quark-gluon plasma

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## Idea

The _quark-gluon plasma_ is the [[phase of matter]] of [[quantum chromodynamics]] at extremely high [[temperature]]. At high temperature [[quarks]] are not [[confinement|confined]] to [[hadron]] [[bound states]] but propagate freely together with the [[gluons]], forming a "quark-gluon soup". Since this is analogous to an ordinary [[plasma]] which is a [[phase of matter|phase]] where [[electrons]] and [[protons]] are no longer [[bound state|bound]] to [[atoms]] but propagate freely, one speaks of _quark-gluon plasma_.

> **Schematic [[phase diagram]] of [[QCD]].** The vertical axis indicates [[temperature]] $T$, the horizontal axis indicates [[baryon]] [[density]]. At low enough temperature [[quarks]] and [[gluons]] only appear as [[hadron]] [[bound states]] ([[confinement]]). But above a critical temperature these [[hadron]] [[bound states]] break apart (deconfinement) and [[quarks]] and [[gluons]] may exist freely. This [[phase of matter|phase]] of [[QCD]] is the _quark-gluon plasma_.

> graphics grabbed from [Blaizot 03](#Blaizot03).

### Non-perturbative regime

Despite the [[confinement|deconfinement]] beyond [[temperature]] $T_c$, the quark-gluon plasma at temperature $\sim 4-5 T_c$ as produced in [[experiment]] ([Adams et al. 05](#AdamsEtAl05), [Adcox et al. 05](#AdcoxEtAl05)) is apparently strongly [[coupling constant|coupled]], meaning that its properties are [[non-perturbative effects]] requiring discussion of [[QCD]] as a [[non-perturbative field theory]]. With an exact such theory largely missing, much of the theoretical discussion of the quark-gluon plasma involves [[lattice QCD]] computer simulation. Indication for strong coupling of the QG-plasma comes from the nature of the _elliptic flow_ seen both in [[experiment]] as well as in these computer simulations, which shows [[hydrodynamic]] behaviour with extremely small [[shear viscosity]] (e.g. [Shuryak 01](#Shuryak01), [Chakraborty 12](#Chakraborty12)).

It has been proposed ([Policastro-Son-Starinets 01](#PolicastroSonStarinets01)) that, therefore, an analytic approach to a description of the quark-gluon plasma (i.e. not just via [[lattice QCD]] computer experiment) might be given by approximate [[AdS-CFT duality]], hence by the [[AdS/QCD correspondence]] or [[fluid/gravity correspondence]] (see e.g. [Biagazzi-Cotrone 12](#BiagazziCotrone12)). The difficulty with this approach is that for [[QCD]] (as opposed to [[N=4 D=4 SYM]]) [[AdS-CFT duality]] applies only to some approximation, see at _[[AdS/QCD correspondence]]_ for more.

### Perturbative regime

At yet higher energies, the quark-gluon plasma is eventually supposed to be weakly [[coupling constant|coupled]] again, due to [[asymptotic freedom]] (e.g. [Blaizot 03](#Blaizot03))

## Properties

### Early cosmology and nucleosynthesis
 {#EarlyCosmologyAndNucleosynthesis}

In the [[standard model of cosmology]] a quark gluon plasma filled the universe about $10^{-6}$ comoving seconds after the [[Big Bang]]. Decrease of [[temperature]] then led to the quark-gluon plasma condensing out to form [[hadrons]] and in particular [[baryons]] and hence in particular [[nucleons]] ([[protons]] and [[neutrons]]) and eventually [[atomic nuclei]] (cosmic [[nucleosynthesis]]).

> graphics grabbed from [Haseeb 09](#Haseeb09)

See the references [below](#ReferencesInEarlyUniverseCosmology).

## Related concepts

* The quark-gluon plasma is naturally studied in terms of [[infinite-temperature thermal field theory]] ([Blaizot-Iancu-Rebhan 03, section 2.2.4](#BlaizotIancuRebhan03), [Blaizot 04, around p. 17](#Blaizot04))

* the [[QCD trace anomaly]] affects the [[equation of state]] of the quark-gluon plasma.

* In as far as the quark-gluon plasma is described by [[fluid dynamics]], the [[fluid/gravity correspondence]] applies.

* [[QCD cosmology]]

* [[quarkonium]]

## References

### General

* Stanislaw Mrowczynski, _Quark-Gluon Plasma_, Acta Phys.Polon.B29:3711, 1998 ([arXiv:nucl-th/9905005](https://arxiv.org/abs/nucl-th/9905005))

* {#Haseeb09} Mahnaz Q. Haseeb, _Introduction to Quark Gluon Plasma_, 2009 ([[HaseebQuarkGluonPlasma.pdf:file]])

* Roman Pasechnik, Michal Šumbera, _Phenomenological Review on Quark-Gluon Plasma: Concepts vs. Observations_, Universe 2017, 3(1), 7 ([arXiv:1611.01533](https://arxiv.org/abs/1611.01533))

* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))

* {#Blaizot04} Jean-Paul Blaizot, around p. 17 of _Thermodynamics of the high temperature Quark-Gluon Plasma_, AIP Conf. Proc. 739, 63-96 (2004) ([doi:10.1063/1.1843592](https://doi.org/10.1063/1.1843592))

* Jacopo Ghiglieri, Aleksi Kurkela, Michael Strickland, Aleksi Vuorinen, _Perturbative Thermal QCD: Formalism and Applications_ ([arXiv:2002.10188](https://arxiv.org/abs/2002.10188))

See also

* Wikipedia, _[Quark-Gluon plasma](http://en.wikipedia.org/wiki/Quark%E2%80%93gluon_plasma)_

On the [[rho-meson]]  and its [[chiral partner]] [[a1-meson]] in the [[quark-gluon plasma]]:

* [[Robert Pisarski]], _Where does the Rho Go? Chirally Symmetric Vector Mesons in the Quark-Gluon Plasma_, Phys. Rev. D52 (1995) 3773-3776 ([arXiv:hep-ph/9503328](https://arxiv.org/abs/hep-ph/9503328))

### Non-perturbative aspects

Discussion of [[non-perturbative effects]] in [[QCD]]:

* {#Shuryak01} [[Edward Shuryak]], _Nonperturbative QCD and Quark-Gluon Plasma_, lectures at Trieste, 2001 ([pdf](http://users.ictp.it/~pub_off/lectures/lns010/Shuryak/Shuryak.pdf))

* {#Chakraborty12} Purnendu Chakraborty, _Non-perturbative aspects of quark gluon plasma above deconfinement temperature_, Proceedings of the DAE Symp. on Nucl. Phys. 57 (2012) ([pdf](http://www.sympnp.org/proceedings/57/I20.pdf))

Further discussion in relation to [[instantons in QCD]] includes

* Nikolai Kochelev, _Ultra-light Glueballs in Quark-Gluon Plasma_ ([arXiv:1501.07002](http://arxiv.org/abs/1501.07002))

### Experimental realization

Realization of the quark-gluon plasma at the [[RHIC]] [[experiment]] has tentatively been claimed in

* {#AdamsEtAl05} John Adams, et al. _Experimental and theoretical challenges in the search for the quark–gluon plasma: The STAR Collaboration's critical assessment of the evidence from RHIC collisions_. Nuclear Physics A 757.1-2 (2005): 102-183.

* {#AdcoxEtAl05} K. Adcox, et al., _Formation of dense partonic matter in relativistic nucleus–nucleus collisions at RHIC: experimental evaluation by the PHENIX collaboration_ Nuclear Physics A 757.1-2 (2005): 184-283.

Exposition and review:

* Maria Elena Tejeda-Yeomans, _Heavy-ion physics: freedom to do hot, dense, exciting QCD_ ([arXiv:2004.13812](https://arxiv.org/abs/2004.13812))

Status report:

* *Dynamics of Hot QCD Matter -- Current Status and Developments* [[arXiv:2208.13440](https://arxiv.org/abs/2208.13440)]

### In early universe cosmology
 {#ReferencesInEarlyUniverseCosmology}

Discussion of the quark gluon-plasma in [[cosmology]] as the [[phase of matter]] shortly after the [[standard model of cosmology|big bang]]:

* [[Edward Witten]], _Cosmic Separation Of Phases_, Phys. Rev. D30, 272 (1984) ([doi:10.1103/PhysRevD.30.272](https://doi.org/10.1103/PhysRevD.30.272))

* Dominik J. Schwarz, _The first second of the Universe_, Annalen Phys.12:220-270, 2003 ([arXiv:astro-ph/0303574](https://arxiv.org/abs/astro-ph/0303574))

* K. Sakthi Murugesan, G. Janhavi, and P. R. Subramanian, _Can the phase transition from quark-gluon plasma to hadron-resonance gas affect primordial nucleosynthesis?_, Phys. Rev. D 41, 2384 ([doi:10.1103/PhysRevD.41.2384](https://doi.org/10.1103/PhysRevD.41.2384))

* K. Sakthi Murugesan, G. Janhavi, and P. R. Subramanian, _Importance of interactions in quark-gluon plasma: Baryon-number density contrast and primordial nucleosynthesis_, Phys. Rev. D 42, 3576 (1990) ([arXiv:10.1103/PhysRevD.42.3576](https://doi.org/10.1103/PhysRevD.42.3576))

* [[Keith Olive]], _The Quark - hadron transition in cosmology and astrophysics_, Science 251, 1194 (1991)

* A.A. Coley, T. Trappenberg, _The Quark-Hadron Phase Transition, QCD Lattice Calculations and Inhomogeneous Big-Bang Nucleosynthesis_, Phys.Rev. D50 (1994) 4881-4885 ([arXiv:astro-ph/9307031](https://arxiv.org/abs/astro-ph/9307031))

* Hideo Suganuma, Hiroko Ichie, Hideko Monden, Shoichi Sasaki, Manabu Orito, Tadahiro Yamamoto, Toshitaka Kajino, _QCD Phase Transition at high Temperature in Cosmology_ ([arXiv:hep-ph/9608333](https://arxiv.org/abs/hep-ph/9608333))

* Deepak Chandra, Ashok Goyal, _Dynamical evolution of the Universe in the quark-hadron phase transition and possible nugget formation_, Phys.Rev. D62 (2000) 063505 ([arXiv:hep-ph/9903466](https://arxiv.org/abs/hep-ph/9903466))

* N. Borghini, W. N. Cottingham, R. Vinh Mau, _Possible Cosmological Implications of the Quark-Hadron Phase Transition_, J.Phys.G26:771, 2000 ([arXiv:hep-ph/0001284](https://arxiv.org/abs/hep-ph/0001284))

* {#Kapusta01} Joseph I Kapusta, _Quark-Gluon Plasma in the Early Universe_, in _Phase Transitions in the Early Universe: Theory and Observations_ Springer, Dordrecht, 2001. 103-121 ([arXiv:astro-ph/0101516](https://arxiv.org/abs/astro-ph/0101516))

* Michael McGuigan, Wolfgang Söldner, _QCD Cosmology from the Lattice Equation of State_ ([arXiv:0810.0265](https://arxiv.org/abs/0810.0265), [spire:798226](http://inspirehep.net/record/798226))

> The effect of the strong interactions on cosmology was considered early on [10, 11] but the [[non-perturbative effect|nonperturbative]] nature of the strong interactions at low energy limited the progress of the subject.

* S. M. Blinder, _[Evolution of Matter from a Quark-Gluon Plasma](http://demonstrations.wolfram.com/EvolutionOfMatterFromAQuarkGluonPlasma/)_, Wolfram Demonstration Project, 2011

* _The liquid universe -- quark-gluon plasma and the creation of matter_ ([pdf](http://rhig.physics.wayne.edu/REU/new_talks/nuclear.pdf))

* Sz. Borsanyi, Z. Fodor, K. H. Kampert, S. D. Katz, T. Kawanai, T. G. Kovacs, S. W. Mages, A. Pasztor, F. Pittler, J. Redondo, A. Ringwald, K. K. Szabo, _Lattice QCD for Cosmology_ ([arXiv:1606.07494](https://arxiv.org/abs/1606.07494))

### Description via geometric engineering of QCD on intersecting branes
 {#ReferencesViaAdSCFT}

Description via [[geometric engineering of QFTs|geometric engineering]] of [[QCD]] on [[intersecting branes]] ("[[holographic QCD]]" see also at [[string theory results applied elsewhere]]):

Expositions and reviews include

* Pavel Kovtun, _Quark-Gluon Plasma and String Theory_, RHIC news (2009) ([blog entry](http://www.bnl.gov/rhic/news/091107/story2.asp))

* Makoto Natsuume, _String theory and quark-gluon plasma_ ([arXiv:hep-ph/0701201](http://arxiv.org/abs/hep-ph/0701201))

* [[Steven Gubser]], _Using string theory to study the quark-gluon plasma: progress and perils_ ([arXiv:0907.4808](http://arxiv.org/abs/0907.4808))

* {#BiagazziCotrone12} Francesco Biagazzi, A. Cotrone, _Holography and the quark-gluon plasma_, AIP Conference Proceedings 1492, 307 (2012) ([doi:10.1063/1.4763537]( https://doi.org/10.1063/1.4763537), [slides pdf](http://cp3-origins.dk/content/movies/2013-01-14-bigazzi.pdf))

* {#Brambilla14} Brambilla et al., section 9.2.2 of _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981 ([doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Jorge Casalderrey-Solana, Hong Liu, [[David Mateos]], Krishna Rajagopal, Urs Achim Wiedemann, _Gauge/string duality, hot QCD and heavy ion collisions_, Cambridge University Press, 2014 ([arXiv:1101.0618](https://arxiv.org/abs/1101.0618))

and specifically via "[[improved holographic QCD]]"

* {#GKMMN10} Umut Gürsoy, [[Elias Kiritsis]], Liuba Mazzanti, Georgios Michalogiorgakis, Francesco Nitti, _Improved Holographic QCD_, Lect.Notes Phys.828:79-146,2011 ([arXiv:1006.5461](https://arxiv.org/abs/1006.5461))

Holographic discussion of the [[shear viscosity]] of the quark-gluon plasma goes back to

* {#PolicastroSonStarinets01} [[Giuseppe Policastro]], D.T. Son, A.O. Starinets, _Shear viscosity of strongly coupled $N=4$ supersymmetric Yang-Mills plasma_, Phys. Rev. Lett.87:081601, 2001 ([arXiv:hep-th/0104066](http://arxiv.org/abs/hep-th/0104066))

Other original articles include:

* Brett McInnes, _Holography of the Quark Matter Triple Point_ ([arXiv:0910.4456](http://arxiv.org/abs/0910.4456))

* Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, _Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma_ ([arXiv:1003.1138](http://arxiv.org/abs/1003.1138))

* Mansi Dhuria, Aalok Misra, _Towards MQGP_, JHEP 1311 (2013) 001 ([arXiv:1306.4339](https://arxiv.org/abs/1306.4339))

* Sara Heshmatian, Razieh Morad, _Jet suppression in non-conformal plasma using AdS/CFT_ ([arXiv:1812.09374](https://arxiv.org/abs/1812.09374))

* Irina Ya. Aref'eva, Kristina Rannu, Pavel Slepov, _Energy Loss in Holographic Anisotropic Model for Heavy Quarks in External Magnetic Field_ ([arXiv:2012.05758](https://arxiv.org/abs/2012.05758))

With emphasis on application to [[neutron stars]]:

* Carlos Hoyos, Niko Jokela, Matti Jarvinen, Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, _Transport in strongly coupled quark matter_ ([arXiv:2005.14205](https://arxiv.org/abs/2005.14205))

* Carlos Hoyos, Niko Jokela, Matti Järvinen, Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, *Holographic approach to transport in dense QCD matter* ([arXiv:2109.12122](https://arxiv.org/abs/2109.12122))

See also:

* Alfonso Ballon-Bayona, Luis A. H. Mamani, Alex S. Miranda, Vilson T. Zanchin, *Effective holographic models for QCD: Thermodynamics and viscosity coefficients* ([arXiv:2103.14188](https://arxiv.org/abs/2103.14188))

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[[!redirects quark gluon plasma]]
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quark-hadron duality

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## Idea

What came to be called _quark-hadron duality_ is a [[duality in physics|duality]] between [[scattering amplitudes]] of, on the one hand [[quarks]] ([[color charge]] in [[quantum chromodynamics]]) and, on the other hand, [[hadrons]] ([[flavor physics|flavor charge]] in [[quantum hadrodynamics]]), valid in a certain parameter range.

## References

The original article:

* E. C. Poggio, H. R. Quinn, and [[Steven Weinberg]], _Smearing method in the quark model_, Phys. Rev. D 13, 1958 (1976) ([doi:10.1103/PhysRevD.13.1958](https://doi.org/10.1103/PhysRevD.13.1958))

Review:

* [[Mikhail Shifman]], _Quark-Hadron Duality_, in: _At The Frontier of Particle Physics_, pp. 1447-1494 (2001) ([doi:10.1142/9789812810458_0032](https://doi.org/10.1142/9789812810458_0032))

* [[Mikhail Shifman]], _Lectures on quark-hadron duality_, Czech J Phys 52, B102–B135 (2002) ([doi:10.1007/s10582-002-0080-6](https://doi.org/10.1007/s10582-002-0080-6))

See also:

* [[Mannque Rho]], _Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter_ ([arXiv:2004.09082](https://arxiv.org/abs/2004.09082))

* Hiroyuki Umeeda, *Quark-hadron duality for heavy meson mixings in the 't Hooft model* ([arXiv:2106.06215](https://arxiv.org/abs/2106.06215))


quark-meson coupling model

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## Idea

In [[nuclear physics]], the _quark-meson coupling model_ is an [[effective field theory]] of [[nuclei]] where effective [[meson]]-[[field (physics)|fields]] are [[interaction|coupled]] to [[quark]] [[field (physics)|fields]].

This is on contrast to [[quantum hadrodynamics]] of the form of [[baryon chiral perturbation theory]], such as the [[Walecka model]], where the effective [[fermion]] fields are instead the [[baryons]] (i.e. the [[confinement|confined]] [[bound states]] of three [[constituent quarks]]) with no isolated [[quarks]] appearing explicitly.

## Related concepts

[[!include effective field theories of nuclear physics -- contents]]

## References

* D. Ebert, T. Feldmann, R. Friedrich, H. Reinhardt, _Effective Meson Lagrangian with Chiral and Heavy Quark Symmetries from Quark Flavor Dynamics_, Nucl.Phys. B434 (1995) 619-646 ([arXiv:hep-ph/9406220](https://arxiv.org/abs/hep-ph/9406220))

* P. A. M. Guichon, K. Saito, E. Rodionov, A. W. Thomas, _The role of nucleon structure in finite nuclei_, Nucl. Phys. A601:349-379, 1996 ([arXiv:nucl-th/9509034](https://arxiv.org/abs/nucl-th/9509034))

* K. Saito, K. Tsushima, A. W. Thomas, _Variation of hadron masses in finite nuclei_, Phys. Rev. C 55, 2637 (1997) ([doi:10.1103/PhysRevC.55.2637](https://doi.org/10.1103/PhysRevC.55.2637))

Relation to [[quantum hadrodynamics]]:

* Koichi Saito, _Relationship between Quark-Meson Coupling Model and Quantum Hadrodynamics_, Prog. Theor. Phys. 108 (2002) 609-614 ([arXiv:nucl-th/0207053](https://arxiv.org/abs/nucl-th/0207053))

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quarkonium

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## Idea

In [[quantum hadrodynamics]] a _quarkonium_ particle is a [[meson]] which is a [[bound state]] of one of the [[heavy flavor|heavy]] [[quarks]] with its own [[antiparticle]].

## Examples

Examples of quarkonia:

* [[charmonium]] ([[charm quark|charm]] [[quarkonium]], [[D-meson]])

* [[upsilon-meson]] ([[bottom quark|bottom]] [[quarkonium]], [[B-meson]])

## References

* R. Faccini, A. Pilloni, A.D. Polosa, _Exotic Heavy Quarkonium Spectroscopy: A Mini-review_ ([arXiv:1209.0107](https://arxiv.org/abs/1209.0107))

* {#Brambilla14} Brambilla et al., Section 4.3 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Zebo Tang, _Quarkonium production: An experimental overview_ ([arXiv:2002.10793](https://arxiv.org/abs/2002.10793))

* An-Ping Chen, Yan-Qing Ma, Hong Zhang, *A short theoretical review of charmonium production* ([arXiv:2109.04028](https://arxiv.org/abs/2109.04028))

See also

* Wikipedia, _[Quarkonium](https://en.m.wikipedia.org/wiki/Quarkonium)_

* Wikipedia, _[XYZ particle](https://en.m.wikipedia.org/wiki/XYZ_particle)_

Dicussion in the context of [[quark gluon plasma]]:

* Xiaojun Yao, Weiyao Ke, Yingru Xu, Ste, _Quarkonium Production in Heavy Ion Collisions: From Open Quantum System to Transport Equation_ ([arXiv:2002.04079](https://arxiv.org/abs/2002.04079))

The [[Dalitz decay]] of quarkonium:

* Yu Jia, Wen-Long Sang, _Observation prospects of leptonic and Dalitz decays of pseudoscalar quarkonia_, JHEP 0910:090, 2009 ([arXiv:0906.4782](https://arxiv.org/abs/0906.4782))

Discussion in [[AdS/QCD]]:

* Hovhannes R. Grigoryan, Paul M. Hohler, Mikhail A. Stephanov, _Towards the Gravity Dual of Quarkonium in the Strongly Coupled QCD Plasma_ ([arXiv:1003.1138](http://arxiv.org/abs/1003.1138))

* Rico Zöllner, Burkhard Kampfer, _Holographic vector meson melting in a thermal gravity-dilaton background related to QCD_ ([arXiv:2002.07200](https://arxiv.org/abs/2002.07200))

* Miguel Angel Martin Contreras, Saulo Diles, Alfredo Vega, _Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up AdS/QCD_ ([arXiv:2101.06212](https://arxiv.org/abs/2101.06212))

[[!redirects quarkonia]]


quasi-abelian category

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## Definition

A __quasi-abelian category__ is an [[additive category]] admitting
the kernels and the cokernels which satisfies the following conditions:

(i) the strict epimorphisms are stable by base changes,

(ii) the strict monomorphisms are stable by co-base changes

This is definition 2.1.1 in ([Kashiwara](#Kashiwara)).

## Examples

* The category of [[bornological vector spaces]] over the [[complex numbers]] is quasi-abelian ([Prosmans-Schneiders 00](#ProsmansSchneiders00))

* The category of [[bornological abelian groups]] is quasi-abelian ([Bambozzi 14](#Bambozzi14)).

## Related concepts

* [[abelian category]]

* [[pseudo-abelian category]]

* [[semi-abelian category]]

## References

* [[Jean-Pierre Schneiders]], _Quasi-abelian categories and sheaves_., Mémoires de la Société Mathématique de France **76** (1999), 144 pp. [[doi:10.24033/msmf.389](http://dx.doi.org/10.24033/msmf.389)]

* {#ProsmansSchneiders00} [[Fabienne Prosmans]], [[Jean-Pierre Schneiders]], _A homological study of bornological spaces_,  Prepublications Mathematiques de l'Universite Paris **13** 46 (2000) [[pdf](http://www.analg.ulg.ac.be/jps/rec/hsbs.pdf)]

* [[Wolfgang Rump]], _Almost abelian categories_, Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001) 163–225. [[numdam:CTGDC_2001__42_3_163_0](http://www.numdam.org/item?id=CTGDC_2001__42_3_163_0)]

* {#Bambozzi14} [[Federico Bambozzi]], section 1 of: _On a generalization of affinoid varieties_ [[arXiv:1401.5702](http://arxiv.org/abs/1401.5702)]

* {#Kashiwara} [[Masaki Kashiwara]], _Equivariant derived category and representation of real semisimple Lie groups_,  Lecture Notes in Mathematics **1931** (2008) 137–234  [[doi:10.1007/978-3-540-76892-0](https://doi.org/10.1007/978-3-540-76892-0_3),  [pdf](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/sd.pdf)]

* Rhiannon Savage, _Koszul monoids in quasi-abelian categories_ [[arXiv:2203.08789](https://arxiv.org/abs/2203.08789)]

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quasi-Borel space

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## Idea

A **quasi-Borel space** (QBS) is a [[set]] equipped with a notion of _[[random variable]]_, providing a model of [[measurable space]] suitable for [[probability theory]]. The advantage of quasi-Borel spaces over traditional formulations is that they provide a nice [[category]] of [[measurable spaces]]: it is [[cartesian closed category|cartesian closed]], and the set of [[probability measures]] of a QBS forms a QBS.

## Definition

A quasi-Borel space $X$ consists of an underlying [[set]] $|X|$ and a set of [[functions]] $M_X \subseteq (\mathbb{R} \to |X|)$ satisfying:

1. $M_X$ contains all [[constant functions]].
2. $M_X$ is closed under [[composition]] with [[measurable functions]]: if $f : \mathbb{R} \to \mathbb{R}$ is measurable and $\alpha \in M_X$, $\alpha \circ f \in M_X$.
3. $M_X$ is closed under gluing functions with [[disjoint union|disjoint]] [[Borel subset|Borel]] domains: for any partition $\mathbb{R} = \biguplus_{i\in\mathbb{N}} S_i$ by Borel $S_i$, and $\{\alpha_i \in M_X\}_{i\in\mathbb{N}}$, then the function $\beta(x) = \alpha_i(x)$ when $x \in S_i$ is in $M_X$.

A morphism of quasi-Borel spaces is a function that respects composition with these functions.

For example, $\mathbb{R}$ is a quasi-Borel space, with the Borel functions. The two-element set $2$ is a quasi-Borel space, with the functions that are characteristic functions of [[Borel subsets]].

The category of quasi-Borel spaces is [[cartesian closed]], unlike the category of [[measurable spaces]].

## Applications

The category of quasi-Borel spaces can be used as a [[denotational semantics]] for higher-order [[probabilistic programming languages]].

## As a quasitopos of concrete sheaves

The category of quasi-Borel spaces is the category of [[concrete sheaves]] on the category of standard Borel spaces considered with the [[extensive category#Superextensive sites|extensive coverage]].  As such, quasi-Borel spaces form a Grothendieck [[quasitopos]].
(A standard Borel space is a [[measurable space]] that is a retract of $\mathbb{R}$, equivalently, it is a measurable space that comes from a [[Polish space]], equivalently, it is either isomorphic to $\mathbb{R}$ or countable, discrete and non-empty.)

## Connection to measurable spaces

There is an [[adjunction]] between quasi-Borel spaces and [[measurable spaces]] (related to the [[nerve and realization]] construction).

$$
   Meas \stackrel{\leftarrow}{\rightarrow} Qbs,
$$

The right adjoint takes a measurable space $X$ to the quasi-Borel space where $M_X$ comprises the measurable functions. The left adjoint regards a quasi-Borel space $X$ with the sigma-algebra comprising all those sets $U\subseteq X$ for which the characteristic function $X\to 2$ is a morphism.

## Probability distributions

A _probability measure_ on a quasi-Borel space is defined to be a function $\mathbb{R}\to X$ in $M_X$. This is regarded as a [[random variable]]. In particular we regard two probability measures as equal if the [[law of a random variable|laws]] are the same [[probability distributions]] on the underlying measurable spaces, when $\mathbb{R}$ is regarded with the uniform distribution on $[0,1]$.

This leads to an affine, [[commutative monad]] on the category of quasi-Borel spaces. Restricted to standard Borel spaces, it agrees with the [[Giry monad]].

The monad can be regarded as a [[probability monad]], and  its [[Kleisli category]] is a [[Markov category]].

## Related Concepts

* [[quasi-topological space]]
* [[subsequential space]]
* [[concrete sheaf]]
* [[standard Borel space]]

## References

Quasi-Borel spaces were introduced in

* [[Chris Heunen]], Ohad Kammar, [[Sam Staton]] and Hongseok Yang, _A convenient category for higher-order probability theory_, Logic in Computer Science 2017 ([arXiv:1701.02547](https://arxiv.org/abs/1701.02547))

[[!redirects quasi-Borel spaces]]
[[!redirects quasi-Borel structure]]
[[!redirects quasi-Borel structures]]


quasi-C groupoid

+--{: .query}
[[David Roberts]]: Warning - this page probably constitutes original research and should used with caution. Help and comments are welcomed.
=--

A **quasi-$C$ groupoid** is a generalisation of a [[quasitopological groupoid]] to a category $C$ other than $Top$. We need conditions on $C$ so that this makes sense, like being concrete or a subcanonical site.


quasi-category

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _quasi-category_ is a [[geometric definition of higher categories|geometric model]] for [[(∞,1)-category]].

In analogy to how a [[Kan complex]] is a model in terms of [[simplicial set]]s of an [[∞-groupoid]] -- also called an [[(∞,0)-category]] --  a [[quasi-category]] is a model in terms of [[simplicial set]]s of an [[(∞,1)-category]].

+-- {: .un_remark}
###### Warning

In older literature, such as [[The Joy of Cats]], the term "quasicategory" was sometimes used for a "very large" category whose objects are [[large categories]] or otherwise built out of [[proper classes]], but nowadays this usage is fairly archaic.  See also [[metacategory]].

=--

## Definition

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

A **quasi-category** or **[[weak Kan complex]]** is a [[simplicial set]] $C$ satisfying the following equivalent conditions

* all _inner_ [[horn]]s in $C$ have fillers. This means that the lifting condition given at [[Kan complex]] is imposed only for horns $\Lambda^i[n]$ with $0 \lt i \lt n$.

* the morphism of simplicial sets

  $$
    sSet(\Delta[2],C) \to sSet(\Lambda^1[2],C)
  $$

  (induced from the inner [[horn]] inclusion $\Lambda^1[2] \to \Delta[2]$)
  is an acyclic [[Kan fibration]].

=--

The equivalence of these two definitions is due to [[Andre Joyal]] and recalled as [[Higher Topos Theory|HTT, corollary 2.3.2.2]].  Quasi-categories are the [[fibrant objects]] in the [[model structure for quasi-categories]].

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

The second condition says manifestly that a quasi-category is a simplicial set in which composition of any two composable edges is defined up to a contractible space of choices. This is the [[coherence law]] on composition.

=--

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

An **[[algebraic quasi-category]]** is a quasi-category equipped with a _choice_ of inner horn fillers.

=--

While quasi-categories provide a [[geometric definition of higher categories]], algebraic quasi-categories provide an [[algebraic definition of higher categories]]. For more details on this see [[model structure on algebraic fibrant objects]].

## Properties

### Relation to simplicially enriched categories

The [[homotopy coherent nerve]] relates quasi-categories with another model for $(\infty,1)$-categories: [[simplicially enriched categories]].

See [[relation between quasi-categories and simplicial categories]] for more.

### Higher associahedra in quasi-categories

While the geometric definition of [[(∞,1)-category]] in terms of quasi-categories elegantly captures all the higher categorical data automatically, it may be of interest in applications to explicitly extract the associators and higher associators encoded by this structure, that would show up in any [[algebraic definition of higher categories|algebraic definition of the same categorical structure]], such as [[algebraic quasi-categories]].

For a discussion of this see

* [[Emily Riehl]], _Associativity data in an $(\infty,1)$-category_ ([pdf](http://www.math.jhu.edu/~eriehl/associativity.pdf) [blog](http://golem.ph.utexas.edu/category/2009/10/associativity_data_in_an_1cate.html))

## Examples

The two basic examples for quasi-categories are

* Every [[Kan complex]] is, in particular, a quasi-category.

* The [[nerve]] of a [[category]] is a quasi-category.

Since the nerve of a category is a [[Kan complex]] iff the category is a [[groupoid]], quasi-categories are a minimal common generalization of Kan complexes and nerves of categories.

By the [[homotopy hypothesis]]-theorem every Kan complex arises, up to equivalence, as the [[fundamental ∞-groupoid]] of a [[topological space]].

Analogously, every [[directed topological space]] $X$ has naturally a [[fundamental (∞,1)-category]] given by a quasi-category whose $k$-cells are maps $\Delta^k_{Top} \to X$ that map the 1-[[simplicial skeleton|skeleton]] of the topological simplex in an order-preserving way to directed paths in $X$.

The [[directed homotopy theory]] that would state that this or a similar construction exhausts all quasicategories up to equivalence, does not quite exist yet.

## Constructions in quasi-categories

The point of quasi-categories is that they are supposed to provide a fully [[homotopy theory|homotopy-theoretic]] refinement of the ordinary notion of [[category]]. In particular, all the familiar constructions of [[category theory]] have natural analogs in the context of quasi-categories. See for instance

* [[hom-object in a quasi-category]]

* [[equivalence in a quasi-category]]

* [[equivalence of quasi-categories]]

* [[join of quasi-categories]]

* [[over quasi-category]]

* [[terminal object in a quasi-category]]

* [[monomorphism in an (∞,1)-category]]

* [[limit in quasi-categories]]

* [[sub-quasi-category]]

* [[opposite quasi-category]]

* [[fibrations of quasi-categories]]

  * [[inner Kan fibration]]

  * [[Cartesian fibration]]

  * [[left Kan fibration]]/[[right Kan fibration]]

  * [[minimal Joyal fibration]]

## Related concepts
 {#RelatedConcepts}

* [[relation between quasi-categories and sSet-enriched categories]]

  [[homotopy coherent nerve]]

  [[rigidification of quasi-categories]]

* One may try to further weaken the filler conditions in order to describe [[(∞,n)-categories]] for $n \gt 1$. One approach along these lines is the theory of *[[weak complicial sets]]*.

* Or one may change the shape category to pass from [[simplicial sets]] to [[cellular sets]]. A quasi-category-like definition of [[(∞,n)-categories]] on these -- _[[n-quasicategories]]_ -- is discussed at *[[model structure on cellular sets]]*.

## References

The notion of quasi-categories were originally defined, under the name *[[weak Kan complexes]]* in:

* [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics **347** Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]

* {#Vogt73} [[Rainer Vogt]], _Homotopy limits and colimits_, Math. Z., **134** (1973) 11-52 [[doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090), [eudml:171965](https://eudml.org/doc/171965)]

The main theorem of [Vogt (1973)](#Vogt73) involved a category of [[homotopy coherent diagrams]] defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.

* [[J.-M. Cordier]], _Sur la notion de diagramme homotopiquement cohérent_, Cahiers de Top. Géom. Diff., 23, (1982), 93 –112,

defined the [[homotopy coherent nerve]] of any [[simplicially enriched category]]. This generalised the [[nerve]] of an ordinary category. In

* [[J.-M. Cordier]] and [[Tim Porter]], _Vogt's theorem on categories of homotopy coherent diagrams_, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65–90,

it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.

A systematic study of SSet-enriched categories in this context is in

* [[J.-M. Cordier]], [[Tim Porter]] _Homotopy coherent category theory_ Trans. Amer. Math. Soc. 349 (1997), no. 1, 1-54. ([pdf](http://www.ams.org/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf))

The importance of quasi-categories as a basis for [[category theory]] has been particularly emphasized in  work by [[André Joyal]]

* [[André Joyal]], *Quasi-categories and Kan complexes*, J. Pure Appl. Algebra **175** (2002) 207-222 [doi:10.1016/S0022-4049(02)00135-4]

For several years Joyal has been preparing a textbook on the subject which never became publically available, but an extensive writeup of lecture notes is:

* {#Joyal08} [[André Joyal]], *[[The Theory of Quasi-Categories and its Applications]]*, lectures at *[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)*, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]

* [[André Joyal]], *Notes on quasi-categories* (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf), [[JoyalNotesOnQuasiCategories.pdf:file]]]

Meanwhile [[Jacob Lurie]], building on Joyal's work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models [[quasi-category]] and [[simplicially enriched category]] is in

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

An overview of the material there is contained in

* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 21.3.2015 [[pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf)]

Textbook accounts:

* [[Emily Riehl]], Part IV 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)]

* [[Denis-Charles Cisinski]], _[[Higher Categories and Homotopical Algebra]]_,  Cambridge University Press (2019) [[doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf)]

The relation between [[quasi-category|quasi-categories]] and [[simplicially enriched categories]] was discussed in detail in

* [[Dan Dugger]], [[David Spivak]], _Rigidification of quasi-categories_ ([arXiv:0910.0814](http://arxiv.org/abs/0910.0814))

* [[Dan Dugger]], [[David Spivak]], _Mapping spaces in quasi-categories_ ([arXiv:0911.0469](http://arxiv.org/abs/0911.0469))

Further survey:

* {#Rezk16} [[Charles Rezk]], _Stuff about quasicategories_, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 ([pdf](http://math.uiuc.edu/~rezk/595-fal16/quasicats.pdf), [[RezkQuasicategories.pdf:file]])

* [[Charles Rezk]], *Introduction to quasicategories* (2022) [[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf), [[Rezk-IntroToQuasicategories.pdf:file]]]

* [[Moritz Groth]], _A short course on ∞-categories_ ([arXiv:1007.2925](https://arxiv.org/abs/1007.2925))

An in-depth study of adjunctions between quasi-categories and the monadicity theorem is given in

* [[Emily Riehl]], [[Dominic Verity]] _The 2-category theory of quasi-categories_ ([arXiv](http://arxiv.org/abs/1306.5144)), _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv](http://arxiv.org/abs/1310.8279))

[[!redirects quasi-categories]]
[[!redirects quasicategory]]
[[!redirects quasicategories]]
[[!redirects quasicategorical]]

[[!redirects inner Kan complex]]
[[!redirects inner Kan complexes]]


quasi-category theory contents

**[[(∞,1)-category theory]]**

**Background**

* [[category theory]]

* [[higher category theory]]

* [[(n,r)-category]]

**Basic concepts**

* [[(∞,1)-category]]

* [[hom-object in a quasi-category|hom-objects]]

* [[equivalence in a quasi-category|equivalences in]]/[[equivalence of quasi-categories|of]] $(\infty,1)$-categories

* [[sub-quasi-category|sub-(∞,1)-category]]

  * [[reflective sub-(∞,1)-category]]

  * [[localization of an (∞,1)-category|reflective localization]]

* [[opposite quasi-category|opposite (∞,1)-category]]

* [[over quasi-category|over (∞,1)-category]]

  * [[join of quasi-categories]]

* [[(∞,1)-functor]]

  * [[exact (∞,1)-functor]]

  * [[(∞,1)-category of (∞,1)-functors]]

  * [[(∞,1)-category of (∞,1)-presheaves]]

* **[[fibrations of quasi-categories|fibrations]]**

  * [[inner fibration]]

  * [[left fibration|left/right fibration]]

  * [[Cartesian fibration]]

    * [[Cartesian morphism]]

**Universal constructions**

* [[limit in quasi-categories|limit]]

  * [[terminal object in a quasi-category|terminal object]]

* [[adjoint (∞,1)-functor|adjoint functors]]

**Local presentation**

* [[locally presentable (∞,1)-category|locally presentable]]

  * [[essentially small (∞,1)-category|essentially small]]

  * [[locally small (∞,1)-category|locally small]]

  * [[accessible (∞,1)-category|accessible]]

  * [[idempotent-complete (∞,1)-category|idempotent-complete]]

**Theorems**

* [[(∞,1)-Yoneda lemma]]

* [[(∞,1)-Grothendieck construction]]

* [[adjoint (∞,1)-functor theorem]]

* [[monadicity theorem|(∞,1)-monadicity theorem]]

**Extra stuff, structure, properties**

* [[stable (∞,1)-category]]

* [[(∞,1)-topos]]

**Models**

* [[category with weak equivalences]]

* [[model category]]

* [[derivator]]

* [[quasi-category]]

  * [[model structure for quasi-categories]]

  * [[model structure for Cartesian fibrations]]

* [[relation between quasi-categories and simplicial categories|relation to simplicial categories]]

  * [[homotopy coherent nerve]]

  * [[simplicial model category]]

  * [[presentable (∞,1)-category|presentable quasi-category]]

* [[Kan complex]]

  * [[model structure on simplicial sets|model structure for Kan complexes]]

[[!redirects (infinity,1)-category theory - contents]]


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

This page provides links related to

* [[Jacob Lurie]], _Quasi-Coherent Sheaves and Tannaka Duality Theorems
_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VIII.pdf))

On [[quasi-coherent sheaves]] and [[Tannaka duality for geometric stacks]] in the context of [[derived algebraic geometry]] over [[E-infinity rings]] -- [[E-∞ geometry]]

#Contents#
* table of contents
{:toc}

## 1. Generalities on spectral Deligne-Mumford stacks

### 1.4 Quasi-compactness of Spectral Deligne-Mumford stacks

+-- {: .num_example }
###### Example
**cor. 1.4.3**

A [[spectral scheme]] or [[spectral Deligne-Mumford stack]], regarded as a [[structured (∞,1)-topos]] is locally coherent.

=--

## 2. Quasi-coherent sheaves

### 2.1 Sheaves on a spectrally ringed $\infty$-topos

### 2.2 Module geometries

* [[tangent (∞,1)-category]]

* [[geometry (for structured (∞,1)-toposes)]]

### 2.3 Quasi-coherent sheaves

* [[quasi-coherent sheaf]]

### 2.4 Quasi-affine spectral Deligne-Mumford stacks

### 2.5 Pullbacks and pushforwards of quasi-coherent sheaves

* [[scallop decomposition]]

* [[Grothendieck context]] [[six operations]]

### 2.6 Local properties of quasi-coherent sheaves

* [[coherent sheaf]]

### 2.7 Quasi-coherent sheaves on a functor

## 3. Geometric stacks and Tannaka duality

## 4. Coaffine stacks

## 5. Tannaka duality for generalized algebraic gerbes

category: reference


quasi-coproduct

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

#Contents#
* table of contents
{:toc}

## Idea

Noticing hat

1. every [[set]] is the [[coproduct]] over its [[singleton]] elements,

1. every [[skeletal groupoid]] is the [[coproduct]] over its [[connected component]] [[delooping groupoids]], hence is a "set of points with [[automorphism groups]]"

1. a [[coproduct]] is a [[colimit]] over a [[diagram]] of the shape of a [[discrete category]] (a [[set]]) [[Set]] $\hookrightarrow$ [[Cat]]

it makes sense, in [[homotopy theory|homotopy theoretic]] generalization of coproducts, to consider *quasi-coproducts* [[Hu & Tholen (1996)](#HuTholen96)] to be colimits over diagrams of the shape of [[skeletal groupoids]] $Grpd_{skl} \hookrightarrow Cat$ which are degreewise [[free actions]], in a suitable sense, such that these colimits tend to behave like [[homotopy colimits]].

## Definition

### Quasi-coproducts

\begin{definition}\label{QuasiCoproduct}
  A *quasi-coproduct* in a [[category]] $\mathcal{C}$ is a [[colimit]] over a [[diagram]]
$$
  \mathscr{V}_{(-)}
    \,\colon\,
  \mathcal{G} \longrightarrow \mathcal{C}
$$
of the shape of a [[skeletal groupoid]] $\mathcal{G} \,\in\, Grpd_{skl} \hookrightarrow$ such that the following condition is satisfied

* for all [[connected components]] $\mathbf{B}G_i \hookrightarrow \mathcal{G}$ and all non-[[initial objects]] $\mathscr{W} \,\in\, \mathcal{C}$ the [[group action]] in [[Set]] given by the composite with the [[hom-functor]] $\mathcal{C}(\mathscr{W},-)$
$$
 \mathbf{B}G_i
 \hookrightarrow
 \mathcal{G}
  \overset{
    \mathscr{V}_{(-)}
  }{\longrightarrow}
 \mathcal{C}
  \overset{\mathcal{C}(\mathscr{W},-)}{\longrightarrow}
 Set
$$
is a [[free action]].

\end{definition}
[[Hu & Tholen (1996), §1.3](#HuTholen96)]

### Homotopy quasi-coproducts

The following Def. \ref{CategoryWithHomotopicalQuasiCoproducts} is a slight variant/specialization of Def. \ref{QuasiCoproduct}, meant to adapt the notion of quasi-coproducts properly to the context of [[homotopy theory]].

(One may consider other variants, for instance with respect to [[homotopical categories]], and in particular one could consider more general variants, such as with [[groupoids]] generalized to [[sSet]]-[[enriched groupoids]] aka "[[simplicial groupoids]]" regarded as models for [[infinity-groupoids|$\infty$-groupoids]]. Maybe later.)

Here, in the vein of speaking obout [[category theory|category theoretic]] models for [[homotopy types]], throughout by *[[groupoids]]* we mean *[[small category|small]] [[strict category|strict]]* groupoids (as usual), and we regard the collection [[Grpd]] as a [[1-category]] (with [[functors]] as the morphisms).

\begin{definition}
\label{DeloopingGroupoids}
**(conventions for delooping groupoids)**
\linebreak
For $(G, \mu, \mathrm{e})$ a [[group]], we write

* $\mathbf{B}G \,\equiv\, (G \rightrightarrows pt)$

  for its [[delooping groupoid]] with [[composition]] given  by reverse multiplication in $G$:

  $$
    \array{
      &&
      pt
      \\
      &
      \mathllap{{}^{g_{12}}}\nearrow
      &&
      \searrow\mathrlap{{}^{g_{23}}}
      \\
      pt
      &&
      \underset{\mu(g_{23},\,g_{12})}{\longrightarrow}
      &&
      pt
    }
  $$

  so that (ordinary, left) $G$-[[group action|actions]] ([[group representations]])

  $$
    \rho
       \,\colon\,
    G
      \longrightarrow
    Hom_{\mathcal{C}}(\mathscr{V}, \mathscr{V})
  $$

  are identified with [[functors]] of the form

  \[
    \label{GroupRepresentationsAsFunctors}
    \array{
      \mathllap{
        \rho
          \,\colon\,
      }
      \mathbf{B}G &\longrightarrow& \mathcal{C}
      \\
      pt &\mapsto& \mathscr{V}
      \\
      \Big\downarrow\mathrlap{{}^{g}}
        &&
      \Big\downarrow\mathrlap{{}^{ \rho(g) }}
      \\
      pt &\mapsto& \mathscr{V}
    }
  \]

* $\mathbf{E}G \,\equiv\, (G \times G \rightrightarrows G)$

  for the [[action groupoid]] of $G$ [[action|acting]] on itself by left multiplication, so that we have the evident [[forgetful functor]]

  $$
    \array{
      \mathbf{E}G &\longrightarrow& \mathbf{B}G
      \\
      g &\mapsto& pt
      \\
      \Big\downarrow\mathrlap{{}^{ g_{12} }}
        &&
      \Big\downarrow\mathrlap{{}^{ g_{12} }}
      \\
      \mu(g_{12},g)
        &\mapsto&
      pt
    }
  $$

and the remaining *right inverse multiplication* action of $G$ on $\mathbf{E}G$ defines a [[group action]]
$$
  \array{
    \mathbf{B}G
    &\longrightarrow&
    Grpd
    \\
    pt &\mapsto& \mathbf{E}G
    \\
    \Big\downarrow\mathrlap{{}^{ g }}
      &&
    \Big\downarrow\mathrlap{{}^{ \mu(\text{-},g^{-1}) }}
    \\
    pt &\mapsto& \mathbf{E}G
  }
$$
whose [[quotient object]] is
$$
  \mathbf{B}G
  \;\simeq\;
  \big(\mathbf{E}G\big)/G
  \,.
$$
(See also at *[[universal principal bundle]]* and at *[[simplicial classifying space]]* and at *[[Borel construction]]*.)
\end{definition}

\begin{definition}
\label{CategoryWithHomotopicalQuasiCoproducts}
A category with *homotopy quasi-coproducts* is a [[category]] $\mathcal{C}$

* equipped with a [[tensoring]] over the [[1-category]] [[Grpd]]

  \[
    \label{GrpdTensoring}
    Grpd \times \mathcal{C}
      \xrightarrow{(-)\cdot(-)}
    \mathcal{C}
  \]

* which has all quasi-coproducts (Def. \ref{QuasiCoproduct}) of the form

  \[
    \label{HomotopicalQuasiCoproducts}
    \array{
      \mathcal{G} &\longrightarrow& \mathcal{C}
      \\
      \mathllap{{}^{=}}\Big\uparrow
        &&
      \Big\uparrow\mathrlap{
        (-)\cdot(-)
      }
      \\
      \underset{i}{\coprod}
      \mathbf{B}G_i
      &\underset{
        \big(
          \mathbf{E}G_i  ,\, \mathscr{V}_{(-)}
        \big)_{i\in I}
      }{\longrightarrow}&
      Grpd \times \mathcal{C}
    }
  \]

\end{definition}

\begin{definition}
\label{FreeHomotopyQuasiCoproductCompletion}
**(free homotopy quasi-coproduct completion)**
\linebreak
  For $\mathcal{C}$ a category, its *free homotopy quasi-coproduct completion* is

* a category $QC(\mathcal{C})$ with homotopy quasi-products (Def. \ref{CategoryWithHomotopicalQuasiCoproducts})

equipped with

* a [[full subcategory]] inclusion $\mathcal{C} \hookrightarrow QC(\mathcal{C})$,

such that

* [[functors]] $QC(\mathcal{C}) \longrightarrow \mathcal{D}$ to another category $\mathcal{D}$ with homotopy quasi-coproducts which [[preserved colimit|preserve]]

  1. the $Grpd$-tensoring (eq:GrpdTensoring)

  1. the quasi-coproducts (eq:HomotopicalQuasiCoproducts)

  are fixed by the restriction of the [[underlying]] functor along $\mathcal{C} \hookrightarrow QC(\mathcal{C})$.

\end{definition}

## Properties

### Free quasi-coproduct-completion

In generalization of how

* the [[free coproduct completion]] of a category $\mathcal{C}$ is the [[Grothendieck construction]]

  $$
    Fam_{\mathcal{C}}
    \;\;
      \coloneqq
    \;\;
    \underset
      {X \in Set}
      {\textstyle{\int}}
    \mathcal{C}^X
  $$

  (see [there](free+coproduct+completion#AsAGrothendieckConstruction))

it ought to be the case that

* the free homotopy quasi-coproduct completion of $\mathcal{C}$ is the [[Grothendieck construction]]

  $$
    Loc_{\mathcal{C}}
    \;\;
      \coloneqq
    \;\;
    \underset{\mathcal{X} \in Grpd}{\textstyle{\int}}
    \mathcal{C}^{\mathcal{X}}
  $$

This we make precise now.

\begin{definition}
\label{CategoryOfLocalSystems}
 For $\mathcal{C}$ a [[category]], we write
\[
  \label{Loc}
  Loc_{\mathcal{C}}
  \;\;
  \coloneqq
  \;\;
  \underset{\mathcal{X} \in Grpd}{\textstyle{\int}}
  \mathcal{C}^{\mathcal{X}}
\]
for the [[Grothendieck construction]] on the [[pseudofunctor]] which sends [[groupoids]] $\mathcal{X} \in $ [[Grpd]] to the [[functor category]] $\mathcal{C}^{\mathcal{X}} \,\coloneqq\,Func(\mathcal{X}, \mathcal{C})$ and [[functors]] $f \,\colon\,\mathcal{X} \to \mathcal{X}'$ to the [[precomposition]] operation $f^\ast \,\coloneqq\, (-) \circ f$:
\[
  \label{PseudofunctorOfFunctorCategories}
  \array{
    Grpd &\longrightarrow& Cat
    \\
    \mathcal{X} &\mapsto& \mathcal{C}^{\mathcal{X}}
    \\
    \Big\downarrow\mathrlap{{}^{f}}
    &&
    \Big\uparrow\mathrlap{{}^{f^\ast}}
    \\
    \mathcal{X}' &\mapsto& \mathcal{C}^{\mathcal{X}'}
  }
\]

There is a canonical [[full subcategory]]-empbedding
\[
  \label{ConstantLocalSystem}
  \array{
    \mathcal{C}
      &\longrightarrow&
    Loc_{\mathcal{C}}
    \\
    \mathscr{V}
      &\mapsto&
    \mathscr{V}_{pt}
  }
\]
by regarding objects of $\mathcal{C}$ as (necessarily [[constant functors|constant]]) [[functors]] on the [[terminal groupoid]] $pt$.
\end{definition}

\begin{example}
For $\mathbb{K}$ a [[field]] and $\mathcal{C} \,\equiv\, $ [[Mod|$Mod_{\mathbb{K}}$]] $\equiv$ [[Vect|$Vect_{\mathbb{K}}$]] its category of [[vector spaces]], a functor $\mathcal{X} \longrightarrow Mod_{\mathbb{C}}$ --- for $\mathcal{X}$ thought of as the [[fundamental groupoid]] of some [[topological space]] $X$ ---, is (equivalently a [[flat vector bundle]] on $X$ but) also known as a *[[local system]]* $X$. Therefore in this case the category (eq:Loc)
$$
  Loc_{\mathbb{K}}
  \;\coloneqq\;
  Loc_{Mod_{\mathbb{K}}}
$$
may be thought of as the category of "local systems over varying base spaces", whence the notation in Def. \ref{CategoryOfLocalSystems}.
\end{example}

\begin{remark}
\label{BifibrantCategoryOfLocalSystems}
If $\mathscr{C}$ has all [[small diagram|small]] [[colimits]],  then the [[Grothendieck construction]] of Def. \ref{CategoryOfLocalSystems}
$$
  \array{
    Loc_{\mathcal{C}}
    \\
    \Big\downarrow
    \\
    Grpd
  }
$$
is actually a [[bifibration]], induced from the [[CatAdj|$Cat_{adj}$]]-valued [[pseudofunctor]]
\[
  \label{CatAdjValuedPseudofunctorOfFunctorCategories}
  \array{
    Grpd_{skl} &\longrightarrow& Cat_{adj}
    \\
    \mathcal{X} &\mapsto& \mathcal{C}^{\mathcal{X}}
    \\
    \Big\downarrow\mathrlap{{}^{f}}
    &&
    \mathllap{{}^{f_!}}\Big\downarrow
    {}^{\dashv}
    \Big\uparrow\mathrlap{{}^{f^\ast}}
    \\
    \mathcal{X}' &\mapsto& \mathcal{C}^{\mathcal{X}'}
  }
\]
whose [[left adjoint]]-components $f_!$ are given by [[left Kan extension]].

In this case, all [[colimits]] over [[diagrams]] $\mathscr{V}_{\mathcal{X}} \,\colon\, \mathcal{I} \longrightarrow Loc_{\mathcal{C}}$ in the [[Grothendieck construction]] $Loc_{\mathcal{C}}$ (eq:Loc) exist and (by the discussion [there](Grothendieck+construction#CoLimitsInAGrothendieckConstruction)) are given on [[underlying]] groupoids as the corresponding underlying colimit $\underset{\longrightarrow}{lim} \mathcal{X}$ in [[Grpd]] and on components in $\mathcal{C}$ the colimit in $\mathcal{C}^{ \underset{\longrightarrow}{lim} \mathcal{X}}$ of the diagram of pushforwards $q_!$ (eq:CatAdjValuedPseudofunctorOfFunctorCategories) of morphisms along the underlying [[coprojections]] $\mathcal{X}_i \xrightarrow{q_i} \underset{\longrightarrow}{lim} \mathcal{X}$:
$$
  \underset{\longrightarrow}{\lim}
  \big(\mathscr{V}_{\mathcal{X}}\big)
  \;\;
  \simeq
  \;\;
  \big(
    \underset{\longrightarrow}{lim}
    q_!\mathscr{V}
  \big)_{ \underset{\longrightarrow}{lim} \mathcal{X} }
  \;\;\;\;
  \in
  \;
  Loc_{\mathcal{C}}
  \,.
$$
\end{remark}

\begin{remark}
\label{TensoringOfLocalSystemsOverGroupoids}
**(tensoring of local systems over groupoids)**
\linebreak
If $\mathcal{C}$ carries the [[structure]] of a [[symmetric monoidal category]] such that the pullback functors $f^\ast$ (eq:PseudofunctorOfFunctorCategories) are [[strong monoidal functors]], then $Loc_{\mathcal{C}}$ (eq:Loc) inherits the corresponding [[external tensor product]], which we denote

$$
  Loc_{\mathcal{C}}
  \times
  Loc_{\mathcal{C}}
  \xrightarrow{\;\; \boxtimes \;\;}
  Loc_{\mathcal{C}}
  \,.
$$

Notice that for any $\mathcal{X} \,\in\, Grpd$ the external tensor product with $\mathbb{1}_{\mathcal{X}} \,\coloneqq\, (p_{\mathcal{X}})^\ast \mathbb{1}_{pt}$ (i.e. with the [[tensor unit]] over $\mathcal{X}$, which is the [[constant functor]] $\mathcal{X} \to \ast \xrightarrow{\mathbb{1}} \mathcal{C}$) is equivalently the [[base change]] operation along the [[projection]] out of the [[product groupoid]] $pr_{\mathcal{Y}} \,\colon\, \mathcal{X} \times \mathcal{Y} \longrightarrow \mathcal{Y}$

\[
  \label{GrpdTensoringAsExternalTensorProduct}
  \mathbb{1}_{\mathcal{X}}
  \,\boxtimes\,
  \mathscr{V}_{\mathcal{Y}}
  \;\;\;
    \simeq
  \;\;\;
  \big(
    (pr_{\mathcal{Y}})^\ast \mathscr{V}
  \big)_{\mathcal{X} \times \mathcal{Y}}
  \,.
\]
Since here the expression on the right hand side does not actually invoke the [[monoidal category|monoidal structure]] of $\mathcal{C}$ it is expedient to allow ourselves to use the notation on the left even when $\mathcal{C}$ is not equipped with monoidal structure.

This is the canonical [[tensoring]] of $Loc_{\mathcal{C}}$ over [[Grpd]]:
\[
  \label{CanonicalTensoringOfLocOverGrpd}
  \array{
    Grpd \times Loc_{\mathcal{C}}
    &\overset{(-)\cdot(-)}{\longrightarrow}&
    Loc_{\mathcal{C}}
    \\
    \big(
      \mathcal{X}
      ,\,
      \mathscr{V}_{\mathcal{Y}}
    \big)
    &\mapsto&
    \big(
      (pr_{\mathcal{Y}})^\ast
      \mathscr{V}
    \big)_{
      \mathcal{X} \times \mathcal{Y}
    }
  }
\]

\end{remark}

\begin{lemma}
\label{AllLocalSystemsAreGrpdTensoringsOfCoproductsOfSkeletalLocalSystems}
  Given an object $\mathscr{V}_{\mathcal{X}} \,\in\, Loc_{\mathbb{C}}$ (eq:Loc) it is [[isomorphism|isomorphic]] to a [[coproduct]] of [[tensorings]] with [[codiscrete groupoids]] of objects over [[delooping groupoids]]
$$
  \mathscr{V}_{\mathcal{X}}
  \;\;
  \simeq
  \;\;
  \underset{i \in \pi_0(\mathcal{C})}{\coprod}
    \Big(
      CoDisc\big(
        Obj(\mathcal{X}_i)
      \big)
      \cdot
      \big(
       \iota_{\mathbf{B}G_i}^\ast
       \mathscr{V}
    \big)_{\mathbf{B}G_i}
  \Big)
  \,.
$$
and hence to the [[domain]] of a morphism obtained as a [[coproduct]] of [[tensorings]] (eq:CanonicalTensoringOfLocOverGrpd) of local system over a [[skeletal groupoid]] with the terminal maps out of [[codiscrete groupoids]]:
$$
  \mathscr{V}_{\mathcal{X}}
  \,\in\,
  Loc_{\mathcal{C}}
  \;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;
  \mathscr{V}_{\mathcal{X}}
  \simeq
  \bigg(
    \underset{i \in \mathcal{X}_i}{\amalg}
    \Big(
      CoDisc\big(\mathrm{Obj}(\mathcal{X}_i)\big)
      \cdot
      \big(
        \iota_{\mathbf{B}G_i}^\ast
        \mathscr{V}
      \big)_{ \mathbf{B}G_i }
    \Big)
  \Bigg)
  \overset{
    \underset{i \in I}{\amalg}
    \Big(
      p_i \cdot
      \big(
        \iota_{\mathbf{B}G_i}^\ast
        \mathscr{V}
      \big)_{ \mathbf{B}G_i }
    \Big)
  }{\longrightarrow}
  \Big(
    \underset{i \in \mathcal{X}_i}{\amalg}
    \big(
      \iota_{\mathbf{B}G_i}^\ast
      \mathscr{V}
    \big)_{ \mathbf{B}G_i }
  \Big)
$$
\end{lemma}
\begin{proof}
  On [[underlying]] groupoids the isomorphism is induced by a choice of [[adjoint equivalence|adjoint]] [[deformation retraction]] onto a [[skeletal groupoid]] (spelled out [here](Introduction+to+Topology+--+2#EveryGroupoidIsomorphicToQuasiSkeletalGroupoid)).

It remains to observe that every local system on $CoDisc\big(Obj(\mathcal{X}_i)\big) \times \mathbf{B}G_i$ is isomorphic to one pulled back from $\mathbf{B}G_i$, i.e. to one which constant along $CoDisc\big(Obj(\mathcal{X}_i)\big)$. For this choose any $x_i \in Obj(\mathcal{X}_i)$ and consider the evident [[natural isomorphism]]

$$
  \array{
    CoDisc\big(Obj(\mathcal{X}_i)\big)
    &\overset{}{\longrightarrow}&
    \mathcal{C}
    \\
    x
    &\mapsto&
    \mathscr{V}_x
    &\overset{\mathscr{V}_{(x,x_i)}}{\longrightarrow}&
    \mathscr{V}_{x_i}
    \\
    \Big\downarrow
    &&
    \Big\downarrow\mathrlap{{}^{ \mathscr{V}_{(x,x')} }}
    &&
    \Big\downarrow\mathrlap{{}^{id}}
    \\
    x'
    &\mapsto&
    \mathscr{V}_{x'}
    &\underset{\mathscr{V}_{(x',x_i)}}{\longrightarrow}&
    \mathscr{V}_{x_i}
    \mathrlap{\,.}
  }
$$
\end{proof}

\begin{lemma}
\label{GroupRepresentationsAreQuasiCoproductsOfConstantLocalSystems}
**(group representations are quasi-coproducts of constant local systems)**
\linebreak
If $\mathcal{C}$ has all [[colimits]]
them for $G \in Grp$, any $G$-[[group representation|representation]] $\mathscr{V}_{\mathbf{B}G} \,\colon\, \mathbf{B}G \overset{\rho}{\longrightarrow} \mathcal{C}$ (eq:GroupRepresentationsAsFunctors), regarded as an object of $Loc_{\mathcal{C}}$ (eq:Loc), is the [[colimit]] over the [[diagram]]
$$
  \array{
    \mathbf{B}G
    &\overset{\;}{\longrightarrow}&
    Loc_{\mathcal{C}}
    \\
    pt
      &\mapsto&
    \mathbf{E}G
      \cdot
    \mathscr{V}_{pt}
    \\
    \Big\downarrow\mathrlap{{}^g}
      &&
    \Big\downarrow\mathrlap{{}^{
      \mu(\text{-},g^{-1})
      \,\cdot\,
      {\rho(g)}_{pt}
    }}
    \\
    pt
    &\mapsto&
    \mathbf{E}G
      \cdot
    \mathscr{V}_{pt}
  }
$$
\end{lemma}
\begin{proof}
We need to exhibit a colimiting cocone of the following form, where we find it helpful to now use the $\boxtimes$-notation (eq:GrpdTensoringAsExternalTensorProduct) for the $Grpd$-tensoring:
$$
  \array{
    \mathbb{1}_{\mathbf{E}G}
    \boxtimes
    \mathscr{V}_{pt}
    &&\overset{
      \mathbb{1}_{\mu(\text{-},g^{-1})} \,\boxtimes\, \rho(g)_{pt}
    }{\longrightarrow}&&
    \mathbb{1}_{\mathbf{E}G}
    \boxtimes
    \mathscr{V}_{pt}
    \\
    &
    \mathllap{{}_{\mathscr{q}_q}}\searrow
      &&
    \swarrow\mathrlap{{\mathfrak{q}}_{q}}
    \\
    &&
    \mathscr{V}_{\mathbf{B}G}
  }
$$
Due to the assumption that $\mathcal{C}$ has colimits, this exists and (Rem. \ref{BifibrantCategoryOfLocalSystems})
by the general formula for colimits in Grothendieck constructions ([here](Grothendieck+construction#CoLimitsInAGrothendieckConstruction)) the [[underlying]] object in [[Grpd]] the is colimiting cocone of the underlying diagram in [[Grpd]]:
$$
  \array{
    \mathbf{E}G
    &&
      \overset{\mu(\text{-},g^{-1})}{\longrightarrow}
    &&
    \mathbf{E}G
    \\
    &
    \mathllap{{}_q}\searrow
      &&
    \swarrow\mathrlap{{}_q}
    \\
    && \mathbf{B}G
  }
$$
and the $\mathcal{C}$-component over that is the colimit in $\mathcal{C}$ of the diagram of images of morphism under pushforward $q_!$ along the underlying [[coprojection]] $q$.

Now observe that
$$
  \array{
  q_!
  \big(
    \mathbb{1}_{\mathbf{E}G}
    \,\boxtimes\,
    \mathscr{V}_{pt}
  \big)
  &
  \simeq
  &
  (G\cdot\mathbb{1})_{\mathbf{B}G} \boxtimes \mathscr{V}_{pt}
  \\
  \mathllap{{}^{
    q_!\big(
      \mathbb{1}_{\mu(\text{-},g^{-1})} \,\boxtimes\, \rho(g)_{pt}
    \big)
  }}
  \Big\downarrow
  &&
  \Big\downarrow\mathrlap{{}^{
    \big(
      \mu(\text{-},g^{-1})
      \cdot
      \mathbb{1}
    \big)_{\mathbf{B}G}
    \,\boxtimes\,
    {\rho(g)}_{pt}
  }}
  \\
  q_!
  \big(
    \mathbb{1}_{\mathbf{E}G}
    \,\boxtimes\,
    \mathscr{V}_{pt}
  \big)
  &
  \simeq
  &
  (G\cdot\mathbb{1})_{\mathbf{B}G} \boxtimes \mathscr{V}_{pt}
  }
$$
and the colimit over this system of morphisms is clearly $\mathscr{V}$ with its $G$-action, with [[coprojection]]
$$
  \array{
    G \cdot \mathscr{V}
    &\xrightarrow{\;\; \mathscr{q} \;\;}&
    \mathscr{V}
    \\
    (g,v) &\mapsto& \rho(g)(v)
  }
$$

\end{proof}

It follows that:
\begin{theorem}
 \label{LocalSystemsAreFreeHomotopyQuasiCoproductCompletion}
  If $\mathcal{C}$ has all [[colimits]] then $Loc_{\mathcal{C}}$ (Def. \ref{CategoryOfLocalSystems}) is its free homotopy quasi-coproduct completion (Def. \ref{FreeHomotopyQuasiCoproductCompletion}).
\end{theorem}
\begin{proof}
  First, $Loc_{\mathcal{C}}$ is canonically tensored over $Grpd$ (by Rem. \ref{TensoringOfLocalSystemsOverGroupoids})
and has all colimits (by Rem. \ref{BifibrantCategoryOfLocalSystems}) and as such is in particular a category with homotopy quasi-coproducts (Def. \ref{CategoryWithHomotopicalQuasiCoproducts}).

Moreover, by Lemma \ref{AllLocalSystemsAreGrpdTensoringsOfCoproductsOfSkeletalLocalSystems} and Lemma \ref{GroupRepresentationsAreQuasiCoproductsOfConstantLocalSystems} every object of $Loc_{\mathcal{C}}$ is a coproduct of $Grpd$-tensorings of quasi-coproducts of constant local systems (eq:ConstantLocalSystem). Therefore, a functor out of $Loc_{\mathcal{C}}$ which preserves the $Grpd$-tensoring and the quasi-coproducts is fixed by its restriction to constant local systems.
\end{proof}

## References

The notion of quasi-coproducts (Def. \ref{QuasiCoproduct}) is due to:

* {#HuTholen96} [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 [[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)]

[[!redirects quasi-coproducts]]

[[!redirects homotopy quasi-coproduct]]
[[!redirects homotopy quasi-coproducts]]

[[!redirects quasicoproduct]]
[[!redirects quasicoproducts]]


quasi-finite CW-complex

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

#Contents#
* table of contents
{:toc}

## Definition

A countable [[CW-complex]] $K$ is **quasi-finite** if for any finite subcomplex $M\subset K$, there is (possibly larger) finite subcomplex $e(M)\subset K$, such that for every [[separable space|separable]] [[metric space]] $X$ satisfying

* ($K$ is an [[absolute extensor]] of $X$:) for every [[closed subspace]] $A\subset X$ and a function $f:A\to K$ there is an [[extension]] $\tilde{f}:X\to K$ (i.e. $\tilde{f}=f\circ i$, where $i:A\hookrightarrow X$ is the closed embedding)

one has a similar property

* for every closed subspace $A\subset X$ and a function $f:A\to M$ there is an [[extension]] $g:X\to e(M)$ (i.e. $g\circ i = f$).

There is a characterization: a coutable CW-complex $K$ is quasi-finite iff for all separable metric spaces $X$, if $K$ is an absolute extensor of $X$ implies then it is an absolute extensor of its Stone-Čech compactification $\beta(X)$ as well.

In fact, the original definition asks that one has a function $e:M\to e(M)$ (the same under the [[axiom of choice]]).

## References

* A.Karasev, _On two problems in extension theory_, arXiv:math.GT/0312269

* M.Cencelj, J.Dydak, J.Smrekar, A.Vavpetic, Ž.Virk, _Algebraic properties of quasi-finite complexes_, Fund. Math. 197 (2007), 67-80 [math/0509582](http://arxiv.org/abs/math/0509582)

[[!redirects quasi-finite CW-complexes]]


quasi-free algebra

#Contents#
* table of contents
{:toc}

## Idea

An [[associative algebra]] (over a [[field]] $k$) is quasi-free if dually regarded as a [[noncommutative scheme]] it is [[formally smooth]].

## Definition

Given an [[associative algebra]] $A$ let $\Omega A$ be its [[universal differential envelope]].

An associative unital $k$-algebra $A$ is __quasi-free__ (or __formally smooth__) if one of the following equivalent conditions is satisfied

* Given an extension $0\to N\to E\stackrel{q}\to B\to 0$ of algebras where the ideal $N$ is
nilpotent and $a:A\to B$ an algebra map.
Then there exists a homomorphism $a' : A \to E$ such that $q \circ a' = a$.

* $A$ has cohomological dimension $\leq 1$ with respect to [[Hochschild cohomology]];

* $\Omega^1 A$ is a projective $A$-bimodule;

* the universal Hochschild 2-cocycle $c : A\otimes A\to\Omega^2 A$, $c : a\otimes b\mapsto d a d b$ is a coboundary, i.e. $c = b\phi$ for some $\phi:A\to\Omega^2 A$ satisfying the cocycle condition $\phi(a b)=a\phi(b)+\phi(a)b-d a d b$;

* there exists a "right connection" $\nabla:\Omega^1 A\to \Omega^2 A$ i.e. a $k$-linear map satisfying $\nabla(a w)=a\nabla(w)$ and $\nabla (w a) = \nabla(w)a+w d a$ where $w\in\Omega^1 A$ and $a\in A$.

This is due to ([CuntzQuillen](#CuntzQuillen)).

## Properties

For $A$ an associative algebra, the object $Spec A \in [Alg_k, Set]$ is [[formally smooth]] with respect to the standard infinitesimal cohesive structure over non-commutative algebras (see there for details) precisely if it is quasi-free.

Notice that the characterization via nilpotent extensions is similar to the definition of commutative [[formally smooth]] algebras as in [[EGAIV]]4 17.1.1. However most commutative formally smooth algebras are _not_ formally smooth in the associative noncommutative sense.

## Examples

[[path algebra|Path algebras]] of [[quivers]] and free algebras are some of the (few classes of) examples.

## References

* J. Cuntz and [[Daniel Quillen]], _Algebra extensions and nonsingularity_ , J.Amer. Math.
Soc. 8 (1995), 251-289.
 {#CuntzQuillen}

* J. Cuntz and D. Quillen: Cyclic homology and nonsingularity, J. Amer. Math.
Soc. 8 (1995), 373-442.

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative smooth spaces, ([arXiv:math/9812158](http://arxiv.org/abs/math/9812158))


quasi-free quantum state

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] a _[[quantum state]]_ is called _quasi-free_ if

1. its [[n-point functions]] are non-vanishing only for [[even number|even]] $n$;

1. its [[2-point function]] determines all its $n$-point functions.

This reflects the structure of the [[Wick algebra]] of [[quantum observables]] for the [[free field]].

## Examples

* [[quasi-free Hadamard states]]

## Related concepts

[[!include states and observables -- content]]

## References

* {#Radzikowski96} [[Marek Radzikowski]], p. 4 of _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))

* {#KhavkineMoretti14} [[Igor Khavkine]], [[Valter Moretti]], section 2.4 of _Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction_, Chapter 5 in [[Romeo Brunetti]] et al. (eds.) _Advances in Algebraic Quantum Field Theory_, Springer, 2015 ([arXiv:1412.5945](https://arxiv.org/abs/1412.5945))

[[!redirects quasi-free quantum states]]

[[!redirects quasi-free state]]
[[!redirects quasi-free states]]


quasi-Frobenius algebra

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

# Quasi-Frobenius algebra
* table of contents
{: toc}

## Definition

Recall that an [[algebra]] is a [[Frobenius algebra]] if its left [[regular representation]] $L(A)$ is isomorphic to its right regular representation $R(A)$. For *quasi*-Frobenius algebras, this isomorphism is weakened.

+-- {: .num_defn}
###### Definition
An algebra $A$ is called a quasi-Frobenius algebra if the isomorphism classes of idecomposable components of its left-regular representation $L(A)$ matches those of its right-regular representation $R(A)$.
=---

## References ##

The notion of quasi-Frobenius algebras (first?) appears in

* [[Tadasi Nakayama]]. *On Frobeniusean Algebras. I*. Annals of Mathematics, Second Series, Vol. 40, No. 3 (Jul., 1939), pp. 611-633 (23 pages). ([doi](https://doi.org/10.2307/1968946))


quasi-Hopf algebra

+-- {: .rightHandSide}
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#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

The notion of a quasibialgebra generalizes that of a
[[bialgebra]] [[Hopf algebra]] by introducing a nontrivial [[associativity]] [[coherence]] ([Drinfeld 89](#Drinfeld89)) [[isomorphisms]] (representable by multiplication with an element in triple tensor product) into axioms; a _quasi-Hopf algebra_ is a quasi-bialgebra with an [[antipode]] satisfying [[axioms]] which also involve nontrivial left and right unit coherences.

In particular, quasi-Hopf algebras may be obtained from ordinary [[Hopf algebras]] via twisting by a [[Drinfeld associator]], i.e. a nonabelian [[bialgebra cocycle|bialgebra 3-cocycle]].

### Motivation from quantum field theory

Drinfel'd was motivated by study of [[monoidal categories]] in [[rational CFT|rational]] [[2d conformal field theory]] (RCFT) as well as by an idea from [[Grothendieck]]'s _[[Esquisse d'un programme|Esquisse]]_ namely the [[Grothendieck-Teichmüller tower]] and its modular properties. In RCFT, the [[monoidal categories]] appearing can be, by [[Tannaka duality|Tannaka reconstruction]] considered as [[categories of modules]] of [[Hopf algebra]]-like objects where the flexibility of associativity coherence in building a theory were natural thus leading to quasi-Hopf algebras.

A special case of the motivation in RCFT has a toy example of [[Dijkgraaf-Witten theory]] which can be quite geometrically
explained. Namely, where the [[groupoid convolution algebra]] of the [[delooping]] [[groupoid]] $\mathbf{B}G$ of a [[finite group]] $G$ naturally has the structure of a Hopf algebra, the [[twisted groupoid convolution algebra]] of $\mathbf{B}G$ equipped with a 3-[[cocycle]] $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ is naturally a quasi-Hopf algebra.
Since such a 3-cocycle is precisely the [[background gauge field]] of the 3d [[TFT]] called [[Dijkgraaf-Witten theory]], and hence quasi-Hopf algebras arise there ([Dijkgraaf-Pasquier-Roche 91](#DijkgraafPasquierRoche)).

## Definition (Drinfeld)

A __quasibialgebra__ is a unital [[associative algebra]] $(A,m,\eta)$ with a structure of not necessarily [[coassociative coalgebra]] $(A,\Delta,\epsilon)$, with multiplicative
comultiplication $\Delta$ and counit $\epsilon$, and an invertible element $\phi \in A\otimes A\otimes A$ such that

(i) the coassociativity is modified by conjugation by $\phi$ in the sense

$$
(\Delta \otimes 1)\Delta(a) = \phi\left((1\otimes\Delta)\Delta(a)\right)\phi^{-1},\,\,\,\,\,\forall a\in A,
$$

(ii) the following pentagon identity holds

$$
(1\otimes 1\otimes\Delta)(\phi)(\Delta\otimes 1\otimes 1)(\phi) =
(1\otimes\phi)(1\otimes\Delta\otimes 1)(\phi)(\phi\otimes 1)
$$

(iii) some identities involving unit $\eta$ and counit $\epsilon$ hold:

$$ (\epsilon\otimes A)\Delta(a) = a = (A\otimes\epsilon)\Delta(a), \,\,\,\,\,\,a\in A;$$
$$
(A\otimes\epsilon\otimes A)\phi = 1.
$$
It follows that $(\epsilon\otimes A\otimes A)\phi
 = 1 = (A\otimes A\otimes\epsilon)\phi$.

The category of left $A$-modules is a monoidal category, namely the coproduct is used to define the action of $A$ on the tensor product of modules $(M,\nu^M)$, $(N,\nu^N)$:

$$
A \otimes (M\otimes N) \stackrel{\Delta\otimes M\otimes N}\longrightarrow (A\otimes A)\otimes(M\otimes N)
\rightarrow (A\otimes M)\otimes (A\otimes N)\stackrel{\nu_M\otimes\nu_N}\longrightarrow M\otimes N
$$

Using the [[Sweedler notation|Sweedler-like notation]] $\phi = \sum \phi^1\otimes \phi^2\otimes \phi^3$, formulas

$$
\Phi_{M,N,P}: (M\otimes N)\otimes P\stackrel\cong\longrightarrow M\otimes (N\otimes P)
$$
$$
(m\otimes n)\otimes p\mapsto \sum (\phi^1\triangleright m)
\otimes ((\phi^2\triangleright n)\otimes (\phi^3\triangleright p))
$$
define a [[natural transformation]] $\Phi$
and the pentagon for $\phi$ yields the [[pentagon identity|MacLane's pentagon]] for $\Phi$ understood as a new [[associator]],
$$
(M\otimes\Phi_{N,P,Q})\Phi_{M,N\otimes P,Q}(\Phi_{M,N,P}\otimes Q)=\Phi_{M,N,P\otimes Q}\Phi_{M\otimes N,P,Q}
$$
For this reason, $\phi$ is sometimes called the [[associator]] of the quasibialgebra. While it is due to Drinfeld, another variant of it, written as a [[formal power series]] and used in [[knot theory]] is often called the [[Drinfeld associator]] (see there).

A __quasi-Hopf algebra__ is a quasibialgebra $(A, \Delta, \varepsilon, \phi)$ equipped with elements $\alpha,\beta \in A$ and an antiautomorhphism $S$ of $A$ (a suitable kind of [[antipode]]) such that:
$$\sum_i S(b_i)\alpha c_i = \varepsilon (a) \alpha, \sum_i b_i\beta S(c_i) = \varepsilon(a)\beta$$
for $a \in A$ with $\Delta(a) = \sum_i b_i \otimes c_i$ in [[Sweedler notation]]. Further we require:
$$\sum_i X_i\beta S(Y_i)\alpha Z_i = 1, \quad where \sum_i X_i \otimes Y_i\otimes Z_i = \phi,$$
$$\sum_j S(P_j)\alpha Q_j\beta S(R_j) =1, \quad where \sum_j P_j \otimes Q_j \otimes R_j = \phi^{-1}.$$

### Twisting quasibialgebras by 2-cochains

The [[associator]] $\phi$ is a counital 3-cocycle in the sense of bialgebra cohomology theory of [[Shahn Majid|Majid]]. The 3-cocycle condition is the [[pentagon identity|pentagon]] for $\phi$. The abelian cohomology would add a coboundary of 2-cochain to get a cohomologous 3-cocycle. In nonabelian case, however, the twist by an invertible 2-cochain is done in a nonabelian way, described by Drinfeld and generalized by Majid to $n$-cochains.

Thus, for a bialgebra $A$, and fixed $n$, the
$i$-th coface
$$
\partial^i  = id_{A^{\otimes (i-1)}}\otimes \Delta \otimes \id_{A^{\otimes (n-i)}} : A^{\otimes n}\to A^{\otimes (n+1)},
$$
for $1\leq i\leq n$, and $\partial^0 = 1\otimes id_{A^{\otimes n}}$, $\partial^{n+1} =
id_{A^{\otimes n}}\otimes 1$. For $F\in A^{\otimes n}$, Majid defines
$$
\partial^+ F = \prod_{i\,\,\,\,even} (\partial^i F),\,\,\,\,\,\partial^- F = \prod_{i\,\,\,\,odd} (\partial^i F),
$$
where the products are in the order of ascending $i$. If $F\in A^{\otimes n}$ is a cochain then its coboundary is $\delta F = (\partial^+ F)(\partial^- F^{-1})$, which is automatically an $(n+1)$-cochain.
If $F \in A^{\otimes n}$ is an $n$-cochain and $\phi\in A^{\otimes (n+1)}$ is an $(n+1)$-cochain then one defines a cochain twist $\phi^F$ of $\phi$ by $F$ by the formula
$$
\phi^F = (\partial^+ F)\phi(\partial^- F^{-1}).
$$

Drinfeld proved that for $n=2$ the following is true. Given a quasiabialgebra $A = (A,m,\eta,\Delta,\epsilon,\phi)$ and a 2-cochain $F$, the data
$A^F = (A,m,\eta,F\Delta(-)F^{-1},\epsilon,\phi^F)$ is also a quasibialgebra. Furthermore, [[categories of modules]] $A-mod$ and $A^F-mod$ are [[monoidal category|monoidally]] [[equivalence of categories|equivalent]] reflecting the idea that cohomologous cocycles lead to nonessential categorical effects. If $(A,R)$ is quasitriangular quasibialgebra then we can twist the
R-element $R\in H\otimes H$ to $R^F = F_{21} R F$ to obtain quasitriangular quasibialgebra $(A^F,R^F)$ and their [[braided monoidal categories]] of representations are braided monoidally equivalent.

## Related concepts

* [[hopfish algebra]]

* [[quasitriangulated quasi-Hopf algebra]]

## References

The notion was introduced in

* {#Drinfeld89} [[Vladimir Drinfel'd]], _Квазихопфовы алгебры_, Algebra i Analiz __1__ (1989), no. 6, 114--148, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=53&what=fullt&option_lang=rus); translation _Quasi-Hopf algebras_, Leningrad Math. J. __1__ (1990), no. 6, 1419–1457 [MR1047964](http://www.ams.org/mathscinet-getitem?mr=1047964)

The relation to [[Dijkgraaf-Witten theory]] appeared in

* [[Robbert Dijkgraaf]], V. Pasquier, P. Roche, _QuasiHopf algebras, group cohomology and orbifold models_, Nucl. Phys. B Proc. Suppl. __18B__ (1990), 60-72; _Quasi-quantum groups related to orbifold models_, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991
 {#DijkgraafPasquierRoche}

and some arguments about the general relevance of quasi-Hopf algebras is in

* Gerhard Mack, [[Volker Schomerus]], _Quasi Hopf quantum symmetry in quantum theory_,  Nuclear Physics B 370:1 (1992) 185--230 doi

Recently a monograph appeared

* Daniel Bulacu, Stefaan Caenepeel, Florin Panaite, Freddy Van Oystaeyen, _Quasi-Hopf algebras: a categorical approach_, 544 pp., EMA 174 (2019)

Wikipedia article: [Quasi-Hopf algebra](https://en.wikipedia.org/wiki/Quasi-Hopf_algebra)

Other articles include

* В. Г. Дринфельд, _О структуре квазитреугольных квазихопфовых алгебр_, Функц. анализ и его прил. __26__:1 (1992), 78–80, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=768&what=fullt&option_lang=rus); transl. V. G. Drinfeld, _Structure of quasitriangular quasi-hopf algebras_, Funct. Anal. Appl., 26:1 (1992), 63–65
* V. G. Drinfelʹd, _О квазитреугольных квазихопфовых алгебрах и одной группе, тесно связанной с $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$_, Algebra i Analiz 2 (1990), no. 4, 149--181, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=199&volume=2&year=1990&issue=4&fpage=149&what=fullt&option_lang=eng); translation _On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf {Q})$_, Leningrad Math. J. __2__ (1991), no. 4, 829–860, [MR1080203](http://www.ams.org/mathscinet-getitem?mr=1080203)
* V. G. Drinfelʹd, _Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations_, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.

* [[Shahn Majid]], _Quantum double for quasi-Hopf algebras_, Lett. Math. Phys. __45__ (1998), no. 1, 1–9, [MR2000b:16077](http://www.ams.org/mathscinet-getitem?mr=1631648), [doi](http://dx.doi.org/10.1023/A:1007450123281), [q-alg/9701002](http://arxiv.org/abs/q-alg/9701002)
* Peter Schauenburg, _Hopf modules and the double of a quasi-Hopf algebra_, Trans. Amer. Math. Soc. __354__ (2002), 3349-3378 [pdf](http://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf)
* M. Jimbo, H. Konno, S. Odake, J. Shiraishi, _Quasi-Hopf twistors for elliptic quantum groups_, Transformation Groups 4(4), 303–327 (1999) [doi](http://sci-hub.tw/10.1007/BF01238562)
* Ivan Kobyzev, Ilya Shapiro, _A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids_, Applied Categorical Structures, __27__:1 (2019) 85–109 [doi](https://doi.org/10.1007/s10485-018-9544-0)
* L Frappat, D Issing, E Ragoucy, _The quantum determinant of the elliptic algebra $\mathcal{A}_{q, p}(\widehat{gl}_N)$_, J. Phys. __A51__:44, [doi](https://doi.org/10.1088/1751-8121/aae296)

[[!redirects quasibialgebra]]
[[!redirects quasihopf algebra]]
[[!redirects quasiHopf algebra]]
[[!redirects quasi-bialgebra]]
[[!redirects quasi-Hopf algebras]]


quasi-isomorphism

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

#Contents#
* table of contents
{:toc}

## Idea

A _quasi-isomorphism_ is a [[chain map]] that induces [[isomorphisms]] on all [[homology groups]]. These are the natural choice of [[weak equivalences]] between [[chain complexes]] in the context of ([[stable homotopy theory|stable]]) [[homotopy theory]].

The [[localization]] of a [[category of chain complexes]] at the quasi-isomorphisms is called the _[[derived category]]_ of the underlying [[abelian category]].

Under the relation between [[topological spaces]] and [[chain complexes]] established by forming [[singular simplicial complexes]], quasi-isomorphism can be understood as the [[abelianization]] of [[weak homotopy equivalences]] (see the [[Hurewicz theorem]] for more on this).

## Definition

Let $\mathcal{A}$ be an [[abelian category]] and write
$Ch_\bullet(\mathcal{A})$ for its [[category of chain complexes]].

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

A [[chain map]] $f_\bullet : C_\bullet \to D_\bullet$ in $Ch_\bullet(\mathcal{A})$ is called a  **quasi-isomorphism** if for each $n \in \mathbb{N}$ the induced morphisms on [[chain homology]] groups

$$
  H_n(f) \colon H_n(C) \to H_n(D)
$$

is an [[isomorphism]].

=--

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

Quasi-isomorphisms are also called, more descriptively, **homology isomorphisms** or **$H_\bullet$-isomorphisms**.  See at _[[homology localization]]_ for more on this.

=--

## Properties

### Relation to chain homology type
 {#RelationToChainHomologyType}

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

The [[relation]] "There exists a quasi-isomorphism from $C_\bullet$ to $D_\bullet$." is a [[reflexive relation|reflexive]] and [[transitive relation]], but it is not a [[symmetric relation]].

=--

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

Reflexivity and transitivity are evident. An explicit counter-example
showing the non-symmetry is the [[chain map]]

$$
  \array{
     \cdots &\to& 0 &\to& \mathbb{Z} &\stackrel{\cdot 2}{\to}& \mathbb{Z} &\to& 0 &\to& \cdots
     \\
     \cdots && \downarrow && \downarrow && \downarrow && \downarrow && \cdots
     \\
     \cdots &\to& 0 &\to& 0 &\to& \mathbb{Z}/2\mathbb{Z} &\to& 0 &\to& \cdots
  }
$$

from the chain complex concentrated on the morphism of multiplication by 2 on integers, to the chain complex concentrated on the [[cyclic group of order 2]].

This clearly induces an isomorphism on all homology groups. But there is not even a non-zero chain map in the other direction, since there is no non-zero group homomorphism $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$.

=--

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

This is as for [[weak homotopy equivalences]], see the discussion at _[Relation to homotopy types](weak+homotopy+equivalence#RelationToHomotopyTypes)_ there.

=--

### Relation to mapping cones and homotopy (co)fibers
 {#RelationToMappingCone}

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

A [[chain map]] is a quasi-isomorphism precisely if its [[homotopy cofiber]] in the [[(∞,1)-category of chain complexes]] has trivial [[homology groups]].

=--

By basic properties discussed at _[[truncated object in an (∞,1)-category]]_.

Concretely this means in particular the following.

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

A [[chain map]] $f_\bullet : C_\bullet \to D_\bullet$ is a quasi-isomorphism precisely if its [[mapping cone]] $cone(f)_\bullet \in Ch_\bullet(\mathcal{A})$ has all trivial [[chain homology]] groups.

=--

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

This follows for instance from the [[homology long exact sequence]]

$$
  \cdots
  \to
   H_{n+1}(c)\to H_{n+1}(D) \to H_{n+1}(cone(f))
  \to
  H_n(C) \to H_n(D) \to H_n(cone(f))
  \to
  H_{n-1}(C) \to H_{n-1}(D) \to H_{n-1}(cone(f))
  \to
  \cdots
  \,.
$$

If here by assumption $H_n(cone(f)) = 0$ for all $n$, then this involves [[exact sequences]] of the form

$$
  0 \to H_n(C) \stackrel{H_n(f)}{\to} H_n(D) \to 0
$$

for all $n$. But this says that the [[kernel]] and [[cokernel]] of $H_n(f)$ are trivial for all $n$, hence that $H_n(f)$ is an [[isomorphism]] for all $n$, hence that $f_\bullet$ is a quasi-isomorphism.

=--

## In homotopy theory

Quasi-isomorphisms are the [[weak equivalences]] in the most common [[model category|model category structures]] on the [[category of chain complexes]]. See at _[[model structure on chain complexes]]_ and _[[derived category]]_.

## Related concepts

* [[weak homotopy equivalence]]

* [[injective resolution]], [[projective resolution]]

* [[category of chain complexes]]

  * [[chain complex]]

  * [[chain map]]

  * [[chain homotopy]]

## References

A basic introduction is around definition 1.1.2 in

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

A more systematic discussion is in section 12 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

[[!redirects quasi-isomorphisms]]
[[!redirects quasiisomorphism]]
[[!redirects quasiisomorphisms]]
[[!redirects quasi-isomorphism of chain complexes]]
[[!redirects quasi-isomorphisms of chain complexes]]
[[!redirects quasiisomorphism of chain complexes]]
[[!redirects quasiisomorphisms of chain complexes]]

[[!redirects homology isomorphism]]
[[!redirects H-*-isomorphism]]
[[!redirects H*-isomorphism]]


quasi-limit

+-- {: .rightHandSide}
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[2-category theory]], the notion called *quasi-limits* [[Gray (1997)](#Gray74)] or *soft limits* [[MacDonald & Stone (1989)](#MS89)] is a weakening of the notion of [[bilimit]] so as to make them satisfy a 'quasi-universal' property.
This means that instead of being singled out up to unique [[isomorphism]], quasi-limits are identified only up to unique _[[adjunction]]_.

## Definition

Recall that [[limits]] and [[colimits]] of shape $\mathbf K$ are computed by the [[right adjoint|right]] and [[left adjoint]], respectively, to the [[diagonal]] [[functor]] $\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C]$ (the one which [[constant functors]]):

\[
   \colim \dashv \Delta \dashv \lim
   \,.
\]

Suppose now $\mathbf C$ is a [[2-category]] and $\mathbf K$ is a [[small category|small 2-category]]. Gray defines quasi(co)limits by relaxing the previous adjunction to be only a [[local adjunction]]:
\[
   \mathrm{qcolim}
   \dashv_{\mathrm{loc}}
   \Delta
   \dashv_{\mathrm{loc}}
   \mathrm{qlim}
\]
This means that, naturally in $X : \mathbf C$, there are adjunctions (instead of equivalences, like in a [[2-adjunction]]) between hom-categories:
\[
   \mathbf C(X, \mathrm{qlim} F) \rightleftarrows \mathrm{Nat}(\Delta_X, F)
\]

Alternatively, one can give a definition in terms of a weakened representability property.

Let $F : \mathbf K \to \mathbf C$ be a small [[diagram]] in $\mathbf C$. Then recall that a [[bilimit]] for $F$ is an [[object]] $\lim F:\mathbf C$ equipped with a natural family of [[equivalence in a 2-category|equivalences]]
\[
  \phi_X
  \,\colon\,
  \mathbf C(X, \lim F)
    \cong
  \mathrm{Nat}\Big(
    1,
    \mathbf C\big(X, F(-)\big)
  \Big)
\]
where $\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C]$ is the diagonal functor.

A **quasi-limit** for $F$ is then an object $\mathrm{qlim} F:\mathbf C$ equipped with a family of adjunctions
\[
  \psi_X
  \,\colon\,
  \mathbf C(X, \mathrm{qlim}F)
  \rightleftarrows
  \mathrm{Nat}\Big(
    1,
    \mathbf C\big(X, F(-)\big)
  \Big)
\]
naturally in $X:\mathbf C$.

Notice that by replacing $1 : \mathbf C \to \mathrm{Cat}$ with an arbitrary weight $W$ we can easily give the definition of _weighted quasilimit_.

## Examples

* A **quasi-terminal** object in a 2-category $\mathbf C$ is an object $Q$ such that, for each other object $X: \mathbf C$, there exists a 'quasi-universal map' $t : X \to Q$ such that every other map $f : X \to Q$ admits a unique 2-cell $\phi_f : f \Rightarrow t$ to it. In $\mathbf C = \mathbf{Cat}$ quasi-terminal objects are categories with a [[terminal object]].

From this example alone one can see quasi-limits are, in general, not unique up to isomorphism, like limits and other [[universal constructions]]. The universality of the objects is replaced by universality of their quasi-universal maps. A consequence of this fact is that we can't say _the_ quasi-limit but _a_ quasi-limit.

The following is an example of quasi-colimit from [(Gray '74, p.199)](#Gray74):

* A **quasi-coproduct** of $A_1$ and $A_2$ in a 2-category $\mathbf C$ is an object $A_1 +_q A_2$ together with injections $i_j : A_j \to A_1 +_q A_2$ ($j=1,2$) which satisfies a relaxed version of the universal property of [[coproducts]]. Instead, we require that for any other object $X$ with maps $h_j : A_j \to X$ there is a map $h : A_1 +_q A_2 \to X$ and 2-cells $\lambda_j : h\circ i_j \Rightarrow h_j$ ($j=1,2$) which are terminal as such. This means that given any other $g : A_1 +_q A_2 \to X$ and 2-cells $\gamma_j : g \circ i_j \Rightarrow h_j$ ($j=1,2$), there is a 2-cell $\phi_{(g,\gamma_j)} : g \to h$ that factorizes $\gamma$ through $\lambda$: $(\tau \cdot i_j) \circ \lambda_j = \gamma_j$.

In [Capucci (2022)](#Capucci22) it is shown that this (with [[2-cells]] reversed though) is the [[universal property]] of a certain example of monoidal product common in game theory, called _external choice_.

* A **quasi-product** is dual to a quasi-coproduct in the sense of reversing 1-cells. Reversing 2-cells is also possible, but there doesn't seem to be an agreed terminological scheme to distinguish between the two kinds of quasi-limits.

Notably, the product of sets, which in the 2-category of [[relations]] loses its usual universal property, becomes a quasi-product there. Notice both this and the above example of quasi-coproduct are unique up to iso.

## References

Quasi-limits and their duals have been defined in §I.7.9 of:

* {#Gray74} [[John Gray]], _[[Adjointness for 2-Categories|Formal category theory: adjointness for 2-categories]]_, Lecture Notes in Mathematics **391**
Springer (1974) [[doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280)]

See also:

* _["Very lax" 2
-dimensional co/limits](https://mathoverflow.net/questions/386414/very-lax-2-dimensional-co-limits)_, MathOverflow

* {#MS89} MacDonald, Stone, _[Soft adjunction between 2-categories](https://www.sciencedirect.com/science/article/pii/0022404989901278)_, Journal of Pure and Applied Algebra, **60** (1989)

* {#Capucci22} [[Matteo Capucci]], _The Universal Property of External Choice_ (2022) [[pdf](https://matteocapucci.files.wordpress.com/2022/03/main.pdf)]


quasi-net

#Contents#
* table of contents
{:toc}

## Definition

In [[non-archimedean analytic geometry]], one considers:

Given a [[topological space]], a _quasinet_ in $X$ is a [[set]] $\tau = \{V_i\}$ of [[subsets]] of $X$ such that for each point $x\in X$ there exists a finite number of such subsets $V_{i_1}, \cdots, V_{i_n} \in \tau$ such that

1. $x\in V_{i_1} \cap V_{i_2} \cap \cdots \cap V_{i_n}$ (the point is in their [[intersection]]);

1. $V_{i_1} \cup \cdots \cup V_{i_n}$ is a [[neighbourhood]] of $x$.

(e.g. [Berkovich 09, def. 3.1.1.](#Berkovich09))

A quasinet $\tau$ is called a _net_ if for all $U,V \in \tau$ then the restriction $\tau|_{U \cap V}$ is a quasinet of $U \cap V$.

## Examples

* A [[Berkovich analytic space]] is a [[locally Hausdorff topological space|locally Hausdorff]] [[topological space]] equipped with an [[atlas]] by [[affinoid domains]] such that the underlying [[analytic spectrum]] [[topological spaces]] of these form a net of [[compact subsets]] on $X$.

## References

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

[[!redirects quasi-nets]]

[[!redirects quasinet]]
[[!redirects quasinets]]


quasi-pointed category

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

#Contents#
* table of contents
{:toc}

## Idea

While a [[category]] is a called a [[pointed category]] if it has a [[zero object]], i.e. if it has an [[initial object]] and a [[terminal]] object and both are [[isomorphism|isomorphic]], for a _quasi-pointed category_ the last condition is relaxed.

#Definition#

A category is **quasi-pointed** if it has an [[initial object]] $0$, a [[terminal object|final object]] $1$ and its unique [[morphism]] $0\to 1$ is a [[monomorphism]].

#References#

* D. Bourn, _$3\times 3$ lemma and protomodularity_, J. algebra 236 (2001), 778--795


quasi-state

+-- {: .rightHandSide}
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###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum mechanics]], a _quasi-state_ on an [[algebra of observables]] $A$ is a [[function]] $\rho : A \to \mathbb{C}$ that is required to satisfy the axioms of a genuine [[state]] (linearity and positivity) only on the [[poset of commutative subalgebras]] of $A$.

While therefore the condition on quasi-states is much weaker than that for states, [[Gleason's theorem]] asserts that if $A = B(H)$ for $dim H \gt 2$, then all quasi-states are in fact already genuine [[quantum states]].

Notice that a quasi-state is naturally regarded as an actual state, but [[internalization|internal]] to the [[ringed topos]] over the [[poset of commutative subalgebras]] of $A$ --  the "[[Bohr topos]]". Therefore [[Gleason's theorem]] is one of the motivations for regarding this ringed topos as the quantum [[phase space]] ("[[Bohrification]]".) The other is the [[Kochen-Specker theorem]].

## Related concepts

[[!include states and observables -- content]]

[[!redirects quasi-states]]


quasi-symmetric function

# Quasi-symmetric functions
* table of contents
{: toc}

## Idea

Quasi symmetric functions are a generalisation of [[symmetric functions]] and are closely related to [[noncommutative symmetric functions]].

## Definition

+--{: .num_defn #qsymmfn}
###### Definition
Let $X$ be a totally ordered set of indeterminants.  Let $R$ be a ring.  A [[polynomial]] in $R[X]$ or a [[power series]] in $R[ [X] ]$ is said to be **quasi-symmetric** if whenever $X_1 \lt X_2 \lt \dots \lt X_n$ and $Y_1 \lt Y_2 \lt \dots \lt Y_n$ are finite sets of indeterminants then the coefficients of $X_1^{i_1} X_2^{i_2} \cdots X_n^{i_n}$ and $Y_1^{i_1} Y_2^{i_2} \cdots Y_n^{i_n}$ are the same.
=--

+-- {: .num_defn #qsymm}
###### Definition
The ring $\QSymm^{\hat{}}$ is defined as the ring of quasi-symmetric power series over $\mathbb{Z}$ in countably many variables.  Its subring $\QSymm$ is defined as the ring of quasi-symmetric polynomials (meaning, power series of bounded degree).
=--

## References

(Copied from [[noncommutative symmetric function]] as the two concepts are often studied together.)

### Research articles

* G. Duchamp, F. Hivert, J.-Y. Thibon, _Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras_, Internat. J. Alg. Comput. 12 (2002), 671–717.
* [[I. M. Gelfand]], D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, _Noncommutative symmetric functions_, Adv. in Math. __112__ (1995), 218–348, [hep-th/9407124](http://arxiv.org/abs/hep-th/9407124)
* Jean-Christophe Novelli, Jean-Yves Thibon, _Noncommutative symmetric functions and Lagrange inversion_, [math.CO/0512570](http://arxiv.org/abs/math/0512570); _Noncommutative symmetric functions and an amazing matrix_ [arxiv/1109.1184](http://arxiv.org/abs/1109.1184)
* Lenny Tevlin, _Noncommutative Monomial Symmetric Functions_, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings [pdf](http://www.fpsac.org/FPSAC07/SITE07/PDF-Proceedings/Talks/83.pdf)
* D. Krob, J.-Y. Thibon, _Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at $q = 0$_, [pdf](http://hal.inria.fr/docs/00/05/79/10/PDF/ncsf4.pdf)
* Christos A. Athanasiadis, _Power sum expansion of chromatic quasisymmetric functions_, [arxiv/1409.2595](http://arxiv.org/abs/1409.2595)
* Claudia Malvenuto, [[Christophe Reutenauer]], _Plethysm and conjugation of quasi-symmetric functions_, Discrete Mathematics __193__ (1998) 225-233 [pdf](https://sites.math.washington.edu/~billey/colombia/references/malvenuto.reutenauer.1998.pdf)

### Long surveys and lecture notes

* Michael Hazewinkel, _Symmetric functions, noncommutative symmetric functions and quasisymmetric functions_, [pdf](http://oai.cwi.nl/oai/asset/10344/10344B.pdf)
* [[V. Retakh]], R. Wilson, Advanced Course on Quasideterminants and Universal Localization: [pdf](http://castellet.cat/Publications/quaderns/Quadern41.pdf) (see the part _Factorization of Noncommutative Polynomials
and Noncommutative Symmetric Functions_)

### Expositions/short summaries

* Mike Zabrocki, _Non-commutative symmetric functions II:
Combinatorics and coinvariants_, slides from a talk [pdf](http://garsia.math.yorku.ca/~zabrocki/talks/coinvariants.pdf), III: A representation theoretical approach [pdf](http://garsia.math.yorku.ca/~zabrocki/talks/reptheory.pdf)
* Lenny Tevlin, _Introduction to quasisymmetric and noncommutative
symmetric functions_, slides, Fields Institute 2010  [pdf](http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/Noncommutative-Symmetric-and-Quasi-Symmetric-Functions-Fields-2010.pdf)

category: combinatorics

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