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Antonio Cegarra	https://ncatlab.org/nlab/source/Antonio+Cegarra	Antonio Martínez Cegarra is a Spanish mathematician, who is in the [Departamento de Álgebra](http://www.ugr.es/~algebra/) of the Universidad de Granada, Spain. He has published extensively in homotopical and homological algebra, and non-abelian cohomology. A (partial) list of his publications is [here](https://www.ugr.es/~acegarra/Publicaciones.html). ##Related entries * [[hypercrossed complex]] * [[Moore complex]] ## Selected writings On [[cat-n-groups]] and [[homotopy types]]: * [[Manuel Bullejos]], [[Antonio M. Cegarra]], [[John W. Duskin]], On cat$^n$ -groups and homotopy types, J. Pure Appl. Alg. 86 (1993) 135-154 \[<a href="https://doi.org/10.1016/0022-4049(93)90099-F">doi:10.1016/0022-4049(93)90099-F</a>] * [[Pilar Carrasco]] and [[Antonio Cegarra\|A. M. Cegarra]], _Group-theoretic Algebraic Models for Homotopy Types_, J. Pure Appl. Alg., 75, (1991), 195 – 235. On [[non-abelian cohomology]] with [[coefficients]] in [[braided 2-groups]] and [[symmetric 2-groups]]: * [[Manuel Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498)) On "classifying spaces" (ie: [[topological realization]] of [[simplicial nerves]]) of [[2-categories]]: * [[Manuel Bullejos]], [[Antonio M. Cegarra]], On the geometry of 2-categories and their classifying spaces, K-Theory 29 3 (2003) 211-229 [[doi:10.1023/B:KTHE.0000006921.50151.00](http://dx.doi.org/10.1023/B:KTHE.0000006921.50151.00), [pdf](http://www.ugr.es/\%7Ebullejos/geometryampl.pdf)] On the [[diagonal of a bisimplicial set\|diagonal]] of [[bisimplicial sets]]: * [[Antonio Cegarra]], [[Josué Remedios]], _The relationship between the diagonal and the bar constructions on a bisimplicial set_, Topology and its applications, volume 153 (1) (2005) ([pdf](http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf), [doi:10.1016/j.topol.2004.12.003](https://doi.org/10.1016/j.topol.2004.12.003)) On [[double groupoids]] and [[homotopy 2-types]]: * [[Antonio Martínez Cegarra]], Benjamın A. Heredia, [[Josué Remedios]], _Double groupoids and homotopy 2-types_, Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also [arXiv:1003.3820](http://arxiv.org/abs/1003.3820). category:people [[!redirects Antonio M. Cegarra]] [[!redirects Antonio Martínez Cegarra]] [[!redirects A. M. Cegarra]] [[!redirects Cegarra]]
Antonio Gómez-Tato	https://ncatlab.org/nlab/source/Antonio+G%C3%B3mez-Tato	* [institute page](https://www.usc.gal/en/department/mathematics/directory/antonio-gomez-tato-1899) ## Selected writings On [[rational homotopy theory]] of non-[[nilpotent spaces]] via [[deck transformation\|deck]]-[[Borel-equivariant rational homotopy theory]] of their [[universal cover\|universal]] [[covering spaces]]: * [[Antonio Gómez-Tato]], [[Stephen Halperin]], [[Daniel Tanré]], Rational homotopy theory for non-simply connected spaces, Trans. Amer. Math. Soc. 352 (2000), 1493-1525 ([doi:10.1090/S0002-9947-99-02463-0](https://doi.org/10.1090/S0002-9947-99-02463-0), [jstor:118074](https://www.jstor.org/stable/118074)) category: people [[!redirects Antonio Gomez-Tato]]
Antonio N. Bernal	https://ncatlab.org/nlab/source/Antonio+N.+Bernal	* [GoogleScholar page](https://scholar.google.com/citations?user=m72DV5EAAAAJ) * [InSpire page](https://inspirehep.net/authors/2276636) ## Selected writings On [[spacetime]] [[causality]], [[globally hyperbolic spacetimes]] and [[Cauchy surfaces]]: * [[Antonio N. Bernal]], [[Miguel Sánchez]], _On smooth Cauchy hypersurfaces and Geroch's splitting theorem_, Commun. Math. Phys. 243 (2003) 461-470 [[arXiv:gr-qc/0306108v2](http://arxiv.org/abs/gr-qc/0306108), [doi:10.1007/s00220-003-0982-6](https://doi.org/10.1007/s00220-003-0982-6)] * {#BernalSánchez05} [[Antonio N. Bernal]], [[Miguel Sánchez]], Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Commun. Math. Phys. 257 (2005) 43-50 [[arXiv:gr-qc/0401112](https://arxiv.org/abs/gr-qc/0401112), [doi:10.1007/s00220-005-1346-1](https://doi.org/10.1007/s00220-005-1346-1)] * [[Antonio N. Bernal]], [[Miguel Sánchez]], Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions, Lett. Math. Phys. 77 (2006) 183-197 [[arXiv:gr-qc/0512095](https://arxiv.org/abs/gr-qc/0512095), [doi:10.1007/s11005-006-0091-5](https://doi.org/10.1007/s11005-006-0091-5)] category: people [[!redirects Antonio Bernal]]
Antonio Pich	https://ncatlab.org/nlab/source/Antonio+Pich	* [webpage](https://webific.ific.uv.es/web/en/content/pich-zardoya-antonio) ## Selected writings On [[chiral perturbation theory]]: * [[Gerhard Ecker]], [[Jürg Gasser]], [[Antonio Pich]], E. DeRafael, Sec 3 and Appendix A in: _The role of resonances in chiral perturbation theory_, Nuclear Physics B Volume 321, Issue 2, 24 July 1989, Pages 311-342 (<a href="https://doi.org/10.1016/0550-3213(89)90346-5">doi:10.1016/0550-3213(89)90346-5</a>) On [[kaon]] [[decay]] in relation to the [[chiral anomaly]]: * [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (<a href="https://doi.org/10.1016/0550-3213(94)90623-8">doi:10.1016/0550-3213(94)90623-8</a>) On [[electroweak symmetry breaking]] and the [[Higgs boson]]/[[Higgs field]]: * [[Antonio Pich]], _Electroweak Symmetry Breaking and the Higgs Boson_, [Acta Phys. Pol. B 47, 151](https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=47&N=1#151) (2016) ([arXiv:1512.08749](https://arxiv.org/abs/1512.08749)) On [[flavour physics]], [[CP violation]] and [[flavour anomalies]]: * [[Antonio Pich]], _Flavour Dynamics and Violations of the CP Symmetry_, Lectures at the 2017 and 2019 CERN - Latin-American Schools of High-Energy Physics ([arXiv:1805.08597](https://arxiv.org/abs/1805.08597)) On [[flavour anomalies]]: * [[Antonio Pich]], _Flavour Anomalies_, PoS LHCP2019 (2019) 078 ([arxiv:1911.06211](https://arxiv.org/abs/1911.06211), [spire:1765034](https://inspirehep.net/literature/1765034)) On [[right-handed neutrinos]] in relation to [[flavor anomalies]]: * Rusa Mandal, Clara Murgui, Ana Peñuelas, [[Antonio Pich]], _The role of right-handed neutrinos in $b \to c \tau \bar \nu$ anomalies_ ([arXiv:2004.06726](https://arxiv.org/abs/2004.06726)) category: people
Antonio R. Garzón	https://ncatlab.org/nlab/source/Antonio+R.+Garz%C3%B3n	Antonio Garzón is a Spanish mathematician, who is in the [Departamento de Álgebra](http://www.ugr.es/~algebra/) of the Universidad de Granada, Spain. He has published extensively in [[homotopical algebra\|homotopical]] and [[homological algebra]], and [[non-abelian cohomology]]. ## Selected writings On [[homotopy theory]] of [[2-groups]] and [[braided 2-groups]]: * [[Antonio R. Garzón]], Jesus G Miranda, Sec. 1 of: Homotopy theory for (braided) cat-groups, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 38 (1997) no. 2, pp. 99-139 ([numdam:CTGDC_1997__38_2_99_0](http://www.numdam.org/item/?id=CTGDC_1997__38_2_99_0)) On [[k-tuply groupal n-groupoids]]: * [[Antonio R. Garzón]], Jesus G. Miranda, Section 6 of: _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (<a href="https://doi.org/10.1016/S0022-4049(98)00143-1">doi:10.1016/S0022-4049(98)00143-1</a>) category:people [[!redirects A.R. Garzón]] [[!redirects Antonio Garzon]] [[!redirects Antonio Garzón]] [[!redirects A. Garzon]]
Antony Wassermann	https://ncatlab.org/nlab/source/Antony+Wassermann	* [course notes](https://www.dpmms.cam.ac.uk/~ajw/) category: people
Antti J. Niemi	https://ncatlab.org/nlab/source/Antti+J.+Niemi	* [institute page](https://www.su.se/english/profiles/aniem-1.306055) * [research group](https://www.su.se/english/research/research-groups/quantum-frontiers?open-collapse-boxes=research-group-members) ([[Frank Wilczek]]) ## Selected writings On [[anyon\|anyonic]] [[vortices]] in [[Bose-Einstein condensates]]: * Julien Garaud, Jin Dai, [[Antti J. Niemi]], Vortex precession and exchange in a Bose-Einstein condensate, J. High Energ. Phys. 2021 157 (2021) $[$[arXiv:2010.04549](https://arxiv.org/abs/2010.04549)$]$ category: people [[!redirects Antti Niemi]]
Antti Kupiainen	https://ncatlab.org/nlab/source/Antti+Kupiainen	* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Antti_Kupiainen) ## Selected writings On the rigorous construction of [[Liouville theory]] * [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318)) * [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics, Pages 81-166 from Volume 191 (2020), Issue 1, ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2)) * [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], The DOZZ formula from the path integral, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) <a href="https://doi.org/10.1007/JHEP05(2018)094">doi:10.1007/JHEP05(2018)094</a>) and via the [[conformal bootstrap]]: * [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530)) and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88): * [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], Segal's axioms and bootstrap for Liouville Theory [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)] category: people
anyonic braiding in momentum space -- references	https://ncatlab.org/nlab/source/anyonic+braiding+in+momentum+space+--+references	### Anyons in momentum-space {#ReferencesAnyonicBraidingInMomentumSpace} On non-trivial [[braid group statistics\|braiding]] of nodal points in the [[Brillouin torus]] of [[topological semi-metals]] ("braiding in momentum space"): * {#AhnParkYang19} Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle, Phys. Rev. X 9 (2019) 021013 $[$[doi:10.1103/PhysRevX.9.021013](https://doi.org/10.1103/PhysRevX.9.021013), [arXiv:1808.05375](https://arxiv.org/abs/1808.05375)$]$ > "here are band crossing points, henceforth called [[vortices]]" * QuanSheng Wu, Alexey A. Soluyanov, [[Tomáš Bzdušek]], Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 $[$[arXiv:1808.07469](https://arxiv.org/abs/1808.07469), [doi:10.1126/science.aau8740](https://doi.org/10.1126/science.aau8740)$]$ > [[fundamental group]] of [[complement]] of nodal points/lines considered above (3) * {#TiwariBzdusek20} Apoorv Tiwari, [[Tomáš Bzdušek]], Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101 (2020) 195130 $[$[doi:10.1103/PhysRevB.101.195130](https://doi.org/10.1103/PhysRevB.101.195130)$]$ > "a new type non-Abelian 'braiding' of nodal-line rings inside the momentum space" * {#BWSWYB20} [[Adrien Bouhon]], QuanSheng Wu, [[Robert-Jan Slager]], Hongming Weng, Oleg V. Yazyev, [[Tomáš Bzdušek]], Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nature Physics 16 (2020) 1137-1143 $[$[arXiv:1907.10611](https://arxiv.org/abs/1907.10611), [doi:10.1038/s41567-020-0967-9](https://doi.org/10.1038/s41567-020-0967-9)$]$ > "Here we report that Weyl points in three-dimensional (3D) systems with $\mathcal{C}_2\mathcal{T}$ symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space." {#InTwistedBilayerGraphene} Braiding of Dirac points in twisted bilayer [[graphene]]: * Jian Kang, Oskar Vafek, Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene, Phys. Rev. B 102 (2020) 035161 $[$[arXiv:2002.10360](https://arxiv.org/abs/2002.10360), [doi:10.1103/PhysRevB.102.035161](https://doi.org/10.1103/PhysRevB.102.035161)$]$ * {#JBLZHLSJ21} Bin Jiang, [[Adrien Bouhon]], Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, [[Robert-Jan Slager]], Jian-Hua Jiang Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nature Physics 17 (2021) 1239-1246 $[$[arXiv:2104.13397](https://arxiv.org/abs/2104.13397), [doi:10.1038/s41567-021-01340-x](https://doi.org/10.1038/s41567-021-01340-x)$]$ (analog realization in [[phononic crystals]]) > Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space. > $[$...$]$ > we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions * {#ParkWongZhangOh21} Haedong Park, Stephan Wong, Xiao Zhang, and Sang Soon Oh, Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal, ACS Photonics 8 (2021) 2746–2754 [[doi:10.1021/acsphotonics.1c00876](https://doi.org/10.1021/acsphotonics.1c00876)] * {#CBSM22} Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions (formerly: Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions), Phys. Rev. B 105 (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$ > "Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding." * {#PBSM22} [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], Multi-gap topology and non-Abelian braiding of phonons from first principles, Phys. Rev. B 105 (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi:10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115)) (analog realization in [[phononic crystals]]) > new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies. > Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding. * {#PBMS22} [[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nature Communications 13 423 (2022) $[$[doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9)$]$ (analog realization in [[phononic crystals]]) > it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field. * {#ParkGaoZhangOh22} Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems, Nanophotonics 11 11 (2022) 2779–2801 $[$[doi:10.1515/nanoph-2021-0692](https://doi.org/10.1515/nanoph-2021-0692)$]$ (analog realization in [[photonic crystals]]) * [[Adrien Bouhon]], [[Robert-Jan Slager]], Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs $[$[arXiv:2203.16741](https://arxiv.org/abs/2203.16741)$]$ See also: * [[Robert-Jan Slager]], [[Adrien Bouhon]], [[Fatma Nur Ünal]], Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase $[$[arXiv:2208.12824](https://arxiv.org/abs/2208.12824)$]$ * Huahui Qiu et al., Minimal non-abelian nodal braiding in ideal metamaterials, Nature Communications 14 1261 (2023) $[$[doi:10.1038/s41467-023-36952-9](https://doi.org/10.1038/s41467-023-36952-9)$]$ * Wojciech J. Jankowski, Mohammedreza Noormandipour, [[Adrien Bouhon]], [[Robert-Jan Slager]], Disorder-induced topological quantum phase transitions in Euler semimetals $[$[arXiv:2306.13084](https://arxiv.org/abs/2306.13084)$]$ Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space [[Brillouin torus]]): [Lan 19, p. 1](Laughlin+state#Lan19), .... Knotted nodal lines in 3d semimetals Beware that various authors consider [[braids]]/[[knots]] formed by nodal lines in 3d semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to [[worldlines]] (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the [[anyon]]-[[braiding]] considered above. An argument that these nodal lines in 3d space, nevertheless, may be controlled by [[Chern-Simons theory]]: * {#LianVafaVafaZhang17} [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], Chern-Simons theory and Wilson loops in the Brillouin zone, Phys. Rev. B 95 (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]
anyonic topological order via braided fusion categories -- references	https://ncatlab.org/nlab/source/anyonic+topological+order+via+braided+fusion+categories+--+references	### Anyonic topological order in terms of braided fusion categories {#AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences} #### Claim and status In [[condensed matter theory]] it is [[folklore]] that species of [[anyon\|anyonic]] [[topological order]] correspond to [[braided monoidal categories\|braided]] [[unitary fusion category\|unitary]] [[fusion categories]]/[[modular tensor categories]]. The origin of the claim is: * {#Kitaev06} [[Alexei Kitaev]], Section 8 and Appendix E of: Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 $[$[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)$]$ Early accounts re-stating this claim (without attribution): * [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], pp. 28 of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) $[$[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)$]$ * [[Zhenghan Wang]], Section 6.3 of: Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS (2010) $[$[ISBN-13: 978-0-8218-4930-9](http://www.ams.org/publications/authors/books/postpub/cbms-112), [pdf](http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf)$]$ Further discussion (mostly review and mostly without attribution): * [[Simon Burton]], A Short Guide to Anyons and Modular Functors $[$[arXiv:1610.05384](https://arxiv.org/abs/1610.05384)$]$ > (this one stands out as still attributing the claim to [Kitaev (2006), Appendix E](#Kitaev06)) * {#RowellWang18} [[Eric C. Rowell]], [[Zhenghan Wang]], _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605)) * [[Tian Lan]], A Classification of (2+1)D Topological Phases with Symmetries $[$[arXiv:1801.01210](https://arxiv.org/abs/1801.01210)$]$ * From categories to anyons: a travelogue $[$[arXiv:1811.06670](https://arxiv.org/abs/1811.06670)$]$ * [[Colleen Delaney]], A categorical perspective on symmetry, topological order, and quantum information (2019) $[$[pdf](https://crdelane.pages.iu.edu/dissertationch1-5.pdf), [uc:5z384290](https://escholarship.org/uc/item/5z384290)$]$ * [[Colleen Delaney]], Lecture notes on modular tensor categories and braid group representations (2019) $[$[pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf), [[DelaneyModularTensorCategories.pdf:file]]$]$ * Liang Wang, [[Zhenghan Wang]], In and around Abelian anyon models, J. Phys. A: Math. Theor. 53 505203 (2020) $[$[doi:10.1088/1751-8121/abc6c0](https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0)$]$ * [[Parsa Bonderson]], Measuring Topological Order, Phys. Rev. Research 3, 033110 (2021) $[$[arXiv:2102.05677](https://arxiv.org/abs/2102.05677), [doi:10.1103/PhysRevResearch.3.033110](https://doi.org/10.1103/PhysRevResearch.3.033110)$]$ * Zhuan Li, Roger S.K. Mong, Detecting topological order from modular transformations of ground states on the torus $[$[arXiv:2203.04329](https://arxiv.org/abs/2203.04329)$]$ * [[Eric C. Rowell]], Braids, Motions and Topological Quantum Computing $[$[arXiv:2208.11762](https://arxiv.org/abs/2208.11762)$]$ * [[Sachin Valera]], A Quick Introduction to the Algebraic Theory of Anyons, talk at [[CQTS]] Initial Researcher Meeting (Sep 2022) $[$[[CQTS-InitialResearcherMeeting-Valera-220914.pdf:file]]$]$ * Willie Aboumrad, Quantum computing with anyons: an F-matrix and braid calculator $[$[arXiv:2212.00831](https://arxiv.org/abs/2212.00831)$]$ Emphasis that the expected description of [[anyons]] by [[braided fusion categories]] had remained [[folklore]], together with a list of minimal assumptions that would need to be shown: * {#Valera21} [[Sachin J. Valera]], Fusion Structure from Exchange Symmetry in (2+1)-Dimensions, Annals of Physics 429 (2021) $[$ [arXiv:2004.06282](https://arxiv.org/abs/2004.06282), [doi:10.1016/j.aop.2021.168471](https://doi.org/10.1016/j.aop.2021.168471)$]$ An argument that the statement at least for [[SU(2)-anyons]] does follow from an enhancement of the [[K-theory classification of topological phases of matter]] to interacting [[topological order]]: * [[Hisham Sati]], [[Urs Schreiber]], [[schreiber:Anyonic topological order in TED K-theory]], Rev. Math. Phys. (20223) $[$[arXiv:2206.13563](https://arxiv.org/abs/2206.13563), [doi:10.1142/S0129055X23500010](https://doi.org/10.1142/S0129055X23500010)$]$ #### Further discussion Relation to [[ZX-calculus]]: * [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$ On detection of [[topological order]] by observing [[modular transformations]] on the [[ground state]]: * [[Zhuan Li]], [[Roger S. K. Mong]], Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 $[$[doi:10.1103/PhysRevB.106.235115](https://doi.org/10.1103/PhysRevB.106.235115), [arXiv:2203.04329](https://arxiv.org/abs/2203.04329)$]$ See also: * [[Liang Kong]], Topological Wick Rotation and Holographic Dualities, [talk at](CQTS#KongOct2022) [[CQTS]] (Oct 2022) $[$[[Kong-TalkAtCQTS-20221019.pdf:file]]$]$
anyons in the quantum Hall effect -- references	https://ncatlab.org/nlab/source/anyons+in+the+quantum+Hall+effect+--+references	### Anyons in the quantum Hall liquids {#AnyonsInTheQuantumHallEffectReferences} References on [[anyon]]-excitations (satisfying [[braid group statistics]]) in the [[quantum Hall effect]] (for more on the application to [[topological quantum computation]] see the references [there](topological+quantum}computation#TopologicalQuantumComputationWithAnyons)): The prediction of [[abelian group\|abelian]] [[anyon]]-excitations in the [[quantum Hall effect]] (i.e. satisfying [[braid group statistics]] in 1-dimensional [[linear representations]] of the [[braid group]]): * B. I. Halperin, _Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States_, Phys. Rev. Lett. 52, 1583 (1984) ([doi:10.1103/PhysRevLett.52.1583](https://doi.org/10.1103/PhysRevLett.52.1583)) Erratum Phys. Rev. Lett. 52, 2390 (1984) ([doi:10.1103/PhysRevLett.52.2390.4](https://doi.org/10.1103/PhysRevLett.52.2390.4)) * Daniel Arovas, J. R. Schrieffer, [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$ The original discussion of [[non-abelian group\|non-abelian]] [[anyon]]-excitations in the [[quantum Hall effect]] (i.e. satisfying [[braid group statistics]] in higher dimensional [[linear representations]] of the [[braid group]], related to [[modular tensor categories]]): * [[Gregory Moore]], [[Nicholas Read]], _Nonabelions in the fractional quantum Hall effect_, Nucl. Phys. 360B(1991)362 ([pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf), <a href="https://doi.org/10.1016/0550-3213(91)90407-O">doi:10.1016/0550-3213(91)90407-O</a>) Review: * [[Ady Stern]], _Anyons and the quantum Hall effect -- A pedagogical review_, Annals of Physics Volume 323, Issue 1, January 2008, Pages 204-249 ([doi:10.1016/j.aop.2007.10.008](https://doi.org/10.1016/j.aop.2007.10.008), [arXiv:0711.4697](https://arxiv.org/abs/0711.4697))
anyons in topological superconductors -- references	https://ncatlab.org/nlab/source/anyons+in+topological+superconductors+--+references	### Anyons in topological superconductors {#AnyonsInTopologicalSuperconductorsReferences} On [[anyon]]-excitations in [[topological superconductors]]. via [[Majorana zero modes]]: Original proposal: * [[Nicholas Read]], Dmitry Green, _Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect_, Phys. Rev. B61:10267, 2000 ([arXiv:cond-mat/9906453](https://arxiv.org/abs/cond-mat/9906453)) Review: * {#DasSarmaFreedmanNayak15} [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], _Majorana Zero Modes and Topological Quantum Computation_, npj Quantum Information 1, 15001 (2015) ([nature:npjqi20151](https://www.nature.com/articles/npjqi20151)) * Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, _The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview_ ([arXiv:2009.07764](https://arxiv.org/abs/2009.07764)) * Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors [[arXiv:2301.11614](https://arxiv.org/abs/2301.11614)] Further developments: * [[Meng Cheng]], Victor Galitski, [[Sankar Das Sarma]], _Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors_, Phys. Rev. B 84, 104529 (2011) ([arXiv:1106.2549](https://arxiv.org/abs/1106.2549)) * {#ShabaniEtAl15} [[Javad Shabani]] et al., Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks, Phys. Rev. B 93 155402 (2016) \[<a href="https://doi.org/10.1103/PhysRevB.93.155402">doi:10.1103/PhysRevB.93.155402</a>, [arXiv:1511.01127](https://arxiv.org/abs/1511.01127)\] * {#ShabaniEtAl17} [[Javad Shabani]] et al., Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform, Phys. Rev. Lett. 119 (2017) 176805 \[<a href="https://doi.org/10.1103/PhysRevLett.119.176805">doi:10.1103/PhysRevLett.119.176805</a>, [arXiv:1703.03699](https://arxiv.org/abs/1703.03699)\] * {#ShabaniEtAl22} [[Javad Shabani]] et al., Fusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems, Nature Communications 13 (2022) 1738-1747 \[<a href="https://doi.org/10.1038/s41467-022-29463-6">doi:10.1038/s41467-022-29463-6</a>, [arXiv:2101.09272](https://arxiv.org/abs/2101.09272)\] * {#ShabaniEtAl23} [[Javad Shabani]] et al., Quasiparticle dynamics in epitaxial Al-InAs planar Josephson junctions, PRX Quantum 4 030339 (2023) \[<a href="https://doi.org/10.1103/PRXQuantum.4.030339">doi:10.1103/PRXQuantum.4.030339</a>, [arXiv:2303.04784](https://arxiv.org/abs/2303.04784)\] via [[Majorana zero modes]] restricted to edges of [[topological insulators]]: * Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, _Topological quantum computation based on chiral Majorana fermions_, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 ([doi:10.1073/pnas.1810003115](https://doi.org/10.1073/pnas.1810003115)) See also: * Yusuke Masaki, Takeshi Mizushima, [[Muneto Nitta]], Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors $[$[arXiv:2301.11614](https://arxiv.org/abs/2301.11614)$]$
apartness relation	https://ncatlab.org/nlab/source/apartness+relation	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Relations +-- {: .hide} [[!include relations - contents]] =-- =-- =-- # Apartness relations * table of contents {: toc} ## Idea An _apartness relation_ is a [[binary relation]] that, instead of saying when two things are the same (as an [[equivalence relation]]), states when two things are different -- an [[inequality relation]]. Apartness relations are most used in [[constructive mathematics]]; in [[classical mathematics]], equivalence relations can take their place (mediated by [[negation]]). The apartness relations that we discuss here are sometimes called __point--point apartness__, to distinguish this from the related concepts of _set--set_ or _point--set_ apartness relations; see [[proximity space]] and [[apartness space]] (respectively) for these. ## Definitions ### Abstract A [[set]] $S$ equipped with an __apartness relation__ is a [[groupoid]] (with $S$ as the set of [[objects]]) [[enriched category\|enriched]] over the [[cartesian monoidal category]] $TV^\op$, that is the [[opposite category\|opposite]] of the [[partial order\|poset]] of [[truth value\|truth values]], made into a [[monoidal category]] using [[disjunction]]. By the law of [[excluded middle]] (which says that $TV$ is self-dual under [[negation]]), this is equivalent to equipping $S$ with an [[equivalence relation]] (which makes $S$ a groupoid [[enriched category\|enriched]] over the cartesian category $TV$ itself). But in [[constructive mathematics]] (or interpreted [[internalization\|internally]]), it is a richer concept with a topological flavour, as $TV^\op$ is a [[co-Heyting algebra]]. ### Concrete Of course, nobody but a [[category theory\|category-theorist]] would use the above as a definition of an apartness relation. Normally, one defines an apartness relation on $S$ as a [[binary relation]] $\#$ satisfying these three properties: * [[irreflexive relation\|irreflexivity]]: for all $x: S$, $x \# x$ is false; * [[symmetric relation\|symmetry]]: for all $x, y: S$, if $y # x$, then $x # y$; * [[comparison]]: for all $x, y, z: S$, if $x # z$, then $x # y$ or $y # z$. (Notice that these are dual to the axioms for an [[equivalence relation]]; like those axioms, these correspond to [[identity morphisms]], [[inverses]], and [[composition]] in a groupoid.) ### Related notions The [[negation]] of an apartness relation is an equivalence relation. (On the other hand, the statement that every equivalence relation is the negation of some apartness relation is equivalent to [[excluded middle]], and the statement that the negation of an equivalence relation is always an apartness relation is equivalent to the nonconstructive [[de Morgan law]].) An apartness relation is a [[tight apartness relation]] if this equivalence relation is [[equality]]; any apartness relation defines a [[tight apartness relation]] on the [[quotient set]]. If $S$ and $T$ are both sets equipped with apartness relations, then a [[function]] $f\colon S \to T$ is __[[strongly extensional function\|strongly extensional]]__ if $x \# y$ whenever $f(x) \# f(y)$; that is, $f$ reflects apartness. The strongly extensional functions are precisely the [[enriched functors]] between $TV^\op$-enriched groupoids, so they are the correct morphisms. (Note that there is no nontrivial notion of enriched [[natural isomorphism]], at least not when the apartness in $T$ is tight.) ## (2,1)-category of sets with apartness relations Sets with apartness relations, strongly extensional functions, and equivalences of strongly extensional functions, which serve to identify unequal but equivalent (that is, not apart) elements of a set, form a locally thin [[(2,1)-category]]; i.e. a [[bicategory]] [[enriched bicategory\|enriched]] in [[thin groupoids]]. This bicategory is [[locally small category\|locally small]] and a [[univalent bicategory]]. ## Topological aspects {#topology} ### The apartness topology Let $S$ be a set equipped with an apartness relation $\ne$. Using $\ne$, many [[topology\|topological]] notions may be defined on $S$. (Often one assumes that the apartness is tight; this corresponds to the $T_0$ [[separation axiom]] in topology.) If $U$ is a [[subset]] of $S$ and $x$ is an element, then $U$ is a $\ne$-[[neighbourhood]] (or $\ne$-neighborhood) of $x$ if, given any $y\colon S$, $x \ne y$ or $y \in U$; note that $x \in U$ by irreflexivity. The neighbourhoods of $x$ form a [[filter]]: a superset of a neighbourhood is a neighbourhood, and the intersection of $0$ or $2$ (hence of any finite number) of neighbourhoods is a neighbourhood. A subset $G$ is $\ne$-[[open subset\|open]] if it\'s a neighbourhood of all of its members. The open subsets form a [[topological space\|topology]] (in the sense of Bourbaki): any union of open subsets is open, and the intersection of $0$ or $2$ (hence of any finite number) of open subsets is open. The $\ne$-complement of $x$ is the subset $\{y\colon S \;\|\; x \ne y\}$; this is open by comparison. More generally, the $\ne$-[[complement]] of any subset $A$ is the set $\tilde{A}$, defined as: $$ \tilde{A} \coloneqq \{y \;\|\; \forall{x} \in A,\; x \ne y\} .$$ This is not in general open, but you would use it where you would classically use the set-theoretic complement. However, if $A$ is open to begin with, then $\tilde{A}$ equals the set-theoretic complement. If $x \ne y$, then $x \in \tilde{y}$ and $y \in \tilde{x}$. Thus, if $\ne$ is tight, then $(S, \ne)$ satisfies the $T_1$ [[separation axiom]]. Symmetry is important here; if we removed symmetry from the axioms of apartness (obtaining a [[quasi-apartness]]) but retained tightness, then we would still get a $T_0$ topology, but it would not be $T_1$. This is a version of the fact that failure of $T_1$ is given by a [[partial order]] (or a [[preorder]] if $T_0$ might also fail). The $\ne$-[[closure]] $\bar{A}$ of a subset $A$ is the complement of its complement. This closure is a [[closure operator]]: $A \subset \bar{A}$, $\overline{\bar{A}} = \bar{A}$ (in fact, $\overline{\tilde{A}} = \tilde{A}$), $\bar{A} \subset \bar{B}$ whenever $A \subset B$, $\overline{S} = S$ (in fact, $\bar{\empty} = \empty$ too), and $\overline{A \cap B} = \bar{A} \cap \bar{B}$ (but not $\overline{A \cup B} = \bar{A} \cup \bar{B}$). The __antigraph__ of a function $f\colon S \to T$ is $$ \{(x,y) \;\|\; x\colon S, y\colon T \;\|\; f(x) \ne y\} .$$ Recall that in ordinary topology, a function between [[Hausdorff spaces]] is continuous iff its [[graph of a function\|graph]] is closed. Similarly, a function $f\colon S \to T$ is strongly extensional iff its antigraph is open. (Then the [[graph of a function\|graph]] of $f$ is the complement of the antigraph.) One important topological concept that doesn\'t appear classically is locatedness; in an inequality space, a subset $A$ is __[[located subspace\|located]]__ if, given any point $x$ and any neighbourhood $U$ of $x$, either $U \cap A$ is [[inhabited set\|inhabited]] (that is, it has a point) or some neighbourhood of $x$ (not necessarily $U$) is contained in $\tilde A$. Note that every point is located. (For an example of a set that need not be located, consider $\{x\colon S \;\|\; p\}$, where $p$ is an arbitrary [[truth value]]. In an inhabited space, this set is located iff $p$ is true or false.) ### Relation to metric spaces Recall that, as [[Bill Lawvere]] taught us, a [[metric space]] is a groupoid (or $\dagger$-[[dagger-category\|category]]) enriched over the category $([0,\infty[^\op,+)$ of nonnegative [[real numbers]], ordered in reverse, and made monoidal under addition. (Actually, you get a metric only if you impose a tightness condition, although again you can recover this up to equivalence from the $2$-morphisms. Furthermore, Lawvere advocated using $[0,\infty]$ instead of $[0,\infty[$, and also dropping the symmetry requirement to get enriched categories instead of groupoids. Thus, he dealt with extended quasipseudometric spaces. These details are not really important here.) There is a [[monoidal functor]] from $([0,\infty[^\op,+)$ to $TV^\op$ that maps a nonnegative real number $x$ to the truth value of the statement that $x \gt 0$. Accordingly, any (symmetric) metric space becomes an inequality space, and any function satisfying $d(f(x),f(y)) \leq d(x,y)$) is strongly extensional. The topological properties of metric spaces fit well with those of inequality spaces if you always work in this direction. For example, a set which is $d$-open will also be $\ne$-open, but not necessarily the other way around. Similarly, a (merely) [[continuous function]] between metric spaces is (still) strongly extensional. ### Relation to gauge spaces and uniform spaces In analysis, many spaces are given as [[gauge spaces]], that is by families of pseudometrics; these also become inequality spaces by declaring that $x \ne y$ iff $d(x,y) \gt 0$ for some pseudometric $d$ in the family. (This will actually be a tight apartness iff the family of pseudometrics is separating.) Classically, any [[uniform space]] may be given by a family of pseudometrics, but this doesn\'t hold constructively. In particular, a topological group may not be an inequality group (as in the next section). However, we can generalize a bit beyond gauge spaces: any [[uniformly regular uniform space]] becomes an inequality space by declaring that $x \ne y$ iff there is an entourage $U$ with $(x,y)\notin U$. (If the uniform space is not uniformly regular, the result is merely an [[inequality relation]], not an apartness.) ### Relation to proximity spaces The constructive theory of [[proximity spaces]] is based on a generalisation of apartness relations (which here go between points) to an apartness relation between sets. These are called _[[apartness space\|apartness spaces]]_; just as apartness relations (between points) are classically equivalent to equivalence relations, so apartness spaces are classically equivalent to proximity spaces, with two sets being proximate if and only if they are not apart. Of course, any apartness space has an apartness relation between points: $x$ and $y$ are apart iff $\{x\}$ and $\{y\}$ are apart. ### Relation to locales Let $X$ be a set, regarded as a [[discrete locale]], whose [[frame]] of opens is $O(X) = P(X)$, the [[power set]] of $X$. That is, the opens in the locale $X$ are precisely the subsets of the set $X$. Since discrete locales are [[locally compact locale\|locally compact]] (every set is the union of its [[K-finite set\|K-finite]] subsets), the locale product $X\times X$ agrees with the spatial product, so that $X\times X$ is also discrete and every subset of $X\times X$ is open. Thus, the opens in the locale $X\times X$ are precisely the subsets of $X\times X$. In particular, an [[equivalence relation]] on the set $X$ can be identified with an open equivalence relation (in [[Loc]]) on the discrete locale $X$. Thus, the following theorem gives a different precise sense in which apartness relations are dual to equivalence relations. +--{: .num_theorem #ClosedLocalicEquivalenceRelation} ###### Theorem An apartness relation on a set $X$ is the same as a (strongly) [[closed subspace\|closed]] equivalence relation on the discrete locale $X$. Moreover, the apartness topology defined above is, as a locale, the quotient of this equivalence relation. =-- +--{: .proof} ###### Proof By definition, a (strongly) closed sublocale of a locale $Y$ is one of the form $\mathsf{C}U$, for some open $U\in O(Y)$. Thus, when $X$ is a discrete locale, a closed sublocale of $X\times X$ is of the form $\mathsf{C}U$ for some subset $U$ of $X\times X$. This subset is the extension of the apartness relation, i.e. $U = \{ (x,y) \mid x\#y \}$. For the first claim, therefore, it remains to show that the three axioms of an equivalence relation for $\mathsf{C}U$ correspond to the apartness axioms for $\#$. Note that pullback along locale maps respects closed complements, i.e. $f^(\mathsf{C}U) = \mathsf{C}(f^U)$. Thus, the pullback of $\mathsf{C}U$ along the twist map $X\times X \to X\times X$ is the closed sublocale corresponding to the twist of $U$, i.e. the set $\{ (x,y) \mid y\#x \}$. Since $\mathsf{C}$ is a contravariant order-isomorphism between the posets of open and closed sublocales, symmetry for $\mathsf{C}U$ is equivalent to symmetry for $\#$. Similarly, pulling $\mathsf{C}U$ back to $X\times X\times X$ along one of the three canonical projections gives the closed sublocale dual to the corresponding pullback of $U$ itself, and $\mathsf{C}$ transforms unions to intersections; thus transitivity for $\mathsf{C}U$ is equivalent to comparison for $\#$. Finally, the pullback of $\mathsf{C}U$ along the diagonal is the closed sublocale dual to the similar pullback of $U$, so to say that the former is all of $X$ is equivalent to saying that the latter is $\emptyset$; thus reflexivity for $\mathsf{C}U$ is equivalent to irreflexivity for $\#$. Now, the quotient in $Loc$ of such an an equivalence relation in particular comes equipped with a surjective locale map from $X$. Thus, it is a spatial locale and can be regarded as a topology on the set $X$. Moreover, quotients in $Loc$ are constructed as [[equalizers]] in $Frm$, so we have to compute the equalizer of the two maps $O(X) = P(X) \to O(\mathsf{C}U)$, where $O(\mathsf{C}U)$ is the frame of opens of $\mathsf{C}U$ regarded as a locale in its own right. Equivalently, this means the equalizer of the two maps $P(X) \xrightarrow{\pi_i} P(X\times X) \xrightarrow{j_{\mathsf{C}U}} P(X\times X)$, where $j_{\mathsf{C}U}$ is the [[nucleus]] corresponding to $\mathsf{C}U$. Now by definition, $j_{\mathsf{C}U}(V) = V\cup U$. Thus, the elements of this equalizer --- which is to say, the opens in the locale quotient --- are subsets $V$ of $X$ such that $(V\times X) \cup U = (X\times V) \cup U$. Reexpressed in terms of $\#$, that means that for any $x,y\in X$ we have $(x\in V \vee x\#y) \iff (y\in V \vee x\#y)$. But since $\#$ is symmetric, this is equivalent to the unidirectional implication $(x\in V \vee x\#y) \to (y\in V \vee x\#y)$, and since $x\#y$ always implies itself, this is equivalent to $x\in V \to (y\in V \vee x\#y)$, which is precisely the condition defining the open sets in the apartness topology above. =-- Recall that the negation of an apartness relation on $X$ is an equivalence relation on the set $X$. This is the spatial part of the above closed localic equivalence relation, which in general (constructively) need not be itself spatial. The apartness relation is tight just when this spatial part is the diagonal. (By contrast, to say that the closed localic equivalence relation is itself the diagonal is to say that the discrete locale $X$ is [[Hausdorff space\|Hausdorff]], which is only true if $X$ has [[decidable equality]].) Another characterization of the $\#$-open sets is that $U$ is $\#$-open if $U\times X \subseteq (X\times U) \cup W_\#$, where $W_\#$ is $\#$ regarded as a subset of $X\times X$. Rephrased in terms of complementary closed sublocales, this says that $\mathsf{C}U$ is "closed under the equivalence relation" dual to $\#$. Thus, the closed sublocales of $X$ with its $\#$-topology (i.e. the formal complements of $\#$-open sets) correspond precisely to the closed sublocales of $X$ (the formal complements of arbitrary subsets of $X$) that respect this equivalence relation. As a partial converse to the above theorem, if $X$ is a [[Hausdorff space\|localically strongly Hausdorff]] topological space, meaning that its diagonal is a strongly closed sublocale, then the pullback of this diagonal to the discrete locale on the set of points of $X$ is a closed localic equivalence relation, hence an apartness, whose $\ne$-topology refines the given topology. See [this theorem](/nlab/show/Hausdorff+space#Apartness). If we are given an apartness relation $\ne$, it is unclear whether the $\ne$-topology is localically strongly Hausdorff; but if it is, then the apartness relation resulting from this topology is stronger than the given $\ne$. ## Related concepts * [[inequality relation]] * [[denial inequality]] * [[antisubalgebra]] * [[inequality space]] [[!include logic symbols -- table]] ## References According to [Troelstra and van Dalen](#TvD): > Brouwer ([1919](#Brouwer1919)) introduced the notion of apartness (örtlich verschieden, Entfernung).... The axioms of the theory of apartness were formulated by Heyting ([1925](#Heyting1925)). * {#Brouwer1919} [[L.E.J. Brouwer]], _Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten II: Theorie der Punktmengen. 1919 * {#Heyting1925} [[Arend Heyting]], _Intuïtionistische Axiomatiek der Projectieve Meetkunde_ (Dutch), Ph.D. Thesis, 1925 * [[Errett Bishop]]'s _Foundations of Constructive Analysis_ (1967) uses apartness for the real numbers and more general metric spaces. * The 1985 edition with Douglas Bridges, _Constructive Analysis_, includes the general definition of apartness relation, there called an "inequality relation" (though in many other sources, as here, an [[inequality relation]] need not satisfy comparison). * {#TvD} Anne Troelstra\'s and Dirk van Dalen\'s _Constructivism in Mathematics_ (1988) uses apartness for the reals (volume 1), and general apartness relations in algebra (volume 2, chapter 8). They say "preapartness" and "apartness" instead of "apartness" and "tight apartness" respectively. * Apartness plays a minimal role in _A Course in Constructive Algebra_ (also 1988), by Ray Mines, Fred Richman, and Wim Ruitenburg. * A great reference for point-set topology in constructive mathematics is the Ph.D. thesis of Frank Waaldijk, _[Modern Intuitionist Topology](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf)_ (1996). 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apartness space	https://ncatlab.org/nlab/source/apartness+space	# Apartness spaces * table of contents {: toc} ## Idea An apartness space is a set equipped with an "apartness relation" that distinguishes between pairs of points or sets. They are particularly interesting in [[constructive mathematics]]; in [[classical mathematics]] they tend to be equivalent to better-known definitions expressed in terms of "closeness" rather than "apartness". ## Definitions There are actually several different notions of "apartness space" depending on whether the objects being compared on each side are points or sets. * A _point--point apartness space_ is a set $X$ equipped with an _[[apartness relation]]_, usually written $x # y$ on elements $x,y\in X$. Sometimes it is required to be [[tight relation\|tight]], or to be only an [[inequality relation]]. * A __point--set apartness space__ is a set $X$ equipped with a relation $x \bowtie A$ between points $x\in X$ and subsets $A\subseteq X$, satisfying appropriate axioms. In [[classical mathematics]], these axioms are obtained by [[contrapositive\|contraposition]] from the definition of a [[topological space]] in terms of a [[exact functor\|right exact]] [[Moore closure]] operator, so that point--set apartness spaces are equivalent to topological spaces. In [[constructive mathematics]] ... well, keep reading. * A _set--set apartness space_ is a set $X$ equipped with a relation $A\bowtie B$ between [[subsets]] $A,B\subseteq X$, satisfying appropriate axioms. A set--set apartness space is one of the ways to describe the [[classical mathematics\|classical]] notion of a _[[proximity space]]_. In [[constructive mathematics]], the definition of a proximity space in terms of $\bowtie$ can be taken as a definition of a set--set apartness space. * A _uniform apartness space_ is a set $X$ equipped with a collection of "co-[[entourages]]", each giving a different notion of when two points are apart, satisfying appropriate axioms. In [[classical mathematics]], the co-entourages are exactly the [[complements]] of the entourages of a [[uniform space]], and the same is true constructively if the space is [[uniformly regular]]. Since point--point apartness spaces are described at [[apartness relation]], set--set apartness spaces at [[proximity space]], and uniform apartness spaces at [[uniformly regular space]], the rest of this page will be about point--set apartness spaces. ## Point--set apartness spaces +-- {: .un_defn} ###### Definition An __apartness space__ is a set $X$ equipped with a relation $\bowtie$ between points $x\in X$ and subsets $A\subseteq X$ such that 1. $x\bowtie \emptyset$ for any $x$. 2. if $x\bowtie A$, then $x\neq y$ for all $y\in A$. 3. $x\bowtie (A\cup B)$ iff $x\bowtie A$ and $x\bowtie B$. 4. If $x\bowtie A$, then $x \bowtie \{ y \mid \forall z, (z\bowtie A \to z\neq y) \}$. =-- The relation $x\bowtie A$ should be regarded as a "positive" way of saying that $x$ does not belong to the [[closure]] of $A$, i.e. $x\notin \overline{A}$. Under this interpretation, the above axioms contrapose to become 1. $\overline{\emptyset} = \emptyset$. 2. If $x\in A$, then $x\in \overline{A}$, i.e. $A\subseteq \overline{A}$. 3. $\overline{A} \cup \overline{B} = \overline{A\cup B}$ (and in particular $(A\subseteq B) \to (\overline{A}\subseteq \overline{B})$). 4. If $B\subseteq \overline{A}$ and $x\in \overline{B}$, then $x\in\overline{A}$, i.e. $\overline{\overline{A}} = \overline{A}$. which are precisely the axioms of a [[topology]] expressed in terms of a closure operator. In constructive mathematics, of course, the law of contraposition does not hold. The axiom $x\bowtie \emptyset$ is almost unnecessary, since the last axiom ensures that if $x\bowtie A$ for any set $A$ then $x\bowtie\emptyset$. In particular, this is the case if $X$ is $T_1$ (see below) and for any $x\in X$ there is a $y\in X$ with $x\neq y$. The above definition is almost exactly that of a "pre-apartness" from [BV11](#BV11). The differences are (1) they require $X$ to be [[inhabited set\|inhabited]] (which category-theoretically is wrong-headed, since it excludes the [[initial object]]), and (2) they assume that $X$ is given with an ambient [[inequality relation]] that is referred to by the symbol $\neq$ in axioms 2 and 4. (If $\neq$ is the [[denial inequality]], then these axioms can be written more simply as "if $x\bowtie A$ then $x\notin A$" and "if $x\bowtie A$ then $x\bowtie \{ y \mid \neg(y\bowtie A) \}$".) Note that if the space is $T_1$ (see below) then this ambient inequality is definable in terms of $\bowtie$ as $x\bowtie \{y\}$. For an "apartness", [BV11](#BV11) also require comparability (see below). Axiom 4 is phrased in [BV11](#BV11) as "if $\forall x, (x\bowtie A \Rightarrow (\forall y\in B, x\neq y))$, then $\forall x, (x\bowtie A \Rightarrow x\bowtie B)$. This is equivalent to our version, since $B = \{ y \mid \forall z, (z\bowtie A \to z\neq y) \}$ is the largest set $B$ satisfying $\forall x, (x\bowtie A \Rightarrow (\forall y\in B, x\neq y))$. The earlier paper [BSV02](#BSV02) omits the axiom $x\bowtie \emptyset$, and phrases axiom 2 with the denial inequality but axiom 4 with an ambient inequality --- although these are probably oversights --- and requires $T_1$ as part of the definition too (see below). An earlier, more predicative presentation of "apartness spaces" can be found in [Waaldijk](#FW96). ## Separation properties Any apartness space comes with an [[irreflexive relation]] $\nleq$ defined by $x \nleq y$ iff $x \bowtie \{y\}$. This is a positive version of the negation of the [[specialization order]]. A topological space is called $T_1$ (see [[separation axioms]]) if its specialization order is discrete, i.e. $(x\le y) \to (x=y)$; thus an apartness space is called $T_1$ if $\neg(x\nleq y) \to (x=y)$. We could contrapose this to obtain $\neg(x=y) \to (x\nleq y)$, but it is usually too strong constructively to have a denial statement imply a positive one. However, if we replace $\neg(x=y)$ with a given [[apartness relation]] or [[inequality relation]] other than the [[denial inequality]], then we obtain an axiom $(x\neq y) \to (x\nleq y)$ that is a purely positive version of $T_1$. Note that if $\neq$ is [[tight relation\|tight]], then this statement implies the negative version of $T_1$, while conversely if $\nleq$ is symmetric (see below) then the negative version of $T_1$ says precisely that $\nleq$ is tight. Note that axiom 2 implies the converse $(x\bowtie \{y\}) \to (x\neq y)$, so that if the axioms and the $T_1$ property are stated with reference to some ambient inequality $\neq$, then $\neq$ can be defined in terms of $\bowtie$. If we define $\neq$ in this way, then axiom 2 should be stated in terms of the denial inequality (thereby asserting that this relation $x\bowtie \{y\}$ is [[irreflexive]]). If one wants the relation $x\bowtie \{y\}$ to be [[symmetric relation\|symmetric]] and thus an "[[inequality relation]]" one needs to assert this separately. An apartness space with this property is naturally called $R_0$, or perhaps strongly $R_0$, since it implies (and classically is equivalent to) the topological property called $R_0$ that the specialization order is symmetric (see [[separation axioms]]). However, stating axiom 4 in terms of the $\bowtie$-derived inequality is weaker, even in [[classical mathematics]], than stating it in terms of the denial inequality. For instance, if $X = \{x,y,z\}$ with the only nonempty apartness relations $x\bowtie \{z\}$ and $z\bowtie \{x\}$, then axiom 4 for the denial inequality fails for $A=\{z\}$, since $\{ y \mid \neg(y\bowtie A)\} = \{y,z\}$ which is not apart from $x$; but stated in terms of the $\bowtie$-derived inequality it holds, since $\{ v \mid \forall u, (u\bowtie A \Rightarrow u \bowtie \{v\})\} = \{z\}$. This is a [[pretopological space]] that is not a topological space; thus only axiom 4 with the denial inequality (or at least a [[tight inequality]], which classically becomes equivalent to denial) gives a notion of point-set apartness space that reduces classically to ordinary toplogical spaces. In [BV11](#BV11), axiom 4 for the denial inequality is called the reverse Kolmogorov property. An apartness space may be called comparable (nonce definition on this page) if $x\bowtie A$ implies $(x\neq y) \vee (y\bowtie A)$ for any $y$, where $\neq$ might also be a given apartness on $X$. This condition is classically trivial, and generalizes the [[comparison]] axiom on a point--point [[apartness relation]]. In particular, if $\neq$ denotes the relation $\nleq$ defined above, then this property implies that $\nleq$ is a [[comparison]], and hence (if the space is also $R_0$, so it is symmetric) an [[apartness relation]]. It is also related topologically to Penon's definition of intrinsic [[open subset]] in [[synthetic topology]] and to the natural topology on a [[point-point apartness space]], and can be viewed as a very weak version of [[regular space\|regularity]]. An apartness space is [[locally decomposable space\|locally decomposable]] if whenever $x\bowtie A$, there exists a set $B$ such that $x\bowtie B$ and every $y$ satisfies either $y\bowtie A$ or $y\in B$. This condition is also classically trivial: take $B = \{ y \mid \neg(y\bowtie A) \}$. It implies comparability (for $\neq$ the [[denial inequality]]). ## Relation to topological spaces If $X$ is a topological space, we define $x\bowtie A$ if there is a neighborhood of $x$ disjoint from $A$, or equivalently if the complement of $A$ is a neighborhood of $x$. This makes $X$ an apartness space. Only the last axiom is somewhat nontrivial: if $x\in U$ with $U$ open and $U\cap A = \emptyset$, and $\forall y, (y\bowtie A \to y\notin B)$, then since $(y\in U) \Rightarrow (y\bowtie A)$ we have $U\cap B = \emptyset$ too, so $x\bowtie B$. Conversely, if $X$ is an apartness space, define $U\subseteq X$ to be a neighborhood of $x\in U$ if there is a set $A$ such that $x\bowtie A$ and $\forall y, (y\bowtie A \Rightarrow y\in U)$. This makes $X$ a topological space. Again the nontrivial part is the "transitivity" of neighborhoods, i.e. that if $U$ is a neighborhood of $x$ then so is $\{ y \mid U$ is a neighborhood of $y \}$. To see this, if $x\bowtie A$ and $\forall y, (y\bowtie A \Rightarrow y\in U)$, then $U$ is a neighborhood of any point $y$ with $y\bowtie A$, so it suffices to show that $\{ y \mid y\bowtie A\}$ is a neighborhood of $x$; but this is obvious. If we order the topologies on $X$ by $\tau_1 \le \tau_2$ if $\tau_2 \subseteq \tau_1$ (i.e. $\tau_1$ is finer than $\tau_2$), and the apartnesses by $\bowtie_1 \le \bowtie_2$ if $(x\bowtie_2 A) \Rightarrow (x\bowtie_1 A)$, then these constructions define [[monotone functions]] $\tau \mapsto \bowtie_\tau$ and $\bowtie \mapsto \tau_\bowtie$ respectively. Moreover, we have: +-- {: .un_theorem} ###### Theorem The above functions exhibit the poset of apartnesses on $X$ as a [[reflective subcategory\|reflective]] sub-poset of the poset of topologies on $X$: we have $\tau \le \tau_{\bowtie_\tau}$ and $\bowtie_{\tau_\bowtie} = \bowtie$. =-- +-- {: .proof} ###### Proof For the former inequality, if $U$ is open in $\tau_{\bowtie_\tau}$, then for every $x\in U$ there is a set $A$ such that $x\bowtie_\tau A$ and $\forall y, (y\bowtie_\tau A \Rightarrow y\in U)$. Since $x\bowtie_\tau A$, there is an open set $V$ with $x\in V$ and $V\cap A = \emptyset$. But then every $y\in V$ satisfies $y\bowtie_\tau A$, hence $y\in U$; so $V\subseteq U$. Thus $U$ is open. For the latter equation, if $x\bowtie A$, then to show $x \bowtie_{\tau_\bowtie} A$ we must construct an open set $U\in \tau_\bowtie$ with $x\in U$ and $U\cap A = \emptyset$; but it suffices to take $U = \{ y \mid y\bowtie A \}$. Conversely, if $x \bowtie_{\tau_\bowtie} A$, then there is an open set $U\in \tau_\bowtie$ with $x\in U$ and $U\cap A = \emptyset$. By definition, $U\in \tau_\bowtie$ and $x\in U$ mean there is a set $B$ with $x\bowtie B$ and $\forall y, (y\bowtie B \Rightarrow y\in U)$. And by the last axiom of an apartness space, to show $x\bowtie A$ it suffices to show $\forall y, (y\bowtie B \to y\notin A)$; but this follows since $y\bowtie B \Rightarrow y\in U$ and $U\cap A = \emptyset$. =-- In other words, an apartness space can be regarded as a well-behaved kind of topological space: one in which for every open neighborhood $U$ of a point $x$ there is a set $A$ and an open neighborhood $V$ of $x$ such that $V\cap A = \emptyset$ and $U$ contains the interior of the complement of $A$. Note that $V$ is then in the interior of the complement of $A$, and if $x$ is in the interior of the complement of $A$ then the latter is such a $V$. Thus, the condition can equivalently be phrased as: for every open neighborhood $U$ of $x$, there is a set $A$ such that $x \in int(\neg A) \subseteq U$. In other words, the interiors of complements form a base for the topology. In [Bridges et. al.](#BSV) this condition is called being topologically consistent. A sufficient condition for topological consistency is local decomposability. This was defined above for apartness spaces; more generally, a topological space is said to be [[locally decomposable space\|locally decomposable]] if for every open neighborhood $U$ of $x$ there is an open neighborhood $V$ of $x$ such that $U \cup \neg V = X$, i.e. every $y\in X$ satisfies $(y\in U)\vee (y\notin V)$. This implies that $x\in int(\neg\neg V) \subseteq \neg\neg V \subseteq U$, giving topological consistency. (Of course, in classical mathematics every space is locally decomposable.) In contrast to the above theorem, it is not quite justified to say that the category of apartness spaces is a subcategory of the category of topological ones. It is most natural to say that a function $f:X\to Y$ between apartness spaces is continuous if for all $x\in X$ and $A\subseteq X$, if $f(x) \bowtie f(A)$ then $x\bowtie A$. It is true that if $X$ and $Y$ are topological spaces and $f$ is topologically continuous, then it is apartness-continuous in this sense for the induced apartnesses. For if $f(x)\bowtie f(A)$, then $f(x) \in U$ for some open set $U$ disjoint from $f(A)$; topological continuity of $f$ then gives that $f^{-1}(U)$ is an open set containing $x$ and disjoint from $A$, so that $x\bowtie A$. Thus we do have a functor $Top \to Apart$. However, a continuous map between apartness spaces in the above sense apparently need not be continuous for the induced topologies; but this is true if $Y$ is locally decomposable. For if $U\subseteq Y$ is open and $x\in f^{-1}(U)$, we have a set $A\subseteq Y$ such that $f(x)\bowtie A$ and $y\bowtie A$ implies $y\in U$, and by local decomposability gives a $B\subseteq Y$ such that $f(x)\bowtie B$ and every $y$ satisfies either $y\bowtie A$ or $y\in B$. Let $C = f^{-1}(B)$; if $x'\bowtie C$, we have $x'\notin C$ and hence $f(x')\notin B$, so $f(x')\bowtie A$ and thus $f(x')\in U$ and $x'\in f^{-1}(U)$. Moreover, since $f(x)\bowtie B$ we have $x\bowtie C$ by apartness-continuity. Thus, $f^{-1}(U)$ is open. Thus, the category of locally decomposable apartness spaces is equivalent to the category of locally decomposable topological spaces. ## References Notions of "apartness space" based on a presentation in terms of basic opens, somewhat akin to [[formal topology]], were developed in: * [[Frank Waaldijk]], Modern intuitionistic topology, Ph.D. thesis, 1996, [link](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf) {#FW96} * [[Frank Waaldijk]], Natural topology, 2011, 2012, [arxiv](https://arxiv.org/abs/1210.6288) The above definition in terms of an apartness relation between points and subsets is slightly adapted from the version given in: * [[Douglas Bridges]], Peter Schuster, and Luminita Vita, Apartness, Topology, and Uniformity: a Constructive View, 2002, [doi](http://dx.doi.org/10.1002/1521-3870%28200210%2948:1%2B%3C16::AID-MALQ16%3E3.0.CO;2-7) {#BSV02} * [[Douglas Bridges]] and Luminita Vita, Apartness and Uniformity: A Constructive Development. 2011, [link](http://link.springer.com/book/10.1007%2F978-3-642-22415-7) {#BV11} [[!redirects apartness space]] [[!redirects apartness spaces]] [[!redirects point-set apartness space]] [[!redirects point-set apartness spaces]] [[!redirects point–set apartness space]] [[!redirects point–set apartness spaces]] [[!redirects point--set apartness space]] [[!redirects point--set apartness spaces]] [[!redirects point-set apartness structure]] [[!redirects point-set apartness structures]] [[!redirects point–set apartness structure]] [[!redirects point–set apartness structures]] [[!redirects point--set apartness structure]] [[!redirects point--set apartness structures]] [[!redirects point-set apartness relation]] [[!redirects point-set apartness relations]] [[!redirects point–set apartness relation]] [[!redirects point–set apartness relations]] [[!redirects point--set apartness relation]] [[!redirects point--set apartness relations]]
apartness-open predicate	https://ncatlab.org/nlab/source/apartness-open+predicate	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +-- {: .hide} [[!include higher algebra - contents]] =-- =-- =-- \tableofcontents ## Idea In [[material set theory]], a [[subset]] $S$ of a [[set]] $A$ with a [[apartness relation]] $\#$ is an apartness-[[open subset]] $S \subseteq A$ if for all $x \in A$ and $y \in A$, $$(x \in S) \implies (y \in S) \vee (x \# y)$$ In [[dependently sorted set theory]], where membership $x \in S$ is not a [[relation]], the above statement that $x \in S$ for every element $x \in A$ in material set theory is equivalently a predicate in the logic $x \in A \vdash P_S(x)$. The given apartness-open subset $S$ is then defined by [[restricted separation]] as the set $S \coloneqq \{x \in A \vert P_S(x)\}$, which in structural set theory automatically comes with an [[injection]] $$i:\{x \in A \vert P_S(x)\} \hookrightarrow A$$ such that $$\exists y \in \{x \in A \vert P_S(x)\}.x = i(y) \iff P_S(x)$$ Hence, the notion of apartness-open predicate, a formulation of the notion of apartness-open as a predicate rather than a subset. ## Definition Given a set $A$ with an [[apartness relation]] $\#$, an $\#$-open predicate is a predicate $x \in A \vdash P_S(x)$ which satisfies $$x \in A, y \in A \vdash P_S(x) \implies P_S(y) \vee (x \# y)$$ The $\#$-open subset $S$ is then defined by [[restricted separation]] as $S \coloneqq \{x \in R \vert P_S(x)\}$ ## See also * [[apartness relation]] * [[predicate]] * [[anti-ideal predicate]] * [[restricted separation]] [[!redirects apartness-open predicate]] [[!redirects apartness-open predicates]]
application of holographic QCD to B-meson physics -- references	https://ncatlab.org/nlab/source/application+of+holographic+QCD+to+B-meson+physics+--+references	### Application of holographic QCD to B-meson physics and flavour anomalies Application of [[holographic QCD]] ([[holographic light front QCD]]) to [[B-meson]] physics and [[flavour anomalies]]: * Ruben Sandapen, [[Mohammad Ahmady]], _Predicting radiative B decays to vector mesons in holographic QCD_ ([arXiv:1306.5352](https://arxiv.org/abs/1306.5352)) * [[Mohammad Ahmady]], R. Campbell, S. Lord, Ruben Sandapen, _Predicting the $B \to \rho$ form factors using AdS/QCD Distribution Amplitudes for the $\rho$ meson_, Phys. Rev. D88 (2013) 074031 ([arXiv:1308.3694](https://arxiv.org/abs/1308.3694)) * [[Mohammad Ahmady]], Dan Hatfield, Sébastien Lord, Ruben Sandapen, _Effect of $c \bar c$ resonances in the branching ratio and forward-backward asymmetry of the decay $B \to K^\ast\mu^+ \mu^-$_ * [[Mohammad Ahmady]], Alexandre Leger, Zoe McIntyre, Alexander Morrison, Ruben Sandapen, _Probing transition form factors in the rare $B \to K^\ast \nu \bar \nu$ decay_, Phys. Rev. D 98, 053002 (2018) ([arXiv:1805.02940](https://arxiv.org/abs/1805.02940)) * [[Mohammad Ahmady]], _Holographic light-front QCD in B meson phenomenology_, PoS DIS2013 (2013) 182 ([arXiv:2001.00266](https://arxiv.org/abs/2001.00266))
application of topology	https://ncatlab.org/nlab/source/application+of+topology	# Applications of Topology There are many applications of [[topology]] as like ... category: topology [[!redirects application of topology]] [[!redirects applications of topology]]
applications of (higher) category theory	https://ncatlab.org/nlab/source/applications+of+%28higher%29+category+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Higher category theory +-- {: .hide} [[!include higher category theory - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea > I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! > A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance. > [[Alexander Grothendieck]], _[[Récoltes et semailles]]_, 1985–1987, pp. 552-3-1 ("[[The Rising Sea]]") > I don't want you to think all this is theory for the sake of it, or rather for the sake of itself. It's theory for the sake of other theory. > ([[Jacob Lurie\|J. Lurie]], [ICM 2010](http://gowers.wordpress.com/2010/08/31/icm2010-spielman-csornyei-lurie/)) The tools of [[category theory]] and [[higher category theory]] serve to organize other structures. There is a plethora of applications that have proven to be much more transparent when employing the [[nPOV]]. Higher category theory has helped foster entire new fields of study that would have been difficult to conceive otherwise. This page lists and discusses examples. ## Examples The following is a (incomplete) list of examples of topics for which higher category theory has proven to be useful. ### In geometry The field of [[differential geometry]] has long managed to avoid the change to an $n$-point of view that had been found to be unavoidable, natural and fruitful in algebraic geometry long ago. But more recently -- not the least due to the recognition of differential [[higher geometry\|higher geometric]] structures in the physics of [[gauge theory]] and [[supergravity]] (such as that of [[orbifold]]s and [[orientifold]]s, of smooth [[gerbe]]s and smooth [[principal ∞-bundle]]s) -- [[sheaf and topos theory\|sheaf and topos theoretic]] concepts, such as [[synthetic differential geometry]], [[diffeological space]]s and [[differentiable stack]]s are gaining wider recognition and appreciation. For instance the ordinary category [[Diff]] of [[smooth manifold]]s fails to have all [[pullback]]s, it only has pullbacks along [[transversal map]]s. This observation is usually the starting point for realizing that differential geometry is in need of a bit of [[category theory]] in the form of [[higher geometry]]. In all notions of [[generalized smooth space]]s all pullbacks do exist. But they may still not be the "right" pullbacks. For instance cohomology of pullback objects may not have the expected properties. This is solved by passing to smooth [[derived stack]]s, such as [[derived smooth manifold]]s. Recent developments in [[higher category theory]], such as the concept of higher [[Structured Spaces]] based on [[Higher Topos Theory]], put all these notions of generalized geometries into a unified picture of [[higher geometry]] that realizes old ideas about how category theory provides a language for [[space and quantity]] in great detail and powerful generality and sheds new light on old [[classical mathematics\|classical]] problems such the description of the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves\|derived moduli stack of derived elliptic curves]] and the construction of the [[tmf]] [[spectrum]] from it. This construction has benefited tremendously from the adoption of the [[nPOV]]. Using this point of view, the general strategy becomes naturally evident. #### In differential equations {#DiffEqu} Much of [[topological vector space]] theory, e.g., the theory of [[distribution]]s, [[nuclear space]]s, etc. has its origins in [[partial differential equation]] theory and is intensely conceptual (categorical) in spirit. It is routine these days to accept distributional solutions, but it wasn't always so, and it was the efficacy of the abstract TVS theory which changed people's minds. Way back Cartan studied differential equations in terms of [[exterior differential system]]s. From the $n$POV, these may be understood naturally as sub [[Lie ∞-algebroid]]s of a [[tangent Lie algebroid]]. [[Bill Lawvere]] noticed in the 1960s that the notion of differential equation makes sense in any [[smooth topos]] (as described [here](http://ncatlab.org/nlab/show/differential+equation#InSynthDiff)). In his highly influential article _Categorical dynamics_ he promoted the point of view that all things [[differential geometry\|differential geometric]] can be formulated in abstract category theory internal to a suitable [[topos]]. This is the origin of [[synthetic differential geometry]]. It may be understood as providing the fundamental characterization of the notion of the [[infinitesimal space\|infinitesimal]]. Closely related to both these perspectives, a modern point of view on differential equations that is proving to be very fruitful regards them as part of the theory of [[D-module]]s. ### In cohomology A multitude of notions of cohomology and its variants are unified from the $n$POV when viewed as [[derived hom space\|∞-categorical hom-spaces]] in [[(∞,1)-topoi]]. See [[cohomology]]. #### Hochschild (co)homology Specifically, the subject of [[Hochschild cohomology]], when generalized to _higher order Hochschild cohomology_ effectively merges into the canonical concept of [[(∞,1)-powering]] of an [[(∞,1)-topos]] over [[∞Grpd]]. See [[Hochschild cohomology]] for details. ### In homotopy theory {#InHomotopyTheory} The study of [[homotopy theory]] originated in the study of categories such as those of [[topological space]]s and other objects such as [[chain complex]]es whose [[morphism]]s were known to admit a notion of [[homotopy]]. Historically, in a sequence of steps formalisms were proposed that would organize the rich interesting structure found in such situations. As a first approximation the notion of [[homotopy category]] and [[derived category]] was introduced in order to deal with structures "up to homotopy". But it was clear that the [[homotopy category]] captured only a very small part of the interesting information. Quillen introduced the notion of [[model category]] as a formalization of the full structure, and this formalization turned out to yield a powerful theory that today provides a powerful toolset for dealing with homotopy theoretic situations. But also the notion of model category was seen to not be the full answer. For instance a model category in a sense retains _too much_ non-intrinsic information. Equivalence classes of model categories under [[Quillen equivalence]] are a more intrinsic characterization of a given [[homotopy theory]]. But this means that one needs some [[higher category theory\|higher categorical]] notion for the collection of all model categories. This problem came to be known as the search for the homotopy theory of homotopy theories. Recently, this problem was fully solved and homotopy theory fully understood as the special case of [[higher category theory]] that deals with [[(∞,1)-category\|(∞,1)-categories]]: * the notion of [[model category]], in particular when refined to that of a [[simplicial model category]] serves as a [[presentable (infinity,1)-category\|presentation]] of the notion of [[(∞,1)-category]]; * the "homotopy theory of homotopy theories" is accordingly the [[(∞,1)-category of (∞,1)-categories]] $(\infty,1)Cat$; better yet: there is an [[(∞,n)-category\|(∞,2)-category]] of all $(\infty,1)$-categories; * in $(\infty,1)Cat$ two $(\infty,1)$-categories presented by model categories are equivalent precisely if the presenting model categories may be connected by a zig-zag sequence of [[Quillen equivalence]]s; * all "homotopy"-constructions in model category theory, such as [[homotopy limit]]s, [[mapping cone]]s etc. are _tools for constructing_ the corresponding higher categorical intrinsic notions, such as [[limit in a quasi-category\|limit in an (∞,1)-category]]. * all variant notions find their intrinsic higher categorical interpretation this way: for instance [[stable homotopy theory]] is the study of [[stable (∞,1)-category\|stable (∞,1)-categories]]; * the [[homotopy category]] of a [[model category]] is simply the [[decategorification]] of the corresponding $(\infty,1)$-category to just a [[1-category]]; * and for instance the notion of homotopy category of a stable $(\infty,1)$-category reproduces the notion of [[triangulated category]], thus incorporating also a large toolset from [[homological algebra]] into the picture. #### In rational homotopy theory {#RationalHomotopyTheory} ... The study of [[rational homotopy theory]] is naturally understood as the study of the [[localization of an (∞,1)-category\|localizations]] of [[(∞,1)-topos]]es at morphism that induce equivalences in [[cohomology]] with certain line-object coefficients. See [[rational homotopy theory in an (∞,1)-topos]]. ... ### In K-theory In full generality, ([[algebraic K-theory\|algebraic]]) [[K-theory]] is a universal assignment of [[spectra]] to [[stable (∞,1)-categories]]. ... ### In Tannaka duality ... see [[Tannaka duality]] ... ### In differential geometry See at * [[higher differential geometry applied to plain differential geometry]] ### In differential cohomology {#InDifferentialCohomology} [[cohesion]] on [[(∞,1)-toposes]] solves the [Simons-Sullivan question](differential%20cohomology%20diagram#SimonsSullivan07) characterization on the characterization of [[generalized (Eilenberg-Steenrod) cohomology\|generalized (Eilenberg-Steenrod-type)]] [[differential cohomology]]. See at _[[differential cohomology hexagon]]_ for details. ### In deformation theory {#DeformationTheory} In deformation theory it was early on recognized that for a good theory the notion of [[Kähler differential]]s has to be generalized to the notion of [[cotangent complex]]. With the advent of the study of derived [[moduli space]]s, such as the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves\|derived moduli space of derived elliptic curves]], this needed to be further generalized to notions of cotangent complexes not just of [[ring]]s, but of [[E-∞-ring]]s. It turns out that all these concepts are special cases of a construction obtained from a simple higher categorical notion, that of [[left adjoint]] [[section]]s of a [[tangent (∞,1)-category]]. ### In logic and type theory {#Logic} While it is common to view logic as the study of absolute truth, in fact logic can have many different interpretations, or [[semantics]]. A particular semantics for logic can be useful both to inform the study of logic, and to prove facts logically about the semantics. One very fruitful semantics of this sort is categorical semantics for logic and [[type theory]], according to which every category (and especially every [[topos]]) has an [internal language](/nlab/show/type+theory#CategoricalSemantics) and [[internal logic]]. Interpreting "ordinary" mathematical statements in the internal language of exotic categories can make it much easier to study those categories, while on the other hand it can provide new insight into otherwise mysterious logical notions. In particular, the internal logic of a category (such as a topos) is, in general, [[constructive mathematics\|constructive]], i.e. the principle of [[excluded middle]] (and also stronger statements, such as the [[axiom of choice]]) are generally false. Thus, in order for a theorem to be interpretable internally in such categories, its proof must be constructive. So while the original "constructivists" believed that classical mathematics was "wrong," nowadays there are good reasons to care about constructive mathematics even if one believes that excluded middle and the axiom of choice are "true," since regardless of their "global" truth they will not be true in the internal logic of many interesting categories. Conversely, category-theoretic models have provided new insight into the independence of various axioms in constructive mathematics, such as differing forms of the axiom of choice. As another example, the [[identity types]] in Martin-Löf's original constructive dependent [[type theory]] construct, from any [[type]] $A$ and terms $a, b \in A$, a new type $Id_A(a, b)$. According to the [[propositions as types]] interpretation, the elements of $Id_A(a,b)$ are proofs that $a$ and $b$ are propositionally equal; thus $Id_A(a,b)$ is a replacement for the [[truth value]] of the [[proposition]] $(a=b)$. There are type-theoretic [[functions]] $1 \to Id(a, a)$, $Id(b, c) \times Id(a, b) \to Id(a, c)$ and $Id(a, b) \to Id(b, a)$ expressing the [[equivalence relation\|reflexivity, transitivity and symmetry]] of this propositional [[equality]], but in general an identity type (even the "reflexive" identity type $Id(a,a)$) can have many distinct elements. This has long been a source of discomfort to type theorists. However, from a higher-categorical point of view, it is natural to view the terms of identity types as isomorphisms in a [[groupoid]]---or, more precisely, an [[∞-groupoid]], since identity types have their own identity types, and all the laws of associativity, exchange, etc. only hold up to terms of these higher identity types. This suggests that the nonuniqueness of identity proofs should be embraced rather than denigrated, producing a theory at least related to the "internal logic" of [[(∞,1)-category]] theory and [[homotopy theory]]; see [[identity type]] for more details. This is now known as _[[homotopy type theory]]_, see there for more. ### In physics {#Physics} See also _[[higher category theory and physics]]_. #### Classical mechanics and its geometric quantization {#ClassMech} By the end of the 19th century a fairly complete, powerful and elegant mathematical formulation of [[classical mechanics]]: in terms of [[symplectic geometry]]. By the middle of the 20th century, the passage to the corresponding quantum theory was pretty well modeled by the [[geometric quantization]] of symplectic geometries. But there were some loose ends. Notably the fully general theory involved [[Poisson manifold]]s, not just symplectic manifolds. And the mechanics of relativistic [[classical field theory]] was realized to be more naturally described by [[multisymplectic geometry]]. Both these generalizations have a natural common higher categorical formulation: that of [[Lie ∞-algebroid]]s: a Poisson geometry is naturally encoded in its corresponding [[Poisson Lie algebroid]]. Its higher categorical versions -- the [[n-symplectic manifold]]s -- encode the corresponding multisymplectic geometry. Moreover, the quantization step of geometric quantization was understood to be effectively the [[Lie integration]] of these [[Lie ∞-algebroid]]s to the corresponding [[Lie ∞-groupoid]]s (currently this is well understood for low $n$). #### Quantum mechanics and quantum information {#QuantumMechanics} The basic structure of [[quantum mechanics]] and [[quantum information theory]] is encoded in the theory of [[dagger-compact categories]]. ... #### Gauge theory {#GaugeTheory} Maxwell realized that the [[electromagnetic field]] is controlled by a degree 2-cocycle in [[de Rham cohomology]]: the electromagnetic [[field strength]]. Later Dirac noticed that this is one part of a degree 2-cocycle in [[differential cohomology]] that characterize a [[connection on a bundle\|connection]] on a [[line bundle]]. Later the [[Yang-Mills field]] was understood to similarly be a [[connection on a bundle]], this time on a $G$-[[principal bundle]] for $G$ some possibly nonabelian [[group]]. While thinking about the mathematical structures possibly underlying [[standard model of particle physics]] and [[gravity]], theoretical physicists considered more general hypothetical [[gauge field]]s, such as the [[Kalb-Ramond field]], the [[RR-field]] or the [[supergravity C-field]]. Today all these gauge fields are understood to be modeled, mathematically, by generalized [[differential cohomology]]. ##### Supergravity {#Supergravity} Theories of [[supergravity]] have been known to require higher [[gauge field]]s in the above sense -- hence the term [[supergravity C-field]]. A powerful formalism for handling these theories is the [[D'Auria-Fre formulation of supergravity]]. As described there, this is secretly (but evidently) nothing but a description of supergravity as a theory of connections on nonabelian $G$-[[principal ∞-bundle]]s for $G$ some super [[Lie ∞-groupoid\|Lie ∞-group]]. For instance Cremmer-Scherk 11-dimensional supergravity theory is governed by the super Lie 3-group $G$ whose [[L-∞-algebra]] is the [[supergravity Lie 3-algebra]]. #### BV-BRST formalism {#BVBRST} The [[BV-BRST formalism]] is secretly a way to talk about the fact that configuration spaces of [[gauge theory\|gauge theories]] are not naive spaces such as [[manifold]]s, but are general [[space]]s in the sense of [[higher geometry]]: the configuration space is really an object $Conf \in Sh_{(\infty,1)}((dgAlg^-)^{op})$ in the [[∞-stack]] [[(∞,1)-topos]] on the [[(∞,1)-site]] $(dgAlg^-)^{op}$ of certain [[algebra in an (∞,1)-category\|∞-algebras]] modeled as [[dg-algebra]]s. The BV-BRST-complex of a physical system is the global [[derived geometry\|derived]] function algebra $$ \mathcal{O}(Conf) \in dgAlg \,. $$ > (many more aspects go here, eventually)... #### Quantum field theory {#QFT} There are essentially two axiomatizations of what [[quantum field theory]] is, both of which are inherently $\infty$-categorical: * in the [[FQFT]] picture -- the _Schrödinger picture_ -- a quantum field theory is described as an [[(∞,n)-functor]] on an [[(∞,n)-category of cobordisms]]. The [[cobordism hypothesis]] -- now a theorem that characterizes central properties of these [[(∞,n)-categories]], has been a major driving force in the development of [[higher category theory]]. * in the [[AQFT]]/[[factorization algebra]] picture -- the _Heisenberg picture_ -- a quantum field theory is described as an $\infty$-copresheaf of observables on its parameter space. ##### Holography * [[holographic principle of higher category theory]] ##### 3d TFT and 2d CFT {#3dTFT2dCFT} 3-dimensional [[TFT]] such as [[Chern-Simons theory]] and [[Dijkgraaf-Witten theory]] and the global aspects of 2-dimensional [[conformal field theory]] are inherently governed by the theory of [[modular tensor categories]]. The local aspects of 2-dimensional conformal field theory are governed by [[vertex operator algebra]]s. A [[vertex operator algebra]] is really the [[algebra over an operad]], for the operad of holomorphic pointed spheres (as described there). ... ### In your favorite topic here ... ## Related pages * [[geometry of physics]] * [[fiber bundles in physics]] * [[higher category theory and physics]] * [[string theory FAQ]] * [[twisted smooth cohomology in string theory]] * [[motives in physics]] * [[Hilbert's sixth problem]] * [[model theory and physics]] * [[L-infinity algebras in physics]] * [[motivation for sheaves, cohomology and higher stacks]] * [[applications of (higher) category theory]] * [[motivation for higher differential geometry]] * [[motivation for cohesion]]
applications of double category theory	https://ncatlab.org/nlab/source/applications+of+double+category+theory	A list of works and resources about applications of [[double category]] theory. See there for introductory material. ## Theory ### [[2-category equipped with proarrows\|Equipments]] * [[David Jaz Myers]], _String diagrams for double categories and (virtual) equipments_, [(arXiv)](https://arxiv.org/abs/1612.02762), 2016 * [[Mike Shulman]], _Equipments_, [(n-Café)](https://golem.ph.utexas.edu/category/2009/11/equipments.html), 2009 * [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, [(arXiv)](https://arxiv.org/abs/0706.1286), 2009 * Max New, Dan Licata, _A Formal Logic for Formal Category Theory_, [(arXiv)](https://arxiv.org/abs/2210.08663), 2022 ### Grothendieck constructions * Evan Patterson, _Grothendieck construction for double categories_, [(Topos Institute)](https://topos.site/blog/2022/05/grothendieck-construction-for-double-categories/), 2022 * Cruttwell, Lambert, Pronk, Szyld, _Double Fibrations_, [(arXiv)](https://arxiv.org/abs/2205.15240), 2022 ## Applications ### Lenses & optics * [[Bryce Clarke]], _The double category of lenses_, [(online)](https://bryceclarke.github.io/The_Double_Category_Of_Lenses_Phd_Thesis.pdf), 2022 * Capucci, _Seeing double through dependent optics_, [(arXiv)](https://arxiv.org/abs/2204.10708), 2022 * Guillaume Boisseau, Chad Nester, Mario Roman, _Cornering optics_,[(arXiv)](https://arxiv.org/abs/2205.00842), 2022 ### Open systems theory * Jared Culbertson, Paul Gustafson, Daniel E. Koditschek, Peter F. Stiller, _Formal composition of hybrid systems_, [(arXiv)](https://arxiv.org/abs/1911.01267) * Kenny Courser, _Open systems: a double categorical perspective_, [(arXiv)](https://arxiv.org/abs/2008.02394), 2020 See also [[structured cospans]] and [[decorated cospans]] * Chad Nester, _Situated Transition Systems_, [(arXiv)](https://arxiv.org/abs/2105.04355), 2021 * [[David Jaz Myers]], _Categorical systems theory_, [(online)](http://davidjaz.com/Papers/DynamicalBook.pdf), 2021 * Evan Patterson, _Decorated cospans via the double Grothendieck construction_, [(Topos Institute)](https://topos.site/blog/2022/05/decorated-cospans-via-the-grothendieck-construction/), 2022 * Evan Patterson, _Structured cospans as a cocartesian equipment_, [(Topos Institute)](https://topos.site/blog/2023/03/structured-cospans-as-a-cocartesian-equipment/), 2023 * [[John Baez]], Kenny Courser, [[Christina Vasilakopoulou]], _Structured versus Decorated Cospans_, [(Compositionality)](https://compositionality-journal.org/papers/compositionality-4-3/pdf), 2022 * Chad Nester, _Concurrent Process Histories and Resource Transducers_, [(arXiv)](https://arxiv.org/abs/2010.08233), 2022 ### Theory of programming languagues * Pierre-Evariste Dagand, [[Conor McBride]], _A Categorical Treatment of Ornaments_, [(acm)](https://dl.acm.org/doi/abs/10.5555/2591370.2591396), 2013 * Max New, Dan Licata, _Call-by-name Gradual Type Theory_, [(arXiv)](https://arxiv.org/abs/1802.00061), 2018 * Max New, Dan Licata, _A Formal Logic for Formal Category Theory_, [(arXiv)](https://arxiv.org/abs/2210.08663), 2022 * Clovis Eberhart, Tom Hirschowitz, _Fibred Pseudo Double Categories for Game Semantics_, [TAC 34(19)](http://www.tac.mta.ca/tac/volumes/34/19/34-19abs.html), 2019. Beware, “fibred” is use there to mean that the codomain functor is a fibration.
applicative functor	https://ncatlab.org/nlab/source/applicative+functor	#Contents# * table of contents {:toc} ##Idea In [[computer science]], applicative functors (also known as idioms) are the programming equivalent of [[lax monoidal functors]] with a [[tensorial strength]] in category theory. A [[monad (in computer science)\|monad]] gives rise to an applicative functor, but not all applicative functors result from monads. Unlike monads, applicative functors are closed under composition. ##Related concepts * [[arrow (in computer science)]] ##References * Conor Mcbride, Ross Paterson, _Applicative programming with effects_, Journal of Functional Programming. 18 (01): 1–13. ([doi:10.1017/S0956796807006326](https://doi.org/10.1017/S0956796807006326); [author’s version](http://www.staff.city.ac.uk/~ross/papers/Applicative.html)) * Ross Paterson, _Constructing Applicative Functors_, in Mathematics of Program Construction, Madrid, 2012, Lecture Notes in Computer Science vol. 7342, pp. 300-323, Springer, 2012. ([paper](http://www.staff.city.ac.uk/~ross/papers/Constructors.html)) [[!redirects applicative functors]]
applied category theory	https://ncatlab.org/nlab/source/applied+category+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ##Idea## The whole point of [[category theory]] is to study fundamental [[general abstract]] patterns and phenomena that (re-)appear throughout [[mathematics]]. Hence applications of [[theorems]] of category theory are ubiquituous in [[mathematics]] and in subjects with a mathematical basis, such as [[physics]] and [[computer science]]. Often this goes without saying. In the introduction to [Bradley 18](#Bradley18) it says: > ... ideas and results from category theory have found applications in computer science and quantum physics (not to mention pure mathematics itself), but these are not the only applications to which the word _applied_ in _applied category theory_ is being applied... > category theory has found applications in a wide range of disciplines outside of pure mathematics—even beyond the closely related fields of computer science and quantum physics. These disciplines include chemistry, neuroscience, systems biology, natural language processing, causality, network theory, dynamical systems, and database theory to name a few. And what do they all have in common? ... In other words, the techniques, tools, and ideas of category theory are being used to identify recurring themes across these various disciplines with the purpose of making them a little more formal. ##References## * {#Bradley18} [[Tai-Danae Bradley]], _What is applied category theory?_, [arXiv:1809.05923](https://arxiv.org/abs/1809.05923) * [[Brendan Fong]] and [[David Spivak]], _Seven Sketches in Compositionality: An Invitation to Applied Category Theory_, [arXiv:1803.05316](https://arxiv.org/abs/1803.05316). * [Applied category theory](https://www.appliedcategorytheory.org/) [[John Baez]] ran a series of lectures based on this book: * lectures on chapter 1 * [Lecture 1 - Introduction](https://forum.azimuthproject.org/discussion/1807/lecture-1-introduction/p1) * [Lecture 2 - What is Applied Category Theory?](https://forum.azimuthproject.org/discussion/1808/lecture-2-what-is-applied-category-theory/p1) * [Lecture 3 - Chapter 1: Preorders](https://forum.azimuthproject.org/discussion/1812/lecture-3-chapter-1-posets/p1) * [Lecture 4 - Chapter 1: Galois Connections](https://forum.azimuthproject.org/discussion/1828/lecture-4-chapter-1-galois-connections/p1) * [Lecture 5 - Chapter 1: Galois Connections](https://forum.azimuthproject.org/discussion/1845/lecture-5-chapter-1-galois-connections/p1) * [Lecture 6 - Chapter 1: Computing Adjoints](https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints/p1) * [Lecture 7 - Chapter 1: Logic](https://forum.azimuthproject.org/discussion/1909/lecture-7-chapter-1-logic/p1) * [Lecture 8 - Chapter 1: The Logic of Subsets](https://forum.azimuthproject.org/discussion/1921/lecture-8-chapter-1-the-logic-of-subsets/p1) * [Lecture 9 - Chapter 1: Adjoints and the Logic of Subsets](https://forum.azimuthproject.org/discussion/1931/lecture-9-chapter-1-adjoints-and-the-logic-of-subsets/p1) * [Lecture 10 - Chapter 1: The Logic of Partitions](https://forum.azimuthproject.org/discussion/1963/lecture-10-the-logic-of-partitions/p1) * [Lecture 11 - Chapter 1: The Poset of Partitions](https://forum.azimuthproject.org/discussion/1991/lecture-11-chapter-1-the-poset-of-partitions/p1) * [Lecture 12 - Chapter 1: Generative Effects](https://forum.azimuthproject.org/discussion/1999/lecture-12-chapter-1-generative-effects/p1) * [Lecture 13 - Chapter 1: Pulling Back Partitions](https://forum.azimuthproject.org/discussion/2008/lecture-13-chapter-1-pulling-back-partitions/p1) * [Lecture 14 - Chapter 1: Adjoints, Joins and Meets](https://forum.azimuthproject.org/discussion/2013/lecture-14-adjoints-joins-and-meets/p1) * [Lecture 15 - Chapter 1: Preserving Joins and Meets](https://forum.azimuthproject.org/discussion/2027/lecture-15-chapter-1-preserving-joins-and-meets/p1) * [Lecture 16 - Chapter 1: The Adjoint Functor Theorem for Posets](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) * [Lecture 17 - Chapter 1: The Grand Synthesis](https://forum.azimuthproject.org/discussion/2037/lecture-17-chapter-1-the-grand-synthesis/p1) * lectures on chapter 2 * [Lecture 18 - Chapter 2: Resource Theories](https://forum.azimuthproject.org/discussion/2075/lecture-18-chapter-2-resource-theories/p1) * [Lecture 19 - Chapter 2: Chemistry and Scheduling](https://forum.azimuthproject.org/discussion/2079/lecture-19-chapter-2-chemistry-and-scheduling/p1) * [Lecture 20 - Chapter 2: Manufacturing](https://forum.azimuthproject.org/discussion/2081/lecture-20-chapter-2-manufacturing/p1) * [Lecture 21 - Chapter 2: Monoidal Preorders](https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders/p1) * [Lecture 22 - Chapter 2: Symmetric Monoidal Preorders](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1) * [Lecture 23 - Chapter 2: Commutative Monoidal Posets](https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1) * [Lecture 24 - Chapter 2: Pricing Resources](https://forum.azimuthproject.org/discussion/2089/lecture-24-chapter-2-pricing-resources/p1) * [Lecture 25 - Chapter 2: Reaction Networks](https://forum.azimuthproject.org/discussion/2090/lecture-25-chapter-2-reaction-networks/p1) * [Lecture 26 - Chapter 2: Monoidal Monotones](https://forum.azimuthproject.org/discussion/2095/lecture-26-chapter-2-monoidal-monotones/p1) * [Lecture 27 - Chapter 2: Adjoints of Monoidal Monotones](https://forum.azimuthproject.org/discussion/2098/lecture-27-chapter-2-adjoints-of-monoidal-monotones/p1) * [Lecture 28 - Chapter 2: Ignoring Externalities](https://forum.azimuthproject.org/discussion/2105/lecture-28-chapter-2-ignoring-externalities/p1) * [Lecture 29 - Chapter 2: Enriched Categories](https://forum.azimuthproject.org/discussion/2121/lecture-29-chapter-2-enriched-categories/p1) * [Lecture 30 - Chapter 2: Preorders as Enriched Categories](https://forum.azimuthproject.org/discussion/2124/lecture-30-chapter-1-preorders-as-enriched-categories/p1) * [Lecture 31 - Chapter 2: Lawvere Metric Spaces](https://forum.azimuthproject.org/discussion/2128/lecture-31-chapter-2-lawvere-metric-spaces/p1) * [Lecture 32 - Chapter 2: Enriched Functors](https://forum.azimuthproject.org/discussion/2169/lecture-32-chapter-2-enriched-functors/p1) * [Lecture 33 - Chapter 2: Tying Up Loose Ends](https://forum.azimuthproject.org/discussion/2192/lecture-33-chapter-2-tying-up-loose-ends/p1) * [[David Spivak]], _Category theory for the sciences._ MIT Press, 2014. * [[Brandon Coya]], _Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective_, [arXiv:1805.08290](https://arxiv.org/abs/1805.08290) * [[John Baez]] and [[Brendan Fong]], _A Compositional Framework for Passive Linear Networks_, [arXiv:1504.05625](https://arxiv.org/abs/1504.05625) * [[Blake Pollard]], _Open Markov processes: A compositional perspective on non-equilibrium steady states in biology_, [arXiv:1601.00711](https://arxiv.org/abs/1601.00711) * [[Brendan Fong]], _The Algebra of Open and Interconnected Systems_, [arXiv:arXiv:1609.05382](https://arxiv.org/abs/1609.05382) * [[John Baez]] and [[Blake Pollard]], _A Compositional Framework for Reaction Networks_, [arXiv:1704.02051](https://arxiv.org/abs/1704.02051) * [[John C. Baez]], [[Brendan Fong]] and [[Blake Pollard]], _A Compositional Framework for Markov Processes_, [arXiv:1508.06448](https://arxiv.org/abs/1508.06448) * [[Jules Hedges]] and [[Martha Lewis]], _Towards Functorial Language-Games_, [arXiv:1807.07828](https://arxiv.org/abs/1807.07828)
applied mathematics	https://ncatlab.org/nlab/source/applied+mathematics	The area conventionally called _applied mathematics_ is not the central focus of the $n$Lab: our main focus is in (higher) category theory, [[sheaves]], [[stack]]s, [[homotopy theory\|homotopy]] and (co)homological methods, foundations, [[topology]], [[algebra]] and modern geometry as well as mathematical physics and philosophical aspects. Unfortunately, there is a problem in defining what applied mathematics is, and many mathematicians disagree about the range. Some, including [[Henri Poincaré]] and [[Vladimir Arnol'd]], said that > there is no applied mathematics, there are only applications of mathematics. Moreover, in many areas and departments the subdivision into pure and applied mathematics (or any similar variants) appears to be a social designation, belong to a group rather than an imperative to apply anything to real world. One can argue that what will be applied can not be nearly fully predicted by the area and intention of the creator of particular mathematical result, but by the internal power of the result and by the future of applications themselves. Hence it is a bit presumptuous that people working on some particular mathematics problems, who do not themselves apply mathematics to real-life problems, declare themselves applied just by the common opinion on classification of their result. For instance it is common to assume that work on [[partial differential equation]]s (especially by analytic and numerical methods) is a subarea of applied mathematics, even if one studies completely unnatural and never applied differential equations, while it is not considered applied mathematics to prove theorems about the homology of spaces, even if they strongly influence [[index theorem]]s widely used in applications to other sciences and "real-life problems". In any case, to well-spirited people who believe they do applied mathematics we may vaguely recommend our moderately developed pages relating * [[mathematical physics]], which is important for us. Unfortunately, in the departments of the applied mathematics world, there is not really any interest in the foundations of [[quantum field theory]] (and siblings like [[superstring theory]], [[statistical field theory]], [[quantum gravity]], ...) which is our central interest in mathematical physics. We have stubs for some other areas intersecting with traditional "applied mathematics" but not many: * [[hydrodynamics]] * [[finite element method]] * [[regular differential operator]] * [[homological algebra in finite element method]] * [[Runge-Kutta method]] * [[symplectic integrator]] * [[preconditioner]] * [[numerical analysis]] Wikipedia: [applied mathematics](http://en.wikipedia.org/wiki/Applied_mathematics) As, if and when applications of the nPOV to areas traditionally called 'applied mathematics', we cordially invite practitioners to contribute so that those developments may be recorded and developed here. Recent developments of category theoretical methods and insights to areas of Chemistry and Biology are being discussed in [Azimuth](https://johncarlosbaez.wordpress.com/). category: applications [[!redirects applied math]] [[!redirects applied maths]] [[!redirects applied mathematics]]
applied topology	https://ncatlab.org/nlab/source/applied+topology	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebraic topology +--{: .hide} [[!include algebraic topology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The application of [[topology]] ([[general topology]] or [[algebraic topology]]) to other fields. ## Related concepts * [[computational topology]] * [[topological data analysis]] ## References Textbook account: * [[Robert Ghrist]], Elementary Applied Topology (2014) [[web](https://www2.math.upenn.edu/~ghrist/notes.html), [ISBN:978-1502880857]()] See also: * [[Dmitry Feichtner-Kozlov]], [Applied Topology Unit](https://groups.oist.jp/atu) On [[combinatorics\|combinatorial]] topology * [[Dmitry N. Kozlov]], Trends in Topological Combinatorics [[arXiv:math/0507390](https://arxiv.org/abs/math/0507390)] * [[Dmitry Kozlov]], Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21, Springer (2008) [[doi:10.1007/978-3-540-71962-5](https://doi.org/10.1007/978-3-540-71962-5)]
approach space	https://ncatlab.org/nlab/source/approach+space	## Idea The concept of approach space generalized the concept of [[metric space]]. The idea is that the distance describes not only the distance between two points but the distance of a point to a subset. This relatively tangible generalization gives the missing link in the triad of the concepts of [[uniform space\|uniformity]], [[topological space\|topology]], and [[metric space\|metric spaces]]. ## Definition An __approach space__ is a set $X$ together with a __distance__ $d\colon X \times \mathcal{P}(X) \to [0,\infty]$ (where $\mathcal{P}(X)$ denotes the [[power set]]) such that the following axioms hold for all $x \in X$ 1. $d(x, \{x\}) = 0$ 1. $d(x, \emptyset) = \infty$ 1. for all $A, B \in \mathcal{P}$: $d(x, A\cup B) = \min\{ d(x, A), d(x, B) \}$ 1. for all $A \in \mathcal{P}$ and $\varepsilon \in [0,\infty]$: $d(x, A ) \leq d(x, A^{\varepsilon]} ) + \varepsilon $ where $A^{\varepsilon]} \coloneqq \{x \in X \mid d(x, A) \leq \varepsilon \}$. ## Properties Every approach space $d$ induces a [[topological space\|topology]] on $X$ via the [[closed subspace \|closure operator]] $Cl_d(A) = \{x \in X \mid d(x, A) = 0 \}$. ## Examples * Every [[topological space]] is induced by a canonical approach structure given by $d(x, A) = 0$ if $x \in Cl(A)$ and $d(x, A) = \infty$ otherwise. * The [[one-point compactification]] of a metric space $d$ can be metrised in a canonical way as an approach space by $$ d^(x, A) = \begin{cases} d(x, A\setminus\{\infty\}) & x \neq \infty \\ 0 & x = \infty\, and\, A\, is\, not\, precompact \\ \infty & x = \infty\, and\, A\, is\, precompact. \end{cases} $$ A [[gauge space\|gauge]], or more generally a gauge base, $G$ on $X$ gives a distance on $X$ by $d_G(x, A) = \sup_{d \in G} \inf_{y\in A} d(x,y)$. ## Related concepts * [[metric space]] * [[gauge space]] * [[relational beta-module]] ## References * Robert Lowen, _Approach spaces: the missing link in the topology-uniformity-metric triad_, Oxford Mathematical Monographs. 1997. ([publisher link](https://global.oup.com/academic/product/approach-spaces-9780198500308)).
approximate fibration	https://ncatlab.org/nlab/source/approximate+fibration	The approximate homotopy lifting property is a weak version of the [[homotopy lifting property]] in the setup of [[metric spaces]]. A [[proper map]] $p:E\to B$ between [[locally compact space\|locally compact]] metric [[absolute neighborhood retract]]s (ANRs) satisfies the approximate homotopy lifting property for a space $X$ if for any [[open covering]] $U$ of B, and any map $h : X\to E$ with a [[homotopy]] $H : X \times I \to B$ such that $p\circ h = H_0$, there exists a homotopy $G : X\times I\to E$ such that $G_0 = h$ and the maps $p\circ G$ and $H$ are $U$-[[close map\|close]]. A proper map $p : E\to B$ between locally compact metric ANRs is an approximate fibration if $p$ has the approximate homotopy lifting property for all metric spaces. It is straightfoward to generalize this notion to the level maps of inverse systems of locally compact metric ANRs. [[!redirects approximate homotopy lifting property]]
approximate integral domain	https://ncatlab.org/nlab/source/approximate+integral+domain	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +-- {: .hide} [[!include higher algebra - contents]] =-- =-- =-- \tableofcontents ## Idea A weaker notion of [[integral domain]] which allows for some [[zero divisors]], but for which one may [[quotient ring\|quotient out]] the zero divisors to obtain an [[integral domain]]. "Approximate integral domain" is a placeholder name for a concept which may or may not have another name in the mathematics literature. The idea however is that approximate integral domains are to integral domains as [[local rings]] are to [[Heyting fields]], and as [[weak local rings]] are to [[weak Heyting fields]]. ## Definition An approximate integral domain is a [[commutative ring]] $R$ such that: * $R$ is nontrivial ($0 \ne 1$); and * The [[zero divisors]] form an [[ideal]]. (equivalently, the non-[[cancellative elements]] form an [[ideal]]). Thus, the [[quotient ring\|quotient]] of an approximate integral domain by its [[ideal]] of [[zero divisors]] is an [[integral domain]]. Every approximate integral domain has an [[equivalence relation]] $\approx$, defined as $x \approx y$ if and only if $x - y$ is a [[zero divisor]]. Hence the name "approximate" integral domain. Then [[integral domains]] are precisely the approximate integral domains for which $\approx$ implies [[equality]]. ## In constructive mathematics In constructive mathematics, similar to the notion of [[local ring]], [[integral domain]], and [[field]], the notion of approximate integral domain bifurcates into multiple distinct notions: A weak approximate integral domain is an approximate integral domain defined as above. Recall that a [[cancellative element]] in a commutative ring $R$ is an element $a \in R$ for which both left and right multiplication by $a$ is an injection, and [[zero divisors]] are precisely the elements in $R$ which are not cancellative. A strict approximate integral domain is a weak approximate integral domain for which additionally the cancellative elements form an [[anti-ideal]]: * $0$ is not cancellative; * if $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative; * if $a \cdot b$ is cancellative, then $a$ is cancellative and $b$ is cancellative (this is trivially true in any commutative ring). The [[quotient object\|quotient ring]] of a strict approximate integral domain by its anti-ideal of cancellative elements is a [[Heyting integral domain]]. One can define an [[apartness relation]] in any strict approximate integral domain: $x \# y$ iff $x - y$ is cancellative. Then the local ring is a Heyting field if and only if this apartness relation is [[tight relation\|tight]]. +-- {: .num_prop #internal} ###### Proposition The addition and multiplication operations on a strict approximate integral domain $R$ are strongly extensional with respect to the canonical apartness relation $\#$ defined by $x \# y$ iff $x - y$ is cancellative. In this way a strict approximate integral domain becomes an internal [[ring object]] in the category $Apart$, consisting of sets with apartness relations and maps (strongly extensional functions) between them. =-- +-- {: .proof} ###### Proof Recall that products $X \times Y$ in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by $(x, y) \# (x', y')$ iff $x \# x'$ in $X$ or $y \# y'$ in $Y$. Recall also that a function $f: X \to Y$ between sets with apartness relations is strongly extensional if $f(x) \# f(y)$ implies $x \# y$. For addition, if $(x + y) \# (x' + y')$, then $x + y - (x' + y') = (x - x') + (y - y')$ is cancellative, so $x - x'$ or $y - y'$ is cancellative since $R$ is a strict approximate integral domain, whence $(x, y) # (x', y')$. Thus addition is strongly extensional. For multiplication, if $x y # x' y'$, then $x y - x' y'$ is cancellative. Write $x y - x' y' = (x - x')y + x'(y - y')$. Since $R$ is a strict approximate integral domain, either $(x - x')y$ is a cancellative element or $x'(y - y')$ is a cancellative element. From this we easily conclude $x - x'$ is a cancellative element or $y - y'$ is, since cancellative elements are closed under multiplicaiton, whence $(x, y) # (x', y')$. So multiplication is also strongly extensional. =-- \begin{theorem} For a strict approximate integral domain, the [[ring of fractions]] obtained by inverting the cancellative elements is a [[local ring]]. \end{theorem} ## Examples and non-examples * The [[integers]] are an approximate integral domain which are an [[integral domain]]. * The [[dual numbers\|dual]] integers $\mathbb{Z}[\epsilon]/\epsilon^2$ is an approximate integral domain where the [[nilpotent]] [[infinitesimal]] $\epsilon \in \mathbb{Z}[\epsilon]/\epsilon^2$ is a non-[[zero]] [[zero divisor]]. * For any [[prime number]] $p$ and any [[positive number\|positive]] [[natural number]] $n$, the [[prime power local ring]] $\mathbb{Z}/p^n\mathbb{Z}$ is an approximate integral domain, whose [[ideal]] of [[zero divisors]] is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of [[zero divisors]] is the [[finite field]] $\mathbb{Z}/p\mathbb{Z}$, indicating that it is also a [[weak local ring]]. * There exist commutative rings which are not approximate integral domains. For example, the [[integers modulo]] 6 $\mathbb{Z}/6\mathbb{Z}$ is not an approximate integral domain, because $3$ and $4$ are both zero divisors, but $3 + 4$ is cancellative. When one tries to quotient out the zero divisors, the resulting ring is [[trivial ring\|trivial]]. ## See also * [[integral domain]] * [[ordered integral domain]] * [[weak local ring]] * [[local ring]] [[!redirects approximate integral domain]] [[!redirects approximate integral domains]] [[!redirects weak approximate integral domain]] [[!redirects weak approximate integral domains]] [[!redirects strict approximate integral domain]] [[!redirects strict approximate integral domains]]
approximation	https://ncatlab.org/nlab/source/approximation	## Related concepts * [[epsilontic analysis]] ## References * Wikipedia, _[Approximation](https://en.wikipedia.org/wiki/Approximation)_ [[!redirects approximations]]
approximation of the identity	https://ncatlab.org/nlab/source/approximation+of+the+identity	"Approximation of the identity" is a rubric for a general technique in functional analysis for proving that certain inclusions of [[topological vector space]]s are dense. It refers to the fact that an identity for a convolution product (aka "Dirac distribution") may not literally exist in a particular TVS, but is virtually there in the sense that it can be approximated by elements in the subspace which is being included. ## Examples ## We illustrate the technique with two examples. +-- {: .num_example} ###### Example Consider first the problem of showing that restrictions of polynomials to $[-1, 1]$ are dense in $L^p([-1, 1])$ (under Lebesgue [[measure space\|measure]]). The idea is to take the formula $$ f(y) = (f * \delta)(y) = \int_{-1}^1 f(x) \delta(y - x) d x $$ (which literally makes no sense because $\delta$ is not an actual integrable function) and then replace $\delta$ by polynomial functions $p_n$ which "approximate" to it (so each $p_n$ has "mass" 1 and is vanishingly small outside a given neighborhood of 0, if $n$ is sufficiently large). Put for example $f_n(x) = (1 - x^2)^n$ and "normalize" it, putting $$p_n = \frac{f_n}{\\|f_n\\|_1}$$ where $\\|g\\|_1$ indicates $L^1$ norm. By "differentiating under the integral sign", we have $$ D^j (f * p_n) = (-1)^j f * D^j(p_n) $$ so that for each $n$, the $j^{th}$ derivative of $f * p_n$ is identically zero for $j$ sufficiently large. Hence $f * p_n$ is polynomial. Next, the claim is that for $f \in L^p$, we have $$lim_{n \to \infty} \\|f - (f * p_n)\\|_p = 0$$ Intuitively, the idea is that $ f - (f * p_n) = f * (\delta - p_n) $ and that (because $L^p$ is a module over the [[Banach algebra]] $L^1$) $$\\|f * (\delta - p_n)\\|_p \leq \\|f\\|_p \\|\delta - p_n\\|_1 \to 0$$ as $n \to \infty$. For a more careful proof, see theorem 9.6 in Wheeden and Zygmund (referenced below). =-- +-- {: .num_example} ###### Example For a second example, consider how to prove that the functions $z^n$, with $n$ ranging over integers, forms an orthonormal basis of the [[Hilbert space]] $L^2(S^1)$ where $S^1$ is the unit circle in the complex plane, where the inner product is given by $$\langle f, g \rangle = \frac1{2\pi i} \int_{S^1} \overline{f(z)} g(z) \frac{d z}{z}$$ The monomials $z^n$ are clearly orthonormal, so again the idea is to use appropriate linear combinations of the $z^n$ (i.e., Laurent polynomials) to approximate a Dirac mass concentrated at the identity $z = 1$ in $S^1$. There are various ways of doing that; one of the most useful is by taking the Féjer kernel $$F_n(z) = \frac1{n+1} (\sum_{-n \leq k \leq n} z^{k/2})^2 = \frac1{n+1}(z^{-n} + 2z^{-n+1} + \ldots + (n+1)z^0 + \ldots + 2z^{n-1} + z^n)$$ Each Laurent polynomial $F_n(z)$ is real-valued, nonnegative, and its $L^1$ norm is 1. Putting $z = e^{i x}$, we have $$F_n(e^{i x}) = \frac1{n+1} \frac{\sin^2((n+1)x/2)}{\sin^2(x/2)}$$ which makes it clear that $F_n(e^{i x})$ becomes very small outside a neighborhood of 0 (in $\mathbb{R}/2\pi\mathbb{Z}$) as $n$ grows large. Thus $F_n$ approximates the identity; therefore for any $L^2$ function $f$ on $S^1$, we have $$\lim_{n \to \infty} \\|(F_n * f) - f\\|_2 = 0$$ Finally, $F_n * f$ is itself a Laurent polynomial; this follows from the fact that for the function $e_n(z) = z^n$, one has $$ (e_n * f)(w) = \frac1{2\pi i}\int_{S^1} (w/z)^n f(z) \frac{d z}{z} = e_n(w)\langle e_n, f \rangle$$ It follows from all this that the Laurent polynomials on $S^1$ are dense in $L^2(S^1)$. A similar technique applies to any compact Hausdorff abelian group $G$ equipped with its normalized Haar measure $d\mu$, in place of the measure space $(S^1, \frac1{2\pi i}\frac{d z}{z})$, and shows that the characters on the group span a dense subspace in $L^2$ norm. In other words, the characters form an orthonormal basis of $L^2(G, d\mu)$. =--
AQFT	https://ncatlab.org/nlab/source/AQFT	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebraic Quantum Field Theory +--{: .hide} [[!include AQFT and operator algebra contents]] =-- #### Quantum systems +--{: .hide} [[!include quantum systems -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Algebraic Quantum Field Theory or Axiomatic Quantum Field Theory or AQFT for short is a formalization of [[quantum field theory]] (and specifically full, hence [[non-perturbative quantum field theory]]) that axiomatizes the assignment of _[[algebras of observables]]_ to patches of parameter space ([[spacetime]], [[worldvolume]]) that one expects a quantum field theory to provide. As such, the approach of AQFT is roughly dual to that of [[FQFT]], where instead _spaces of states_ are assigned to boundaries of [[cobordism]]s and propagation maps between state spaces to cobordisms themselves. One may roughly think of AQFT as being a formalization of what in basic [[quantum mechanics]] textbooks is called the [[Heisenberg picture]] of quantum mechanics. On the other hand [[FQFT]] axiomatizes the _[[Schrödinger picture]]_ . The axioms of traditional AQFT encode the properties of a [[local net]] of observables and are called the [[Haag-Kastler axioms]]. They are one of the oldest systems of axioms that seriously attempt to put [[quantum field theory]] on a solid conceptual footing. From the [[nPOV]] we may think of a [[local net]] as a co-flabby [[presheaf\|copresheaf]] of [[algebra\|algebras]] on spacetime which satisfies a certain _locality_ axiom with respect to the [[smooth Lorentzian manifold\|Lorentzian structure]] of [[spacetime]]: * locality: algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra. This is traditionally formulated (implicitly) as a structure in ordinary [[category theory]]. More recently, with the proof of the [[cobordism hypothesis]] and the corresponding [[(∞,n)-category]]-formulation of [[FQFT]] also [[higher category theory\|higher categorical]] versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms: * [[factorization algebra]]s * [[topological chiral homology]] * [[blob homology]]. Initially, all three of these encoded what in physics are called _Euclidean_ quantum field theories, whereas only the notion of [[local net]] incorporated the fact that the underlying spacetime of a quantum field theory is a [[smooth Lorentzian space]]. Recent developments in the formalism of [[factorization algebra]]s have extended their theory to globally hyperbolic [[Lorentzian manifolds]]. In the context of the Haag-Kastler axioms there is a precise theorem, the [[Osterwalder-Schrader theorem]], relating the Euclidean to the Lorentzian formulation: this is the operation known as [[Wick rotation]]. Sheaves are used explicitly in: * Roberts, John E.: [New light on the mathematical structure of algebraic field theory.](http://books.google.com/books?id=IFjzuLjE43kC&lpg=PA297&ots=5Ld1B3I45m&dq=Operator%20algebras%20and%20applications%2C%20Part%202&pg=PA523#v=onepage&q=Operator%20algebras%20and%20applications,%20Part%202&f=false) Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 523–550, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982. * Roberts, John E.: [Localization in algebraic field theory](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-85/issue-1/Localization-in-algebraic-field-theory/cmp/1103921341.full). Comm. Math. Phys. 85 (1982), no. 1, 87–98. --- much information to be filled in --- ## Axioms * [[Wightman axioms]] * [[Haag-Kastler axioms]] ## Theorems * [[Reeh-Schlieder theorem]] * [[Osterwalder-Schrader theorem]] * [[PCT theorem]] * [[Bisognano-Wichmann theorem]] * [[spin-statistics theorem]] ## Properties Generically the algebra of a relativistic AQFT turns out to be a ([[generalized the\|the]]) hyperfinite type $III_1$ [[von Neumann algebra factor]]. See ([Yngvason](#Yngvason)) ## Examples {#Examples} Examples of AQFT [[local nets of observables]] that encode interacting quantum field theories are not easy to construct. The construction of _free_ field theories is well understood, see the references [below](#ExamplesReferences). In [[perturbation theory]] also interacting theories can be constructed, see the references [here](#ReferencesPerturbationTheory). ### Free scalar field / Klein Gordon field {#FreeScalarField} A survey of the AQFT description of the [[free field theor\|free]] [[scalar field]] on [[Minkowski spacetime]] is in ([Motoya, slides 11-17](#Montoya)). Discussion in more general context of [[AQFT on curved spacetimes]] in ([Brunetti-Fredenhagen, section 5.2](#BrunettiFredenhagen)) ### Free fermion / Dirac field The [[free field theory\|free]] [[Dirac field]] and its deformations is discussed for instance in ([DLM, section 3.2](#DLM)), ([Dimock 83](#Dimock82)). ### Electromagnetic field The quantized [[electromagnetic field]] is discussed for instance in ([Dimock 92](#Dimock92)). ### Proca field ([Furliani](#Furliani)) ## Related concepts * [[quantum mechanics]], [[quantum field theory]], [[perturbative quantum field theory]] * AQFT * [[Haag-Kastler axioms]], [[Wightman axioms]] * [[local net of observables]], [[field net]] * [[time slice axiom]], [[split property]] * [[quantum lattice systems]], [[string-localized quantum field]] * [[locally covariant AQFT]] * [[perturbative AQFT]] * [[homotopical algebraic quantum field theory]] * [[Haag-Ruelle scattering theory]] * [[FQFT]] * [[constructive quantum field theory]] [[!include Isbell duality - table]] ## References ### Axioms The original article that introduced the [[Haag-Kastler axioms]] is * {#Haag64} [[Rudolf Haag]], [[Daniel Kastler]], An algebraic approach to quantum field theory, Journal of Mathematical Physics, 5 (1964) 848-861 [[doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124)] following * {#Haag59} [[Rudolf Haag]], _Discussion des "axiomes" et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs_, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151. translated to English as: * [[Rudolf Haag]], Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles, EPJ H 35, 243–253 (2010) ([doi:10.1140/epjh/e2010-10041-3](https://doi.org/10.1140/epjh/e2010-10041-3)) The generalization of the [[spacetime]] [[site]] from open in [[Minkowski space]] to more general and [[curvature\|curved]] spacetimes (see [[AQFT on curved spacetimes]]) is due to * {#BrunettiFredenhagen} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum field theory on curved spacetimes_ [arXiv:0901.2063](http://arxiv.org/abs/0901.2063) * [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], _The generally covariant locality principle -- A new paradigm for local quantum physics_ Commun. Math. Phys. 237:31-68 (2003) ([arXiv:math-ph/0112041](http://arxiv.org/abs/math-ph/0112041)) * [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Backgrounds_ , Proceedings of the Kompaktkurs "Quantenfeldtheorie auf gekruemmten Raumzeiten" held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen See also _[[AQFT on curved spacetimes]]_ . ### Lecture notes and Textbooks Introductory lecture notes: * {#Fredenhagen03} [[Klaus Fredenhagen]], _Algebraische Quantenfeldtheorie_, lecture notes, 2003 ([[FredenhagenAQFT2003.pdf:file]]) * {#FewsterRejzner19} [[Christopher Fewster]], [[Kasia Rejzner]], _Algebraic Quantum Field Theory - an introduction_ ([arXiv:1904.04051](https://arxiv.org/abs/1904.04051)) and for just [[quantum mechanics]] in the algebraic perspective: * {#Gleason09} [[Jonathan Gleason]], The $C^$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]] * {#Gleason11} [[Jonathan Gleason]], From Classical to Quantum: The $F^\ast$-Algebraic Approach, contribution to [VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html), Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]] Textbook accounts: * {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], Operator Algebras and Quantum Statistical Mechanics -- vol 1: $C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States., Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)] * [[Raymond F. Streater]], [[Arthur S. Wightman]], PCT, Spin and Statistics, and All That, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)] * [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)] * [[Rudolf Haag]], _[[Local Quantum Physics -- Fields, Particles, Algebras]]_, Texts and Monographs in Physics, Springer (1992) * [[Huzihiro Araki]], _[[Mathematical Theory of Quantum Fields]]_ (1999) * [[Franco Strocchi]], _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)] * [[Hans Halvorson]], [[Michael Müger]], _Algebraic Quantum Field Theory_ [[arXiv:math-ph/0602036](http://arxiv.org/abs/math-ph/0602036)] An account written by mathematicians for mathematicians: * [[Hellmut Baumgärtel]], Manfred Wollenberg, _Causal nets of operator algebras._ Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete)) * [[Hellmut Baumgärtel]], _Operator algebraic Methods in Quantum Field Theory. A series of lectures._ Akademie Verlag 1995 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete)) ### Reviews Recent account of the principle of locality in AQFT from the point of view of traditional school * [[Franco Strocchi]], _Relativistic Quantum Mechanics and Field Theory_, Found. Phys. 34 (2004) 501-527 [[arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143)] * [[Sergio Doplicher]], _The principle of locality: Effectiveness, fate, and challenges_, J. Math. Phys. __51__, 015218 (2010), [doi](http://dx.doi.org/10.1063/1.3276100) Talk slides include * {#Montoya} Edison Montoya, _Algebraic quantum field theory_ (2009) ([pdf](http://www.matmor.unam.mx/~robert/sem/20091021_Montoya.pdf)) More on the role of [[von Neumann algebra factors]] in AQFT * {#Yngvason} J. Yngvason, The role of type III factors in quantum field theory [[arXiv:math-ph/0411058](http://arxiv.org/abs/math-ph/0411058)] ### Examples {#ExamplesReferences} Construction of examples is considered for instance in * [[Jonathan Dimock]], _Dirac quantum fields on a manifold_, Trans. Amer. Math. Soc. 269 (1982), 133-147. ([web](http://www.ams.org/journals/tran/1982-269-01/S0002-9947-1982-0637032-8/home.html)) {#Dimock82} * [[Jonathan Dimock]], _Quantized electromagnetic field on a manifdold_, Reviews in mathematical physics, Volume 4, Issue 2 (1992) ([web](http://www.worldscinet.com/rmp/04/0402/S0129055X92000078.html)) {#Dimock92} * Edward Furliani, _Quantization of massive vector fields in curved space–time_, J. Math. Phys. 40, 2611 (1999) ([web](http://jmp.aip.org/resource/1/jmapaq/v40/i6/p2611_s1?isAuthorized=no)) {#Furliani} General discussion of AQFT quantization of [[free quantum fields]] is in * [[Christian Bär]], N. Ginoux, [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, (EMS, 2007) ([arXiv:0806.1036](http://arxiv.org/abs/0806.1036)) * [[Christian Bär]], N. Ginoux, _Classical and quantum fields on lorentzian manifolds_ (2011) ([arXiv:1104.1158](http://arxiv.org/abs/1104.1158)) Examples of [[non-perturbative quantum field theory\|non-perturbative]] [[interacting quantum field theory\|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] (in particular in $d \geq 4$) are claimed in * {#BuchholtzFredenhagen20} [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C^\ast$-algebraic approach to interacting quantum field theories_, Commun. Math. Phys. 377 (2020) 947–969 [[arXiv:1902.06062](https://arxiv.org/abs/1902.06062), [doi:10.1007/s00220-020-03700-9](https://doi.org/10.1007/s00220-020-03700-9)] ### Local gauge theory {#LocalGaugeTheory} Discussion of aspects of [[gauge theory]] includes * Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, _Causal posets, loops and the construction of nets of local algebras for QFT_ ([arXiv:1109.4824](http://arxiv.org/abs/1109.4824)) * Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, _QED Representation for the Net of Causal Loops_ ([arXiv:1305.7059](http://arxiv.org/abs/1305.7059)) * [[Giuseppe Ruzzi]], _Nets of local algebras and gauge theories_, 2014 ([pdf slides](http://www.aqft14.eu/wp-content/uploads/2014/05/Ruzzi.pdf)) Construction and axiomatization of gauge field AQFT via [[homotopy theory]] and [[homotopical algebra]] (see also at _[[field bundle]]_) is being developed in * {#BDS} [[Marco Benini]], [[Claudio Dappiaggi]], [[Alexander Schenkel]], _Quantized Abelian principal connections on Lorentzian manifolds_, Communications in Mathematical Physics 2013 ([arXiv:1303.2515](http://arxiv.org/abs/1303.2515)) * {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839)) * {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_ ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071)) The issue of the tension between local gauge invariance and locality and the need to pass to [[stacks]]/[[higher geometry]] is made explicit in * {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14)) Further development of this [[homotopical algebraic quantum field theory]] includes * {#BeniniSchenkel16} [[Marco Benini]], [[Alexander Schenkel]], _Quantum field theories on categories fibered in groupoids_ ([arXiv:1610.06071](https://arxiv.org/abs/1610.06071)) ### Perturbation theory and renormalization {#ReferencesPerturbationTheory} [[perturbation theory\|Perturbation theory]] and [[renormalization]] in the context of AQFT and is discussed in the following articles. The observation that in [[perturbation theory]] the [[renormalization\|Stückelberg-Bogoliubov-Epstein-Glaser]] local [[S-matrix\|S-matrices]] yield a [[local net of observables]] was first made in * V. Il'in, D. Slavnov, _Observable algebras in the S-matrix approach_ Theor. Math. Phys. 36 , 32 (1978) which was however mostly ignored and forgotten. It is taken up again in * [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_ Commun.Math.Phys.208:623-661 (2000) ([arXiv](http://arxiv.org/abs/math-ph/9903028)) (a quick survey is in section 8, details are in section 2). Further developments along these lines are in * [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079)) (relation to [[deformation quantization]]) * [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_ Commun.Math.Phys.208:623-661 (2000) ([arXiv](http://arxiv.org/abs/math-ph/9903028)) * [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_ Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](http://arxiv.org/abs/0901.2038)) (relation to [[renormalization]]) * [[Michael Dütsch]], [[Klaus Fredenhagen]], _A local (perturbative) construction of observables in gauge theores: the example of qed_ , Commun. Math. Phys. 203 (1999), no.1, 71-105, ([arXiv:hep-th/9807078](http://xxx.uni-augsburg.de/abs/hep-th/9807078)). (relation to [[gauge theory]] and [[QED]]) Lecture notes are in * [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428)) * [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](https://arxiv.org/abs/1503.07814)) and a textbook acount is in * [[Katarzyna Rejzner]], _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7)) ### Further developments * {#DLM} [[Claudio Dappiaggi]], [[Gandalf Lechner]], E. Morfa-Morales, _Deformations of quantum field theories on spacetimes with Killing vector fields_, Commun.Math.Phys.305:99-130, (2011), ([arXiv:1006.3548](http://arxiv.org/abs/1006.3548)) Relation to [[factorization algebras]]: * [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], _Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds_, Communications in Mathematical Physics volume 377, pages 971–997 (2020) ([arXiv:1903.03396v2](https://arxiv.org/abs/1903.03396), [doi:10.1007/s00220-019-03561-x](https://doi.org/10.1007/s00220-019-03561-x)) Relation to [[smooth stacks]]: * [[Marco Benini]], [[Marco Perin]], [[Alexander Schenkel]], Smooth 1-dimensional algebraic quantum field theories ([arXiv:2010.13808](https://arxiv.org/abs/2010.13808)) ### Relation to holographic entanglement entropy Discussion of [[local nets of observables]] in [[AQFT]] as the natural language for grasping [[holographic entanglement entropy]]: * [[Edward Witten]], _Notes on Some Entanglement Properties of Quantum Field Theory_, Rev. Mod. Phys. 90, 45003 (2018) ([arXiv:1803.04993](https://arxiv.org/abs/1803.04993)) * [[Thomas Faulkner]], _The holographic map as a conditional expectation_ ([arXiv:2008.04810](https://arxiv.org/abs/2008.04810)) [[!include relation between algebraic and functorial field theory -- references]] [[!redirects algebraic quantum field theory]] [[!redirects algebraic quantum field theories]]
AQFT and operator algebra contents	https://ncatlab.org/nlab/source/AQFT+and+operator+algebra+contents	[[algebraic quantum field theory]] ([[perturbative AQFT\|perturbative]], [[AQFT on curved spacetime\|on curved spacetimes]], [[homotopical algebraic quantum field theory\|homotopical]]) [[A first idea of quantum field theory\|Introduction]] ## Concepts [[field theory]]: * [[classical field theory\|classical]], [[prequantum field theory\|pre-quantum]], [[quantum field theory\|quantum]], [[perturbative quantum field theory\|perturbative quantum]] * [[relativistic field theory\|relativistic]], [[Euclidean field theory\|Euclidean]], [[thermal quantum field theory\|thermal]] [[Lagrangian field theory]] * [[field (physics)]] * [[field bundle]] * [[field history]] * [[space of field histories]] * [[Lagrangian density]] * [[Euler-Lagrange form]], [[presymplectic current]] * [[Euler-Lagrange equations\|Euler-Lagrange]] [[equations of motion]] * [[locally variational field theory]] * [[covariant phase space]] * [[Peierls-Poisson bracket]] * [[advanced and retarded propagator]], * [[causal propagator]] [[quantization]] * [[geometric quantization]] [[geometric quantization of symplectic groupoids\|of symplectic groupoids]] * [[algebraic deformation quantization]], [[star algebra]] [[quantum mechanical system]], [[quantum probability]] * [[subsystem]] * [[observables]] * [[field observables]] * [[local observables]] * [[polynomial observables]] * [[microcausal observables]] * [[operator algebra]], [[C-algebra]], [[von Neumann algebra]] [[local net of observables]] * [[causal locality]] * [[Haag-Kastler axioms]] * [[Wightman axioms]] * [[field net]] * [[conformal net]] * [[state on a star-algebra]], [[expectation value]] * [[pure state]] [[wave function]] [[collapse of the wave function]]/[[conditional expectation value]] * [[mixed state]], [[density matrix]] * [[space of quantum states]] * [[vacuum state]] * [[quasi-free state]], * [[Hadamard state]] * [[Wightman propagator]] * [[picture of quantum mechanics]] [[free field]] [[quantization]] * [[star algebra]], [[Moyal deformation quantization]] * [[Wick algebra]] * [[canonical commutation relations]], [[Weyl relations]] * [[normal ordered product]] * [[Fock space]] [[gauge theories]] * [[gauge symmetry]] * [[BRST complex]], [[BV-BRST formalism]] * [[local BV-BRST complex]] * [[BV-operator]] * [[quantum master equation]] * [[master Ward identity]] * [[gauge anomaly]] [[interacting field theory\|interacting field]] [[quantization]] * [[causal perturbation theory]], [[perturbative AQFT]] * [[interaction]] * [[S-matrix]], [[scattering amplitude]] * [[causal additivity]] * [[time-ordered product]], [[Feynman propagator]] * [[Feynman diagram]], [[Feynman perturbation series]] * [[effective action]] * [[vacuum stability]] * [[interacting field algebra]] * [[Bogoliubov's formula]] * [[quantum Møller operator]] * [[adiabatic limit]] * [[infrared divergence]] * [[interacting vacuum]] [[renormalization]] * [[renormalization scheme\|("re-")normalization scheme]] * [[extension of distributions]] * [[renormalization condition\|("re"-)normalization condition]] * [[quantum anomaly]] * [[renormalization group]] * [[interaction vertex redefinition]] * [[Stückelberg-Petermann renormalization group]] * [[renormalization group flow]]/[[running coupling constants]] * [[effective quantum field theory]] * [[UV cutoff]] * [[counterterms]] * [[relative effective action]] * [[Wilsonian RG]], [[Polchinski flow equation]] ## Theorems {#Theorems} ### States and observables * [[order-theoretic structure in quantum mechanics]] * [[Alfsen-Shultz theorem]] * [[Harding-Döring-Hamhalter theorem]] * [[Kochen-Specker theorem]] * [[Bell's theorem]] * [[Fell's theorem]] * [[Gleason's theorem]] * [[Wigner theorem]] * [[Bub-Clifton theorem]] * [[Kadison-Singer problem]] ### Operator algebra * [[Wick's theorem]] * [[GNS construction]] * [[cyclic vector]], [[separating vector]] * [[modular theory]] * [[Fell's theorem]] * [[Stone-von Neumann theorem]] * [[Haag's theorem]] ### Local QFT * [[Reeh-Schlieder theorem]] * [[Bisognano-Wichmann theorem]] * [[PCT theorem]] * [[spin-statistics theorem]] * [[DHR superselection theory]] * [[Osterwalder-Schrader theorem]] ([[Wick rotation]]) ### Perturbative QFT * [[Schwinger-Dyson equation]] * [[main theorem of perturbative renormalization]]
AQFT on curved spacetimes	https://ncatlab.org/nlab/source/AQFT+on+curved+spacetimes	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### AQFT +--{: .hide} [[!include AQFT and operator algebra contents]] =-- #### Gravity +--{: .hide} [[!include gravity contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Where the [[Haag-Kastler axioms]] formulate [[quantum field theory]] on [[Minkowski spacetime]], known as _[[algebraic quantum field theory]]_ (AQFT) there is a generalization of these axioms to [[curved spacetimes]] ([Brunetti-Fredenhagen 01](#BrunettiFredenhagen01)), also known as _locally covariant algebraic quantum field theory_. For the case of [[perturbative quantum field theory]] this is _[[locally covariant perturbative quantum field theory]]_, see there for more. (This falls short of being a theory of [[quantum gravity]], instead it describes [[quantum field theory]] on classical [[background field]] configurations of [[gravity]].) This is the mathematically rigorous framework for studying subjects such as the [[cosmological constant]] (see [there](cosmological+constant#InPerturbativeQuantumGravity)), [[Hawking raditation]] or the [[cosmic microwave background]] ([Fredenhagen-Hack 13](#FredenhagenHack13)). ## Applications ### Vacuum energy and Cosmological constant The [[renormalization]] freedom in [[perturbative QFT\|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]]. Review includes ([Hack 15, section 3.2.1](#Hack15)). For more see at _[[cosmological constant]]_ [here](cosmological+constant#InPerturbativeQuantumGravity). ## Related concepts * [[general covariance]] * [[causal structure]] ## References To some extent the problem of AQFT on curved spacetime was formulated in * {#Dyson72} [[Freeman Dyson]], Missed opportunities, Bulletin of the AMS, Volume 78, Number 5, September 1972 ([pdf](https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf)) > $\,$ > $[$ the [[Haag-Kastler axioms]] $]$ taken together with the axioms defining a [[C-algebra]] are a distillation into abstract mathematical language of all the general truths that we have learned about the physics of microscopic systems during the last 50 years. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. In some sense, the axioms represent the most serious attempt that has yet been made to define precisely what physicists mean by the words "observability, causality, locality, relativistic invariance," which they are constantly using or abusing in their everyday speech. $[$...$]$ I therefore propose as an outstanding opportunity still open to the pure mathematicians, to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance. $P$ here denotes the [[Poincaré group]], while $E$ denotes what Dyson calls the 'Einstein group', which is now called the [[diffeomorphism group]]. General accounts of (perturbative, algebraic) quantum field theory on curved spacetimes include N. Birrell, P. Davies, _Quantum Fields in Curved Space_, Cambridge: Cambridge University Press, 1982 * [[Robert Wald]], _Quantum field theory in curved spacetime and black hole thermodynamics_. Univ. of Chicago Press 1994 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0842.53052&format=complete)). * {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001)) * [[Christopher Fewster]], [[Rainer Verch]], _Algebraic quantum field theory in curved spacetimes_ ([arXiv:1504.00586](https://arxiv.org/abs/1504.00586)) See also: * [[Bernard S. Kay]], Quantum Field Theory in Curved Spacetime, Encyclopaedia of Mathematical Physics (2023) [[arXiv:2308.14517](https://arxiv.org/abs/2308.14517)] Foundations for [[perturbative quantum field theory]] on curved spacetimes in terms of [[causal perturbation theory]] were laid in * {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028)) * {#HollandsWald01} [[Stefan Hollands]], [[Robert Wald]], _Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime_, Commun. Math. Phys. 223:289-326,2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074)) * {#HollandsWald02} [[Stefan Hollands]], [[Robert Wald]], _On the Renormalization Group in Curved Spacetime_, Commun. Math. Phys. 237 (2003) 123-160 ([arXiv:gr-qc/0209029](https://arxiv.org/abs/gr-qc/0209029)) * {#HollandsWald04} [[Stefan Hollands]], [[Robert Wald]], _Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes_, Rev.Math.Phys. 17 (2005) 227-312 ([arXiv:gr-qc/0404074](https://arxiv.org/abs/gr-qc/0404074)) The [[AQFT]]-style axiomatization via [[local nets]] on a category of [[Lorentzian manifolds]] ([[locally covariant perturbative quantum field theory]]) is due to: * {#BrunettiFredenhagen01} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], _The generally covariant locality principle -- A new paradigm for local quantum physics_, Commun. Math. Phys. 237:31-68 (2003) ([arXiv:math-ph/0112041](http://arxiv.org/abs/math-ph/0112041)) * [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Backgrounds_ , Proceedings of the Kompaktkurs "Quantenfeldtheorie auf gekruemmten Raumzeiten" held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen ([arXiv:0901.2063](http://arxiv.org/abs/0901.2063)) Reviews with emphasis on the AQFT-local-nets point of view: * [[Robert Wald]], _The Formulation of Quantum Field Theory in Curved Spacetime_ ([arXiv:0907.0416](https://arxiv.org/abs/0907.0416)) * [[Robert Wald]], _The History and Present Status of Quantum Field Theory in Curved Spacetime_ ([arXiv:gr-qc/0608018](https://arxiv.org/abs/gr-qc/0608018)) * [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _QFT on curved spacetimes: axiomatic framework and examples_ ([arXiv:1412.5125](http://arxiv.org/abs/1412.5125)) * Níckolas de Aguiar Alves, Nonperturbative Aspects of Quantum Field Theory in Curved Spacetime [[arXiv:2305.17453](https://arxiv.org/abs/2305.17453)] On the [[perturbative algebraic quantum field theory\|locally covariant pAQFT approach]] to effective [[quantum gravity]] and applications to experiment: * [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], Locally covariant approach to effective quantum gravity, in [[Handbook of Quantum Gravity]], Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/2212.07800)] There is also a complementary approach via [[OPEs]]: * [[Stefan Hollands]], [[Robert Wald]], _Axiomatic quantum field theory in curved spacetime_, Commun. Math. Phys. 293:85-125, 2010 ([arXiv:0803.2003](https://arxiv.org/abs/0803.2003)) On the application of [[microlocal analysis]]: * Alexander Strohmaier, [[Rainer Verch]], Manfred Wollenberg: _Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems_ ([arXiv] (http://xxx.uni-augsburg.de/abs/math-ph/0202003)). Discussion of [[renormalization]] in AQFT on curved spacetimes includes * {#KhavkineMoretti16} [[Igor Khavkine]], [[Valter Moretti]], _Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT_, Communications in Mathematical Physics, March 2016 ([arXiv:1411.1302](http://arxiv.org/abs/1411.1302), [publisher](http://link.springer.com/article/10.1007%2Fs00220-016-2618-7)) Discussion of the [[cosmology]] in the context of AQFT on curved spacetimes includes * {#FredenhagenHack13} [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], _Quantum field theory on curved spacetime and the standard cosmological model_ ([arXiv:1308.6773](http://arxiv.org/abs/1308.6773)) * {#BrunettiFredenhagenHackPinamontoRejzner16} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], [[Nicola Pinamonti]], [[Katarzyna Rejzner]], _Cosmological perturbation theory and quantum gravity_ ([arXiv:1605.02573](https://arxiv.org/abs/1605.02573)) * {#Hack15} [[Thomas-Paul Hack]], _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6)) Relation to [[2d CFT]]: * [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], A skeletal model for 2d conformal AQFTs ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837)) [[!redirects AQFT on curved spacetime]] [[!redirects quantum field theory on curved spacetime]] [[!redirects AQFT on curved backgrounds]] [[!redirects locally covariant algebraic quantum field theory]] [[!redirects locally covariant AQFT]] [[!redirects quantum field theory on curved spacetime]] [[!redirects quantum field theory on curved spacetimes]] [[!redirects quantum field theories on curved spacetime]] [[!redirects quantum field theories on curved spacetimes]] [[!redirects QFT on curved spacetime]] [[!redirects QFT on curved spacetimes]] [[!redirects QFTs on curved spacetime]] [[!redirects QFTs on curved spacetimes]]
Arakelov geometry	https://ncatlab.org/nlab/source/Arakelov+geometry	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Complex geometry +--{: .hide} [[!include complex geometry - contents]] =-- #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Motivation The aim of _Arakelov geometry_ is to extend [[intersection theory]] to the case of [[algebraic curves]] over $Spec(\mathbb{Z})$, hence in [[arithmetic geometry]]. Arakelov complemented the algebraic geometry at finite primes with a holomorphic piece at a [[place at infinity]]. Then using [[complex analytic geometry]] and [[Green's functions]] he defined the [[intersection numbers]] using the complementary piece at infinity. ## Definitions ... e.g. ([Durov 07](#Durov07)) ## Properties ### Arithmetic Riemann-Roch theorem * [[arithmetic Riemann-Roch theorem]] ## Related concepts * [[Mordell conjecture]] * [[arithmetic Chow group]] * [[Vojta's conjecture]] * [[global analytic geometry]] * [[function field analogy]] ## References ### Introductions and surveys {#IntroductionsAndSurveys} * Wikipedia: [Arakelov theory](http://en.wikipedia.org/wiki/Arakelov_theory) * {#Lang88} [[Serge Lang]], _Introduction to Arakelov theory_ Springer-Verlag, New York, 1988. * {#SABK91} [[Christophe Soulé]], D. Abramovich, , J.-F. Burnol, J. Kramer, _Lectures on Arakelov Geometry_, Cambridge University Press 1991 * {#deJong04} [[Robin de Jong]], _Explicit Arakelov geometry_, PhD thesis 2004 ([pdf](http://www.math.leidenuniv.nl/~rdejong/publications/thesis.pdf)) * Alberto Camara, _Notes on Arakelov theory_, 2011 ([[CamaraOnArakelov11.pdf:file]]) ### Original articles {#ReferencesOriginal} The theory originates in * [[Suren Arakelov]], _Intersection theory of divisors on an arithmetic surface_, Math. USSR Izv. 8 (6): 1167–1180, 1974, [doi](http://dx.doi.org/10.1070%2FIM1974v008n06ABEH002141); _Theory of intersections on an arithmetic surface_, Proc. ICM Vancouver 1975, vol. 1, 405–408, Amer. Math. Soc. 1975, [djvu](http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0405.0408.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0405.0408.ocr.pdf) After Arakelov there were main improvements by Faltings and Gillet and Soulé. * [[Gerd Faltings]], _Calculus on arithmetic surfaces_, Ann. of Math. (2) 119 (1984), no. 2, 387–424, [MR86e:14009](http://www.ams.org/mathscinet-getitem?mr=740897), [doi](http://dx.doi.org/10.2307/2007043); _Arakelov's theorem for abelian varieties_, Invent. Math. __73__ (1983), no. 3, 337–347, [MR85m:14061](http://www.ams.org/mathscinet-getitem?mr=718934), [doi](http://dx.doi.org/10.1007/BF01388431) The [[arithmetic Riemann-Roch theorem]] is due to * [[Henri Gillet]], [[Christophe Soulé]], _An arithmetic Riemann–Roch Theorem_, Invent. Math. __110__: 473–543, 1992, [doi](http://dx.doi.org/10.1007/BF01231343) * Shou-Wu Zhang, _Small points and Arakelov theory_, Proc. ICM 1998, vol. 2, [djvu](http://www.mathunion.org/ICM/ICM1998.2/Main/03/Zhang.MAN.ocr.djvu), [pdf](http://www.mathunion.org/ICM/ICM1998.2/Main/03/Zhang.MAN.ocr.pdf) In a recent Bonn thesis under Faltings' supervision, * {#Durov07} [[Nikolai Durov]], _A new approach to Arakelov geometry_, [arxiv/0704.2030](http://arxiv.org/abs/0704.2030) a completely algebraic replacement (using generalized [[schemes]] whose local models are spectra of commutative [[algebraic monad]]s) for the original mixed approach is proposed; it is not known if that approach can be closely and precisely compared with the traditional. See also * [[Jean-Benoit Bost]], [[Klaus Künnemann]], _Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers_ ([arXiv:math/0701343](http://arxiv.org/abs/math/0701343)) * [[Jean-Benoit Bost]], [[Klaus Künnemann]], _Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension_ ([arXiv:0807.4374](http://arxiv.org/abs/0807.4374)) ### Relation to AdS3/CFT2 Relation of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]: * [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036) [[!redirects Arakelov theory]]
Araminta Amabel	https://ncatlab.org/nlab/source/Araminta+Amabel	* [personal page](https://math.mit.edu/~araminta/) ## Selected writings On [[Whitehead-generalized cohomology theory\|Whitehead-generalized]] [[differential cohomology]] via [[sheaves of spectra]]: * {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250)) On [[factorization algebras]] in relation to [[functorial field theory]]: * [[Araminta Amabel]], Notes on Factorization Algebras and TQFTs [[arXiv:2307.01306](https://arxiv.org/abs/2307.01306)] category: people
Aravind Asok	https://ncatlab.org/nlab/source/Aravind+Asok	* [institute page](https://dornsife.usc.edu/aravind-asok/) * [ResearchGate page](https://www.researchgate.net/profile/Aravind-Asok) ## Selected writings On [[motivic homotopy theory]] and [[h-cobordism]]: * {#AsokMorel} [[Aravind Asok]], [[Fabien Morel]], _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_, Adv. Math. 227 (5) (2011) 1990-2058 [[arXiv:0810.0324](http://arxiv.org/abs/0810.0324), [doi:10.1016/j.aim.2011.04.009](https://doi.org/10.1016/j.aim.2011.04.009)] and [[representable functor\|representability]]: * [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], Affine representability results in $\mathbb{A}^1$-homotopy theory I: Vector bundles, Duke Math. J. 166 10 (2017) 1923-1953 [[arXiv:1506.07093](http://arxiv.org/abs/1506.07093), [doi:10.1215/00127094-0000014X](https://doi.org/10.1215/00127094-0000014X)] * [[Aravind Asok]], [[Marc Hoyois]], [[Matthias Wendt]], _Affine representability results in $\mathbb{A} ^1$-homotopy theory II: Principal bundles and homogeneous spaces_, Geom. Topol. 22 (2018) 1181-1225 [[arXiv:1507.08020](http://arxiv.org/abs/1507.08020), [doi:10.2140/gt.2018.22.1181](https://doi.org/10.2140/gt.2018.22.1181)] On ([[stable cohomotopy\|stable]]) [[motivic cohomology\|motivic]] [[Cohomotopy]] of [[schemes]] (as [[motivic homotopy theory\|motivic homotopy classes]] of maps into [[motivic sphere\|motivic]] [[Tate spheres]]): * [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], Euler class groups and motivic stable cohomotopy, Journal of the EMS 24 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)] category: people
arc space	https://ncatlab.org/nlab/source/arc+space	#Contents# * table of contents {:toc} ## Idea While for finite $m$, $m$-jets of a scheme of finite type (over an algebraically closed field of characteristic $0$) are represented by a scheme, the $\infty$-[[jet scheme]], the (inverse) limit of $m$-jet schemes is not of finite type; this is the arc space. ## Motivation The arc space (and the jet schemes) of a variety $X$ gives information about the singular locus $X_{sing}$. ## Definition Let $k$ be the algebraically closed field, $Sch/k$ the category of schemes over $k$ and $X$ an object in $Sch/k$. The presheaf $$ (Sch/k)^{op}\to Set\,\,\,\,\,\,\,\,\,\, Y\mapsto (Sch/k) (Y\times_k k[t]/t^{m+1},X) $$ is representable by a $k$-scheme of finite type $X_m$ the $m$-jet scheme. For $s\geq 1$, the canonical maps $k[t]/t^{m+1}\to k[t]^{m+s+1}$ induces maps $(Sch/k) (Y\times_k k[t]/t^{m+s+1},X)\to (Sch/k)(Y\times_k k[t]/t^{m+1},X)$, what is $(Sch/k) (Y,X_{m+1})\to (Sch/k) (Y, X_m)$ hence also on representing objects $X_{m+1}\to X_m$. The limit is the __arc space__ $X_\infty = lim_m X_m$ of $X$ and it comes along with natural projections $X_\infty\to X_m\to X$ (under some assumptions each of the maps is locally trivial). ## Properties If $X$ is a scheme of finite type over $k$ then there is a bijection $$ (Sch/k) (Y,X_\infty) \cong (ind-Sch/k) (Y\hat\times_{Spec k} Spec k[[t]],X) $$ natural in $Y$ in $Sch/k$, where $Y\hat\times_k k[[t]]$ is the formal completion of $Y$ along subscheme $Y\times_{Spec k} \{0\}$.1 ## Related entries * [[singularity]] * [[loop space]] * [[jet space]] * [[motivic integration]] * [[Greenberg scheme]] ## Literature Early ideas appeared in * J. Nash Jr., _Arc structure of singularities_, Duke Math. J., 81 (1995), 31–38. and its appearance in motivic integration stems from * [[M. Kontsevich]], lecture on motivic integration, Orsay, December 7, 1995. For basic lectures see * M. Mustaţǎ, _Spaces of arcs in birational geometry_, [pdf](http://www.math.lsa.umich.edu/~mmustata/lectures_arcs.pdf) * M. Popa, 571 Ch. 5. _Jet schemes and arc spaces_, [pdf](http://homepages.math.uic.edu/~mpopa/571/chapter5.pdf) Surveys: * Jan Denef, Francois Loeser, _Geometry on arc spaces of algebraic varieties_, Proceedings of 3rd ECM, Barcelona, July 10-14, 2000, [math.AG/0006050](http://arxiv.org/abs/math/0006050) * L. Ein, M. Mustaţǎ, _Jet schemes and singularities_, Algebraic geometry- Seattle 2005, 505–546, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009 [MR2483946](http://www.ams.org/mathscinet-getitem?mr=2483946) * Tommaso de Fernex, _The space of arcs of an algebraic variety_, [arxiv/1604.02728](http://arxiv.org/abs/1604.02728) On connections to combinatorics and representation theory: * Clemens Bruschek, Hussein Mourtada, Jan Schepers, _Arc spaces and the Rogers–Ramanujan identities_, The Ramanujan Journal 30:1 (2013) 9-38 Other papers * J. Denef, F. Loeser, _Germs of arcs on singular algebraic varieties and motivic integration_, Invent. Math. 135 (1999), 201–232. * S Ishii, J Kollár, _The Nash problem on arc families of singularities, _ Duke Math. J., 120 (2003) 601–620 [math.AG/0207171](http://arxiv.org/abs/math/0207171) * Shihoko Ishii, _The arc space of a toric variety_, [doi](http://dx.doi.org/10.1016/j.jalgebra.2003.12.015) [arxiv/0312324](http://arxiv.org/abs/math/0312324) * L Ein, R Lazarsfeld, M Mustaţǎ, _Contact loci in arc spaces_, Comput. Math. and [math.AG/0303268](http://arxiv.org/abs/math/0303268) * M Mustaţǎ, _Jet schemes of locally complete intersection canonical singularities_, with an appendix by David Eisenbud and Edward Frenkel, Invent. Math., 145 (2001) 397–424; _Singularities of pairs via jet schemes_, J. Amer. Math. Soc., 15 (2002) 599–615 * Cobo Pablos, H. and González Pérez, Pedro Daniel (2012) Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals. Journal of algebraic geometry, 21 (3) 495-529 [pdf](http://www.ams.org/journals/jag/2012-21-03/S1056-3911-2011-00567-5/S1056-3911-2011-00567-5.pdf) * Dave Anderson, Alan Stapledon, _Arc spaces and equivariant cohomology_, Transformation Groups 18:4 (2013) 931-969 * J. Nicaise, _Arcs and resolution of singularities_, Manuscr. Math. 116: pp. 297-322 (2005) * W. Veys, _Arc spaces, motivic integration and stringy invariants_, in: Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo (2006) 529-572 See also Corollary 4.4 in * [[Bhargav Bhatt]], _Algebraization and Tannaka duality_, Cambridge J. Math. __4__: 4 (2016) 403-461 [doi](https://doi.org/10.4310/CJM.2016.v4.n4.a1) [arXiv:404.7483](https://arxiv.org/abs/1404.7483) A formal version (ind-scheme) of free loop space for a complex algebraic variety containing the Kontsevich-Denef-Loeser arc scheme is studied in * [[Mikhail Kapranov]], [[Eric Vasserot]], _Vertex algebras and formal loop space_, Publ. Math., Inst. Hautes Étud. Sci. 100 (2004) 209--269 (2004) [doi](https://doi.org/10.1007/s10240-004-0023-9) [arXiv:math/0107143](https://arxiv.org/pdf/math/0107143.pdf) [[!redirects arc spaces]] [[!redirects arc schemes]] [[!redirects arc scheme]]
arccos	https://ncatlab.org/nlab/source/arccos	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Trigonometry +-- {: .hide} [[!include trigonometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In [[trigonometry]] the arccosine or "inverse cosine" is one of the [[inverse trigonometric functions]]. ## Definition The arccosine is a [[section]] (or [[right inverse]]) to the cosine function [[cos]], which in turn is one of the [[trigonometric functions]]. More precisely, it is the unique [[continuous function\|continuous]] section of the [[epimorphism]] in the [[(epi, mono) factorization]] of the cosine such that its value at $0$ is $\pi/2$ (see [[pi]]). ## Related concepts * [[cos]] * [[arctan]] * [[pi]] ## References See also * Wikipedia, _<a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions">Inverse trigonometric functions</a>_ [[!redirects arccosine]]
Archana Morye	https://ncatlab.org/nlab/source/Archana+Morye	* [webpage](http://mathstat.uohyd.ernet.in/people/profile/archana-s-morye) category: people
archetypal example	https://ncatlab.org/nlab/source/archetypal+example	An __archetypal example__ of a notion in mathematics is a class of examples such that every instance of the notion in some sense reduces to it: by being isomorphic to it, equivalent in some other sense, or at least that the archetypal example has all essential features (for the problem at hand) found in general case. It is often the case that the fact that some example is in fact archetypal is an insight which comes only after nontrivial study of the subject. Its special role is often a conjecture in the development of the subject. The terminology is not meant so much to label the subclasses of objects, as to emphasise on the exhaustive status of some examples in studying the general case of the notion. It is often used not only for objects but often for procedures, algorithms and alike. Pedagogically it is often good to introduce some definition by introducing the archetypal example first. However the general definition often has appeal of often more invariant definition, and allows possibly some new but nonsubstantial examples. ## Examples Each sheaf is isomorphic to a sheaf of sections of some etale space. Hence sheaves of sections of etale spaces are the archetypal example of a sheaf. In the axiomatic (synthetic) theory of projective spaces, a well known results is that every projective space of dimension different from $2$ is isomorphic to a projective space defined in the algebraic way from a vector space over a division ring. Hence in dimension $n$ different from $2$ (as well as e.g. all finite planes) the algebraic construction of a projective space from the vector space $k^n$ (where $k$ is the ground division ring) amounts to the archetypal example of projective space in synthetic sense. All [[homogeneous spaces]] for Hausdorff paracompact topological groups are _isomorphic_ to [[coset space]]s of that group. Hence the coset spaces are the archetypal example of a homogeneous space, provided we take rather weak conditions on a topological group. Exponential function can be defined for finite matrices (over say real numbers) by the exponential power series. In practice, one uses similarity transformations to reduce the matrix to a similar Jordan form matrix, calculates the exponential for that case and then by the inverse similarity transformation obtains the answer. The calculation for the Jordan forms reduces to calculating for each Jordan block separately and then using the rule that for every block matrix one calculates the exponential and inserts it. In particular for the diagonal matrix one can just take the exponentials of the diagonal entries. Thus, the two examples of the calculation of the exponential function, that is calculating it for the block matrix and for the single Jordan block, are the acrhetypal examples, at least provided we consider the similarity transformation nonessential. An arbitrary full subcategory of the category of all modules over a ring is an archetypal example of an [[abelian category]] by the Freyd-Mitchell embedding theorem. The theorem has conditions of set versus class size which are nonessential for almost all practical purposes in mathematics. Hence we can consider categories of modules over rings as the archetypal examples of abelian categories. This motivated many techniques, for example the method of elements in the study of abelian categories. According to some mathematicians the embedding theorem has its usefulness also used in converse sense. Namely, finding the abelian category proof of some fact on categories of modules is often beneficial as it may bring clarity and it may surface deeper essential ideas (and sometimes even lead to simplifications). Why the embedding theorem helps and not just the fact that the category of modules is an abelian category: if it were not an almost general example, then we would be unsure that if the internal proof even exists, hence we would less likely work hard on it and find it. ## Non-examples While [[vector space]] is a basic example of a [[module]] (namely, when the underlying ring is a [[field]] or [[division ring]]) it is not an archetypal example. Indeed, most interesting phenomena in modules become trivial or nonexistent in the case of vector spaces. For example, in every dimension there is only one isomorphism class of vector spaces, while this is far from so in the case of general modules. Similarly, every vector space is semisimple (and there is only one isomorphism class of simple vector spaces) while for many rings there are nonsemisimple modules. [[!redirects archetypal examples]]
Archibald Richardson	https://ncatlab.org/nlab/source/Archibald+Richardson	[[!redirects Richardson]] Archibald Richardson (1881-1954) was a British algebraist, who was a professor of mathematics at Swansea from 1920. He collaborated with [[D. E. Littlewood]] on invariants and the theory of group representations. They introduced the immanant of a matrix, studied [[Schur functions]] and developed the [[Littlewood-Richardson rule]] for their multiplication. * [St. Andrews history entry](mathshistory.st-andrews.ac.uk/Biographies/Richardson_Archibald/) category: people
archimedean difference protoring	https://ncatlab.org/nlab/source/archimedean+difference+protoring	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition ## A [[protoring]] $M$ is an archimedean difference protoring it is both an [[archimedean protoring]] and a [[difference protoring]]. ## Examples ## * Every [[archimedean integral domain]] is an archimedean difference protoring. * In [[impredicative mathematics]], the [[Dedekind real numbers]] are the [[terminal object\|terminal]] archimedean difference protoring. ## See also ## * [[protoring]] * [[archimedean protoring]] * [[difference protoring]] * [[archimedean integral domain]] * [[streak]] ## References * [[Davorin Lešnik]], Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399)) [[!redirects archimedean difference protorings]] [[!redirects Archimedean difference protoring]] [[!redirects Archimedean difference protorings]]
Archimedean field	https://ncatlab.org/nlab/source/Archimedean+field	## Disambiguation The term Archimedean field may refer to either: * [[Archimedean ordered fields]] * [[Archimedean valued fields]] category: disambiguation [[!redirects archimedean field]] [[!redirects archimedean fields]] [[!redirects non-archimedean field]] [[!redirects non-archimedean fields]] [[!redirects nonarchimedean field]] [[!redirects nonarchimedean fields]] [[!redirects Archimedean field]] [[!redirects Archimedean fields]] [[!redirects non-Archimedean field]] [[!redirects non-Archimedean fields]]
archimedean group	https://ncatlab.org/nlab/source/archimedean+group	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition An _archimedean group_ is a [[strict total order\|strictly]] [[ordered group]] which satisfies the [[Archimedean property]], in which every positive element is bounded above by a [[natural number]]. So an archimedean group has no infinite elements (and thus no non-zero [[infinitesimal object\|infinitesimal]] elements). ## Properties * Every archimedean group is an [[abelian group]] and has no bounded [[cyclic group\|cyclic]] [[subgroups]]. Every archimedean group admits an [[embedding]] into the group of [[real numbers]]. * Every archimedean group is a [[flat module]] and a torsion-free group. * Every Dedekind complete archimedean group is isomorphic to the [[integers]], if the group is not dense, or the [[Dedekind real numbers]], if the group is dense. ## Examples Archimedean groups include * [[integers]] * [[half integers]] * [[rational numbers]] * [[dyadic rationals]] * [[decimal rationals]] * [[real numbers]] Non-archimedean groups include * [[p-adic integers]] * [[p-adic numbers]] ## See also * [[archimedean property]] * [[archimedean protoring]] ## External links * Wikipedia, _<a href="https://en.wikipedia.org/wiki/Archimedean_group">Archimedean group</a>_ [[!redirects archimedean group]] [[!redirects archimedean groups]] [[!redirects non-archimedean group]] [[!redirects non-archimedean groups]] [[!redirects nonarchimedean group]] [[!redirects nonarchimedean groups]] [[!redirects Archimedean group]] [[!redirects Archimedean groups]] [[!redirects non-Archimedean group]] [[!redirects non-Archimedean groups]]
Archimedean ordered Artinian local ring	https://ncatlab.org/nlab/source/Archimedean+ordered+Artinian+local+ring	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Differential geometry +-- {: .hide} [[!include synthetic differential geometry - contents]] =-- #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- #### Formal geometry +--{: .hide} [[!include formal geometry -- contents]] =-- =-- =-- \tableofcontents ## Idea Since every [[ordered Artinian local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered Artinian local ring is an [[ordered Artinian local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$. Archimedean ordered Artinian local rings are important because they are the [[ordered local rings]] with [[nilpotent]] [[infinitesimals]] but no [[infinite elements]] or invertible infinitesimals, and thus play an important role in synthetic approaches to [[analysis]] and [[differential geometry]]. [[Archimedean ordered fields]] are the Archimedean ordered Artinian local rings in which the [[nilradical]] is the [[trivial\|trivial ideal]], or equivalently, in which every non-invertible element is equal to zero. ## Properties ### Purely real and purely infinitesimal elements Suppose that $K$ is an [[Archimedean ordered field]] and $A$ is an Archimedean ordered Artinian local $K$-algebra. Since $A$ is a [[local ring]], the quotient of $A$ by its ideal of non-invertible elements $D$ is the [[residue field]] $K$ itself, and the canonical function used in defining the quotient is the function $\Re:A \to K$ which takes a number $a \in A$ to its purely real component $\Re(a) \in K$. Since $A$ is an ordered $K$-algebra, there is a [[strictly monotone]] [[ring homomorphism]] $h:K \to A$. An element $a \in A$ is purely real if $h(\Re(a)) = a$, and an element $a \in A$ is purely [[infinitesimal]] if it is in the [[fiber]] of $\Re$ at $0 \in K$. Zero is the only element in $A$ which is both purely real and purely infinitesimal. ### Prelattice structure Now, suppose that the [[Archimedean ordered field]] $K$ has [[lattice]] structure $\min:K \times K \to K$ and $\max:K \times K \to K$. Then the Archimedean ordered local $K$-algebra $A$ has a [[prelattice]] structure given by functions $(-)\wedge(-):A \times A \to A$ and $(-)\vee(-):A \times A \to A$, defined by $$a \wedge b \coloneqq h(\min(\Re(a), \Re(b)))$$ and $$a \vee b \coloneqq h(\max(\Re(a), \Re(b)))$$ ### Pseudometric and seminorm structure The [[distance]] function on $A$ is given by the function $\rho:A \times A \to K$, defined as $$\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))$$ and the [[absolute value]] on $A$ is given by the function $\vert-\vert:A \times A \to K$, defined as $$\vert a \vert \coloneqq \rho(a, 0)$$ The distance function and absolute value are [[pseudometrics]] and multiplicative [[seminorms]], because every [[Archimedean ordered field]] $K$ embeds into the [[real numbers]] $\mathbb{R}$, and since $\min(a, b) \leq \max(a, b)$, the pseudometric and seminorm are always non-negative. This also implies that in every Archimedean ordered field with lattice structure, the pseudometric defined above is a metric. ### Smooth and differentiable function structure Suppose that $K$ is an [[Archimedean ordered field]] with lattice structure, and $A$ is an [[Archimedean ordered local ring\|Archimedean ordered]] [[Artinian local algebra\|Artinian local $K$-algebra]]. Then [[continuous functions]], [[differentiable functions]], and [[smooth functions]] are each definable on $K$ using the algebraic, order, and metric structure on $K$. The ring homomorphism $h:K \to A$ preserves [[smooth functions]]: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$, then for every [[smooth function]] $f \in C^\omega(K)$, there is a function $f_A:A \to A$ such that for all elements $x \in K$, $f_A(h(x)) = h(f(x))$ and $$f_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i$$ If we restrict to Archimedean ordered Artinian local $K$-algebras $A$ where every element of the nilradical $D$ is a nilsquare element, where for all $\epsilon \in D$, $\epsilon^2 = 0$, then the ring homomorphism $h:K \to A$ preserves [[differentiable functions]]; for every [[differentiable function]] $f:K \to K$ with given [[derivative]] $f':K \to K$, there is a function $f_A:A \to A$ such that for all elements $x \in K$ and nilpotent elements $\epsilon \in D$, $f_A(h(x)) = h(f(x))$ and $$f_A(h(x) + \epsilon) = h(f(x)) + h(f'(x)) \epsilon$$ In particular, every [[polynomial function]] $p:K \to K$ lifts to a polynomial function $p_A:A \to A$. Alternatively, one could use this property to define differentiable and smooth functions in $K$, such as the [[exponential function]], [[natural logarithm]], [[sine function]], and [[cosine function]]. One could also work with [[partial functions]] instead. Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A [[partial function]] $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$. A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $(D^0 j)(a) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$ $$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$ A function $f:I \to \mathbb{R}$ is smooth at an element $a \in I$ if it is smooth at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is smooth if it is smooth at the [[improper subset]] of $I$. A function $f:I \to \mathbb{R}$ is differentiable at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d f}{d x}:S \to \mathbb{R}$ such that for all Archimedean ordered Artinian local $K$-algebras $A$ with ring homomorphism $h:K \to A$ such that for all $\epsilon \in D$, $\epsilon^2 = 0$, for all nilpotent elements $\epsilon \in D$, $$f_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon$$ A function $f:I \to \mathbb{R}$ is differentiable at an element $a \in I$ if it is differentiable at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is differentiable if it is differentiable at the [[improper subset]] of $I$. ### Square roots and Euclidean pseudometric structure Now, assume that $K$ is an [[Euclidean field]] as well, in addition to being an [[Archimedean ordered field]]. While $K$ has a [[principal square root function]] $\sqrt{-}:[0, \infty) \to [0, \infty)$, not every Archimedean ordered local $K$-algebra $A$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, because purely infinitesimal elements in $A$ are not guaranteed to have [[square roots]]. An Archimedean ordered Artinian local $K$-algebra is Euclidean if every nilpotent element has a square root. However, given an Archimedean ordered Artinian local $K$-algebra $A$, every [[rank]] $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ has a [[Euclidean pseudometric]] $\rho_V:V \times V \to K$, given by $$\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}$$ for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where $$a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i$$ If $A$ is an [[ordered field]], then the Euclidean pseudometric on $V$ is a [[metric]]. ## See also * [[Archimedean ordered field]] * [[Archimedean ordered local ring]] * [[ordered Artinian local ring]] [[!redirects Archimedean ordered Artinian local ring]] [[!redirects Archimedean ordered Artinian local rings]] [[!redirects Archimedean ordered local Artinian ring]] [[!redirects Archimedean ordered local Artinian rings]] [[!redirects Archimedean ordered Artinian local algebra]] [[!redirects Archimedean ordered Artinian local algebras]] [[!redirects Archimedean ordered local Artinian algebra]] [[!redirects Archimedean ordered local Artinian algebras]] [[!redirects Archimedean ordered Artin local ring]] [[!redirects Archimedean ordered Artin local rings]] [[!redirects Archimedean ordered local Artin ring]] [[!redirects Archimedean ordered local Artin rings]] [[!redirects Archimedean ordered Artin local algebra]] [[!redirects Archimedean ordered Artin local algebras]] [[!redirects Archimedean ordered local Artin algebra]] [[!redirects Archimedean ordered local Artin algebras]]
Archimedean ordered field	https://ncatlab.org/nlab/source/Archimedean+ordered+field	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea An _Archimedean ordered field_ is an [[ordered field]] that satisfies the [[archimedean property]]. ## Definition ### Using field homomorphisms from the rationals The [[rational numbers]] are the [[initial]] [[ordered field]], so for every ordered field $F$ there is a field homomorphism $h:\mathbb{Q}\to F$. Since every field homomorphism between ordered fields is an [[injection]], the rational numbers $\mathbb{Q}$ is a [[subset]] of the ordered field $F$, and we can suppress the field homomorphism via [[coercion]], such that given $q \in \mathbb{Q}$ one can derive that $q \in F$. Thus, $F$ is _Archimedean_ if for all elements $x \in F$ and $y \in F$, if $x \lt y$, then there exists a rational number $q\in \mathbb{Q}$ such that $x \lt q$ and $q \lt y$. ### Using Dedekind cut-like conditions A [[real number]] in an [[ordered field]] $F$ is an element $x \in F$ which satisfies the [[Dedekind cut]] axioms: 1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$ 1. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$ 1. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$ 1. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$ We have the following results: * The first and second conditions say that every element $x \in F$ is bounded below and above by rational numbers, and thus strictly not an [[infinite]] element. This also implies that there are no infinitesimal elements, because there are no element $x \in F$ whose multiplicative inverse is an infinite element. * The fifth and sixth conditions independently imply that every element $x \in F$ is strictly not an [[infinitesimal]] element. These four conditions together imply the [[archimedean property]] for the ordered field $F$. * The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the [[strict order]] relation. * Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the [[pushout]] of the [[open intervals]] $(q, \infty)$ and $(-\infty, r)$ with canonical inclusions $(q, r) \to (q, \infty)$ and $(q, r) \to (-\infty, r)$ is equivalent to $F$ itself. Thus, an Archimedean ordered field is an ordered field $F$ where every element $x \in F$ is a real number. Note that this definition is not the same as saying that $F$ contains every [[real number]] - the latter definition results in the [[Dedekind real numbers]], which is the union of all Archimedean ordered fields and the [[terminal object\|terminal]] Archimedean ordered field. ## Properties Every Archimedean ordered field is a [[dense linear order]]. This means that the [[Dedekind completion]] of every Archimedean ordered field is the field of all [[real numbers]]. ### Dedekind cuts Every element $x \in F$ in an Archimedean ordered field satisfies the axioms of [[Dedekind cuts]]: 1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$ 1. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$ 1. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$ 1. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$ 1. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$ We have the following results: * The first condition is always true because for all $x \in F$, we have $x - 1 \in F$, and by the Archimedean principle there exists a rational number $q \in \mathbb{Q}$ such that $x - 1 \lt q \lt x$. * The second condition is always true because for all $x \in F$, we have $x + 1 \in F$, and by the Archimedean principle there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r \lt x + 1$. * The fifth condition is always true because for all $x \in F$ and $q \in \mathbb{Q}$, if $q \lt x$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $q \lt q' \lt x$. * The sixth condition is always true because for all $x \in F$ and $r \in \mathbb{Q}$, if $x \lt r$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $x \lt r' \lt r$. * The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the [[strict order]] relation. * Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the [[union]] $(q, \infty) \cup (-\infty, r)$ of the [[open intervals]] $(q, \infty)$ and $(-\infty, r)$ is the [[improper subset]] of $F$. ### Continuous and differentiable structure Every Archimedean ordered field is a [[differentiable space]]: #### Pointwise continuous functions Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is __continuous at a point__ $c \in F$ if $$isContinuousAt(f, c) \coloneqq \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). (\vert x - c \vert \lt \delta) \implies (\vert f(x) - f(c) \vert \lt \epsilon)$$ $f$ is __pointwise continuous__ in $F$ if it is continuous at all points $c$: $$isPointwiseContinuous(f) \coloneqq \forall c \in F. isContinuousAt(f, c)$$ The set of all pointwise continuous functions is defined as $$C^0(F) \coloneqq \{f \in F \to F \vert isPointwiseContinuous(f)\}$$ #### Pointwise differentiable functions Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is __differentiable at a point__ $c \in F$ if $$isDifferentiableAt(f, c) \coloneqq isContinuousAt(f, c) \times \exists L \in F. \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). \forall h \in (-\delta, 0) \cup (0, \delta). \left\| \frac{f(c + h) - f(c)}{h} - L \right\| \lt \epsilon$$ $f$ is __pointwise differentiable__ in $F$ if it is differentiable at all points $c$: $$isPointwiseDifferentiable(f) \coloneqq \forall c \in F. isDifferentiableAt(f, c)$$ The set of all pointwise differentiable functions is defined as $$D^0(F) \coloneqq \{f \in F \to F \vert isPointwiseDifferentiable(f)\}$$ ## Category of Archimedean ordered fields The category of Archimedean ordered fields is the [[category]] whose [[objects]] are Archimedean ordered fields and whose [[morphisms]] are [[strictly monotonic]] field [[homomorphisms]] between Archimedean ordered fields. The category of Archimedean ordered fields is a [[thin category]]. It is also a [[skeletal category]] and a [[gaunt category]], and impredicatively is the subset of the [[power set]] of [[real numbers]] which consists of all the Archimedean ordered [[subfields]] of the real numbers. The [[initial object]] in the category of Archimedean ordered fields is the [[rational numbers]] and the [[terminal object]] in the category of Archimedean ordered fields is the ([[Dedekind real number\|Dedekind]]) [[real numbers]]. ## Examples Archimedean ordered fields include * [[rational numbers]] * [[real closed fields]] * [[real numbers]] In [[constructive mathematics]], one has the different notions of real numbers * [[Cauchy real numbers]] (terminal Archimedean ordered field where every element merely has a [[locator]]) * [[HoTT book real numbers]] (initial [[Cauchy complete]] Archimedean ordered field) * [[Dedekind real numbers]] (terminal Archimedean ordered field, also initial [[Dedekind complete]] Archimedean ordered field) These notions of real numbers are the same if every Dedekind real number merely has a [[locator]], so that the Cauchy real numbers are Dedekind complete. Both [[excluded middle]] and [[countable choice]] imply that every Dedekind real number has a [[locator]]. Non-Archimedean ordered fields include * [[finite fields]] * [[complex numbers]] * [[p-adic numbers]] * [[surreal numbers]] ## Related concepts * [[archimedean valued field]] * [[archimedean integral domain]] * [[archimedean group]] * [[Archimedean ordered Artinian local ring]] * [[Archimedean ordered reduced local ring]] * [[Archimedean ordered local integral domain]] * [[differentiable space]] ## References * [[Univalent Foundations Project]], [[HoTT book\|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013) The definition of the Archimedean property for an ordered field is given in section 4.3 of * Auke B. Booij, Analysis in univalent type theory (2020) $[$[etheses:10411]( http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$ The [[real numbers]] are defined as [[generalized the\|the]] [[terminal object\|terminal]] [[Archimedean ordered field]] and the [[complete space\|complete]] [[Archimedean ordered field]] in: * {#Richman08} [[Fred Richman]], Real numbers and other completions, Mathematical Logic Quarterly 54 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)] [[!redirects archimedean ordered field]] [[!redirects archimedean ordered fields]] [[!redirects Archimedean ordered field]] [[!redirects Archimedean ordered fields]] [[!redirects category of archimedean ordered fields]] [[!redirects category of Archimedean ordered fields]]
Archimedean ordered integral domain	https://ncatlab.org/nlab/source/Archimedean+ordered+integral+domain	# Contents * table of contents {: toc} ## Idea An _Archimedean ordered integral domain_ is an [[ordered integral domain]] that satisfies the [[Archimedean property]]. ## Properties Every Archimedean ordered integral domain extension of the integers $\mathbb{Z}[x]$ is a [[dense linear order]]. Since $\mathbb{Z}[x]$ is Archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer $a$ and natural number $b$ such that $0 \lt x - a \lt 1$ and $1 \lt b(x - a)$. $0 \lt x - a \lt 1$ implies $0 \lt (d - c)(x - a) \lt d - c$ and $c \lt (d - c)(x - a) + c \lt d$ for all $c$ and $d$ in $\mathbb{Z}[x]$. Let $y = (d - c)(x - a) + c$. Since there exists an element $y$ such that $c \lt y \lt d$ for all $c$ and $d$, $\mathbb{Z}[x]$ is a [[dense linear order]]. This means that the [[Dedekind completion]] of every Archimedean ordered integral domain extension of the integers is the integral domain of [[real numbers]]. ## Examples Archimedean ordered integral domains include * [[integers]] * [[rational numbers]] * [[decimal rational numbers]] * [[dyadic rational numbers]] * [[real numbers]] Non-Archimedean ordered integral domains include * [[p-adic integers]]. ## Related concepts * [[Archimedean ordered field]] * [[Archimedean group]] * [[archimedean difference protoring]] * [[enriched set]] [[!redirects archimedean ordered integral domain]] [[!redirects archimedean ordered integral domains]] [[!redirects Archimedean ordered integral domain]] [[!redirects Archimedean ordered integral domains]] [[!redirects archimedean integral domain]] [[!redirects archimedean integral domains]] [[!redirects Archimedean integral domain]] [[!redirects Archimedean integral domains]]
Archimedean ordered local integral domain	https://ncatlab.org/nlab/source/Archimedean+ordered+local+integral+domain	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- \tableofcontents ## Idea Since every [[ordered local integral domain]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered local integral domain is an [[ordered local integral domain]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$. Archimedean ordered local integral domains are important because they are the [[ordered local rings]] with [[infinitesimals]] but no [[infinite elements]], infinitesimals for which the zero-product property fails, or invertible infinitesimals. These play an important role in some approaches to [[analysis]] and [[differential geometry]], such as the approaches where the ring of [[formal power series]] with $n$ commuting variables on the [[real numbers]] is used to model [[infinitesimals]] in the real line. [[Archimedean ordered fields]] are the Archimedean ordered local integral domains in which every non-invertible element is equal to zero. However, unlike Archimedean ordered fields, Archimedean ordered local integral domains can be defined inside of an arbitrary [[arithmetic pretopos]] in [[coherent logic]]. ## See also * [[Archimedean ordered field]] * [[Archimedean ordered local ring]] * [[Archimedean ordered reduced local ring]] * [[ordered local integral domain]] * [[formal power series]] [[!redirects Archimedean ordered local integral domain]] [[!redirects Archimedean ordered local integral domains]]
Archimedean ordered local ring	https://ncatlab.org/nlab/source/Archimedean+ordered+local+ring	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Differential geometry +-- {: .hide} [[!include synthetic differential geometry - contents]] =-- #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- #### Formal geometry +--{: .hide} [[!include formal geometry -- contents]] =-- =-- =-- \tableofcontents ## Definition Since every [[ordered local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered local ring is an [[ordered local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$. Unlike [[Archimedean ordered fields]], which require arithmetic [[Heyting pretoposes]], Archimedean ordered local rings are definable in any [[arithmetic pretopos]]. ## In analysis Archimedean ordered local rings are important for modeling notions of [[infinitesimals]]. These include the dual numbers, which represent nilsquare infinitesimals and are used to synthetically define [[differentiable functions]] in the real numbers, Archimedean ordered Weil rings, which represent nilpotent infintiesimals and are used to synthetically define [[smooth functions]] in the real numbers, as well as [[formal power series]] on the [[ground ring]], which represent infinitesimals which are not nilpotent and are used to synthetically define [[analytic functions]] in the real numbers. ### Kock-Lawvere axiom An Archimedean ordered local ring $R$ satisfies the [[Kock-Lawvere axiom]] if and only if given any Weil $R$-algebra $W$, the canonical function from $W$ to $R^{\mathrm{spec}_R^W}$ the [[function algebra]] with [[domain]] the [[formal spectrum]] of $W$ and codomain $R$ is an $R$-algebra [[isomorphism]]. ## See also * [[Archimedean ordered field]] * [[ordered local ring]] * [[Archimedean ordered Artinian local ring]] * [[Archimedean ordered reduced local ring]] * [[Archimedean ordered local integral domain]] ## References * [[Ieke Moerdijk]], [[Gonzalo E. Reyes]]: Models for Smooth Infinitesimal Analysis, Springer (1991), [doi:10.1007/978-1-4757-4143-8](https://link.springer.com/book/10.1007/978-1-4757-4143-8) [[!redirects Archimedean ordered local ring]] [[!redirects Archimedean ordered local rings]] [[!redirects archimedean ordered local ring]] [[!redirects archimedean ordered local rings]] [[!redirects Archimedean local ring]] [[!redirects Archimedean local rings]] [[!redirects archimedean local ring]] [[!redirects archimedean local rings]]
Archimedean ordered reduced local ring	https://ncatlab.org/nlab/source/Archimedean+ordered+reduced+local+ring	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- \tableofcontents ## Idea Since every [[ordered reduced local ring]] $R$ has [[characteristic zero]], the positive integers $\mathbb{Z}_+$ are a subset of $R$, with [[injection]] $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered reduced local ring is an [[ordered reduced local ring]] which satisfies the [[archimedean property]]: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$. An important class of Archimedean ordered reduced local rings are [[Archimedean ordered local integral domains]], which are used in [[differential geometry]] and [[analysis]] to define [[analytic functions]] in the same way that [[Archimedean ordered Artinian local rings]] are used to define [[smooth functions]]. [[Archimedean ordered fields]] are the Archimedean ordered reduced local rings in which every non-invertible element is nilpotent, or equivalently, in which every non-invertible element is equal to zero. ## See also * [[Archimedean ordered field]] * [[Archimedean ordered local ring]] * [[Archimedean ordered local integral domain]] * [[ordered reduced local ring]] * [[formal power series]] [[!redirects Archimedean ordered reduced local ring]] [[!redirects Archimedean ordered reduced local rings]] [[!redirects Archimedean ordered reduced local algebra]] [[!redirects Archimedean ordered reduced local algebras]]
Archimedean property	https://ncatlab.org/nlab/source/Archimedean+property	\tableofcontents ## Idea The Archimedean property states that every positive element in a [[strictly weakly ordered]] [[cancellative monoid\|cancellative]] [[commutative monoid]] is bounded above by a [[natural number]]. So an object satisfying the Archimedean property has no infinite elements. If the [[strict weak order]] is additionally a [[connected relation]] and thus a [[pseudo-order]], every [[infinitesimal]] element is equal to zero. ## Definition ### In general Let $(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)$ be the set of positive integers. Let $(A,\lt, +, 0)$ be a [[strictly ordered]] [[cancellative monoid\|cancellative]] [[commutative monoid]]. The positive integers are embedded into the [[function algebra\|function monoid]] $A \to A$; there is an injection $inj:\mathbb{N}^+\to (A \to A)$ such that $inj(1) = id_A$ and $inj(s(n)) = inj(n) + id_A$ for all $n:\mathbb{N}^+$. The __archimedean property__ states that for every $a,b:A$ such that $0 \lt a$ and $0 \lt b$, then there exist $n:\mathbb{N}^+$ such that $a \lt inj(n)(b)$. By [[uncurrying]] $inj$ one gets an [[action]] $act: (\mathbb{N}^+\times A) \to A$ such that $act(1,a) = a$ and $act(s(n),a) = act(n,a) + a$ for all $n:\mathbb{N}^+$ and $a:A$. The archimedean property then states that for all $a,b:A$ such that $0 \lt a$ and $0 \lt b$, there exist $n:\mathbb{N}^+$ such that $a \lt act(n,b)$. ### For ordered fields For [[ordered fields]] $F$, since $\mathbb{Q}$ is the initial ordered field, there exists a unique ordered field homomorphism $i:\mathbb{Q} \to F$ for all ordered fields $F$. Then the archimedean property for $F$ states that the rational numbers are [[dense]] in $F$: * for all $x \in F$ and $y \in F$ such that $x \lt y$, there exists a rational number $q:\mathbb{Q}$ such that $x \lt i(q) \lt y$. ## Archimedean structure In [[dependent type theory]], both definitions of the usual archimedean property above use the phrase "there exists...". This existence in the definitions is mere existence; i.e. using the [[existential quantifier]] rather than the [[dependent sum type]]. The untruncated version of the archimedean property using dependent sum types can be called archimedean structure. ### In general Archimedean structure for [[strictly ordered]] [[cancellative monoid\|cancellative]] [[commutative monoids]] $(A,\lt, +, 0)$ using the positive natural numbers says that * for every $x:A$, $y:A$ such that $0 \lt x$ and $0 \lt y$, then there exist as structure a positive natural number $n:\mathbb{N}^+$ such that $x \lt \mathrm{inj}(n)(y)$. $$\prod_{x:A} \prod_{y:A} (0 \lt x) \times (0 \lt y) \to \sum_{n:\mathbb{N}^+} x \lt \mathrm{inj}(n)(y)$$ Defining the type of positive elements in $A$ as $A^+ \coloneqq \sum_{x:A} 0 \lt x$, by [[uncurrying]] and using the [[type theoretic axiom of choice]], one gets the equivalent statement $$\prod_{x:A^+} \prod_{y:A^+} \sum_{n:\mathbb{N}^+} x \lt \mathrm{inj}(n)(y)$$ which by the type theoretic axiom of choice is the same as saying that * there exists as structure a function $M:(A^+ \times A^+) \to \mathbb{N}^+$ such that for every $x:A^+$ and $y:A^+$, $x \lt \mathrm{inj}(M(x, y))(y)$ $$\sum_{M:(A^+ \times A^+) \to \mathbb{N}^+} \prod_{x:A^+} \prod_{y:A^+} x \lt \mathrm{inj}(M(x, y))(y)$$ These functions can be called the archimedean modulus in parallel to the [[modulus of convergence]] and [[modulus of continuity]] which are defined for the untruncated versions of the definitions of [[Cauchy sequence]] and [[continuous function]]. ### For ordered fields Similarly, archimedean structure for ordered fields states that * for all $x:F$ and $y:F$ such that $x \lt y$, there exists as structure a rational number $q:\mathbb{Q}$ such that $x \lt i(q) \lt y$. $$\prod_{x:F} \prod_{y:F} x \lt y \to \sum_{q:\mathbb{Q}} (x \lt i(q)) \times (i(q) \lt y)$$ One could construct archimedean structure for any Archimedean ordered field where every element has a [[locator]]. See lemma 6.7.3 of [Booij20](#Booij20). In particular, one could construct archimedean structure for any Archimedean ordered field satisfying [[trichotomy]], that for all $x:F$ and $y:F$, exactly one of $x \gt y$, $x \lt y$, $x = y$ is true. Suppose the Archimedean ordered field $F$ satisfies [[trichotomy]]. Then, the [[ceiling]] $x \mapsto \lceil x \rceil$ and [[floor]] $x \mapsto \lfloor x \rfloor$ are well defined functions over the entire domain of $F$. By definition of the ceiling and the fact that $x \lt y$, we have that $$\left\lceil\frac{2}{y - x}\right\rceil \geq \frac{2}{y - x}$$ and thus $$\left\lceil\frac{2}{y - x}\right\rceil (y - x) = \left\lceil\frac{2}{y - x}\right\rceil y - \left\lceil\frac{2}{y - x}\right\rceil x \geq 2$$ so we have $$\left\lceil\frac{2}{y - x}\right\rceil x \lt \frac{\left\lceil \left\lceil\frac{2}{y - x}\right\rceil x \right\rceil + \left\lfloor \left\lceil\frac{2}{y - x}\right\rceil y \right\rfloor}{2} \lt \left\lceil\frac{2}{y - x}\right\rceil y$$ Since $x \lt y$, $0 \lt \left\lceil\frac{2}{y - x}\right\rceil$, and so it is possible to divide all sides by $\left\lceil\frac{2}{y - x}\right\rceil$ and keep the elements in the inequality in order. $$x \lt \frac{\left\lceil \left\lceil\frac{2}{y - x}\right\rceil x \right\rceil + \left\lfloor \left\lceil\frac{2}{y - x}\right\rceil y \right\rfloor}{2 \left\lceil\frac{2}{y - x}\right\rceil} \lt y$$ Thus, if the Archimedean ordered field satisfies [[trichotomy]], then the Archimedean ordered field has Archimedean structure. In [[classical mathematics]], as well as in [[constructive mathematics]] in which the [[analytic LPO]] for the [[terminal object\|terminal]] Archimedean ordered field is true, every Archimedean ordered field has Archimedean structure. ## Examples * [[Archimedean group]] * [[Archimedean ordered field]] * [[Archimedean ordered integral domain]] * [[Archimedean ordered local ring]] ## References * Wikipedia, _<a href="https://en.wikipedia.org/wiki/Archimedean_property">Archimedean property</a>_ The alternative definition of the Archimedean property for ordered fields using the rational numbers is defined in section 11.2.1 of * [[Univalent Foundations Project]], [[HoTT book\|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013) and in section 4.3 of * {#Booij20} [[Auke Booij]], Analysis in univalent type theory (2020) $[$[etheses:10411]( http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)$]$ [[!redirects archimedean property]] [[!redirects Archimedean property]] [[!redirects archimedean structure]] [[!redirects Archimedean structure]]
archimedean protoring	https://ncatlab.org/nlab/source/archimedean+protoring	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition ## A [[protoring]] $M$ is an archimedean protoring if for all terms $a, b, c, d \in M$, $c \lt d$ implies that there exists a positive [[natural number]] $n \in \mathbb{N}_+$ such that $$b + \sum_{i=1}^n c \lt a + \sum_{i=1}^n d$$ where $$\sum_{i=1}^{(-)} (-):\mathbb{N}_+ \times M \to M$$ is the canonical left non-unital $\mathbb{N}_+$-[[action]] for [[commutative semigroups]] defined inductively by $$\sum_{i=1}^{1} c \coloneqq c$$ $$\sum_{i=1}^{n + 1} c \coloneqq c + \sum_{i=1}^{n}$$ ## See also ## * [[archimedean group]] * [[archimedean integral domain]] * [[archimedean difference protoring]] * [[protoring]] ## Examples ## * Every [[archimedean integral domain]] is a archimedean protoring. ## References * Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399)) [[!redirects archimedean protorings]] [[!redirects Archimedean protoring]] [[!redirects Archimedean protorings]]
archimedean valued field	https://ncatlab.org/nlab/source/archimedean+valued+field	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition A (complete) _Archimedean valued field_ is a [[field]] equipped with an [[archimedean absolute value]] (and [[complete field\|complete]] with respect to it). A non-Archimedean valued field is one that is not, hence one whose [[norm]] satisfies the [[ultrametric]] [[triangle inequality]]. ## Properties One of [[Ostrowski's theorems]] says that for $k$ a field [[complete field\|complete]] with respect to an [[absolute value]] ${\vert - \vert}$ either the absolute value is archimedean valued, in which case $k$ is either the field of [[real numbers]] or of [[complex numbers]], or the absolute value is non-archimedean. ### Non-Archimedean valued fields For $k$ a non-Archimedean valued field for some non-Archimedean [[absolute value]] ${\vert -\vert}$ one defines * its ring of integers to be $$ k^\circ := \{x \in k \,\|\, {\vert x\vert} \leq 1\} \,. $$ This is a [[local ring]] with maximal ideal $$ k^{\circ\circ} := \{x \in k \,\|\, {\vert x\vert} \lt 1\} \,. $$ * The residue field of $k$ is the [[quotient]] $$ \tilde k := k^\circ / k^{\circ \circ} \,. $$ ## Examples Archimedean valued fields include * [[real numbers]] * [[complex numbers]] Non-Archimedean valued fields include * [[p-adic numbers]]. ## Related concepts * [[archimedean field]] * [[absolute value]] * [[non-Archimedean analytic geometry]] [[!redirects archimedean valued field]] [[!redirects archimedean valued fields]] [[!redirects Archimedean valued field]] [[!redirects Archimedean valued fields]] [[!redirects archimedean-valued field]] [[!redirects archimedean-valued fields]] [[!redirects Archimedean-valued field]] [[!redirects Archimedean-valued fields]] [[!redirects archimedean valuation]] [[!redirects archimedean valuations]] [[!redirects non-archimedean valued field]] [[!redirects non-archimedean valued fields]] [[!redirects non-Archimedean valued field]] [[!redirects non-Archimedean valued fields]] [[!redirects non-archimedean-valued field]] [[!redirects non-archimedean-valued fields]] [[!redirects non-Archimedean-valued field]] [[!redirects non-Archimedean-valued fields]]
Archimedes	https://ncatlab.org/nlab/source/Archimedes	* [Wikiedpia entry](https://en.wikipedia.org/wiki/Archimedes) ## related $n$Lab entries * [[pi]] category: people
Archive for Mathematical Sciences & Philosophy	https://ncatlab.org/nlab/source/Archive+for+Mathematical+Sciences+%26+Philosophy	The "Archive for Mathematical Sciences & Philosophy" * [webpage](http://www.archmathsci.org) run by [[Michael Wright]] claims to have a considerable collection of recordings (video, also pure audio) of talks on [[topos theory]] and [[category theory]] , such as notably talks and private discussion by [[William Lawvere]]. But few or none of these are publically available. Michael Wright actively keeps traveling around to attend conferences and other meetings on topos theory and category theory, video recording talks. He says he is looking for funding that would eventually allow to make his archived recordings publically available. category: reference [[!redirects Archive for Mathematical Sciences and Philosophy]]
Archive of Formal Proofs	https://ncatlab.org/nlab/source/Archive+of+Formal+Proofs	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Constructivism, Realizability, Computability +-- {: .hide} [[!include constructivism - contents]] =-- #### Type theory +-- {: .hide} [[!include type theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The _Archive of Formal Proofs_ (AFP) is an online library of [[formal proofs]], examples, and projects formalized in the [[proof assistant]] [[Isabelle]]. In contrast to other current projects, AFP and Isabelle are based on classical [[set theory\|set theoretic]] [[foundation of mathematics\|foundations]]. It already comprises a large amount of concepts like [smooth manifolds](https://www.isa-afp.org/entries/Smooth_Manifolds.html), [Markov models](https://www.isa-afp.org/entries/Markov_Models.html), [Monoidal Categories](https://www.isa-afp.org/entries/MonoidalCategory.html), or [ordinals and cardinals](https://www.isa-afp.org/entries/Ordinals_and_Cardinals.html) to mention a few. ## Related projects * [[Xena project]] * [[ForMath project]] * [[The QED project]] * [[UniMath project]] ## References * [project page](https://www.isa-afp.org/) [[!redirects AFP]]
arcsin	https://ncatlab.org/nlab/source/arcsin	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Trigonometry +-- {: .hide} [[!include trigonometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In [[trigonometry]] the arcsine or "inverse sine" is one of the [[inverse trigonometric functions]]. ## Definition The arcsine is a [[section]] (or [[right inverse]]) to the sine function [[sin]], which in turn is one of the [[trigonometric functions]]. More precisely, it is the unique [[continuous function\|continuous]] section of the [[epimorphism]] in the [[(epi, mono) factorization]] of the sine such that its value at $0$ is $0$ (see [[pi]]). ## Related concepts * [[sin]] * [[arctan]] * [[pi]] ## References See also * Wikipedia, _<a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions">Inverse trigonometric functions</a>_ [[!redirects arcsine]]
arctangent	https://ncatlab.org/nlab/source/arctangent	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Trigonometry +-- {: .hide} [[!include trigonometry -- contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea In [[trigonometry]], the _arctangent_ or _inverse tangent_ is one of the basic [[trigonometric functions]]. It is the unique [[continuous function\|continuous]] [[section]] of the [[tangent function]] whose value at $0 \in \mathbb{R}$ is $0$, and it is an [[analytic function]] on the [[real line]]. ## Related concepts * [[trigonometric identity]] ## References * Wikipedia, _[Inverse trigonometric function](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions)_ [[!redirects inverse tangent function]] [[!redirects inverse tangent functions]] [[!redirects arctangent]] [[!redirects arctangents]] [[!redirects arctan]] [[!redirects artan]] [[!redirects atan]] [[!redirects tan⁻¹]]
area	https://ncatlab.org/nlab/source/area	#Contents# * table of contents {:toc} ## Idea The [[volume]] of a [[surface]]. ## Definition ### Of polygons Let $\mathrm{Polygons}$ be the set of all [[polygons]] in the [[Euclidean plane]] $\mathbb{R}^2$. Then the area is a function $A:\mathrm{Polygons} \to \mathbb{R}$ such that for all polygons $P \in \mathrm{Polygons}$, * $A$ is invariant under [[translations]]: * Given a [[linear transformation]] $L$ and a polygon $P$, $A(L P) = \det(L) A(P)$ * Given two vertices $p$ and $q$ of $P$, ... ### In terms of Jordan content Given a [[large set]] $M$ of [[Jordan content\|Jordan-measurable]] subsets of $\mathbb{R}^2$ bounded by a [[Jordan curve]] called shapes, the area of a shape $S \in M$ is the [[Jordan content]] of $S$. ## Related concepts * [[Jordan content]] * [[length]] * [[volume]] * [[surface]] * [[plane]] ## See also * Wikipedia, [Area](https://en.wikipedia.org/wiki/Area) ## References * [[Frank Quinn]], Proof Projects for Teachers ([pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf)) * Apostol, Tom (1967). Calculus. Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra. pp. 58–59. ISBN 9780471000051. * Moise, Edwin (1963). Elementary Geometry from an Advanced Standpoint. Addison-Wesley Pub. Co. ([web](https://archive.org/details/elementarygeomet0000mois)) [[!redirects areas]]
area enclosed by a circle	https://ncatlab.org/nlab/source/area+enclosed+by+a+circle	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- #### Differential geometry +--{: .hide} [[!include synthetic differential geometry - contents]] =-- #### Variational calculus +-- {: .hide} [[!include variational calculus - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea ## The [[area]] enclosed by a [[circle]] as discussed in [[Euclidean geometry]]. ## Definition/proposition and proofs Strictly speaking, we are talking about the area of the [[disk]] whose [[boundary]] is the circle; however, the average person usually identifies the interior of a geometric shape with its boundary. +-- {: .num_prop} ###### Proposition Depending on which [[circle constant]] you use, given a radius $r$ of a circle $\mathcal{C}$ in the [[Euclidean space\|Euclidean]] [[plane]] $\mathbb{R}^2$, the area of a circle is expressed either as $A(r) = \frac{1}{2} \tau r^2$ or as $A(r) = \pi r^2$. =-- ### Proof by double integration +-- {: .proof} ###### Proof In this proof, we are using the circle constant $\tau = 2 \pi$. Given any [[Euclidean space\|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \times [0, r] \to \mathbb{R}^2$ defined as $$\overrightarrow{r}(\rho, \theta) \coloneqq \rho \cos(\theta) \hat{i} + \rho \sin(\theta) \hat{j}$$ Then the area of $\mathcal{C}$ is given by the following [[double integral]]: $$A(r) = \int_{0}^{r} \int_{0}^{\tau} \vert \overrightarrow{r}(\rho, \theta) \vert d \theta d \rho$$ which evaluates to $$A(r) = \int_{0}^{r} \int_{0}^{\tau} \vert \rho \cos(\theta) \hat{i} + \rho \sin(\theta) \hat{j} \vert d \theta d \rho = \int_{0}^{r} \int_{0}^{\tau} \rho((\cos(\theta))^2 + (\sin(\theta))^2) d \theta d \rho = \int_{0}^{r} \int_{0}^{\tau} \rho d \theta d \rho = \int_{0}^{r} \tau \rho d \rho = \frac{1}{2} \tau r$$ =-- ### Proof by areal velocity +-- {: .proof} ###### Proof In this proof, we are using the circle constant $\tau = 2 \pi$. Given any [[Euclidean space\|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. There is an [[geometric algebra]] $\mathbb{G}^2$ on the vector space defined by the equations $\hat{i}^2 = 1$, $\hat{j}^2 = 1$, and $\hat{i} \hat{j} = -\hat{j} \hat{i}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as $$\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}$$ Then the area of $\mathcal{C}$ is given by integrating the magnitude of the [[areal velocity]]: $$A(r) = \int_{0}^{\tau} \left\|\frac{\overrightarrow{r}(\theta) \wedge \overrightarrow{v}(\theta)}{2}\right\| d \theta$$ where $a \wedge b$ is the wedge product of two multivectors $a$ and $b$ and $\overrightarrow{v}$ is the [[velocity]] of a point in $\mathcal{C}$. This expression evaluates to $$A(r) = \int_{0}^{\tau} \left\|\frac{(r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}) \wedge \partial_\theta (r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j})}{2}\right\| d \theta$$ $$A(r) = \int_{0}^{\tau} \left\|\frac{(r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}) \wedge (-r \sin(\theta) \hat{i} + r \cos(\theta) \hat{j})}{2}\right\| d \theta$$ $$A(r) = \int_{0}^{\tau} \left\|\frac{(r (\cos(\theta))^2 \hat{i} \hat{j} + r (\sin(\theta))^2 \hat{i} \hat{j})}{2}\right\| d \theta$$ $$A(r) = \int_{0}^{\tau} \left\|\frac{r \hat{i} \hat{j}}{2}\right\| d \theta = \int_{0}^{\tau} \frac{r}{2} d \theta = \frac{1}{2} \tau r$$ =-- ### Proof by action functionals +-- {: .proof} ###### Proof In this proof, we are using the circle constant $\tau = 2 \pi$. Given any [[Euclidean space\|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as $$\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}$$ Then the area of $\mathcal{C}$ is given by the [[action functional]] of the parameterized curve: $$A(r) = \frac{1}{2} \int_{0}^{\tau} {\vert \overrightarrow{r}(\theta) \vert}^2 d \theta$$ which evaluates to $$A(r) = \frac{1}{2} \int_{0}^{\tau} {\vert r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j} \vert}^2 d \theta = \frac{1}{2} \int_{0}^{\tau} (r((\cos(\theta))^2 + (\sin(\theta))^2))^2 d \theta = \frac{1}{2} \int_{0}^{\tau} r^2 d \theta = \frac{1}{2} \tau r$$ =-- ### Proof by limits of regular polygons +-- {: .proof} ###### Proof In this proof, we are using the circle constant $\tau = 2 \pi$. The area of a [[regular polygon]] $\mathcal{P}_n$ with $n$ sides and [[circumradius]] $r$ is given by the sequence of functions $P:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ $$A_\mathcal{P}(n)(r) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2 n}\right)$$ which [[embedding\|embeds]] in the $\mathbb{R}_+$-[[action]] $A_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R})$, defined as $$A_\mathcal{P}^\prime(n)(r) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2n}\right)$$ The [[limit of a function\|limit]] of $A_\mathcal{P}^\prime$ as $n$ goes to infinity is the area of a circle with radius $r$: $$A(r) = \lim_{n \to \infty} A_\mathcal{P}^\prime(n)(r) = \lim_{n \to \infty} \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{n}\right) = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\sin(\tau m)}{m} = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\partial_m \sin(\tau m)}{\partial_m m} = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\tau \cos(\tau m)}{1} = \frac{1}{2} \tau r^2$$ =-- ## See also * [[circle]] * [[unit circle]] * [[area]] * [[circumference of a circle]] * [[Euclidean space]] [[!redirects area of a circle]]
areal velocity	https://ncatlab.org/nlab/source/areal+velocity	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Physics +-- {: .hide} [[!include physicscontents]] =-- #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- #### Differential geometry +--{: .hide} [[!include synthetic differential geometry - contents]] =-- #### Variational calculus +-- {: .hide} [[!include variational calculus - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea ## The [[bivector]] whose magnitude equals the rate of change at which [[area]] is swept out by a [[point]]/[[particle]] as it moves along a [[curve]]. ## Definition Given an $n$-[[dimension\|dimensional]] [[Euclidean space]] $\mathbb{R}^n$, one could select an [[orthonormal basis]] on $\mathbb{R}^n$ by postulating an origin $0$ and a function $\hat{i}:[1, n] \to \mathbb{R}^n$ such that for all $m, p \in [1, n]$ the pair of vectors $\hat{i}_m$ and $\hat{i}_p$ is mutually [[orthonormal]]. There is an [[geometric algebra]] $\mathbb{G}^n$ on $\mathbb{R}^n$ defined by the equations $\hat{i}_m^2 = 1$ for all $m \in [1, n]$, and $\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m$ for all $m, p \in [1, n]$. A [[smooth curve]] $\mathcal{C}$ in $\mathbb{R}^n$ could be parameterized by a [[smooth function]] $\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n$. Then the areal velocity, sector velocity, or sectorial velocity of a point in $\mathcal{C}$ in $\mathbb{R}^n$ is given by the bivector-valued function $A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2$ $$A(t) = \frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}$$ where $a \wedge b$ is the wedge product of two vectors $a$ and $b$, and $\overrightarrow{v}$ is the [[velocity]]. ## In 3 dimensions In 3 dimensions, the vector areal velocity $\overrightarrow{a}:\mathbb{R} \to \langle \mathbb{G}^n \rangle_1$ is the [[Hodge dual]] of the areal velocity, which is the product of the pseudoscalar $$I = \prod_{i:[1, n]} \hat{i}_i$$ with the areal velocity: $$\overrightarrow{a}(t) = I A(t) = I\left(\frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}\right) = \frac{I(\overrightarrow{r}(t) \wedge \overrightarrow{v}(t))}{2} = \frac{\overrightarrow{r}(t) \times \overrightarrow{v}(t)}{2}$$ ## Conservation of areal velocity Conservation of areal velocity is the same as the conservation of [[angular momentum]]. ## See also * [[area]], [[velocity]] * [[area of a circle]] * [[Euclidean space]] * [[geometric algebra]] * [[classical mechanics]] * [[angular momentum]] * [[torque]] ## References See also: * Wikipedia, _[Areal velocity](https://en.wikipedia.org/wiki/Areal_velocity)_ [[!redirects areal velocities]] [[!redirects sector velocity]] [[!redirects sector velocities]] [[!redirects sectorial velocity]] [[!redirects sectorial velocities]]
Arend	https://ncatlab.org/nlab/source/Arend	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Constructivism, Realizability, Computability +-- {: .hide} [[!include constructivism - contents]] =-- #### Type theory +-- {: .hide} [[!include type theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea _Arend_ is a [[proof assistant]] system for [[homotopy type theory]] with native support for [[higher inductive types]] and some [[cubical type theory]]. ## Related entries [[!include proof assistants and formalization projects -- list]] ## References * [Arend home page](https://arend-lang.github.io/)
Arend Heyting	https://ncatlab.org/nlab/source/Arend+Heyting	* [Wikipedia entry](http://en.wikipedia.org/wiki/Arend_Heyting) ## Selected writings On [[intuitionistic logic]]: * {#Heyting1930} [[Arend Heyting]], Die formalen Regeln der intuitionistischen Logik. I, II, III. Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169 abridged reprint in: Karel Berka, Lothar Kreiser (eds.), Logik-Texte, De Gruyter (1986) 188-192 [[doi:10.1515/9783112645826](https://doi.org/10.1515/9783112645826)] * [[Arend Heyting]], Die intuitionistische Grundlegung der Mathematik, Erkenntnis 2 (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)] * [[Arend Heyting]], Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln", Comp. Math. 4 (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)] and making explicit the [[BHK interpretation]]: * {#Heyting56} [[Arend Heyting]], Intuitionism: An introduction, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()] ## Related entries * [[Heyting algebra]] * [[intuitionistic logic]] category: people [[!redirects Heyting]]
Arf invariant	https://ncatlab.org/nlab/source/Arf+invariant	## Related concepts * [[Arf-Kervaire invariant]] * [[Arf-Kervaire invariant problem]] ## References * Wikipedia, _[Arf invariant](https://en.wikipedia.org/wiki/Arf_invariant)_
Arf-Kervaire invariant problem	https://ncatlab.org/nlab/source/Arf-Kervaire+invariant+problem	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Stable Homotopy theory +--{: .hide} [[!include stable homotopy theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The Arf-Kervaire invariant problem asks whether certain hypothetical elements $\theta_j \in \pi^s_{2^{j+1} - 2}$ of the stable [[homotopy groups of spheres]] exist. In fact the definition of these elements works in all dimensions but due to a theorem of Browder they do not exist in dimensions $d\neq 2^{j+1} - 2$. Prior to ([Hill-Hopkins-Ravenel 09](#HillHopkinsRavenel09)), all that was known rested on the explicit construction of such elements for $j=1,...,5$ (so in dimensions 2,6,14,30 and 62). HHR established that these elements do not exist for $j \gt 6$, so the only dimension in which existence remains unknown in 126 (ie $j=6$). The proof is by construction of a 256-periodic [[spectrum]] $\Omega$ and a [[spectral sequence]] for it that can detect the elements $\theta_j$ as elements of $\pi_(\Omega)$. HHR then show that $\pi_n(\Omega)=0$ for $-4\lt n\lt 0$, which by the periodicity, implies that the images of $\theta_j$ must be elements of the trivial group, and hence are themselves trivial. ## Key properties of the $C_8$ fixed point spectrum $\Xi$ We write $\Xi$ to mean the spectrum "$\Omega$" discussed in [Hill-Hopkins-Ravenel 09](#HillHopkinsRavenel09). ### Detection theorem It has an [[Adams-Novikov spectral sequence]] in which the image of each $\theta_j$ is non-trivial. This means if $\theta_j\in \pi_(\mathbb{S})$ exists then it can be seen in $\pi_(\Xi)$. ### Periodicity Theorem The spectrum is 256-periodic, as in $\Omega^{256}\Xi \simeq \Xi$. ### Gap Theorem We have $\pi_k(\Xi) = 0$ for $-4 \lt k \lt 0$. Its proof uses the [[slice spectral sequence]]. ### Result Suppose $\theta_7 \in \pi_{254}(\mathbb{S})$ exists then the detection theorem implies that it has a non-trivial image in $\pi_{254}(\Xi)$. But by the periodicity and gap theorems we see that $\pi_{254}(\Xi)$ is trivial. The argument for $j \ge 7$ is similar since $\|\theta_j\| = 2^{j+1} \equiv -2 \mod 256$. ## Related concepts [[Kervaire invariant]] * [[Hopf invariant one problem]] ## References A solution of the problem in the negative, except for one outstanding dimension (namely 126), using methods of [[equivariant stable homotopy theory]]: * {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)] * {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)] * {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_, Current Developments in Mathematics, 2010: 1-44 (2011) ([[HHRKervaire.pdf:file]], [doi:10.4310/CDM.2010.v2010.n1.a1](https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1)) On the [[equivariant stable homotopy theory]] involved: * [Hill-Hopkins-Ravenel 09, Appendix](#HillHopkinsRavenel09) * [HillHopkinsRavenel, section 4](#HillHopkinsRavenel) More resources are collected at * [[Douglas Ravenel]], _[A solution to the Arf-Kervaire invariant problem](http://www.math.rochester.edu/people/faculty/doug/kervaire.html)_, web resources 2009 * {#HillHopkinsRavenelIntroduction} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_ (2016) ([pdf](http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf)) [[!redirects Kervaire invariant one problem]] [[!redirects Kervaire invariant problem]]
argument shift method	https://ncatlab.org/nlab/source/argument+shift+method	Argument shift method is a method to construct commutative subalgebras of the symmetric algebra on a Lie algebra with respect to the Poisson-Lie bracket. It is of importance in the study of [[integrable system]]s. * A. S. Mishchenko, A. T. Fomenko, _Euler equations on finite-dimensional Lie groups_, Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 42 (1978), no. 2, 386--415; _Integrability of Euler equations on semisimple Lie algebras_, in: Proc. Seminars on Vector and Tensor Analysis [in Russian], no. 19, Moskov. Gos. Univ., Moscow, 1979, 3--94 * S. V. Manakov, _Remark on integrability of Euler equations of the dynamics of an n-dimensional rigid body_, Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 10 (1976), no. 4, 93--94 * L. G. Rybnikov, _The argument shift method and the Gaudin model_, Funct. Anal. Appl. 40, 188--199 (2006) * V. V. Shuvalov, _On limits of Mishchenko--Fomenko subalgebras in Poisson algebras of semisimple Lie algebras_, Funct. Anal. Appl. 36, 298–305 (2002) [doi](https://doi.org/10.1023/A:1021713927119) * Y. Ikeda, _Quasidifferential operator and quantum argument shift method_, Theor Math Phys 212, 918--924 (2022) [doi](https://doi.org/10.1134/S0040577922070030)
Ari M. Turner	https://ncatlab.org/nlab/source/Ari+M.+Turner	## Selected writings On [[symmetry-protected topological order]]: * {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], Symmetry protection of topological order in one-dimensional quantum spin systems, Phys. Rev. B 85 075125 (2012) $[$[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)$]$ On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order\|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy\|topological]]) [[entanglement entropy]] of subregions: * [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85 (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$ On [[topological semi-metals]]: * [[Ari M. Turner]], [[Ashvin Vishwanath]], Part I of: Beyond Band Insulators: Topology of Semi-metals and Interacting Phases, in: Topological Insulators, Contemporary Concepts of Condensed Matter Science 6 (2013) 293-324 $[$[arXiv:1301.0330](https://arxiv.org/abs/1301.0330), [ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$ category: people [[!redirects Ari Turner]]
Aristotle	https://ncatlab.org/nlab/source/Aristotle	* [Wikipedia entry](http://en.wikipedia.org/wiki/Aristotle) ## Selected writings On [[physics]]: * {#Aristotle} [[Aristotle]], Physics, in: Jonathan Barnes (ed.), The Complete Works of Aristotle Vol 1 (1984) [[pdf](https://sites.unimi.it/zucchi/NuoviFile/Barnes%20%20-%20Physics.pdf), [ISBN:9780691016504](https://press.princeton.edu/books/hardcover/9780691016504/complete-works-of-aristotle-volume-1)] ## Related entries * [[Metaphysics (Aristotle)]] * [[syllogism]] * [[potentiality and actuality]] * [[Science of Logic]] category: people
arithmetic	https://ncatlab.org/nlab/source/arithmetic	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic (Greek ἀριθμός: number) is, roughly speaking, the study of [[numbers]] in their various forms, and the structure and properties of the operations defined on them, including at least [[addition]] and [[multiplication]], and sometimes also [[subtraction]], [[division]], and [[exponentiation]]. Notions of "[[number]]" are very broad and not at all easy to encapsulate. There are [[natural numbers]], [[integers]], [[rational numbers]], [[real numbers]], and [[complex numbers]], [[quaternions]] (Hamiltonian numbers), [[octonions]] (Cayley numbers). There are [[dual numbers]] and [[hyperbolic numbers]], and [[geometric algebra]]s (geometric numbers). There are [[cyclic group#Ring structure\|integers modulo $n$]]. There are [[algebraic numbers]] and [[algebraic integers]], and individual [[fields]] of such ([[number fields]]). There are $p$-[[adic numbers]]. Then there are [[cardinal numbers]], [[ordinal numbers]], and [[surreal numbers]]. There are even [[closed intervals]] and [[open intervals]]. For each one of these one can (and does!) speak of its arithmetic. This article will provide links to other articles in which these various cases are discussed. * [[number theory]] * [[Peano arithmetic]] * [[second-order arithmetic]] * [[arithmetic geometry]], [[arithmetic scheme]], [[arithmetic variety]], [[arithmetic curve]], [[arithmetic genus]], [[arithmetic Chern-Simons theory]] * [[arithmetic topology]] * [[arithmetic pretopos]] * [[modular arithmetic]] * [[transfinite arithmetic]] * [[cardinal arithmetic]] * [[ordinal arithmetic]] ## References See also * Wikipedia, _[Arithmetic](https://en.wikipedia.org/wiki/Arithmetic)_ [[!redirects arithmetics]]
arithmetic and noncommutative geometry	https://ncatlab.org/nlab/source/arithmetic+and+noncommutative+geometry	A relation between [[noncommutative geometry]] and [[arithmetic]] (= number theory) has been explored much in the work of [[Alain Connes]] and his collaborators, especially Marcolli and Consani. Some surveys include * P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics __16__, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030) * [[Matilde Marcolli]], _Lectures on arithmetic noncommutative geometry_, [math/0409520](http://arxiv.org/abs/math/0409520) There is also another line of thought in the work Manin-Marcolli on the relation of [[Arakelov geometry]] and noncommutative geometry. Arakelov's geometry is of course, motivated by number theory. Independetly, one should also notice that the noncommutative geometry over non-archimedean fields is relevant for [[homological mirror symmetry]] as explored in the works of [[Maxim Kontsevich]] and [[Yan Soibelman]]. [[!redirects noncommutative geometry and arithmetic]] [[!redirects arithmetic noncommutative geometry]]
arithmetic Chern-Simons theory	https://ncatlab.org/nlab/source/arithmetic+Chern-Simons+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- #### Quantum Field Theory +--{: .hide} [[!include AQFT and operator algebra contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic Chern-Simons theory names the attempt to apply constructions from [[Chern-Simons theory]] to the field of [[arithmetic]], in view of patterns in the [[function field analogy]]. In particular, the papers ([Kim1](#Kim1), [Kim2](#Kim2)) apply ideas of [[Dijkgraaf-Witten theory]] on 2+1 dimensional [[topological quantum field theory]] to [[arithmetic curves]], that is, the [[ring spectrum\|spectra]] of rings of [[algebraic integer\|integers]] in [[algebraic number fields]]. This theory pursues the surprising analogies between 3-dimensional topology and number theory, where knots embedded in a 3-manifold behave like prime ideals in a ring of algebraic integers, known as [[arithmetic topology]]. ## Related concepts * [[number theory and physics]] * [[arithmetic gauge theory]] * [[p-adic physics]] * [[p-adic string theory]] * [[p-adic AdS/CFT correspondence]] ## References * {#Kim1} [[Minhyong Kim]], Arithmetic Chern-Simons Theory I, in: Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants, Proceedings in Mathematics & Statistics 330, Spinger (2020) [[arXiv:1510.05818](http://arxiv.org/abs/1510.05818), [doi:10.1007/978-3-030-51795-3_8](https://doi.org/10.1007/978-3-030-51795-3_8)] * {#Kim2} Hee-Joong Chung, Dohyeong Kim, [[Minhyong Kim]], Jeehoon Park, Hwajong Yoo, _Arithmetic Chern-Simons Theory II_, ([arXiv:1609.03012](http://arxiv.org/abs/1609.03012)) * Frauke M. Bleher, Ted Chinburg, Ralph Greenberg, Mahesh Kakde, George Pappas, Martin J. Taylor, _Unramified arithmetic Chern-Simons invariants_, ([arXiv:1705.07110](https://arxiv.org/abs/1705.07110)) * Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, George Pappas, Jeehoon Park, Hwajong Yoo, _Abelian arithmetic Chern-Simons theory and arithmetic linking numbers_, ([arXiv:1706.03336](https://arxiv.org/abs/1706.03336)) * Hikaru Hirano, _On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields_, ([arXiv:1911.12964](https://arxiv.org/abs/1911.12964)) * Jungin Lee, Jeehoon Park, _Arithmetic Chern-Simons theory with real places_, ([arXiv:1905.13610](https://arxiv.org/abs/1905.13610)) An introductory talk" * [[Minhyong Kim]], _Arithmetic topological quantum field theory?_, ([slides](https://simonsfoundation.s3.amazonaws.com/share/mps/conferences/2017_Conference_on_Number_Theory_Geometry_Moonshine_and_Strings/Kim.pdf)) Introduction in the broader context of [[arithmetic gauge theory]]: * {#Kim18} [[Minhyong Kim]], Arithmetic Gauge Theory: A Brief Introduction, Modern Physics Letters A 33 29 (2018) 1830012 [[arxiv:1712.07602](https://arxiv.org/abs/1712.07602), [doi:10.1142/S0217732318300124](https://doi.org/10.1142/S0217732318300124)]
arithmetic Chow group	https://ncatlab.org/nlab/source/arithmetic+Chow+group	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Differential cohomology +--{: .hide} [[!include differential cohomology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic Chow groups are refinements of ordinary [[Chow groups]] analogous to how [[ordinary differential cohomology]] refines [[ordinary cohomology]]. Let $X$ be an [[arithmetic variety]], that is: a [[quasi-projective variety\|quasi-projective]] [[flat scheme\|flat]] [[regular scheme]] over an [[arithmetic ring]]. In ([Gillet-Soule](#GilletSoule)) the _arithmetic Chow groups_ of $X$, denoted $\hat CH^p(X)$, are defined as groups whose elements are [[equivalence classes]] of pairs consisting of a codimension $p$ subvariety of $X$ together with a [[Green current]] for it. Later, in ([Burgos Gil 97](#BurgosGil97)), an alternative definition was given in terms of a [[Deligne complex]] of [[differential forms]] with logarithmic singularities along infinity, that computes a version of [[ordinary differential cohomology]] groups. When $X$ is [[proper scheme\|proper]], the two definitions are [[natural isomorphism\|naturally isomorphic]]. ## Related concepts * [[Chow group]], [[Arakelov geometry]] ## References ### General Arithmetic intersection theory was introduced in * Henri Gillet, Christoph Soulé, _Arithmetic intersection theory_ IHES Preprint (1988) {#GilletSoule} Generalization are discussed in * J. I. Burgos Gil, J. Kramer, U. Kühn, _Cohomological arithmetic Chow rings_ ([arXiv:math/0404122v2](http://arxiv.org/abs/math/0404122v2)) {#BurgosGilKramerKuehn} * J. I. Burgos Gil, _Higher arithmetic Chow groups_ ([pdf](http://www.crm.es/Publications/10/Pr925.pdf)) ### Relation to differential cohomology {#ReferencesRelationToDifferentialCohomology} Articles that discuss the relation of arithmetic Chow groups to [[ordinary differential cohomology]] include * Henri Gillet, Christoph Soulé, _Arithmetic Chow groups and differential characters_ in [[Rick Jardine]] (ed.) _Algebraic K-theory: Connections with Geometry and Topology_, Springer (1989) * J. I. Burgos Gil, _Arithmetic Chow rings_, Ph.D. thesis, University of Barcelona, (1994). * J. I. Burgos Gil, _Arithmetic Chow rings and Deligne-Beilinson cohomology_, J. Alg. Geom. 6 (1997), 335–377. {#BurgosGil97} [[!redirects arithmetic Chow groups]] [[!redirects Arithmetic Chow groups]]
arithmetic cohesion -- table	https://ncatlab.org/nlab/source/arithmetic+cohesion+--+table	[[cohesion]] in [[E-∞ arithmetic geometry]]: \| [[cohesion]] [[modality]] \| symbol \| interpretation \| \|---\|---\|---\| \| [[flat modality]] \| $\flat$ \| [[formal completion]] at \| \| [[shape modality]] \| $ʃ$ \| [[torsion approximation]] \| \| [[dR-shape modality]] \| $ʃ_{dR}$ \| [[localization]] away \| \| [[dR-flat modality]] \| $\flat_{dR}$ \| [[adic residual]] \| the [[differential cohomology hexagon]]/[[arithmetic fracture squares]]: $$ \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,, $$
arithmetic cryptography	https://ncatlab.org/nlab/source/arithmetic+cryptography	[[!redirects Arithmetic cryptography]] +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic cryptography +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic cryptography is the developing subject that describes [public key](http://en.wikipedia.org/wiki/Public-key_cryptography) [[cryptography]] systems based on the use of arithmetic geometry of schemes (or [[global analytic spaces]]) over $\mathbb{Z}$. There are two well known examples of such systems: 1. The first and most used one, which is very efficient because it is conformal to the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle), is based on the fact that it is very difficult (from the computational viewpoint) to factorize a natural number into a product of two big prime numbers. 1. The second one is based on the [discrete logarithm](http://en.wikipedia.org/wiki/Discrete_logarithm) problem on elliptic curves (or more generally abelian varieties) over finite fields. It has the advantage on the first algorithm of allowing to make shorter keys, without compromising security (NSA has generalized the use of [elliptic curve cryptography](http://en.wikipedia.org/wiki/Elliptic_curve_cryptography) algorithm both for commercial and classified use, as explained on the wikipedia page on elliptic curve cryptography). The basic idea of arithmetic cryptography is to use a finite family $X$ of polynomials with integer coefficients $P_1,\dots,P_m\in \mathbb{Z}[X_1,\dots,X_n]$ (or more generally a quasi-projective scheme $X$ of finite type over $\mathbb{Z}$, or even maybe a [[global analytic space]] $X$ over a convenient Banach ring), encoded in a finite number of integers (the coefficients and degrees of the corresponding polynomials), together with some additional data (such as a way to cut a part of the associated motive) to define a [public key cryptosystem](http://en.wikipedia.org/wiki/Public-key_cryptography). Some computational aspects of general motives have been investigated in the case of motives of modular forms by Bass Edixhoven and Jean-Marc Couveignes, using étale cohomological methods. Kedlaya and Lauder-Wan also studied the Dwork approach from a computational viewpoint. ## Necessary tools Since the space $X$ to be used is given by a non-linear equation, it is not directly adapted to the use of computational methods. One will thus need to extract linear invariants from $X$, by the use of a kind of 'differential calculus' (in Quillen's sense, i.e., using a convenient [[tangent (infinity,1)-category]]) and/or the definition of convenient 'cohomological invariants. It seems that étale cohomological methods, that were originally used by Grothendieck and Deligne (and more recently, Laumon) to prove the full Weil conjectures, are not so easy to implement on a computer (see however the book of Edixhoven and Couveignes). It seems that "[[p-adic]] methods", based on [[p-adic differential equation\|p-adic differential calculus]] and Fourier transform, and now completely developed by Berthelot, Lestum, Caro and Kedlaya (p-adic proof of the Weil-conjectures) are better adapted to computations (e.g., of L-functions over finite fields, i.e., characteristic polynomials of frobenius acting on cohomology). It thus looks like an important project to develop "[[p-adic]] methods" in [[global analytic geometry]], starting with the definition of two types of cohomology theories for [[global analytic spaces]]: an [[absolute cohomology]] theory (related to the Chern character from K-theory to negative cyclic cohomology) and a [[geometric cohomology]] theory (with characteristic $0$ coefficients, e.g., in the full ring $\mathbb{A}$) of adèles). It may also be interesting to develop a notion of [[global Fourier transform]]. One of these cohomology theories (the absolute one) should give a natural way to study various conjectures on [[special values of L-functions]] and the other (the geometric one) should give a natural way to study the position of zeroes and poles of L-functions by spectral methods. The [[geometric cohomology theory]] should also be (since it has characteristic $0$ coefficients) related to the theory of motives and [[global automorphic representations]], and thus give more generally a spectral way of studying zeroes and poles of the L-function of the rational motive of an [[arithmetic variety]]. ## Aims The aim of arithmetic cryptography is to define a good [[geometric cohomology]] theory for [[global analytic spaces]] based on analytic methods and differential calculus that would allow the definition of [public key cryptography](http://en.wikipedia.org/wiki/Public-key_cryptography) systems based on the datum of a [[global analytic space]] $X$ and of (say) a part $M$ of the associated (maybe absolute) rational motive $M(X)$. The definition of such methods would not give any new attacks to the previous ones, but may give a bigger class of public keys, that may allow the use of shorter ones. However, it is not yet clear that such a general approach will be conformal to the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle). In particular, as explained by Edixhoven in a preprint, it seems to be quite a hard computational task to determine if two cohomology classes are equal, at least in the $\ell$-adic setting. ## Possible constraints on the theory The constraints on such a theory would be the following: 1. get back the classical discrete logarithm elliptic public key cryptography method when one starts from an elliptic curve over a finite field (or maybe a corresponding weak formal scheme over the ring of p-typical Witt vectors). 2. get back the original prime factorization public key cryptography method when one starts from $\mathbb{Z}$. 3. It must be clearly stressed here that the proper theoretical setting for such a general theory may be very hard to develop, since it should involve the definition of a proper cohomology for the Riemann zeta function, that would allow a spectral proof of its functional equation (as in Tate's thesis) `and` of the [[Riemann hypothesis]]. The full project thus looks like a very far reaching one, since there is, up to now, no precise idea of a way of constructing such a [[geometric cohomology]] theory (even if ideas on the constraints that it should fulfill are widespread in the mathematical litterature, e.g., in Deninger's work, in the [[field with one element]] litterature, in [[Langlands program]] and in the study of [[Weil-étale cohomology]]). Moreover, the use of adelic coefficients instead of complex ones seems necessary to treat the $p$-adic and archimedean cohomologies on equal footing. 4. One may use the classical GRH conjectures (without proving them) to get the estimates analogous to the classical estimate of the number of primes smaller than a given prime. From this point of view, it seems that representation theoretic approaches may be more interesting. ## Possible methods It is not at all clear that the following propositions will be conformal to the KISS principle. They may also be attacked by quantum computing methods, by higher dimensional generalizations of Shor's algorithm, but one may hope that allowing the use of arbitrary arithmetic schemes may make the quantum computing methods more difficult to apply and/or implement in general. This also means that these propositions may be very hard to implement in practice. A starting point for crash-testing the compatibility of higher dimensional arithmetic cryptography with the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle) may be to test it in the finite characteristic case (with p-adic methods, say derived analytic spaces over $\mathbb{Z}_p$, for computational purposes). First remark that the discrete logarithm problem for an elliptic curve over $\mathbb{F}_p$ may be understood as a discrete logarithm problem in the first (étale torsion) cohomology group of the curve, given by torsion points. One may try to generalize this problem to the higher dimensional situation by giving an algebraic cohomology class $[c]$, and computing $[d]=n.[c]$. The public key is given by the pair $([c],[d])$ and the message is the number $n$. As pointed out by Edixhoven in a preprint, computing a cohomology class may be a very hard computational task, so that this idea may not be conformal to the KISS principle. One may also devise another "product type" approach to the definition of a public key: given two prime cohomology classes $[c]$ and $[d]$ (classes of irreductible subvarieties of a given codimension), compute their product class $[e]=[c].[d]$ in the cohomology ring. One may try, given $[e]$, to find back $[c]$ and $[d]$, and this may be a very hard problem. ## Related references Computing zeta functions of arithmetic schemes, David Harvey, [arXiv](http://arxiv.org/abs/1402.3439)
arithmetic curve	https://ncatlab.org/nlab/source/arithmetic+curve	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea A [[curve]] in [[arithmetic geometry]], hence an [[arithmetic scheme]] of suitable [[dimension]] 1 etc. ## Properties ### Function field analogy [[!include function field analogy -- table]] [[!redirects arithmetic curves]]
arithmetic D-module	https://ncatlab.org/nlab/source/arithmetic+D-module	#Contents# * table of contents {:toc} ## Idea The theory of _arithmetic D-modules_ was primarily developped by Berthelot to better understand the [[functor\|functoriality]] properties of [[rigid cohomology]]. It gives a theory of coefficients for the [[cohomology]] of [[quasi-projective variety\|quasi-projective algebraic varieties]] over finite fields that are stable by the [[six operations\|six Grothendieck operations]], after Kedlaya and Caro. This allows a purely p-adic proof of Deligne's Weil II theorem, that generalized the [[Riemann hypothesis]] over finite fields to the category of coefficients for cohomology (i.e., motivic sheaves). ## Related concepts * [[D-module]] * [[coherent D-module]], [[holonomic D-module]] ## References * Pierre Berthelot: [D-modules arithmetiques I, II](http://perso.univ-rennes1.fr/pierre.berthelot/). * Kiran S. Kedlaya, [Semistable reduction for overconvergent F-isocrystals I, II, III](http://arxiv.org/find/all/1/AND+au:+kedlaya+ti:+AND+semistable+reduction/0/1/0/all/0/1?skip=0&query_id=99b522850f00b5aa). * Daniel Caro: [Stability of holonomicity over quasi-projective varieties](http://arxiv.org/abs/0810.0304) [[!redirects arithmetic D-modules]]
arithmetic differential geometry	https://ncatlab.org/nlab/source/arithmetic+differential+geometry	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- #### Differential geometry +--{: .hide} [[!include synthetic differential geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic differential geometry is an approach to [[arithmetic]] which looks to find analogs of constructions in [[differential geometry]], for example, [[arithmetic jet spaces]]. It has been developed primarily by [[Alexandru Buium]]. According to this approach, the classical derivatives of differential geometry are replaced by [[p-derivations]] for prime $p$, such as the [[Fermat quotient]] $$ \delta_p: \mathbb{Z} \to \mathbb{Z}, a \mapsto \delta_p a = \frac{a - a^p}{p}. $$ ##Comparison with Borger's absolute geometry Buium explains that when working with a single prime his approach is consistent with [[Borger's absolute geometry]], which is described as "an algebraization of our analytic theory" ([Buium 17, p. 24](#Buium17)). However, in the case of multiple primes Borger requires [[Frobenius lifts]] to commute, and this diverges from the non-vanishing 'curvature' Buium derives from non-commuting lifts. For him, the ("manifold" corresponding to) the integers, $\mathbb{Z}$, is "intrinsically curved". ## Related pages * [[arithmetic jet space]] * [[Joyal delta-ring]] * [[Borger's absolute geometry]] * [[D-geometry]] * [[arithmetic geometry]] ## References * {#Buium17} [[Alexandru Buium]], _Foundations of arithmetic differential geometry_, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, ([AMS](http://bookstore.ams.org/surv-222/), [Preface and Introduction](http://www.math.unm.edu/~buium/foundationsintro.pdf)). * [[Alexandru Buium]], _Arithmetic differential geometry_, May 19, 2017, [talk slides](http://www.math.unm.edu/~buium/adgtalk.pdf) [[!redirects arithmetic derivative]] [[!redirects arithmetic derivatives]]
arithmetic gauge theory	https://ncatlab.org/nlab/source/arithmetic+gauge+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- #### Chern-Weil theory +--{: .hide} [[!include infinity-Chern-Weil theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea _Arithmetic gauge theory_ is a theory proposed by [Kim 2018 ](#Kim18) to look at [[Galois representations]] with an action of another group (the [[gauge group]]) as being analogous to [[gauge fields]] in [[gauge theory]]. It is an approach to Diophantine problems such as the effective [[Mordell conjecture]] and is influenced by the theory of the [[Selmer group]] and Chabauty's method. ## Definitions Let $K$ be a field of characteristic zero and let $G_{K}$ be its [[absolute Galois group]]. A _gauge group over $K$_ is a [[topological group]] $U$ with a continuous action of $G_{K}$. A _$U$-gauge field_, or _$U$-principal bundle_, is a topological space $P$ with compatible continuous left $G_{K}$-action and simply transitive continuous right $U$ action. ## Examples This section follows section 8 of [Kim 2018](#Kim18). Let $V$ be a variety over $\mathbb{Q}$ equipped with a base point $b\in V(\mathbb{Q})$. Our motivation is determining the set $V(\mathbb{Q})$ of rational points of $V$. For our gauge group $U$ we let $$U=\pi_{1}(\overline{V},b)_{\mathbb{Q}},$$ the $\mathbb{Q}_{p}$-pro-unipotent fundamental group of $V$. For our gauge field $P$ we let $$P(x)=\pi_{1}(\overline{V},b,x),$$ the $U$-torsor of pro-unipotent paths from $b$ to $x$. Let $S$ be a finite set of primes. Let $H_{f}^{1}(\mathbb{Z}_{S},U)$ be the set of $U$-torsors over $\mathbb{Q}$ which are unramified outside $S$ and crystalline at $p$ (see [[p-adic Hodge theory]] for the meaning of crystalline). Similarly let $H^{1}(\mathbb{Q}_{p},U)$ be the set of $U$-torsors over $\mathbb{Q}_{p}$. We have a localization map $$H_{f}^{1}(\mathbb{Z}_{S},U)\to\prod^{'}H^{1}(\mathbb{Q}_{v},U)$$ where the restricted product on the right means that all but finitely many of the components are unramified, and at $p$ the corresponding component is crystalline. Recall that we are interested in $V(\mathbb{Q})$, the set of rational points of $V$. Now we have a map $A:V(\mathbb{Q})\to H_{f}^{1}(\mathbb{Z}_{S},U)$ given by $$x\mapsto P(x).$$ where $P(x)=\pi_{1}(\overline{V},b,x)$ as above. This map fits into the diagram $$ \array{& V(\mathbb{Q}) & \rightarrow & V(\mathbb{Q}_{p}) & \\ A & \downarrow &&\downarrow & A_{p} \\ & H_{f}^{1}(\mathbb{Z}_{S},U) & \underset{loc_{p}}\rightarrow& H_{f}^{1}(\mathbb{Q}_{p},U) & }$$ Kim conjectures the following: \begin{conjecture} Suppose $V$ is a smooth projective curve of genus $g\geq 2$. Then $$V(\mathbb{Q})=A_{p}^{-1}(Im(loc_{p})).$$ \end{conjecture} ## Relation to L-functions ## Related ideas * [[gauge theory]] * [[Mordell conjecture]] * [[arithmetic Chern-Simons theory]] ## References * {#Kim18} [[Minhyong Kim]], Arithmetic Gauge Theory: A Brief Introduction, Modern Physics Letters A 33 29 (2018) 1830012 [[arxiv:1712.07602](https://arxiv.org/abs/1712.07602), [doi:10.1142/S0217732318300124](https://doi.org/10.1142/S0217732318300124)] * [[Minhyong Kim]], _Recent Progress on the Effective Mordell Problem_, lecture at the Sydney Mathematical Research Institute [YouTube](https://www.youtube.com/watch?v=NDQ_aO7QDEU) * [[Minhyong Kim]], _Foundations of nonabelian Chabauty_, lectures at the Arizona Winter School 2020 [Slides1](https://swc-math.github.io/aws/2020/2020KimNotes1.pdf), [Slides2] (http://swc-alpha.math.arizona.edu/video/2020/2020KimLecture2Slides.pdf) [Slides3] (https://swc-math.github.io/aws/2020/2020KimNotes3.pdf) [Slides4] (https://swc-math.github.io/aws/2020/2020KimNotes4.pdf) [[!redirects arithmetic gauge theories]]
arithmetic genus	https://ncatlab.org/nlab/source/arithmetic+genus	#Contents# * table of contents {:toc} ## Idea A generalization of the concept of [[genus of a surface]] from [[Riemann surfaces]]/[[complex curves]] to [[algebraic varieties]], hence to [[algebraic curves]]. Further base-chaning to [[arithmetic curves]] there is the _[[genus of a number field]]_ (in the sense of the [[function field analogy]]). ## Definition (...) essentially the [[Euler characteristic]] $\chi(\mathcal{O}_{\Sigma})$ of the [[structure sheaf]]: $$ g_\Sigma = - (\chi(\mathcal{O}_\Sigma) - 1) $$ (...) ## Properties ### Function field analogy [[!include function field analogy -- table]] ## References * {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_, Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf)) * Wikipedia, _[Arithmetic genus](https://en.wikipedia.org/wiki/Arithmetic_genus)_ * _[How to see the genus](http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/)_ [[!redirects genus of a curve]] [[!redirects genus of an algebraic curve]]
arithmetic geometry	https://ncatlab.org/nlab/source/arithmetic+geometry	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Arithmetic geometry is a branch of [[algebraic geometry]] studying [[schemes]] (usually of [[morphism of finite type\|finite type]]) over the [[spectrum (geometry)\|spectrum]] [[Spec(Z)]] of the [[commutative ring]] of [[integers]]. More generally, [[algebraic geometry]] over non-[[algebraically closed fields]] or fields of [[positive characteristic]] is also referred to as "arithmetic algebraic geometry". Since an [[affine variety]] in this context is given by solutions to [[Diophantine equations]], this is also called _Diophantine geometry_. An archetypical application of arithmetic geometry is the study of [[elliptic curves]] over the [[integers]] and the [[rational numbers]]. For [[number theory\|number theoretic]] purposes, i.e. in actual [[arithmetic]]; usually one complements this with some data "at the prime at infinity" leading to a more modern notion of an _[[arithmetic scheme]]_ (cf. [[Arakelov geometry]]). The refinement to [[higher geometry]] is [[E-infinity geometry]] ([[spectral geometry]]). ## Properties ### Base over $\mathbb{F}_1$ Arithmetic geometry naturally has as [[base topos]] the topos over [[F1]] in the sense of [[Borger's absolute geometry]], which gives an [[essential geometric morphism]] of [[etale toposes]] $$ Et(Spec(\mathbb{Z})) \longrightarrow Et(Spec(\mathbb{F}_1)) \,. $$ ### Function field analogy [[!include function field analogy -- table]] ## Related concepts * [[arithmetic variety]], [[arithmetic scheme]] * [[arithmetic curve]] * [[arithmetic Chow group]] * [[analytic geometry]] * [[Weil-étale topology for arithmetic schemes]] * [[absolute cohomology]] * [[function field analogy]] * [[Weil conjecture on Tamagawa numbers]] * [[Borger's absolute geometry]] * [[Mordell conjecture]] * [[Iwasawa-Tate theory]] * [[arithmetic jet space]] * [[arithmetic differential geometry]] * [[adelic integration]] * [[shtuka]] * [[higher arithmetic geometry]] * [[E-∞ arithmetic geometry]] * [[arithmetic topology]] * [[Frobenioid]] * [[p-adic physics]] * [[p-adic string theory]] * [[p-adic AdS/CFT correspondence]] ## References * Dino Lorenzini, _An Invitation to Arithmetic Geometry_ (Graduate Studies in Mathematics, Vol 9) GSM/9 An almost entirely self-contained introduction and (according to Werner Kleinert) "the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form (after Grothendieck's [[EGA]])": * Qing Liu, _Algebraic Geometry and Arithmetic Curves_, Oxford Graduate Texts in Mathematics __6__, 2002/2006, 600pp. Lecture notes include * {#Sutherland13} [[Andrew Sutherland]], _Introduction to Arithmetic Geometry_, 2013 ([web](http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/index.htm)) * [[Christophe Soulé\|C. Soulé]], D. Abramovich, J. F. Burnol, J. K. Kramer, _Lectures on Arakelov Geometry_, Cambridge Studies in Advanced Mathematics __33__, 188 pp. and with an eye towards [[anabelian geometry]]: * {#MK09} [[Minhyong Kim]], _Galois Theory and Diophantine geometry_, 2009 ([pdf](http://people.maths.ox.ac.uk/kimm/papers/cambridgews.pdf)) Further resources: * Wikipedia: [Glossary of arithmetic and Diophantine geometry](http://en.wikipedia.org/wiki/Glossary_of_arithmetic_and_Diophantine_geometry), Wikipedi, [Arakelov geometry](http://en.wikipedia.org/wiki/Arakelov_theory) * _Arakelov geometry preprint arxiv_, list of [links](http://people.math.jussieu.fr/~vmaillot/Arakelov) * conferences in arithmetic geometry, at [[Kiran Kedlaya]]'s [wiki](http://scripts.mit.edu/~kedlaya/wiki/index.php?title=Conferences_in_Arithmetic_Geometry) Some relation to [[modular tensor categories]]: * {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)] [[!redirects arithmetic geometries]] [[!redirects Diophantine geometry]] [[!redirects Diophantine geometries]]
arithmetic geometry - contents	https://ncatlab.org/nlab/source/arithmetic+geometry+-+contents	[[number theory]] * [[arithmetic]] * [[arithmetic geometry]], [[arithmetic topology]] * [[higher arithmetic geometry]], [[E-∞ arithmetic geometry]] [[number]] * [[natural number]], [[integer number]], [[rational number]], [[real number]], [[irrational number]], [[complex number]], [[quaternion]], [[octonion]], [[adic number]], [[cardinal number]], [[ordinal number]], [[surreal number]] [[arithmetic]] * [[Peano arithmetic]], [[second-order arithmetic]] * [[transfinite arithmetic]], [[cardinal arithmetic]], [[ordinal arithmetic]] * [[prime field]], [[p-adic integer]], [[p-adic rational number]], [[p-adic complex number]] [[arithmetic geometry]], [[function field analogy]] * [[arithmetic scheme]] * [[arithmetic curve]], [[elliptic curve]] * [[arithmetic genus]] * [[arithmetic Chern-Simons theory]] * [[arithmetic Chow group]] * [[Weil-étale topology for arithmetic schemes]] * [[absolute cohomology]] * [[Weil conjecture on Tamagawa numbers]] * [[Borger's absolute geometry]] * [[Iwasawa-Tate theory]] * [[arithmetic jet space]] * [[adelic integration]] * [[shtuka]] * [[Frobenioid]] [[Arakelov geometry]] * [[arithmetic Riemann-Roch theorem]] * [[differential algebraic K-theory]]
arithmetic geometry -- contents	https://ncatlab.org/nlab/source/arithmetic+geometry+--+contents	[[arithmetic]]/[[number theory]], [[geometry]] [[arithmetic geometry]], [[E-∞ geometry]] * [[Arakelov geometry]] * [[arithmetic scheme]] * [[arithmetic Chow group]]
arithmetic jet space	https://ncatlab.org/nlab/source/arithmetic+jet+space	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- #### Synthetic differential geometry +--{: .hide} [[!include synthetic differential geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An analog of [[jet spaces]] in [[arithmetic geometry]]. ## Definition {#Definition} Notice that the [[p-adic integers]] $\mathbb{Z}_p$ are (by the discussion at [p-adic integer -- as formal power series](p-adic+integer#AsFormalNeighbourhoodOfPrime)) the analog in [[arithmetic geometry]] of a [[formal power series]] ring (around the point $p \in$ [[Spec(Z)]]), hence their [[formal spectrum]] $Spf(\mathbb{Z}_p)$ is an incarnation in [[arithmetic geometry]] of an abstract [[formal disk]]. Therefore in the sense of [[synthetic differential geometry]] the $p$-[[formal neighbourhood]] of any [[arithmetic scheme]] $X$ around a global point $x \colon Spec(\mathbb{Z}) \to X$ is the space of lifts $$ \array{ Spf(\mathbb{Z}_p) && \stackrel{\hat x}{\longrightarrow}&& X \\ & \searrow && \swarrow \\ && Spec(\mathbb{Z}) } \,. $$ Moreover the map that sends an commutative ring, hence an [[arithmetic variety]], to its $p$-formal power series in this sense is the construction of the [[ring of Witt vectors]] ($p$-typical Witt vectors if one fixes one prime, and "big Witt vectors" if one considers all at once) - see e.g. [Hartl 06, section 1.1](#Hartl06). The following definition says essentially this, but further sends the resulting space to [[F1]]-geometry in the sense of [[Borger's absolute geometry]]: For $X= Spec(R)$ an [[affine scheme]] over [[Spec(Z)]] (hence the formal dual of a [[ring]]), then the _arithmetic jet space_ of $X$ at [[prime]] $p$ is $(W_n)_\ast$ applied to the $p$-adic completion of $X$, where $(W_n)_\ast$ is the [[ring of Witt vectors]]-construction, the [[direct image]] of [[Borger's absolute geometry]] $Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$. The definition is originally due to ([Buium 96, section 2](#Buium96), [Buium 05, section 3.1](#Buium05)), reviewed in ([Buium 13, 1.2.3](#Buium13)) as part of his [[arithmetic differential geometry]] program. The above formulation is in ([Borger 10, (12.8.2)](#Borger10)). Buium and Borger have also defined the notion of an arithmetic jet space for a finite set of primes in ([BB09](#BB09)). ## Related concepts * [[arithmetic differential geometry]] ## References The original articles are * {#Buium96} [[Alexandru Buium]], _Geometry of $p$-jets. Duke Math. J., 82(2):349–367, 1996. ([Euclid](http://projecteuclid.org/euclid.dmj/1077245037)) * {#Buium05} [[Alexandru Buium]], _Arithmetic differential equations_, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. ([pdf](http://www.math.unm.edu/~buium/prebook.pdf)) Introduction and survey is in * {#Buium13} [[Alexandru Buium]], _Differential calculus with integers_ ([arXiv:1308.5194](http://arxiv.org/abs/1308.5194), [slightly differing pdf](http://www.math.unm.edu/~buium/statupdated.pdf)) * [[Alexandru Buium]], _Lectures on arithmetic differential equations_ ([pdf](http://www.lorentzcenter.nl/lc/web/2009/342/presentations/lectures%20A.%20Buium.pdf)) * [[Alexandru Buium]], _Foundations of arithmetic differential geometry_, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, ([AMS](http://bookstore.ams.org/surv-222/), [Preface and Introduction](http://www.math.unm.edu/~buium/foundationsintro.pdf)). Discussion in the context of the [[function field analogy]] is in * {#Hartl06} [[Urs Hartl]], _A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic_ ([arXiv:math/0607182](http://arxiv.org/abs/math/0607182)) Discussion in the context of [[Borger's absolute geometry]] over [[F1]] is in * {#Borger10} [[James Borger]], _The basic geometry of Witt vectors, II: Spaces_ ([arXiv:1006.0092](http://arxiv.org/abs/1006.0092)) See also * [[Alexandru Buium]], Taylor Dupuy, _Arithmetic differential equations on $GL_n$, I: differential cocycles_ ([arXiv:1308.0748](http://arxiv.org/abs/1308.0748)) * {#BB09} [[James Borger]], [[Alexandru Buium]], _Differential forms on arithmetic jet spaces_, ([arXiv:0908.2512](https://arxiv.org/abs/0908.2512)) [[!redirects arithmetic jet spaces]] [[!redirects arithmetic differential equation]] [[!redirects arithmetic differential equations]]
arithmetic pretopos	https://ncatlab.org/nlab/source/arithmetic+pretopos	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Topos Theory +-- {: .hide} [[!include topos theory - contents]] =-- #### Arithmetic +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An arithmetic pretopos is a [[pretopos]] $C$ with a _parameterized [[natural numbers object]]_. A list-arithmetic pretopos is a pretopos with all _parameterized list objects_ ([Maietti 10, 2.6](#Maietti10)). Using the (equivalent) definition given in [Cockett 1990](#Cockett90), a parameterized list object is a [[W-type]] for the [[polynomial functor]] $B+ A\times(-)\colon C\to C$. This definition makes sense since a pretopos has [[finite products]] and [[disjoint coproducts]] (here denoted "$+$"). Discussion via its [[internal language]], which is a [[dependent type theory]]... ([Maietti 05](#Maietti05), [Maietti 10, p.6](#Maietti10)). Maietti ([05](#Maetti05),[10](#Maetti10)) proposed that list-arithmetic pretoposes serve as the arithmetic universes that [[André Joyal]] (cf. [Joyal 05](#Joyal05)) once suggested to use for discussion of [[incompleteness theorems]] (cf. [van Dijk/Oldenziel 2020](#VanDijkOldenziel)); they are used directly as the definition of arithmetic universes e.g. in ([Maietti-Vickers 2012](#MaiettiVickers12)). ## Related entries * [[free monoid]] * [[Gödel's incompleteness theorem]] * [[Lawvere's fixed point theorem]] * [[classifying topos for the theory of objects]] ## References * {#Cockett90} [[Robin Cockett]], _List-arithmetic distributive categories: Locoi_, JPAA 66 no.1 (1990) pp.1-29. * {#Cockett97} [[Robin Cockett]], _Finite objects in a locos_, JPAA 116 (1997) pp.169-183. * {#vanDijkOldenziel20} Joost van Dijk, Alexander Gietelink Oldenziel, _Gödel's Incompleteness after Joyal_, arXiv:2004.10482 (2020). ([abstract](https://arxiv.org/abs/2004.10482)) * {#Joyal05} [[André Joyal]], _The Gödel incompleteness theorem, a categorical approach_, (abstract) Amiens 2005, Cah. Top. Géom. Diff. Cat. 46 no.3 (2005) p.202. ([numdam](http://www.numdam.org/item/CTGDC_2005__46_3_163_0)) * {#Maietti05a} [[Maria Maietti]], _Reflection Into Models of Finite Decidable FP-sketches in an Arithmetic Universe_, Electronic Notes in Theoretical Computer Science 122 (2005) 105-126 [[doi:10.1016/j.entcs.2004.06.054](https://doi.org/10.1016/j.entcs.2004.06.054)] * {#Maietti05} [[Maria Maietti]], _Modular correspondence between dependent type theories and categories including pretopoi and topoi_, Mathematical Structures in Computer Science 15 6 (2005) 1089-1149 [[doi:10.1017/S0960129505004962](https://doi.org/10.1017/S0960129505004962), [pdf](https://www.math.unipd.it/~maietti/papers/tumscs.pdf)] * {#Maietti10} [[Maria E. Maietti]], _Joyal's arithmetic universe as list-arithmetic pretopos_, TAC 24 3 (2010) 39-83 [[tac:24-03](http://www.tac.mta.ca/tac/volumes/24/3/24-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/24/3/24-03.pdf)] * {#MaettiVickers12} [[Maria E. Maietti]], [[Steve Vickers]], _An induction principle for consequence in arithmetic universes_, JPAA 216 (2012) pp.2049-2067. [[doi:10.1016/j.jpaa.2012.02.040](https://doi.org/10.1016/j.jpaa.2012.02.040), [pdf](http://www.math.unipd.it/~maietti/papers/aumv12.pdf)] * [[Paul Taylor]], _Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe_, Electronic Notes in Theoretical Computer Science 122 (2005) 247-296 [[doi:10.1016/j.entcs.2004.06.059](https://doi.org/10.1016/j.entcs.2004.06.059)] * {#Vickers16} [[Steve Vickers]], _Sketches for arithmetic universes_, arXiv:1608.0159 (2016) &lbrack[arXiv:1608.01559](http://arxiv.org/abs/1608.01559)] * {#Vickers17} [[Steve Vickers]], _Arithmetic universes and classifying toposes_ [[arXiv:1701.04611](https://arxiv.org/abs/1701.04611)] [[!redirects arithmetic pretoposes]] [[!redirects arithmetic pretopoi]] [[!redirects list-arithmetic pretopos]] [[!redirects list-arithmetic pretoposes]] [[!redirects list-arithmetic pretopoi]] [[!redirects arithmetic universe]] [[!redirects arithmetic universes]]
arithmetic Riemann-Roch theorem	https://ncatlab.org/nlab/source/arithmetic+Riemann-Roch+theorem	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The generalization of the [[Grothendieck-Riemann-Roch theorem]] to [[Arakelov geometry]]. ## References * [[Henri Gillet]], Damian Rössler, [[Christophe Soulé]], _An arithmetic Riemann-Roch theorem in higher degrees_ (2007) [pdf](http://www.ihes.fr/~soule/arrfin.pdf) [[!redirects arithmetic Grothendieck-Riemann-Roch theorem]]
arithmetic scheme	https://ncatlab.org/nlab/source/arithmetic+scheme	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An _arithmetic scheme_ is a [[scheme]] in [[arithmetic geometry]], hence a scheme over [[Isbell duality\|formal dual]]/[[spectrum of a commutative ring\|spectrum]] $Spec \mathbb{Z}$ of the [[ring]] of [[integers]]. Typically one takes an arithmetic scheme to be ([[regular scheme\|regular]]) [[separated scheme\|separated]] and of [[scheme of finite type\|of finite type]]. ## Properties ### Differential K-theory A [[differential algebraic K-theory]] over arithmetic schemes is considered in ([Bunke-Tamme 12](#BunkeTamme12)). ## Related concepts * [[arithmetic curve]] ## References * {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451)) [[!redirects arithmetic schemes]]
arithmetic topology	https://ncatlab.org/nlab/source/arithmetic+topology	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Knot theory +--{: .hide} [[!include knot theory - contents]] =-- #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- ## Contents * table of contents {:toc} ## Idea Arithmetic topology is a theory describing some surprising [[analogies]] between [[low-dimensional topology\|3-dimensional topology]] and [[number theory]] ([[arithmetic]]), where [[knots]] embedded in a [[3-manifold]] behave like [[prime ideals]] in a [[ring of algebraic integers]]. See also at _[Spec(Z) -- As a 3d space containing knots](Spec%28Z%29#As3dSpaceContainingKnots)_. More broadly, the scope of arithmetic topology is now taken to include the intersection of [[arithmetic geometry]], [[algebraic topology]] and [[low-dimensional topology]] (see [GGW20](#GGW20)). Under the original analogy, the [[3-sphere]], $S^3$ corresponds to the ring of [[rational numbers]] $\mathbb{Q}$, or rather (the closure of) $spec(\mathcal{O}_{\mathbb{Q}})$ (i.e., $spec(\mathbb{Z})$), since the 3-sphere has no non-trivial ([[branched cover\|unbranched]]) covers while $\mathbb{Q}$ has no non-trivial [[unramified extensions]]. The [[linking number]] between two embedded knots in the 3-sphere then corresponds to the [[Legendre symbol]] between two primes in the ordinary integers. The so-called _M^2KR dictionary_ (Mazur-Morishita-Kapranov-Reznikov) relates terms from each side of the analogy (see sec 2.2 of [Sikora](#Sikora)). ## The dictionary ### Version of Mazur-Morishita-Kapranov-Reznikov (M^2KR) {#VersionOfM^2KR} 1. [[closed manifold\|Closed]], [[orientation\|orientable]], [[connected topological space\|connected]] [[3-manifolds]] correspond to (the closure of) schemes $Spec \mathcal{O}_K$ for number fields $K$. 1. Links correspond to ideals in $\mathcal{O}_K$ and knots correspond to prime ideals (tame in both cases). Knots can be represented by immersions of $S^1$ into $M$, and prime ideals in $\mathcal{O}_K$ can be identified with closed immersions $Spec \mathbb{F} \to Spec \mathcal{O}_K$, where $\mathbb{F}$'s are finite fields. Each link decomposes uniquely as a union of knots and each ideal decomposes uniquely as a product of primes. 1. An algebraic integer corresponds to an embedded surface (possibly with boundary), and the operation $a \to (a)$ corresponds to taking its boundary. Closed embedded surfaces correspond to units in $\mathcal{O}_K$. Ideals of the form $(a)$ represent the identity in $Cl(K)$, and the links of the form $\partial S$ represent the identity in $H_1(M,\mathbb{Z})$. 1. $Cl(K)$ corresponds to the torsion component of first integral homology. The free component of $H_1(M,\mathbb{Z})$ corresponds to the group of units in $\mathcal{O}_K$ after removing the torsion (roots of unity). 1. Finite extensions of number fields correspond to finite branched coverings. 1. $S^3$ is supposed to correspond to $\mathbb{Q}$. Notice that $S^3$ has no nontrivial unbranched covers, and similarly $\mathbb{Q}$ has no nontrivial unramified extensions. 1. A Galois extension $L/K$ with Galois group $G$ induces a morphism $Spec \mathcal{O}_L \to (Spec \mathcal{O}_L)/G = Spec \mathcal{O}_K$. Such maps correspond to the quotient maps $M \to M/G$ induced by orientation preserving actions of finite groups $G$ on 3-manifolds $M$. One can show that $M/G$ is always a 3-manifold and that the maps $M \to M/G$ are branched coverings. 1. Let $q = p^n$. Consider the cyclotomic extension $\mathbb{Q}(\zeta_q)$. It is ramified only at $p$. These correspond to cyclic branched covers of knots in $S^3$. The union of these as $q$ ranges over all powers of $p$ should correspond to the universal abelian cover of $S^3 \setminus K$. There is a natural action of $\mathbb{Z}$ on the first homology group of the infinite cyclic cover of the knot complement corresponding to the natural action of the $p$-adic integers on the $p$-torsion of $Cl(\mathbb{Q}(\zeta_{p^{\infty}}))$. This concerns the [[Alexander polynomial]] of the knot and [[Iwasawa theory]]. ([Sikora, pp. 5-6](#Sikora), [Koberda08, pp. 32-33](#Koberda08)) Note: Regarding (4), some have argued that $Cl(K)$ should correspond to the full first integral homology group, (see, e.g., [Goundaroulis & Kontogeorgis](#Goundaroulis)). The correspondence between $\pi^{et}_1(\mathbb{Z} -\{p\})$ and $\pi_1(S^3 \setminus K)$ can be developed to relate the Legendre symbol for two primes to the linking number of two knots, and further to the Rédei symbol for three primes and Milner's triple linking number. Thus we can find a 'Borromean link' of primes, such as $(13, 61, 397)$, where each pair is unlinked. #### Disanalogies 1. The algebraic translation of the Poincaré Conjecture is false. $\mathbb{Q}$ is not the only number field with no unramified extensions. Nevertheless, $\hat{H}^i(Spec \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) = \hat{H}^i(Spec \mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) (i \in \mathbb{Z}, n \geq 2)$ if and only if $\mathcal{O}_K = \mathbb{Z}$. 1. Let $M_1 \to M$ be a covering of 3-manifolds. A knot $K$ in $M$ does not necessarily lift to a knot in $M_1$, while every prime ideal $p \triangleleft \mathcal{O}_K$ gives rise to an ideal $p \mathcal{O}_L$, where $L/K$ is a Galois number field extension. ([Goundaroulis & Kontogeorgis](#Goundaroulis)). ### Version of Deninger Similar to M^2KR, but with the introduction of a 2-dimensional foliation on the 3-manifold and a flow such that finite primes $p$ correspond to periodic orbits of length $log N p$ and the infinite primes correspond to the fixed points of the flow ([Deninger02](#Deninger02)). (See also the work of [Baptiste Morin](http://www.math.uni-muenster.de/reine/u/baptiste.morin/) on the Weil-étale topos.) ### Version of Reznikov {#ReznikovVariant} Reznikov has modified the dictionary ([Reznikov 00, section 12](#Reznikov00)) so as to associate a number field with what he calls a $3\frac{1}{2}$-manifold, that is a closed three-manifold $M$, bounding a four-manifold $N$, such that the map of fundamental groups $\pi_1(M) \to \pi_1(N)$ is surjective. ##Explanations for the analogy [[Barry Mazur]] observed that for an affine spectrum $X = Spec(D)$ of the ring of integers $D$ in a number field, the groups $H^n_{et}(X, \mathbb{G}_{m, X})$ vanish (up to 2-torsion) for $n \gt 3$, and is equal to $\mathbb{Q}/\mathbb{Z}$ for $n = 3$, where $\mathbb{G}_{m, X}$ is the étale sheaf on $X$ defined by associating to a connected finite étale covering $Spec(B) \to X$ the multiplicative group $\mathbb{G}_{m, X}(Y) = B^{\times}$. Also, there is a non-degenerate pairing for any constructible abelian sheaf $M$, $$ H^r_{et}(X,M^{'}) \times Ext^{3-r}_X(M,\mathbb{G}_{m, X}) \to H^3_{et}(X,\mathbb{G}_{m, X})\simeq \mathbb{Q}/\mathbb{Z}, $$ where $M^{'} = Hom(M, \mathbb{G}_{m, X})$. This resembles Poincaré duality for 3-manifolds. [[Minhyong Kim]] argues that the normal bundle of an embedding of a circle corresponding to a prime in $Spec(\mathbb{Z})$ is 2-dimensional ([Kim](#Kim)). [[Baptiste Morin]] claims to provide a unified treatment via equivariant etale cohomology ([Morin06](#Morin06)). ## Related concepts * [[Alexander polynomial]] * [[function field analogy]] * The [[virtually fibered conjecture]] says that every [[closed manifold\|closed]], [[irreducible manifold\|irreducible]], [[atoroidal 3-manifold\|atoroidal]] [[3-manifold]] with infinite [[fundamental group]] has a [[finite cover]] which is a [[surface]] [[fiber bundle]] over the [[circle]]. * [[arithmetic Chern-Simons theory]] ##References * {#Deninger02} [[Christopher Deninger]], _A note on arithmetic topology and dynamical systems_, ([arxiv:0204274](http://arxiv.org/abs/math/0204274)) * {#Goundaroulis} Dimoklis Goundaroulis, Aristides Kontogeorgis, _On the Principal Ideal Theorem in Arithmetic Topology_, ([talk](http://users.uoa.gr/~kontogar/talks/GkountPSATHA.pdf), [paper](http://arxiv.org/abs/0705.3937)) * {#Kim} Minhyong Kim, [note](http://minhyongkim.files.wordpress.com/2013/05/baez13-12.pdf) * {#Koberda08} Thomas Koberda, _Class Field Theory and the MKR Dictionary for Knots_, ([pdf](http://users.math.yale.edu/users/koberda/minorthesis.pdf)) * {#Morin06} [[Baptiste Morin]], _Applications of an Equivariant Etale Cohomology to Arithmetic Topology_, [arxiv:0602064](http://arxiv.org/abs/math/0602064) and Utilisation d'une cohomologie étale équivariante en topologie arithmétique, Compositio Math. 144 (2008), no. 1, 32-60. * {#Morishita09} Masanori Morishita, _Analogies between Knots and Primes, 3-Manifolds and Number Rings_, ([arxiv:0904.3399](http://arxiv.org/abs/0904.3399)) * {#Reznikov00} Alexander Reznikov, _Embedded incompressible surfaces and homology of ramified coverings of three-manifolds_, Selecta Math. 6(2000), 1–39 * {#Sikora} Adam Sikora, _Analogies between group actions on 3-manifolds and number fields_, ([arxiv](http://arxiv.org/abs/math/0107210)) * {#KohnoMorishita06} [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), _Primes and Knots_, Contemporary Mathematics, AMS 2006 ([conm:416](http://www.ams.org/bookstore-getitem/item=CONM-416)) * {#Morishita12} [[Masanori Morishita]], _Knots and Primes: An Introduction to Arithmetic Topology_, 2012, Springer, ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false)) * {#GGW20} Claudio Gómez-Gonzáles, [[Jesse Wolfson]], _Problems in Arithmetic Topology_ ([arXiv:2012.15434](https://arxiv.org/abs/2012.15434)) [[!redirects MKR dictionary]] [[!redirects MKR analogy]]
arithmetic variety	https://ncatlab.org/nlab/source/arithmetic+variety	## Idea [[variety]] in [[arithmetic geometry]] ## Related entries * [[arithmetic scheme]] * [[arithmetic curve]] [[!redirects arithmetic varieties]]
arithmetic zeta function	https://ncatlab.org/nlab/source/arithmetic+zeta+function	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Arithmetic geometry +--{: .hide} [[!include arithmetic geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Under the [[function field analogy]] one understands the [[Riemann zeta function]] and [[Dedekind zeta function]] as associated to [[arithmetic curves]], hence to [[spaces]] in [[arithmetic geometry]] of [[dimension]] 1. As one passes to [[higher dimensional arithmetic geometry]] the corresponding generalization are the _arithmetic zeta functions_. ## References * Wikipedia, _[Arithmetic zeta function](http://en.wikipedia.org/wiki/Arithmetic_zeta_function)_ * [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf)) [[!redirects arithmetic zeta functions]]
arity	https://ncatlab.org/nlab/source/arity	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- # N-ary operations * table of contents {: toc} ## Definitions Given a [[natural number]] $n$, an __n-ary operation__ on a [[set]] $S$ is a [[function]] $$ \phi \;\colon\; \big( \mathrm{Fin}(n) \to S \big) \longrightarrow S $$ from the [[function set]] $\mathrm{Fin}(n) \to S$ to $S$ itself, where $\mathrm{Fin}(n)$ is the [[finite set]] with $n$ elements. The __arity__ of the operation is $n$. In general, if the natural number $n$ is not specified, these are called finitary operations. Sets equipped with finitary operations are also called finitary [[magmas]] (or "finitary groupoids" in older terminology which now clashes with another meaning of [[groupoid]], see at [[historical notes on quasigroups]]). More generally, a finitary operation in a [[multicategory]] is just a [[multimorphism]]. ### Arbitrary arity More generally, one could use an arbitrary [[set]] instead of a [[finite set]]. However, the generalizations are only definable in [[closed monoidal category\|closed]] [[multicategories]], rather than any [[multicategory]]. ## Properties Every set $S$ with an $n$-ary operation $\phi$ comes with an [[endomorphism]] called the $n$-th [[power operation]] $$ \array{ S & \overset{\;\;(-)^n\;\;}{\longrightarrow} & S \\ x &\mapsto& \phi \circ diag_n(x) \,, } $$ where $ S \overset{diag_n}{\longrightarrow} S^n$ is the [[diagonal morphism]]. ## See also * [[magma]] * [[monoid]] * [[operation]] * [[multicategory]] * [[operad]] ### $n$-ary algebraic structures * [[n-ary semigroup]] * [[n-ary quasigroup]] * [[n-ary group]] * [[group]] ## References * Wikipedia, [n-ary group](https://en.wikipedia.org/wiki/N-ary_group) [[!redirects n-ary operation]] [[!redirects n-ary operations]] [[!redirects n-ary magma]] [[!redirects n-ary magmas]] [[!redirects finitary operation]] [[!redirects finitary operations]] [[!redirects finitary magma]] [[!redirects finitary magmas]]
arity class	https://ncatlab.org/nlab/source/arity+class	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Regular and Exact categories +-- {: .hide} [[!include regular and exact categories - contents]] =-- #### Category Theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- # Arity classes * table of contents {: toc} ## Idea An arity class is a class of [[cardinalities]] which is suitable to be the collection of [[arity\|arities]] for the operations in an [[algebraic theory]]. ## Definition An arity class is a [[class]] $\kappa$ of [[small set\|small]] [[cardinalities]] such that 1. $1\in\kappa$. 2. $\kappa$ is closed under indexed sums: if $\lambda\in\kappa$ and $\alpha: \lambda \to\kappa$, then $\sum_{i\in \lambda} \alpha(i)$ is also in $\kappa$. 3. $\kappa$ is closed under indexed decompositions: if $\lambda\in\kappa$ and $\sum_{i\in \lambda} \alpha(i)\in \kappa$, then each $\alpha(i)$ is also in $\kappa$. A [[set]] or [[family]] is called $\kappa$-small if its cardinality belongs to $\kappa$. A [[theory]] or other object with a collection of "operations" whose inputs are all $\kappa$-small is called $\kappa$-ary. +-- {: .un_remark} ###### Remark By [[induction]], the second condition implies closure under iterated indexed sums, in the sense that for any $n\ge 2$, we have $$\sum_{i_1\in\lambda_1} \; \sum_{i_2\in\lambda_2(i_1)} \cdots \sum_{i_{n-1} \in\lambda_{n-1}(i_1,\dots,i_{n-2})} \lambda_n(i_1,\dots,i_{n-1}) $$ is in $\kappa$ if all the $\lambda$'s are. The first condition may be regarded as the case $n=0$ of this (the case $n=1$ being just "$\lambda\in\kappa$ iff $\lambda\in\kappa$"). =-- +-- {: .un_remark} ###### Remark An alternative, more category-theoretic, way to state the second and third conditions is that for any [[function]] $f:I\to J$, if ${\|J\|}\in\kappa$, then ${\|I\|}\in\kappa$ if and only if all fibers of $f$ are in $\kappa$. =-- ## Examples * The set $\{1\}$ is an arity class. A $\{1\}$-ary object is called unary. * The set $\{0,1\}$ is an arity class. A$\{0,1\}$-ary object is called subunary. * The set $\omega = \mathbb{N} = \{0,1,2,3\dots\}$ is an arity class. An $\omega$-ary object is called finitary. * For any [[regular cardinal]] $\kappa$, the set of all cardinalities strictly less than $\kappa$ is an arity class, which we abusively denote also by $\kappa$. The previous example $\omega$ is a special case of this, as is $\{0,1\}$ if we consider $2$ to be a regular cardinal. * In particular, if $\kappa$ is the "size of the universe" --- e.g., an [[inaccessible cardinal]] for which we have chosen to call sets of cardinality $\lt\kappa$ [[small set\|small]], or literally the proper-class cardinality of the [[universe]], depending on how one thinks of it ---, then it is an arity class. In this case we call $\kappa$-ary objects infinitary or $\infty$-ary. In [[classical mathematics]], these examples in fact exhaust all arity classes. Classically, if $\lambda$ is any cardinal number strictly greater than $1$, then for any cardinal numbers $\mu\le \nu$, we can write $\nu$ as a $\lambda$-indexed sum containing $\mu$. Hence, if an arity class contains any cardinality $\gt 1$, it must be down-closed, and a down-closed arity class must arise from a regular cardinal. In [[constructive mathematics]], however, not every arity class besides $\{1\}$ must be downward-closed, and not every downward-closed arity class must arise from a regular cardinal. Arguably, however, in constructive mathematics one should consider downward-closed arity classes instead of regular cardinals. ## Related pages * [[algebraic theory]] * [[∞-ary exact category]], [[∞-ary site]] * [[arity space]] ## References * [[Michael Shulman]], "Exact completions and small sheaves". Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html) [[!redirects arity class]] [[!redirects arity classes]] [[!redirects unary]] [[!redirects subunary]] [[!redirects finitary]] [[!redirects n-ary]] [[!redirects κ-ary]] [[!redirects kappa-ary]] [[!redirects ∞-ary]] [[!redirects infinitary]] [[!redirects infinity-ary]]
arity space	https://ncatlab.org/nlab/source/arity+space	# Arity spaces * table of contents {: toc} ## Idea An arity space is a common generalization of [[coherence spaces]], [[finiteness spaces]], and [[totality spaces]] to an arbitrary set of "arities". +--{: .standout} This is an original and tentative definition. In particular, it's not clear whether the allowed sets of arities should be restricted in some way. Should they be an [[arity class]]? The only previously studied examples appear to be the cases $\{0,1\}$ (coherence spaces), $\{0,1,2,3,\dots\}$ (finiteness spaces), and $\{1\}$ (totality spaces), which are all arity classes. =-- ## Definition Let $\kappa$ be a set of [[cardinal numbers]]. Given two [[subsets]] $u,v\subseteq X$ of the same [[set]] $X$, we write $u\perp v$ if $\|u\cap v\|\in \kappa$. This relation defines a [[Galois connection]] in the usual way: for $\mathcal{U}\subseteq P(X)$ we have $\mathcal{U}^\perp = \{ v \mid \forall u\in \mathcal{U}. u\perp v \}$. Since $\perp$ is symmetric, $(-)^\perp$ is self-adjoint on the right. We define a $\kappa$-arity space to be a set $X$ together with a $\mathcal{U}\subseteq P(X)$ that is a fixed point of this Galois connection, $\mathcal{U} = \mathcal{U}^{\perp\perp}$. We call the sets in $\mathcal{U}$ $\kappa$-ary and the sets in $\mathcal{U}^{\perp}$ co-$\kappa$-ary. A morphism or relation between $\kappa$-arity spaces is a [[relation]] $R: X ⇸ Y$ such that 1. If $u\subseteq X$ is $\kappa$-ary, then $R[u] = \{ y \mid \exists x\in u, R(x,y) \}$ is $\kappa$-ary. 1. If $v\subseteq Y$ is co-$\kappa$-ary, then $R^{-1}[v] = \{ x \mid \exists y\in v, R(x,y) \}$ is co-$\kappa$-ary. ## Examples * If $\kappa=\{0,1\}$, then a $\kappa$-arity space is precisely a [[coherence space]]. * If $\kappa = \omega = \{0,1,2,3,\dots\}$, then a $\kappa$-arity space is precisely a [[finiteness space]]. * If $\kappa=\{1\}$, then a $\kappa$-arity space is (almost?) precisely a [[totality space]]. ## Properties Conjecture: For any $\kappa$, the category of $\kappa$-arity spaces is [[star-autonomous]]. This might follow from constructing it using [[double gluing]] and orthogonality. ## Construction as a Comma Double Category We can define arity spaces by a variation on the [[double gluing]] construction. Define a double category $Orth$ of orthogonalities 1. Objects are relations $\bot \subseteq X \times Y$ 2. A vertical morphism from $(X_1,Y_1,\perp_1)$ to $(X_2, Y_2, \perp_2)$ exists when $X_1$ and $Y_1$ are orthogonal subsets of $X_2$ and $Y_2$ respectively. 3. A horizontal morphism from $(X_1,Y_1,\perp_1)$ to $(X_2,Y_2,\perp_2)$ is a pair of a function $f_* : X_1 \to X_2$ and $f^* : Y_2 \to Y_1$ 4. A square from $f$ to $g$ exists when $f_$ is the restriction of $g_$ and similarly for $f^$ and $g^$. Then the (2-)category of arity spaces can be defined as the comma double category (where $Rel$ and $Set$ are viewed as vertically discrete double categories: \begin{tikzcd} Arity(\kappa) \ar[d] \ar[r] \ar[dr,phantom,"\Downarrow"] & Set \times Set^o \ar[d] \\ Rel \ar[r,"L_\kappa"'] & Orth \end{tikzcd} Where $L_\kappa$ maps a set $X$ to the orthogonality $\|U \cap V\| \leq \kappa$ on $Subset(X) \times Subset(X)$ and a pair of sets $X, Y$ is given the trivial orthogonality $x \perp y = \bot$ [[!redirects arity spaces]]
Arjan Keurentjes	https://ncatlab.org/nlab/source/Arjan+Keurentjes	* [webpage](http://www.phys.ens.fr/~arjan/home.html) ## Selected writings On [[string theory]] [[perturbative string theory vacuum\|vacua]] [[KK-compactification\|compactified]] on [[orbifolds]] and [[orientifolds]]: * {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170)) On [[U-duality]]: * [[Arjan Keurentjes]], _The topology of U-duality (sub-)groups_, Class.Quant.Grav. 21 (2004) 1695-1708 ([arXiv:hep-th/0309106](https://arxiv.org/abs/hep-th/0309106)) category: people
Arkaday Tseytlin	https://ncatlab.org/nlab/source/Arkaday+Tseytlin	* [webpage](http://www.imperial.ac.uk/people/a.tseytlin) ## related $n$Lab entries * [[AdS-CFT]] category: people
Arkady Berenstein	https://ncatlab.org/nlab/source/Arkady+Berenstein	Professor of Mathematics in the Mathematics Department of University of Oregon "working in representation theory, quantum groups, Schubert calculus, combinatorics, commutative and noncommutative algebraic geometry". * [webpage](https://pages.uoregon.edu/arkadiy) at Univ. of Oregon; [papers](https://pages.uoregon.edu/arkadiy/papers.html) * Arkady Berenstein, [[Andrei Zelevinsky]], _Quantum cluster algebras_, [math.QA/0404446](https://arxiv.org/abs/math/0404446) * D. Alessandrini, A. Berenstein, [[V. Retakh]], E. Rogozinnikov, A. Wienhard, _Symplectic groups over noncommutative algebras_ Sel. Math. New Ser. __28__, 82 (2022) [doi](https://doi.org/10.1007/s00029-022-00787-x) * A. Berenstein, J. Greenstein, _Canonical bases of quantum Schubert cells and their symmetries_, Sel. Math. New Ser. __23__, 2755--2799 (2017) [doi](https://doi.org/10.1007/s00029-017-0316-8) * Yuri Bazlov, Arkady Berenstein, _Noncommutative Dunkl operators and braided Cherednik algebras_, Selecta Math. (N.S.) __14__ (2009), no. 3-4, 325--372 [pdf](https://pages.uoregon.edu/arkadiy/doubles.pdf), [MR2010k:16044](https://mathscinet.ams.org/mathscinet-getitem?mr=2511188) [doi](https://doi.org/10.1007/s00029-009-0525-x) * Yuri Bazlov, Arkady Berenstein, _Braided doubles and rational Cherednik algebras_, Adv. Math. __220__ (2009) 1466--1530 [doi](https://doi.org/10.1016/j.aim.2008.11.004) > We introduce and study a large class of algebras with triangular decomposition which we call __braided doubles__. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of __quasi-Yetter–Drinfeld__ (QYD) __modules__ over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double--this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the __braided Heisenberg double__ attached to the corresponding complex reflection group. * Arkady Berenstein, Sebastian Zwicknagl, _Braided symmetric and exterior algebras_, Trans. Amer. Math. Soc. 360 (2008) 3429--3472 [doi](https://doi.org/10.1090/S0002-9947-08-04373-0) category: people
Arkady Tseytlin	https://ncatlab.org/nlab/source/Arkady+Tseytlin	Arkady Tseytlin is professor for theoretical [[physics]] at Imperial College London. * [webpage](http://www3.imperial.ac.uk/people/a.tseytlin) ## Selected writings Introducing the [[DBI-action]] for [[D-branes]]: * [[Efim Fradkin]], [[Arkady Tseytlin]], _Non-linear electrodynamics from quantized strings_, Physics Letters B Volume 163, Issues 1–4, 21 November 1985 (<a href="https://doi.org/10.1016/0370-2693(85)90205-9">doi:10.1016/0370-2693(85)90205-9</a>) Review: * [[Arkady Tseytlin]], _Born-Infeld action, supersymmetry and string theory_, in: [[Mikhail Shifman]] (ed.) _[[The many faces of the superworld]]_, pp. 417-452, World Scientific (2000) ([arXiv:hep-th/9908105](https://arxiv.org/abs/hep-th/9908105), [doi:10.1142/9789812793850_0025](https://doi.org/10.1142/9789812793850_0025)) On [[higher curvature corrections]] to the (abelian) [[DBI-action]] for (single) [[D-branes]]: * {#AndreevTseytlin88} O. D. Andreev, [[Arkady Tseytlin]], _Partition-function representation for the open superstring effective action:: Cancellation of Möbius infinites and derivative corrections to Born-Infeld lagrangian_, Nuclear Physics B Volume 311, Issue 1, 19 December 1988, Pages 205-252 (<a href="https://doi.org/10.1016/0550-3213(88)90148-4">doi:10.1016/0550-3213(88)90148-4</a>) On the [[3-brane in 6d]] as [[brane intersection\|self-intersection]] of [[M5-branes]]: * [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl. Phys. B475 (1996) 149 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), <a href="https://doi.org/10.1016/0550-3213(96)00328-8">doi:10.1016/0550-3213(96)00328-8</a>) On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system\|integrable]] [[spin chains]] with respect to the [[dilatation operator]], and the correspondence of their spectrum with the [[classical field theory\|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime\|AdS5]] under the [[AdS/CFT correspondence]]: * {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117)) On defect branes in [[D=4 N=4 SYM]] via [[AdS/CFT]]: * Hongliang Jiang, [[Arkady A. Tseytlin]], On co-dimension 2 defect anomalies in $\mathcal{N}=4$ SYM and $(2,0)$ theory via brane probes in AdS/CFT [[arXiv:2402.07881](https://arxiv.org/abs/2402.07881)] ## Related entries * [[worldline formalism]], [[beta function]] * [[Dirac-Born-Infeld action]] * [[AdS-CFT]] category: people [[!redirects A. Tseytlin]] [[!redirects Arkady A. Tseytlin]]
Arkady Vainshtein	https://ncatlab.org/nlab/source/Arkady+Vainshtein	* [webpage](https://www.physics.umn.edu/people/vainshte.html) ## Selected writings On the [[anomalous magnetic moment]] of the [[muon]]: * Kirill Melnikov, [[Arkady Vainshtein]], _Theory of the Muon Anomalous Magnetic Moment_, Springer Tracts in Modern Physics 216, 2006 On [[quantum field theory]] and [[string theory]]: * [[Mikhail Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.), _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_ Ian Kogan Memorial Collection (In 3 Volumes) World Scientific 2005 [doi:10.1142/5621](https://doi.org/10.1142/5621) category: people
Arkady Vaintrob	https://ncatlab.org/nlab/source/Arkady+Vaintrob	* [webpage](http://pages.uoregon.edu/vaintrob/) ## Selected writings On ([[Lie algebra weight system\|Lie algebra-]])[[weight systems]] on [[chord diagrams]]: * [[Arkady Vaintrob]], _Vassiliev knot invariants and Lie S-algebras_, Mathematical Research Letters1, 579–595 (1994) ([pdf](https://pdfs.semanticscholar.org/bdc3/ac1d8da476245e2408e481a70b115b3e9aab.pdf)) * [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197)) More on [[Lie algebra weight systems]] arising from [[super Lie algebras]]: * {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014)) category: people [[!redirects A. Vaintrob]] [[!redirects A. Y. Vaintrob]] [[!redirects A. Yu. Vaintrob]]
Armand Borel	https://ncatlab.org/nlab/source/Armand+Borel	* [Wikipedia entry](http://en.wikipedia.org/wiki/Armand_Borel) category: people
Armin Frei	https://ncatlab.org/nlab/source/Armin+Frei	* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=18372) ## Selected writings On [[monads]] (old term: "triples"), and their [[monad transformations]] in relation to their [[Eilenberg-Moore categories]]: * {#Frei69} [[Armin Frei]], Some remarks on triples, Mathematische Zeitschrift 109 (1969) 269–272 [[doi:10.1007/BF01110118](https://doi.org/10.1007/BF01110118)] category: people
Arnaud Beauville	https://ncatlab.org/nlab/source/Arnaud+Beauville	__Arnaud Beauville__ is an algebraic geometer and Professeur émérite at Université de Nice. In addition to the study of various "classical" problems of algebraic geometry, he applied complex algebraic geometry to [[integrable model]]s, [[conformal field theory]], study of moduli spaces etc. * [webpage](http://math.unice.fr/~beauvill), [publications](http://math.unice.fr/~beauvill/bibli.html) ## Selected writings On [[holomorphic symplectic geometry]]: * [[Arnaud Beauville]], _Holomorphic symplectic geometry_, 2011 ([pdf](https://math.unice.fr/~beauvill/conf/Lisbon.pdf), [[BeauvilleHolomorphicSymplectic.pdf:file]]) On [[compact hyperkähler manifolds]] as [[Hilbert schemes of points]] of [[K3-surfaces]]: * {#Beauville83} [[Arnaud Beauville]], _Variétés Kähleriennes dont la premiere classe de Chern est nulle_, Jour. Diff. Geom. 18 (1983), 755–782 ([euclid.jdg/1214438181](https://projecteuclid.org/euclid.jdg/1214438181)) category: people [[!redirects A. Beauville]]
Arnaud Spiwack	https://ncatlab.org/nlab/source/Arnaud+Spiwack	* [webpage](http://assert-false.science/arnaud/) ## Selected writings On [[homological algebra]] in [[constructive mathematics]] via [[type theory]]: * {#CoquandSpiwack} [[Thierry Coquand]], [[Arnaud Spiwack]], Towards constructive homological algebra in type theory, in: Towards Mechanized Mathematical Assistants. MKM Calculemus 2007, Lecture Notes in Computer Science 4573 Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), [pdf](https://hal.inria.fr/inria-00432525/document)] On notions of [[finite sets]] in [[constructive mathematics]]: * [[Arnaud Spiwack]], [[Thierry Coquand]], Constructively Finite?, in: Contribuciones cientifícas en honor de Mirian Andrés Gómez, Universidad de La Rioja (2010) 217-230 [ISBN:978-84-96487-50-5, [inria-00503917](https://inria.hal.science/inria-00503917)] On a [[dependent linear type theory]]-version of [[system L]]: * {#Spiwack14} [[Arnaud Spiwack]], _A dissection of L_ (2014) [[pdf](http://assert-false.science/arnaud/papers/A%20dissection%20of%20L.pdf)] category: people