic limit_.

(See around [Duch 17, section 4](#Duch17) for review of strong, weak and algebraic adiabaitc limit; and [Duch 17, chapter II](#Duch17) for results on the weak adiabatic limit)

Here

1. the _algebraic adiabatic_ limit defines the _[[quantum observables]]_ in the limit;

1. the weak adiabatic limit may serve to define also the _[[state on a star-algebra|states]]_, hence the [[interacting vacuum]] ([Duch 17, p. 113-114](#Duch17)).

## Related concepts

* [[adiabatic]]

## References

The concept of adiabatic switching in the time direction was introduced in

* {#LippmannSchwinger50} B. A. Lippmann, [[Julian Schwinger]], *Variational Principles for Scattering Processes. I*,  Phys. Rev. 79, 469 (1950) ([doi:10.1103/PhysRev.79.469](https://doi.org/10.1103/PhysRev.79.469))

reviewed for instance in

* {#Strocchi13} [[Franco Strocchi]], section 6.3 of _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press, 2013

and the corresponding formula for the interacting eigenstates in terms of the free ones is due to

* {#GellMannLow51} [[Murray Gell-Mann]], F. Low, _Bound states in quantum field theory_ Phys. Rev. 84, 350 (1951)

* [[Murray Gell-Mann]], M. L. Goldberger, Phys. Rev. 91 398 (1953)

see

* {#Molinari06} Luca Guido Molinari, _Another proof of Gell-Mann and Low's theorem_, Journal of Mathematical Physics 48, 052113, 2007 ([arXiv:math-ph/0612030](https://arxiv.org/abs/math-ph/0612030))

See also:

* C. Lanczos, R. C. Clark, G. H. Derrick (eds.), Ch. 2, Sec 2 in: *Mathematical Methods in Solid State and Superfluid Theory*, Springer (1986) $[$[doi:10.1007/978-1-4899-6435-9](https://doi.org/10.1007/978-1-4899-6435-9)$]$

* Wikipedia, _[Gell-Mann and Low theorem](https://en.wikipedia.org/wiki/Gell-Mann_and_Low_theorem)_

The generalization to switching in all space-time directions was considered for the construction of [[causal perturbation theory]] in

* {#EpsteinGlaser73} [[Henri Epstein]] and [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.

The observation that this in  fact makes causal perturbation theory a tool for constructing [[local nets of observables]] for [[locally covariant perturbative quantum field theory]] is due to

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32.

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661,2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

The term "algebraic adiabatic limit" for the resulting [[local net of observables]] (or its [[inductive limit]]) appears in

* {#FredenhagenLindner13} [[Klaus Fredenhagen]], [[Falk Lindner]], p. 7 of _Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics_, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014 ([arXiv:1306.6519](https://arxiv.org/abs/1306.6519))

The weak adiabatic limit in [[causal perturbation theory]] for massive fields was shown to exists in

* [Epstein-Glaser 73](#EpsteinGlaser73)

Extension of this result to [[quantum electrodynamics]] and [[phi^4 theory]] was given in

* {#BlanchardSeneor75} P. Blanchard and R. Seneor, _Green’s functions for theories with massless particles (in perturbation theory)_, Ann. Inst. H. Poincaré Sec. A
23 (2), 147–209 (1975) ([Numdam](http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0))

See also

* {#Scharf95} [[Günter Scharf]], section 3.11 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

Further extension of the result is due to

* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))

[[!redirects adiabatic switchings]]

[[!redirects Gell-Mann and Low formula]]

[[!redirects adiabatic limit]]
[[!redirects adiabatic limits]]

[[!redirects algebraic adiabatic limit]]
[[!redirects algebraic adiabatic limits]]

[[!redirects strong adiabatic limit]]
[[!redirects strong adiabatic limits]]

[[!redirects weak adiabatic limit]]
[[!redirects weak adiabatic limits]]


adic noetherian ring

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## Idea

A [[topological ring]] $R$ is an __adic noetherian ring__ if it is [[noetherian ring|noetherian]] as a [[ring]] and it has a [[topological basis]] consisting of all translations of the [[neighborhoods]] of zero of the form $I^n$ ($n\gt 0$) where $I\subset R$ is a fixed [[ideal]] of $R$, and $R$ is [[Hausdorff space|Hausdorff]] and [[complete space|complete]] in that topology. A choice of such an ideal is said to be the __defining ideal__ or (more French) the __ideal of definition__ of the topological ring $R$. If $R$ is an adic noetherian ring, an ideal $J\subset R$ is a defining ideal iff it is open and its powers tend to $\{0\}$.

The [[topology]] of an adic noetherian ring $R$ with the defining ideal $I$ is said to be the **$I$-adic topology** and the descending [[filtration]] of $R$ by the powers of $I$ to be the **$I$-adic filtration**.

For an adic noetherian ring $R$ there is a construction of a [[ringed space]], its [[formal spectrum]] $Spf(R)$, which does not depend on the choice of the ideal $I\subset R$ generating its (fixed in advance) topology. The underlying topological space of $Spf(R)$ is $Spec(R/I)$ which is (under the above assumptions on $R$ and $I$) a [[closed subspace]] of the [[spectrum (geometry)|spectrum]] $Spec(R)$ and it contains all closed points of $Spec(R)$.

## Related concepts

* [[completion of a ring]]

* [[linear topological ring]]

* [[pro-ring]]

## References

* PlanetMath _[I-adic topology](http://planetmath.org/iadictopology)_

* Wikipedia, _[Krull topology](https://en.wikipedia.org/wiki/Completion_%28algebra%29#Krull_topology)_

[[!redirects adic topology]]
[[!redirects adic topologies]]
[[!redirects adic filtration]]

[[!redirects adic ring]]
[[!redirects adic rings]]


adic residual

#Contents#
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## Idea

Forming the [[kernel]] of an [[adic completion]] map is sometimes called forming the _adic residual_, for instance the _$p$-residual_ for $p$-adic completion.

## Properties

### Explicit characterization

For $A$ a [[commutative ring]] and $\mathfrak{a} \subset A$ an ideal, and $N$ an $A$-module, then the $\mathfrak{a}$-adic residual of $N$ (hence the kernel of the map to the [[completion of a module]] $N \longrightarrow N^\wedge_{\mathfrak{a}}$) is equivalently the submodule of elements annihilated by $1 + \mathfrak{a}$.

E.g. theorem 4.3.2. here: [[Completion.pdf:file]]

### As a modality in arithmetic cohesion

For suitably well behaved ideals, forming the adic residual may be understood as the [[dR-flat modality]] in the [[cohesion]] of [[E-infinity arithmetic geometry]]:

[[!include arithmetic cohesion -- table]]

## References


adic ring > history


adic space

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## Idea

__Adic spaces__ are the basic objects in Huber's approach to [[non-archimedean analytic geometry]]. They are built by gluing [[valuation spectra]] of a certain class of [[topological rings]]. Unlike [[Berkovich analytic spectra]] the points of adic spaces correspond to valuations of arbitrary rank, not only rank one. If a [[Berkovich space]] is corresponding to a separated [[rigid analytic space]] then it can be obtained as the largest Hausdorff quotient of the corresponding adic space.

The framework of adic spaces are used to build [[perfectoid spaces]] out of perfectoid rings.

## Definitions

\begin{definition}
Let $(A,A^{+})$ be a Huber pair, i.e. $A$ is a Huber ring and $A^{+}\subseteq A$ is a ring of integral elements. The _adic spectrum_ $\mathrm{Spa}(A,A^{+})$ is the set of equivalence classes of continuous valuations $\vert\cdot\vert$ on $A$ such that $\vert A^{+}\vert\leq 1$.

If $x$ is a valuation, and $g\in A$, we also suggestively write $g\mapsto\vert g(x)\vert$ for the valuation $x$ applied to $g$. The topology on $\mathrm{Spa}(A,A^{+})$ is the one generated by open sets of the form

$$\lbrace x:\vert f(x)\vert\leq\vert g(x)\vert\neq 0\rbrace$$

where $f,g\in A$.

\end{definition}

## Examples

* The final object in the category of adic spaces is $\mathrm{Spa}(\mathbb{Z},\mathbb{Z})$.

* The adic closed disc over $\mathbb{Q}_{p}$ is given by $\mathrm{Spa}(A,A^{+})$ where $A=\mathbb{Q}_{p}\langle T\rangle$ and $A^{+}=\mathbb{Z}_{p}\langle T\rangle$.

* The adic open disc over $\mathbb{Q}_{p}$ is the generic fiber of $\mathrm{Spa}(A,A)\to\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$, where $A=\mathbb{Z}_{p}[[T]]$.

## Related concepts

* [[perfectoid space]]

* [[p-adic geometry]]

* [[analytic geometry]]

## References

* R. Huber, _Étale cohomology of rigid analytic varieties and adic spaces_, Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, 1996. x+450 pp. ([MR2001c:14046](http://www.ams.org/mathscinet-getitem?mr=1734903))

* [[Sophie Morel]], _Adic spaces_ ([pdf](https://web.math.princeton.edu/~smorel/adic_notes.pdf))

* Torsten Wedhorn, _Adic spaces_ ([arXiv:1910.05934](https://arxiv.org/abs/1910.05934))

* [[Brian Conrad]], _A brief introduction to adic spaces_, [PDF](http://virtualmath1.stanford.edu/~conrad/papers/Adicnotes.pdf).

[[!redirects adic spaces]]


adic topology > history

< [[adic topology]]


adinkra

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## Idea

In [[super-algebra|super]]-[[representation theory]], what is called _adinkras_ ([Faux-Gates 04](#FauxGates04)) is a graphical tool for denoting those [[representations]] ([[super multiplets]]) of the $\mathcal{N}$-extended [[supersymmetry]] algebras in one dimension ([[supersymmetric quantum mechanics]] with [[number of supersymmetries|N supersymmetries]]) for which the [[supersymmetry]] generators act, up to [[derivatives]] and prefactors, by [[permutation]] of [[superfield]] components. These are called _adinkraic_ representations ([Zhang 13, p. 16](#Zhang13)).

While adinkraic representations are special among all representations of the 1-dimensional $\mathcal{N}$-extended supersymmetry algebra, the idea is that the [[dimensional reduction]] of representations ([[supermultiplets]]) of higher dimensional supersymmetry algebras down to 1d are of this form, at least for dimensional reduction from $d = 4$ and $N = \mathbf{4}$ ([GGMPPRW 09, section 3](#GGMPPRW09), [Gates-Hubsch-Stiffler 14](#GatesHubschStiffler14)).

The classification of adinkras, and hence of adinkraic representations, turns out to be controlled by [[linear codes]] ([Doran & Faux & Gates & HubschIgaLandweberMiller 11](#DoranFauxGatesHubschIgaLandweberMiller11)) and to be related to certain special [[super Riemann surfaces]] via [[dessins d'enfants]] ([Doran & Iga & Landweber & Mendez-Diez 13](#DoranIgaLandweberMendez-Diez13), [Doran & Iga & Kostiuk &Mendes-Diez 16](#DoranIgaKostiukMendes-Diez16)).

## Definition

For background, see at _[[geometry of physics -- supersymmetry]]_.

For $N \in \mathbb{N}$, write $\mathbb{R}^{1 \vert N}$
for the [[super Lie algebra]] over the [[real numbers]] that is spanned by a single generator $P$ in even (i.e., bosonic) degree and $N$ generators $Q_I$, $I \in \{1, 2, \cdots, N\}$ in odd (i.e., fermionic) degree, whose only non-trivial components of the super-[[Lie bracket]] are

$$
  [Q_I, Q_J]  = 2 \delta_{I J} P
  =
  \left\{
   \array{
      2 P & \vert & I = J
      \\
      0 & \vert & \text{otherwise}
   }
  \right.
$$

This is the 1-dimensional $N$-extended [[super translation super Lie algebra]]. We may think of this as the super-translational symmetry of 1-dimensional $N$-extended [[super Minkowski spacetime]].

Consider then super [[Lie algebra representations]] of $\mathbb{R}^{1 \vert N}$ on [[super vector spaces]] of smooth [[superfields]] on $\mathbb{R}^{1 \vert N}$ (regarded as a [[supermanifold]]) and such that the bosonic generator $P$ acts as the [[derivative]] operator on [[smooth functions]] on $\mathbb{R}^1$ in each component. If in addition the representation is such that in the canonical [[linear basis]] the odd generators $Q_I$ send even/odd basis elements $\phi_i$ to single odd/even basis elements $\psi_j$ (as opposed to [[linear combinations]] of them), hence if the $Q_I$ act apart from degree-shift and possibly [[differentiation]] by [[permutations]] on the components of the [[superfields]], then this representation of $\mathbb{R}^{1\vert N}$ is called _adinkraic_. ([Zhang 13, p. 16](#Zhang13)).

The corresponding _adinkra_ is the [[bipartite graph]] which expresses these permutations:

> table grabbed from [Doran & Iga & Landweber & Mendez-Diez 13, p. 7](#DoranIgaLandweberMendez-Diez13)

> graphics grabbed from [Iga-Zhang 15, p. 3](#IgaZhang15)

## Classification

The topology of an adinkra graph together with its edge coloring in $\{1,2, \cdots, N\}$ is called its _chromotopology_.

The set of adinkra chromotopologies is equivalent to the set of colored $N$-cubs modulo doubly even length-$N$ [[linear codes]] ([Doran-Faux-Gates-Hubsch-Iga-Landweber-Miller 11](#DoranFauxgatesHubschIgaLandweberMiller11))

(A [[linear code]] of length $N$ is a [[linear subspace]] of $(\mathbb{F}_2)^N$ for $\mathbb{F}_2$ the [[prime field]] with two elements and it is _doubly even_ if every element has weight a multiple of 4. )

see [Zhang 13, chapter 2](#Zhang13)

> graphics grabbed from [Iga-Zhang 15, p. 4](#IgaZhang15)

## From supermultiplets in higher dimensions

The [[dimensional reduction]] of the smallest [[supermultiplets]] of $d = 4, N = \mathbf{4}$ supersymmetry down to 1d yield adinkraic representations ([GGMPPRW 09, section 3](#GGMPPRW09), [Gates-Hubsch-Stiffler 14](#GatesHubschStiffler14)).

Corresponding adinkras for the chiral scalar supermultiplet (CM), the vector multiplet (VM) and the tensor multiplet (TM) look as follows:

## References

For an introduction to adinkras, see the talk

* [[Lutian Zhao]], _What is an Adinkra?_, 2014 ([slides](http://math.sjtu.edu.cn/conference/Bannai/2014/data/20141213B/slides.pdf))

The concept of adinkras was introduced into [[supersymmetry]] [[representation theory]] in

* {#FauxGates04} [[Michael Faux]], [[Jim Gates]], _Adinkras: A Graphical Technology for Supersymmetric Representation Theory_, Phys.Rev. D71 (2005) 065002 ([hep-th/0408004](https://arxiv.org/abs/hep-th/0408004))

and further developed in the article

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields_, Int. J. Mod. Phys. A22: 869-930, 2007 ([arXiv:math-ph/0512016](https://arxiv.org/abs/math-ph/0512016))

and many more ("DFGHIL collaboration"). For instance the relation to Clifford supermodules is discussed in

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Off-shell supersymmetry and filtered Clifford supermodules_ ([arXiv:math-ph/0603012](https://arxiv.org/abs/math-ph/0603012))

kinetic terms are discussed in

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Adinkras and the Dynamics of Superspace Prepotentials_ ([arXiv:hep-th/0605269](https://arxiv.org/abs/hep-th/0605269))

The classification of adinkras in terms of [[graphs]] and [[linear codes]] is due to

* {#DoranFauxGatesHubschIgaLandweberMiller11} [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], R. L. Miller, _Codes and Supersymmetry in One Dimension_, Adv. in Th. Math. Phys. 15 (2011) 1909-1970 ([arXiv:1108.4124](https://arxiv.org/abs/1108.4124))

and discussed in mathematical detail in

* {#Zhang11} [[Yan X Zhang]], _Adinkras for Mathematicians_ ([arXiv:1111.6055](https://arxiv.org/abs/1111.6055))

* {#Zhang13} [[Yan X Zhang]], _The combinatorics of Adinkras_, PhD thesis, MIT (2013) ([pdf](http://math.mit.edu/~yanzhang/math/thesis_adinkras.pdf))

The [[dimensional reduction]] of the standard [[supermultiplets]] of $D = 4, \mathcal{N} = 1$ supersymmetry to adinkraic representations of $D = 1, \mathcal{N}=4$ is due to

* {#GGMPPRW09} [[Jim Gates]], J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V.G.J. Rodgers, L. Wassink, _$4D$, $N = 1$ Supersymmetry Genomics (I)_, JHEP 0912:008,2009 ([arXiv:0902.3830](https://arxiv.org/abs/0902.3830))

* {#GatesHubschStiffler14} [[Jim Gates]], [[Tristan Hübsch]], Kory Stiffler, _Adinkras and SUSY Holography_, Int. J. Mod. Phys. A29 no. 7, (2014) 1450041 ([arXiv:1208.5999](https://arxiv.org/abs/1208.5999))

See also

* {#IgaZhang15} [[Kevin Iga]], [[Yan Zhang]], _Structural Theory of 2-d Adinkras_ ([arXiv:1508.00491](https://arxiv.org/abs/1508.00491))

Discussion for $D=11$, $\mathcal{N}=1$:

* [[Jim Gates]], Yangrui Hu, S.-N. Hazel Mak, _Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, $\mathcal{N} = 1$ Superspace_ ([arXiv:2002.08502](https://arxiv.org/abs/2002.08502))

The relation of adinkras to special [[super Riemann surfaces]] via [[dessins d'enfants]] is due to

* {#DoranIgaLandweberMendez-Diez13} [[Charles Doran]], [[Kevin Iga]], [[Greg Landweber]], [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras_ ([arXiv:1311.3736](https://arxiv.org/abs/1311.3736))

* {#DoranIgaKostiukMendes-Diez16} [[Charles Doran]], [[Kevin Iga]], Jordan Kostiuk, [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II_ ([arXiv:1610.09983](https://arxiv.org/abs/1610.09983))

Further developments includes

* {#CalkinsGatesGatesStiffler15} Mathew Calkins, D. E. A. Gates, [[Jim Gates]] Jr., Kory Stiffler, _Adinkras, 0-branes, Holoraumy and the SUSY QFT/QM Correspondence_ ([arXiv:1501.00101](https://arxiv.org/abs/1501.00101))

Discussion in the context of [[spectral triples]] is in

* [[Matilde Marcolli]], Nick Zolman, _Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples_ ([arXiv:1606.04463](https://arxiv.org/abs/1606.04463))

See also

* Wes Caldwell, Alejandro Diaz, Isaac Friend, [[Jim Gates]], Jr., Siddhartha Harmalkar, Tamar Lambert-Brown, Daniel Lay, Karina Martirosova, Victor Meszaros, Mayowa Omokanwaye, Shaina Rudman, Daniel Shin, Anthony Vershov, _On the Four Dimensional Holoraumy of the $4D$, $\mathcal{N} = 1$ Complex Linear Supermultiplet_ ([arXiv:1702.05453](https://arxiv.org/abs/1702.05453))

* Kevin Iga, *Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes* ([arXiv:2110.01665](https://arxiv.org/abs/2110.01665))

* Wikipedia, _Adinkra symbols (physics)_

[[!redirects adinkras]]

[[!redirects adinkraic representation]]
[[!redirects adinkraic representations]]


Adj

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## Idea

The [[2-category]] $Adj$ is the free-standing adjunction ([[walking]] adjunction).

A [[2-functor]] $Adj \to K$ is an [[adjunction]] in the 2-category $K$. These 2-functors form one version of the [[2-category of adjunctions]] of $K$.

## Definition

$Adj$ is the 2-category freely generated by

* two objects: $a$ and $b$,

* two morphisms: $L: a \to b$ and $R: b \to a$,

* and two 2-morphisms, called the "unit" and "counit": $i: 1_a \to L R$ and $e: R L \to 1_b$, satisfying two relations, called the "triangle equations".

The restrictions of the free-standing adjunction, $Adj$, to the sub-2-categories spanned by one endpoint, $a$, or the other, $b$, define the free-standing [[monad]] and the free-standing [[comonad]].

##Related entries

* [[2-category of adjunctions]]

* [[adjoint logic]]

* [[Hegelian taco]]

##References

* C. Auderset, _Adjonction et monade au niveau des 2-categories_, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. ([numdam](http://www.numdam.org/item/CTGDC_1974__15_1_3_0/))

* [[John Baez]], _This Week's Finds in Mathematical Physics (Week 174)_, ([TWF174](http://www.math.ucr.edu/home/baez/week174.html))

*  Kevin Coulembier, [[Ross Street]], Michel van den Bergh, _Freely adjoining monoidal duals_, arXiv:2004.09697 (2020).  ([abstract](https://arxiv.org/abs/2004.09697))

* Dieter Pumplün, _Eine Bemerkung über Monaden und adjungierte Funktoren_, Math. Annalen 185 (1970), 329-377.

* [[Stephen Schanuel]] and [[Ross Street]], The free adjunction, Cah. Top. Géom. Diff. 27 (1986), 81-83. ([numdam](http://archive.numdam.org/item/CTGDC_1986__27_1_81_0/))

[[!redirects walking adjunction]]
[[!redirects the walking adjunction]]


adjacency matrix

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## Definition

In [[graph theory]], an **adjacency matrix** for a [[finite set|finite]] [[multigraph]] or [[pseudograph]] with $n$ [[vertices]] is an $n$ by $n$ [[matrix]] of [[natural numbers]] which encodes the number of [[edges]] between each vertex: entry $a_{i, j}$ in the matrix is the number of edges between vertex $v_i$ and $v_j$.

## Properties

Let $G$ be a finite multigraph, and let $A$ be the associated adjacency matrix. Then the matrix power $A^n$ encodes the number of [[walks]] between each vertex: entry $b_{i, j}$ in $A^n$ is the number of walks between vertex $v_i \in G$ and $v_j \in G$.

## Related concepts

* [[matrix]]

## References

* Wikipedia, [Adjacency matrix](https://en.wikipedia.org/wiki/Adjacency_matrix)


adjoint

## In linear algebra

The term _adjoint_ originates in [[linear algebra]], where it may refer, in increasing generality, to:

* [[adjoint matrix]]

* [[Hermitian adjoint]]

* [[adjoint operator]]

## In category theory

In [[category theory]], the term "adjoint" foremost refers to "[[adjoint functors]]", a terminology which is probably inspired by the similarity between the characteristic [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of such functors and the defining [[equality]] $\big\langle A v ,\, w \big\rangle
 \,=\, \big\langle v ,\, A^\dagger w \big\rangle$
for [[Hermitian adjoint]] [[linear operators]].

Proceeding from here, there are various variants of adjointness in [[category theory]]:

* [[adjunction]], [[adjoint morphism]]

* [[adjoint functor]]

  * [[strong adjoint functor]]

  * [[adjoint triple]], [[adjoint quadruple]], [[recollement]]

* [[proadjoint]], [[Hopf adjunction]]

* [[2-adjunction]]

  [[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]

* [[adjoint (infinity,1)-functor]]

* [[(∞,n)-category with adjoints]]

[[!redirects adjoints]]


adjoint (infinity,1)-functor

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## Idea

The notion of adjunction between two [[(∞,1)-functors]] generalizes the notion of [[adjoint functors]] from [[category theory]] to [[(infinity,1)-category|(∞,1)-category theory]].

There are many equivalent definitions of the ordinary notion of [[adjoint functor]]. Some of them have more evident generalizations to some parts of [[higher category theory]] than others.

* One definition of ordinary adjoint functors says that a pair of functors $C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ is an adjunction if there is a [[natural transformation|natural isomorphism]]

  $$
    Hom_C(L(-),-) \simeq Hom_D(-,R(-))
    \,.
  $$

  The analog of this definition makes sense very generally in [[(∞,1)-category theory]], where $Hom_C(-,-) : C^{op} \times C \to \infty Grpd$ is the $(\infty,1)$-categorical hom-object.

* One other characterization of adjoint functors  in terms of their [[cograph of a functor|cographs]]/[[heteromorphisms]]: the [[Cartesian fibrations]] to which the functor is associated. At [[cograph of a functor]] it is discussed how two functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if the cograph of $L$ coincides with the cograph of $R$ up to the obvious reversal of arrows

$$
  (L \dashv R) \Leftrightarrow
  (cograph(L) \simeq cograph(R^{op})^{op})
  \,.
$$

Using the [[(∞,1)-Grothendieck construction]] the notion of cograph of a functor has an evident generalization to $(\infty,1)$-categories.

## Definition

### In terms of hom-equivalences
 {#CharacterizationInTermsOfHomEquivalences}

+-- {: .num_defn #InTermsOfHomEquivalences}
###### Definition
**(in terms of hom equivalence induced by unit map)**

A pair of [[(∞,1)-functors]]

$$
  C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}}
  D
$$

is an adjunction, if there exists a _unit transformation_
$\epsilon : Id_D \to R \circ L$ -- a morphism in the [[(∞,1)-category of (∞,1)-functors]] $Func(D,D)$ -- such that for all $d \in D$ and $c \in C$ the induced morphism

$$
  Hom_C(L(d),c)
   \stackrel{R_{L(d), c}}{\to}
  Hom_D(R(L(d)), R(c))
  \stackrel{Hom_D(\epsilon, R(c))}{\to}
  Hom_D(d,R(c))
$$

is an [[equivalence of ∞-groupoids]].

=--

In terms of the concrete incarnation of the notion of $(\infty,1)$-category by the notion of [[quasi-category]], we have that $Hom_C(L(d),c)$ and $Hom_D(d,R(c))$ are incarnated as [[hom-object in a quasi-category|hom-objects in quasi-categories]], which are [[Kan complexes]], and the above equivalence is a [[homotopy equivalence]] of Kan complexes.

In this form is due to [Lurie 09, Def. 5.2.2.7](#Lurie09).

{#RiehlVerityOnAdjunctionsViaHomEquivalences} Streamlined discussion is in
[Riehl & Verity 15, 4.4.2-4.4.4](#RiehlVerity15)
and [Riehl & Verity 20, 3.3.3-3.5.1](#RiehlVerity20)
and [Riehl & Verity "Elements", Prop. 4.1.1](#RVElements).

### In terms of cographs/heteromorphisms
 {#InTermsOfCographsHeteromorphisms}

We discuss here the quasi-category theoretic analog of _[Adjoint functors in terms of cographs](cograph+of+a+functor#AdjointFunctorsInTermsOfCographs)_ ([[heteromorphisms]]).

We make use here of the explicit realization of the [[(∞,1)-Grothendieck construction]] in its incarnation for [[quasi-categories]]: here an [[(∞,1)-functors]] $L : D \to C$ may be regarded as a map $\Delta[1]^{op} \to $ [[(∞,1)Cat]], which corresponds under the Grothendieck construction to a [[Cartesian fibration]] of [[simplicial sets]] $coGraph(L) \to \Delta[1]$.

+-- {: .num_defn #InTermsOfCoCartesianFibrations}
###### Definition
**(in terms of Cartesian/coCartesian fibrations)**

Let $C$ and $D$ be [[quasi-categories]]. An **adjunction** between $C$ and $D$ is

* a morphism $K \to \Delta[1]$ of [[simplicial sets]], which is both a [[Cartesian fibration]] as well as a coCartesian fibration.

* together with [[equivalence of quasi-categories]] $C \stackrel{\simeq}{\to} K_{\{0\}}$ and $D \stackrel{\simeq}{\to} K_{\{1\}}$.

Two [[(∞,1)-functors]] $L : C \to D$ and $R : D \to C$ are called **adjoint** -- with $L$ _left adjoint_ to $R$ and $R$ _right adjoint_ to $L$ if

* there exists an adjunction $K \to I$ in the above sense

* and $L$ and $K$ are the associated functors to the Cartesian fibation $p \colon K \to \Delta[1]$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta[1]^{op}$, respectively.

=--

### In the homotopy 2-category
  {#InTheHomotopy2Category}

+-- {: .num_defn #InTermsOfTheHomotopy2Category}
###### Definition
**(in terms of the homotopy 2-category)**

Say that a 2-categorical pair of [[adjoint (∞,1)-functors]] is an [[adjunction]] in the [[homotopy 2-category of (∞,1)-categories]].

=--

This concept, in the spirit of [[formal (infinity,1)-category theory|formal $\infty$-category theory]], was mentioned, briefly, in
[Joyal 2008, p. 159 (11 of 348)](#Joyal08) and then expanded on in
[Riehl-Verity 15, Def. 4.0.1](#RiehlVerity15).

Such a 2-categorical adjunctions (Def. \ref{InTermsOfTheHomotopy2Category}) determines an adjoint pair of $\infty$-functors in the sense of [Lurie 2009](#Lurie) ([Riehl-Verity 15, Rem. 4.4.5](#RiehlVerity15)):

\begin{prop}\label{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho}
  An anti-parallal pair of morphisms in [[(∞,1)Cat|$Cat_\infty$]] is a pair of adjoint $\infty$-functors in the sense of [Lurie 2009, Sec. 5.2](#Lurie) if and only its image in the [[homotopy 2-category]] [[homotopy 2-category of (infinity,1)-categories|$Ho_2\big(Cat_\infty\big)$]] forms an [[adjunction]] in the classical sense of [[2-category theory]] (Def. \ref{InTermsOfTheHomotopy2Category}).
\end{prop}
([Riehl & Verity 2022, Sec. F.5, Prop. F.5.6](#RVElements))

The conceptual content of Prop. \ref{InfinityCollageTypeAdjunctionsAreAdjunctionsIn2Ho} may be made manifest as follows:
\begin{proposition}
  Every 2-categorical pair of adjoint $(\infty,1)$-functors in the sense of Def. \ref{InTermsOfTheHomotopy2Category} extends to a "homotopy coherent adjunction" in an essentially unique way.
\end{proposition}

([Riehl & Verity 2016, Thm. 4.3.11, 4.4.11](#RiehlVerity16))

## Properties

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

For $C$ and $D$ [[quasi-categories]], the two definitions of adjunction,

1. in terms of Hom-equivalence induced by unit maps (Def. \ref{InTermsOfHomEquivalences})

1. in terms of Cartesian/coCartesian fibrations (Def. \ref{InTermsOfCoCartesianFibrations})

are equivalent.

=--

This is [[Higher Topos Theory|HTT, prop 5.2.2.8]].

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

First we discuss how to produce the unit for an adjunction from the data of a correspondence $K \to \Delta[1]$ that encodes an $\infty$-adjunction $(f \dashv g)$.

For that, define a morphism $F' : \Lambda[2]_2 \times C \to K$ as follows:

* on $\{0,2\}$ it is the morphism $F : C \times \Delta[1] \to K$ that exhibits $f$ as associated to $K$, being $Id_C$ on $C \times \{0\}$ and $f$ on $C \times \{2\}$;

* on $\{1,2\}$ it is the morphism $C \times \Delta[1] \stackrel{f \times Id}{\to} D \times \Delta[1] \stackrel{G}{\to} K$, where $G$ is the morphism that exhibits $g$ as associated to $K$;

Now observe that $F'$ in particular sends $\{1,2\}$ to [[Cartesian morphism]]s in $K$ (by definition of functor associated to $K$). By one of the equivalent characterizations of [[Cartesian morphism]]s, this means that the lift in the diagram

$$
  \array{
    \Lambda[2]_2 &\stackrel{F'}{\to}& K
    \\
    \downarrow &{}^{F''}\nearrow& \downarrow
    \\
    \Delta[2] \times C &\to & \Delta[1]
  }
$$

exists. This defines a morphism $C \times \{0,1\} \to K$ whose components may be regarded as forming a [[natural transformation]] $u : d_C \to g \circ f$.

To show that this is indeed a unit transformation, we need to show that the maps of [[hom-object in a quasi-category]] for all $c \in C$ and $d \in D$

$$
  Hom_D(f(f), d) \to Hom_C(g(f(c)), g(d)) \to Hom_C(c, g(d))
$$

is an equivalence, hence an isomorphism in the [[homotopy category]]. Once checks that this fits into a commuting diagram

$$
  \array{
    Hom_D(f(c), d) &\to& Hom_C(g(f(c)), g(d)) &\to& Hom_C(c, g(d))
    \\
    \downarrow &&&& \downarrow
    \\
    Hom_K(C,D) &&=&& Hom_K(C,D)
  }
  \,.
$$

For illustration, chasing a morphism $f(c) \to d$ through this diagram yields

$$
  \array{
    (f(c) \to d) &\mapsto& (g(f(c)) \to g(d)) &\mapsto&
    (c \to g(f(c)) \to g(d))
    \\
    \downarrow && && \downarrow
    \\
    (c \to g(f(c)) \to f(c) \to d)
    &&=&&
    (c \to g(f(c)) \to g(d) \to d)
  }
 \,,
$$

where on the left we precomposed with the Cartesian morphism

$$
  \array{
    && g(f(c))
    \\
    & \nearrow &\Downarrow^{\simeq}& \searrow
    \\
    c &&\to&& f(c)
  }
$$

given by $F''|_{c} : \Delta[2] \to K$, by ...

=--

### Uniqueness of adjoints
  {#UniquenessOfAdjoints}

The adjoint of a functor is, if it exists, essentially unique:

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

If the $(\infty,1)$-functor between quasi-categories $L : D \to C$ admits a right adjoint $R : C \to D$, then this is unique up to homotopy.

Moreover, even the choice of homotopy is unique, up to ever higher homotopy, i.e. the collection of all right adjoints to $L$ forms a [[contractible]] [[∞-groupoid]], in the following sense:

Let $Func^L(C,D), Func^R(C,D) \subset Func(C,D)$ be the full sub-quasi-categories on the [[(∞,1)-category of (∞,1)-functors]] between $C$ and $D$ on those functors that are left adjoint and those that are right adjoints, respectively. Then there is a canonical [[equivalence of quasi-categories]]
$$
  Func^L(C,D) \stackrel{\simeq}{\to}
  Func^R(D,C)^{op}
$$

(to the [[opposite quasi-category]]), which takes every left adjoint functor to a corresponding right adjoint.

=--

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

This is [[Higher Topos Theory|HTT, prop 5.2.1.3]] (also remark 5.2.2.2), and [[Higher Topos Theory|HTT, prop. 5.2.6.2]].

The idea is to construct the category of right adjoints as an intersection of full subcategories
$$
  \array{
  Func^R(C,D) &\to& C^D
  \\  \downarrow & & \downarrow
  \\ (D^C)^{op} &\to& \infty Gpd^{C^{op} \times D}
  }
$$
where the inclusions are given by the yoneda embedding. An element of $Func^R(C,D)$
corresponds to a functor $p : C^{op} \times D \to \infty Gpd$
for which there exists a pair of functors $g : D \to C$ and $f : C \to D$ such that $p \simeq D(f-,-) \simeq C(-,g-)$.

=--

### Uniqueness of unit and counit

Given functors $f : C \to D$ and $g : D \to C$, we can use the [[(∞,1)-end]] to determine compute a chain of equivalences
$$
  \begin{aligned}
  C^C(id, gf)
  &\simeq \int_{c \in C} C(c, gf(c))
  \\ &\simeq \int_{c \in C} \infty Gpd^D(D(f(c), -), C(c, g-))
  \\ &\simeq Gpd^{C^{\op} \times D}(D(f-, -), C(-, g-))
  \end{aligned}
$$
dually, we can identify the space of counits as
$$
  D^D(fg, id) \simeq Gpd^{C^{\op} \times D}(C(-, g-), D(f-, -))
$$
So each half of the equivalence $D(f-,-) \simeq C(-,g-)$ corresponds essentially uniquely to a choice of unit and counit transformation.

### Preservation of limits and colimits {#PresOfLims}

Recall that for $(L \dashv R)$ an ordinary pair of [[adjoint functor]]s, the fact that $L$ preserves [[colimit]]s (and that $R$ preserves [[limit]]s) is a formal consequence of

1. the hom-isomorphism $Hom_C(L(-),-) \simeq Hom_D(-,R(-))$;

1. the fact that $Hom_C(-,-) : C^{op} \times C \to Set$ preserves all limits in both arguments;

1. the [[Yoneda lemma]], which says that two objects are isomorphic if all homs out of (into them) are.

Using this one computes for all $c \in C$ and diagram $d : I \to D$

$$
\begin{aligned}
  Hom_C(L(\lim_{\to} d_i), c) & \simeq
  Hom_D(\lim_\to d_i, R(c))
  \\
  & \simeq
  \lim_{\leftarrow} Hom_D(d_i, R(c))
  \\
  & \simeq \lim_{\leftarrow} Hom_C(L(d_i), c)
  \\
  & \simeq Hom_C(\lim_{\to} L(d_i), c)
  \,,
\end{aligned}
$$

which implies that $L(\lim_\to d_i) \simeq \lim_\to L(d_i)$.

Now to see this in $(\infty,1)$-category theory
(...) HTT Proposition 5.2.3.5

### Adjunctions on homotopy categories {#OnHomotopyCat}

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

For $(L \dashv R) : C \stackrel{\leftarrow}{\to} D$ an $(\infty,1)$-adjunction, its image under decategorifying to [[homotopy category of an (infinity,1)-category|homotopy categories]] is a pair of ordinary [[adjoint functor]]s

$$
  (Ho(L) \dashv Ho(R)) : Ho(C) \stackrel{\leftarrow}{\to} Ho(D)
  \,.
$$

=--

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

This is [[Higher Topos Theory|HTT, prop 5.2.2.9]].

This follows from that fact that for $\epsilon : Id_C \to R \circ L$ a unit of the $(\infty,1)$-adjunction, its image $Ho(\epsilon)$ is a unit for an ordinary adjunction.

=--

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

The converse statement is in general false. A near converse is given by [[Higher Topos Theory|HTT, prop 5.2.2.12]] if one instead considers $Ho$-enriched homotopy categories: if $Ho(L)$ has a right adjoint, then so does $L$.

It is important to consider the $Ho$-enriched homotopy category rather than the ordinary one. For a counterexample, when $Ho$ is considered as an ordinary category, $\pi_0 : Ho \to Set$ is both left and right adjoint to the inclusion $Set \subseteq Ho$. However, $\pi_0 : \infty Gpd \to Set$ does not have a left adjoint.

One way to find  that an ordinary adjunction of homotopy categories lifts to an $(\infty,1)$-adjunction is to exhibit it as a [[Quillen adjunction]] between [[simplicial model category]]-structures. This is discussed in the Examples-section [Simplicial and derived adjunction](#SimplicialAndDerived) below.

=--

### Full and faithful adjoints
 {#FullAndFaithfulAdjoints}

As for ordinary [[adjoint functors]] we have the following relations between full and faithful adjoints and idempotent monads.

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

Given an $(\infty,1)$-adjunction $(L \dashv R) : C \to  D$

* $R$ is a [[full and faithful (∞,1)-functor]] precisely is the counit $L R \stackrel{}{\to} Id$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functors]].

  In this case $C$ is a [[reflective (∞,1)-subcategory]] of $D$.

* $L$ is a [[full and faithful (∞,1)-functor]] precisely is the unit $Id \to R L$ is an [[equivalence of quasi-categories|equivalence]] of [[(∞,1)-functors]].

=--

[Lurie, prop. 5.2.7.4](#Lurie), See also top of p. 308.

[[!include sliced adjoint functors -- section]]

### In terms of universal arrows {#UniversalArrows}

+-- {: .num_prop #UnivArr}
###### Proposition
An $(\infty,1)$-functor $G:D\to C$ admits a left adjoint if and only if for each $X\in C$, the [[comma (infinity,1)-category]] $(X \downarrow G)$ has an [[initial object in an (infinity,1)-category|initial object]], i.e. every object $X\in C$ admits a [[universal arrow]] $X\to G F X$ to $G$.
=--

This is stated explicitly as [Riehl-Verity, Corollary 16.2.7](#RVElements), and can be extracted with some work from [[Higher Topos Theory|HTT, Proposition 5.2.4.2]].

### Preservation by exponentiation

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

Let $f : C \to D$ be left adjoint to $g : D \to C$.
Then for any $A$, $f^A$ is left adjoint to $g^A$ and
$A^g$ is left adjoint to $A^f$.

=--

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

Let $\eta : id_C \Rightarrow gf$ be a unit transformation. The property
of being a unit transformation can be detected at the level of enriched homotopy
categories, so $A^\eta: id_{A^C} \Rightarrow A^f A^g$ and $\eta^A : id_{C^A} \Rightarrow g^A f^A$ are also unit transformations.

=--

## Category of adjunctions

The functorality of adjunctions can be organized into the existence of two wide subcategories $LAdj \subseteq (\infty,1)Cat$ and $RAdj \subseteq (\infty,1)Cat$
whose functors are the left adjoints and the right adjoints respectively.

We can then define the functor categories

* $Func^L : LAdj^{op} \times LAdj \to (\infty,1)Cat$
is defined by taking $Func^L(C,D) \subseteq Func(C, D)$ to be the
full subcategory spanned by $LAdj(C, D)$.

* $Func^R : RAdj^{op} \times RAdj \to (\infty,1)Cat$
is defined by taking $Func^R(C,D) \subseteq Func(C, D)$ to be the
full subcategory spanned by $RAdj(C, D)$.

Lurie defines an adjunction to be a functor $X \to [1]$ that is both a cartesian and a cocartesian fibration. We can generalize this to

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

A functor $p : X \to S$ is an _adjunct fibration_ iff it is both a cartesian fibration and a cocartesian fibration

=--

By the [[(∞,1)-Grothendieck construction]] construction, adjunct fibrations over $S$ correspond to category-valued functors on $S$ that send arrows of $S$ to adjoint pairs of categories.

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

For a functor $p : X \to S$ of (∞,1)-categories with small fibers.

* If $p$ is a cartesian fibration classified by $\chi : S^\op \to (\infty,1)Cat$,
$\chi$ factors through $RAdj$ iff $p$ is an adjunct fibration

* If $p$ is a cocartesian fibration classified by $\chi : S \to (\infty,1)Cat$,
$\chi$ factors through $LAdj$ iff $p$ is an adjunct fibration

=--

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

This is a restatement of [[Higher Topos Theory|HTT, corr. 5.2.2.5]].

=--

\begin{lemma}\label{LadjAndRadj}
There are anti-equivalences $ladj \,\colon\, RAdj^{op} \to LAdj$ and $radj \,\colon\, LAdj^{op} \to RAdj$ that are the identity on objects and the action on homspaces $LAdj(C, D) \simeq RAdj(D,C)$ is the equivalence sending a functor to its adjoint.
\end{lemma}

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

By the covariant Grothendieck construction, for any (∞,1)-category C, $Map(C, LAdj)$ can be identified with the ∞-groupoid of $(\infty,1)\widehat{Cat}_{/C}$ spanned by adjunct fibrations over $C$ with small fibers and all equivalences between them. The same is true of $Map(C^{\op}, RAdj)$.

Since the Grothendieck construction is natural in the base category, we obtain the asserted equivalence between $LAdj$ and $RAdj^{op}$. Taking $C = [1]$, this establishes the correspondence between an adjunction and its associated adjoint pair of functors.

=--

As discussed at [Uniqueness of Adjoints](#UniquenessOfAdjoints), this anti-equivalence extends to the (∞,2)-enrichment, in the sense they induce anti-equivalences $radj : Func^L(C, D)^{op} \to Func^R(D, C)$ and
$ladj : Func^R(C, D)^{op} \to Func^L(D, C)$.

The preservation of adjunctions by products and exponentials implies

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

The product and exponential on $(\infty,1)Cat$ restrict to functors

* $- \times - : LAdj \times LAdj \to LAdj$
and $- \times - : RAdj \times RAdj \to RAdj$
* $Func(-,-) : RAdj^{op} \times LAdj \to LAdj$
and $Func(-,-) : LAdj^{op} \times RAdj \to RAdj$

=--

## Examples

A large class of examples of $(\infty,1)$-adjunctions arises from [[Quillen adjunctions]] of [[model categories]], or adjunctions in [[sSet]]-[[enriched category theory]].

### Quillen adjunctions

Any [[Quillen adjunction]] induces an adjunction of [[(infinity,1)-categories]] on the [[simplicial localizations]]. See [Hinich 14](#Hinich14) or [Mazel-Gee 15](#MazelGee15).

### Simplicial and derived adjunctions {#SimplicialAndDerived}

We want to produce Cartesian/coCartesian fibration $K \to \Delta[1]$ from a given [[sSet]]-[[enriched category theory|enriched]] adjunction. For that first consider the following characterization

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

Let $K$ be a [[simplicially enriched category]] whose [[hom-objects]] are all [[Kan complexes]], regard the [[interval category]] $\Delta[1] := \{0 \to 1\}$ as an $sSet$-category in the obvious way using the embedding $const : Set \hookrightarrow sSet$ and consider an $sSet$-enriched functor $K \to \Delta[1]$. Let $C := K_0$ and $D := K_1$ be the $sSet$-enriched categories that are the fibers of this. Then under the [[homotopy coherent nerve]] $N : sSet Cat \to sSet$ the morphism

$$
  N(p) : N(K) \to \Delta[1]
$$

is a [[Cartesian fibration]] precisely if for all objects $d \in D$ there exists a morphism $f : c \to d$ in $K$ such that postcomposition with this morphism

$$
  C(c',f ) :  C(c',c) = K(c',c) \to K(c',d)
$$

is a [[homotopy equivalence]] of [[Kan complex]]es for all objects $c' \in C'$.

=--

This appears as [[Higher Topos Theory|HTT, prop.  5.2.2.4]].

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

The statement follows from the characterization of [[Cartesian morphism]]s under homotopy coherent nerves ([[Higher Topos Theory|HTT, prop.  2.4.1.10]]), which says that for an $sSet$-enriched functor $p : C \to D$ between Kan-complex enriched categories that is [[hom-object]]-wise a [[Kan fibration]], a morphim $f : c' \to c''$ in $C$ is an $N(p)$-[[Cartesian morphism]] if for all objects $c \in C$ the diagram

$$
  \array{
    C(c,c') &\stackrel{C(c,f)}{\to}& C(c,c'')
    \\
    \downarrow^{\mathrlap{p_{c,c'}}} &&
    \downarrow^{\mathrlap{p_{c,c''}}}
    \\
    D(p(c),p(c')) &\stackrel{D(p(c),p(f))}{\to}&
    D(p(c), p(c''))
  }
$$

is a [[homotopy pullback]] in the [[model structure on sSet-categories]].

For the case under consideration the functor in question is $p : K \to \Delta[1]$ and the above diagram becomes

$$
  \array{
    K(c,c') &\stackrel{K(c,f)}{\to}& K(c,c'')
    \\
    \downarrow
     &&
    \downarrow
    \\
    * &\to& *
  }
  \,.
$$

This is clearly a homotopy pullback precisely if the top morphism is an equivalence.

=--

Using this, we get the following.

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

For $C$ and $D$ [[sSet]]-[[enriched categories]] whose hom-objects are all [[Kan complexes]], the image

$$
  N(C)
    \underoverset
      {\underset{N(R)}{\longleftarrow}}
      {\overset{N(L)}{\longrightarrow}}
      {\bot}
  N(D)
$$

under the [[homotopy coherent nerve]] of an [[sSet]]-enriched adjunction between $sSet$-[[enriched categories]]

$$
  C \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
  D
$$

is an adjunction of [[quasi-categories]].

Moreover, if $C$ and $D$ are equipped with the structure of a [[simplicial model category]] then the quasi-categorically [[derived functors]]

$$
  N(C^\circ) \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}}
  N(D^\circ)
$$

form an adjunction of quasi-categories.

=--

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

The first part is [[Higher Topos Theory|HTT, cor.  5.2.4.5]], the second
[[Higher Topos Theory|HTT, prop. 5.2.4.6]].

To get the first part, let $K$ be the $sSet$-category which is the join of $C$ and $D$: its set of objects is the disjoint union of the sets of objects of $C$ and $D$, and the [[hom-object]]s are

* for $c,c' \in C$: $K(c,c') := C(c,c')$;

* for $d,d' \in D$: $K(d,d') := D(d,d')$;

* for $c \in C$ and $d \in D$: $K(c,d) := C(L(c),d) = D(c,R(d))$;

  and

  $K(d,c) = \emptyset$

and equipped with the evident composition operation.

Then for every $d \in D$ there is the morphism $Id_{R(d)} \in K(R(d),d)$, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence $N(K) \to \Delta[1]$ is a [[Cartesian fibration]].

By the analogous dual argument, we find that it is also a coCartesian fibration and hence an adjunction.

For the second statement, we need to refine the above argument just slightly to pass to the full $sSet$-subcategories on fibrant cofibrant objects:

let $K$ be as before and let $K^\circ$ be the full $sSet$-subcategory on objects that are fibrant-cofibrant (in $C$ or in $D$, respectively). Then for any fibrant cofibrant $d \in D$, we cannot just use the identity morphism $Id_{R(d)} \in K(R(d),d)$ since the right Quillen functor $R$ is only guaranteed to respect fibrations, not cofibrations, and so $R(d)$ might not be in $K^\circ$. But we can use the [[small object argument]] to obtain a functorial cofibrant replacement functor $Q : C \to C$, such that $Q(R(d))$ is cofibrant and there is an acyclic fibration $Q(R(d)) \to R(d)$. Take this to be the morphism in $K(Q(R(d)), d)$ that we pick for a given $d$. Then this does induce a homotopy equivalence

$$
  C(c', Q(R(d))) \to C(c',R(d)) = K(c',d)
$$

because in an [[enriched model category]] the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.

=--

### Localizations

A pair of adjoint $(\infty,1)$-functors $(L \dashv R) : C \stackrel{\leftarrow}{\hookrightarrow} D$ where $R$ is a [[full and faithful (∞,1)-functor]] exhibits $C$ as a [[reflective (∞,1)-subcategory]] of $D$. This subcategory and the composite $R \circ L : D \to D$ are a [[localization of an (∞,1)-category|localization]] of $D$.

## Related concepts

* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]

* [[proadjoint]], [[Hopf adjunction]]

* [[2-adjunction]]

  [[biadjunction]], [[lax 2-adjunction]], [[pseudoadjunction]]

* **adjoint $(\infty,1)$-functor

## References

The suggestion that a pair of adjoint $\infty$-functors should just be an [[adjunction]] in the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $\infty$-categories]] was originally stated, briefly, in:

* {#Joyal08} [[André Joyal]], p. 159 (11 of 348) in: _The theory of quasicategories and its applications_, lectures at: _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, Quadern **45** 2, Centre de Recerca Matemàtica, Barcelona 2008 ([[JoyalTheoryOfQuasiCategories.pdf:file]])

The definition as an isofibration of [[quasicategories]] over $\Delta[1]$ is due to:

* {#Lurie} [[Jacob Lurie]], Section 5.2 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

The original suggestion of [Joyal 2008](#Joyal08) was then much expanded on (and generalized to [[∞-cosmoi]]), in the spirit of [[formal (infinity,1)-category theory|formal $\infty$-category theory]]:

* [[Emily Riehl]], §18.6 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

* {#RiehlVerity15} [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021)),

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))

* {#RiehlVerity20} [[Emily Riehl]], [[Dominic Verity]], Def. 1.1.2 in: _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

That the two definitions (of [Joyal 2008](#Joyal08) and [Lurie 2009](#Lurie)) are in fact equivalent is first indicated in [Riehl-Verity 15, Rem. 4.4.5](#RiehlVerity15) and then made fully explicit in:

* {#RVElements} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

A proof that a [[Quillen adjunction]] of [[model categories]] induces an adjunction between [[(∞,1)-categories]] (in the sense of [Lurie 2009](#Lurie)) is recorded in:

* {#Hinich14} [[Vladimir Hinich]], *Dwyer-Kan Localization Revisited*, Homology, Homotopy and Applications Volume 18 (2016) Number 1 ([arXiv:1311.4128](https://arxiv.org/abs/1311.4128), [doi:10.4310/HHA.2016.v18.n1.a3](https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3))

and also in

* {#MazelGee15} [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, New York Journal of Mathematics Volume 22 (2016) 57-93 ([arXiv:1501.03146](https://arxiv.org/abs/1501.03146), [publisher](http://nyjm.albany.edu/j/2016/22-4.html))

[[!redirects adjoint (infinity,1)-functors]]

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[[!redirects adjoint ∞-functor]]
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[[!redirects (infinity,1)-adjunctions]]
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[[!redirects (∞,1)-adjunctions]]

[[!redirects (∞,1)-adjoint functor]]
[[!redirects (∞,1)-adjoint functors]]
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[[!redirects (infinity,1)-adjoint functors]]

[[!redirects adjunction of (infinity,1)-categories]]
[[!redirects adjunctions of (infinity,1)-categories]]

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[[!redirects adjoint infinity1-functor]]


adjoint (infinity,1)-functor theorem

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

## Idea

The analogs in [[(∞,1)-category]] theory of the [[adjoint functor theorems]] in ordinary [[category theory]].

## Statement

+-- {: .num_defn}
###### Definition
* Let $G:\mathcal{D} \to \mathcal{C}$ be a functor between $(\infty, 1)$-categories. We say that $G$ satisfies the **solution set condition** if the $(\infty, 1)$-category $c/G$ admits a small weakly initial set for any $c$ in $\mathcal{C}$.

* $G$ satisfies the **$h$-initial object condition** if $c/G$ admits an [[h-initial object]] for any $c$ in $\mathcal{C}$.

* $\mathcal{C}$ is said to be **2-locally small** if for every pair of objects, $x$,$y$, of $\mathcal{C}$, the mapping space $map_{\mathcal{C}}(x,y)$ is [[locally small]].
=--

+-- {: .num_theorem #GeneralAdjointFunctorTheorem}
###### Theorem

* Let $G:\mathcal{D} \to \mathcal{C}$ be a [[continuous functor]]. Suppose that $\mathcal{D}$ is locally small and complete and $\mathcal{C}$ is 2-locally small. Then $G$ admits a left adjoint if and only if it satisfies the solution set condition.

* Let $G:\mathcal{D} \to \mathcal{C}$ be a [[finitely continuous functor]]. Suppose that $\mathcal{D}$ is finitely complete. Then $G$ admits a left adjoint if and only if it satisfies the $h$-initial object condition.

=--

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

See Section 3 of ([NRS18](#NRS18)).

=--

The following result is a consequence.

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

Let $F : C \to D$ be an [[(∞,1)-functor]] between [[locally presentable (∞,1)-categories]] then

1. it has a right [[adjoint (∞,1)-functor]] precisely if it preserves small [[limit in a quasi-category|colimits]];

1. it has a left [[adjoint (∞,1)-functor]] precisely if it is an [[accessible (∞,1)-functor]] and preserves small [[limit in a quasi-category|limits]].

=--

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

This is [[Higher Topos Theory|HTT, cor. 5.5.2.9]].

=--

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

For the existence of right adjoints, we can weaken the hypotheses to merely requiring $D$ to be a [[locally small (infinity,1)-category]].
([[Higher Topos Theory|HTT, rem. 5.5.2.10]])

=--

## Related concepts

* [[adjoint functor theorem]]

## References

* {#NRS18} Hoang Kim Nguyen, [[George Raptis]], Christoph Schrade, _Adjoint functor theorems for ∞-categories_, Journal of the London Mathematical Society **101** 2 (2019) 659-681
([arXiv:1803.01664](https://arxiv.org/abs/1803.01664))

Section 5.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects adjoint (∞,1)-functor theorem]]
[[!redirects adjoint (∞,1)-functor theorems]]


adjoint action

> This entry is about _conjugation_ in the sense of adjoint actions, as in forming [[conjugacy classes]]. For conjugation in the sense of [[anti-involutions]] on [[star algebras]] see at _[[complex conjugation]]_.

***

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

#Contents#
* table of contents
{:toc}

## Idea

An _adjoint action_ is an [[action]] by _conjugation_ .

## Definition

### Of a group on itself

The _adjoint action_ of a [[group]] $G$ on itself is the [[action]] $Ad : G \times G \to G$ given by

$$
  Ad : (g,h) \mapsto g^{-1} \cdot h \cdot g
  \,.
$$

### Of a Lie group on its Lie algebra

The adjoint action $ad : G \times \mathfrak{g} \to \mathfrak{g}$ of a [[Lie group]] $G$ on its [[Lie algebra]] $\mathfrak{g}$ is for each $g \in G$ the [[derivative]] $d Ad(g) : T_e G \to T_e G$ of this action in the second argument at the neutral element of $G$

$$
  ad : (g,x) \mapsto Ad(g)_*(x)
  \,.
$$

This is often written as $ad(g)(x) = g^{-1} x g$ even though for a general Lie group the expression on the right is not the product of three factors in any way.  But for a [[matrix Lie group]] $G$ it is: in this case both $g$ as well as $x$ are canonically identified with [[matrices]] and the expression on the right is the product of these matrices.

Since this is a linear action, it is called the _[[adjoint representation]]_ of a Lie group. The [[associated bundles]] with respect to this representation are called [[adjoint bundles]].

### Of a Lie algebra on itself

Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself

$$
  ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}
$$

which is simply the Lie bracket

$$
  ad_x : y \mapsto [x,y]
  \,.
$$

### Of a Hopf algebra on itself

Let $k$ be a commutative unital ring and $H = (H,m,\eta,\Delta,\epsilon, S)$ be a [[Hopf algebra|Hopf]] $k$-algebra with multiplication $m$, unit map $\eta$, comultiplication $\Delta$, counit $\epsilon$ and the antipode map $S: H\to H^{op}$. We can use [[Sweedler notation]] $\Delta(h) = \sum h_{(1)}\otimes_k h_{(2)}$. The adjoint action of $H$ on $H$ is given by

$$
h\triangleright g = \sum h_{(1)} g S(h_{(2)})
$$

and it makes $H$ not only an $H$-module, but in fact a monoid in the monoidal category of $H$-modules (usually called $H$-[[module algebra]]).

### Of a simplicial group on itself
 {#OfASimplicialGroupOnItself}

Let

* $\mathcal{G}$ be a [[simplicial group]],

and write

* $\mathcal{G}\Actions(sSet)$ for the [[category]] of $\mathcal{G}$-[[action objects]] [[internalization|internal to]] [[SimplicialSets]]l

* $W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its [[universal principal simplicial complex]];

* $\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the [[simplicial classifying space]];

* $\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the [[adjoint action]] of $\mathcal{G}$ on itself:

  \[
    \label{ConjugationAction}
    \array{
      \mathcal{G}_{ad} \times \mathcal{G}
      &\xrightarrow{\;\;\;}&
      \mathcal{G}_{ad}
      \\
      (g_k,h_k) &\mapsto&  h_k \cdot g_k \cdot h_k^{-1}
    }
  \]

  which we may understand as the restriction along the [[diagonal morphism]] $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the [[direct product group]]:

  $$
    \array{
      \mathcal{G}_{ad}
      \times
      (\mathcal{G} \times \mathcal{G})
      &\xrightarrow{\;\;\;}&
      \mathcal{G}_{ad}
      \\
      (g_k, (h'_k, h_k))
      &\mapsto&
      h'_k \cdot g_k \cdot h^{-1}_k
      \mathrlap{\,.}
    }
  $$

\begin{proposition}\label{FreeLoopSpaceOfSimplicialClassifyingSpaceAsAdQuotient}
  The [[free loop space object]] of the [[simplicial classifying space]] $\overline{W} \mathcal{G}$ is [[isomorphism|isomorphic]] in the [[classical homotopy category]] to the [[Borel construction]] of the [[adjoint  action]] (eq:ConjugationAction):

$$
  \mathcal{L}
  \big(
    \overline{W}\mathcal{G}
  \big)
  \;\;
  \simeq
  \;\;
  \mathcal{G}_{ad} \sslash \mathcal{G}
  \;\;\;\;\;\;
  \in
  \;\;
  Ho\big(
    sSet_{Qu}
  \big)
$$
\end{proposition}

For proof and more background see at *[[free loop space of classifying space]]*.

## Related concepts

* [[coadjoint action]]

* [[adjoint representation]], [[adjoint bundle]]

* [[free loop space of classifying space]]

* [[conjugation action]]

* [[conjugacy class]]

* [[twisted ad-equivariant K-theory]]

## References

* Sigurdur Helgason, _Differential geometry, Lie groups, and symmetric spaces_

* [[Eckhard Meinrenken]], _Clifford algebras and Lie theory_, Springer

* [[eom]]: [adjoint representation of a Lie group](http://www.encyclopediaofmath.org/index.php/Adjoint_representation_of_a_Lie_group), [adjoint group](http://www.encyclopediaofmath.org/index.php/Adjoint_group)

[[!redirects adjoint actions]]

[[!redirects conjugation]]
[[!redirects conjugations]]


adjoint bundle

[[!redirects adjoint bundles]]

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

#Contents#
* table of contents
{:toc}

## Definition

For $G$ a [[Lie group]] and $\mathfrak{g}$ its [[Lie algebra]], and for $P \to X$ a $G$-[[principal bundle]], the corresponding **adjoint bundle** is the [[associated bundle]] $P \times_G \mathfrak{g} \to X$ via the [[adjoint action]]/[[adjoint representation]] of $G$ on $\mathfrak{g}$.

## Related concepts

* [[Atiyah Lie algebroid]]

[[!redirects adjoint bundle]]


adjoint equivalence

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

# Adjoint equivalences
* table of contents
{: toc}

## Idea

In [[category theory]], the notion of *adjoint equivalence* is a more "coherent" or "structured" notion of [[equivalence of categories]], in which the [[2-morphism|2-]][[isomorphisms]] relating composites to identities are required to satisfy [[coherence laws]] (namely the [[zigzag identities]] for an [[adjunction]]).

## Definition

An **adjoint equivalence** between [[categories]] is an pair of [[adjoint functors]] $f\dashv g$ in which the [[unit of an adjunction|unit]] $\eta$ and [[unit of an adjunction|counit]] $\varepsilon$ are [[natural isomorphisms]].  It follows in particular that each of $f,g$ constitute an [[equivalence of categories]].

There is an identical definition internal to any [[2-category]], which reproduces the above notion when applied in [[Cat]].

## Properties

We work in any 2-category.  First, we observe:

+-- {: .num_lemma}
###### Lemma
If $(f,g,\eta,\varepsilon)$ is an adjoint equivalence, then so is $(g,f,\varepsilon^{-1},\eta^{-1})$.
=--

Therefore, in an adjoint equivalence, each functor is both the [[left adjoint]]  and the [[right adjoint]] of the other (i.e. it is an [[ambidextrous adjunction]]).

The definition as given above is also redundant:

+-- {: .num_lemma}
###### Lemma
If $(f,g,\eta,\varepsilon)$ is any equivalence, then it satisfies one [[zigzag identity]] iff it satisfies the other.
=--

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

Using [[string diagram]] notation, with strings progressing up the page and 1-morphisms progressing from left to right, we can draw the data of an equivalence (omitting labels for the regions denoting objects) as follows:

[[!include equivalence data - SVG]]

If we now suppose that one zigzag identity holds:

[[!include zigzag identity 1 - SVG]]

then we can verify the other as follows.  (The first step uses the inverse of the first zigzag identity.)

[[!include zigzag identity 1 implies 2 - SVG]]

=--

Furthermore, although an adjoint equivalence is a "stronger" or "more structured" notion than a mere equivalence, the property of "being adjoint equivalent" is no stronger a condition than "being equivalent," since every equivalence may be refined to an adjoint equivalence by modifying one of the natural isomorphisms involved.  More specifically:

+--{: .num_theorem}
###### Theorem
If $f\colon X\to Y$ is a morphism which is an equivalence, then given any morphism $g\colon Y\to X$ and any isomorphism $\eta\colon 1 \cong g f$, there exists a unique 2-isomorphism $\varepsilon\colon f g \cong 1$ such that $(f,g,\eta,\varepsilon)$ is an adjoint equivalence.
=--

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

Since $f$ is an equivalence, there exists a $g'$ and isomorphisms $f g' \cong 1$ and $1\cong g' f$.  However, we also have $g \cong g f g' \cong g'$, so the isomorphism $f g' \cong 1$ also induces an isomorphism $f g\cong 1$, which we denote $\xi$.  Now $\eta$ and $\xi$ may not satisfy the zigzag identities, but if we define $\varepsilon$ as follows:
$$
f g
\xrightarrow{f g \xi^{-1}} f g f g
\xrightarrow{f \eta^{-1} g} f g
\xrightarrow{\xi} 1
$$
then we can verify, using string diagram notation as above, that $\varepsilon$ satisfies one zigzag identity, and hence (by the previous lemma) also the other:

[[!include adjointification zigzag identity - SVG]]

Finally, if $\varepsilon'\colon f g \to 1$ is any other isomorphism satisfying the zigzag identities with $\eta$, then we have
$$\varepsilon' = \varepsilon' . (\varepsilon f g) . (f \eta g) =
\varepsilon . (f g \varepsilon') . (f \eta g) = \varepsilon$$
using the [[interchange law]] and two zigzag identities.  This shows uniqueness.

=--

In [[Categories Work]], IV.4, there is a different proof of the weaker fact that if a [[functor]] $f$ is part of an equivalence, then it is part of an adjoint equivalence.  This proof is given in [[Cat]], but can be applied representably to any 2-category.

Since adjoints are unique up to unique isomorphism when they exist, it follows that any adjunction involving one functor which is an equivalence must be an adjoint equivalence.  Therefore, for a fixed morphism $f$, the "category of adjoint equivalence data $(f,g,\eta,\varepsilon)$" is either empty (if $f$ is not an equivalence) or equivalent to the [[terminal category]] (if $f$ is an equivalence).  In other words, it is a [[(-1)-category]].

Therefore, in any 2-category, the following data are all equivalent (i.e. form equivalent categories):

* A morphism $f\colon X\to Y$ with the [[property]] of being an equivalence.

* A morphism $f\colon X\to Y$ with the *structure* of a morphism $g\colon Y \to X$ and an isomorphism $\eta\colon 1 \cong g f$, together with the *property* that there exists an isomorphism $f g \cong 1$.

* A morphism $f$ together with the structure of adjoint equivalence data $(f,g,\eta,\varepsilon)$.

In other words, adjoint equivalences are the way to make the property of "being an equivalence" completely into "algebraic" structure.  However, they are *not* equivalent to the category of the following data:

* A morphism $f$ together with the structure of a morphism $g\colon Y \to X$ and arbitrary isomorphisms $\eta\colon 1 \cong g f$ and $\varepsilon\colon f g \cong 1$.

## Applications

### Intervals in homotopy theory

One instance of the usefulness of adjoint equivalences is that the "[[walking structure|walking]] adjoint equivalence" 2-category is equivalent to the [[point]].  Thus, it can be used as an [[interval object]] in $2Cat$, and in fact it is one of the generating cofibrations for the [[canonical model structure|canonical (Lack) model structure]] on $2Cat$.  This is not true of the "walking non-adjoint equivalence."

### Defining tricategories

The original definition of [[tricategory]] by Gordon-Power-Street involved coherence 2-morphisms with the property of being equivalences in the relevant hom-bicategories.  This is fine for most purposes, but for others it is insufficient, such as the following.

* Since "being an equivalence" is not algebraic structure, the GPS definition of tricategory, taken literally, is not an algebraic structure.  In particular, it is not [[monadic functor|monadic]] over 3-[[globular sets]], nor is it the algebras for a [[globular operad]].  Such monadicity is important if one wants to state [[coherence theorems]] as properties of [[free object|free]] structures.

* The definition of 3-functors and higher [[transfors]] between tricategories include data and axioms that involve composites incorporating not just the coherence equivalences, but their pseudo-inverses.  Therefore, strictly speaking these definitions are not well-defined unless the definition of tricategory comes with chosen pseudo-inverses for these coherence equivalences---in which case one should certainly also choose full adjoint equivalence data in order that the space of choices be contractible.

These problems are, of course, easy to remedy by simply requiring adjoint equivalence data rather than merely single equivalence morphisms.  This change was first written down by Gurski.

### Cartesian closed 2-categories

In a [[cartesian closed category]] with [[equalizers]], for any two objects $X$ and $Y$ one can construct the "object of isomorphisms from $X$ to $Y$" as the following equalizer:
$$ Iso(X,Y) \to X^Y \times Y^X \;\rightrightarrows\; X^X \times Y^Y $$
where the top arrow on the right side is (composition, reversed composition) and the bottom arrow factors through $(id,id)\colon 1 \to X^X \times Y^Y$.  One can then prove that the maps $Iso(X,Y)\to X^Y$ and $Iso(X,Y)\to Y^X$ are monic, so that $Iso(X,Y)$ can be regarded either as "the object of maps $X\to Y$ which are isomorphisms" or "the object of maps $Y\to X$ which are isomorphisms" (or, as is most evident from its construction, "the object of pairs of maps $X\to Y$ and $Y\to X$ which are inverse isomorphisms").

In a [[cartesian closed 2-category]], however, the analogous "2-equalizer" $Eqv(X,Y)$, does not have similar properties: the projections $Eqv(X,Y)\to X^Y$ and $Eqv(X,Y)\to Y^X$ will not in general be [[fully faithful morphism|fully faithful]].  Thus, we can only regard $Eqv(X,Y)$ as "the object of not-necessarily-adjoint equivalence data $(f,g,\eta,\varepsilon)$."  However, if we use a further [[equifier]] to construct its "subobject of adjoint equivalence data" $AdjEqv(X,Y)$, then the projections $AdjEqv(X,Y)\to X^Y$ and $AdjEqv(X,Y)\to Y^X$ will be fully faithful, so that $AdjEqv(X,Y)$ can also be regarded as "the object of maps $X\to Y$ which are equivalences" and dually.

## In higher category theory

In [[higher category theory]], one expects to have a similar "fully coherent" notion of "adjoint equivalence" in any [[n-category]] or [[infinity-category]], and one hopes to prove a similar theorem that any [[equivalence]] can be refined to an adjoint equivalence.  This is known to be true at least in the following cases:

* For [[Gray-categories]], the statement and proof is in [[Steve Lack]]'s paper [1001.2366](http://arxiv.org/abs/1001.2366) on the [[model structure for Gray-categories]].  See [[adjoint 2-equivalence]].

* For [[tricategories]], the corresponding statement can be deduced from the Gray-categorical version using the [[coherence theorem for tricategories]].  A direct proof can also be found in [[Nick Gurski]]'s paper [Biequivalences in tricategories](http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html).

* For [[strict omega-categories]], more or less this fact can be found in the study of "generic squares" in the paper [0712.0617](http://arxiv.org/abs/0712.0617) on the [[model structure for strict omega-categories]].

* For [[quasicategories]], the theorem is true, where an "adjoint equivalence" means simply a map out of the [[nerve]] of the [[interval groupoid]]; see [[equivalence in a quasicategory]].

## Related concepts

* [[enriched adjoint equivalence]]

[[!redirects adjoint equivalences]]

[[!redirects adjoint equivalence of categories]]
[[!redirects adjoint equivalences of categories]]