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Amalendu Krishna	https://ncatlab.org/nlab/source/Amalendu+Krishna	* [Wikipedia entry](https://de.wikipedia.org/wiki/Amalendu_Krishna) * [institute home page](http://www.tifr.res.in/People_Finder/compcode.php?param1=39) ## related $n$Lab entries * [[algebraic K-theory]] category: people
amalgamation	https://ncatlab.org/nlab/source/amalgamation	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Limits and colimits +--{: .hide} [[!include infinity-limits - contents]] =-- =-- =-- Traditionally: * in [[group theory]] 1. the term [[amalgamated free product of groups]] refers to what really are [[pushouts]] in the [[category]] [[Grp]], while 1. the plain term "[[free product]]" really refers to [[coproducts]] in the [[category]] [[Grpd]]; therefore it makes sense to generally agree that: * in [[category theory]] the term "amalgamated sum" [[Gabriel & Zisman (1967), p. 1](pushout#GabrielZisman67)] and variants like "amalgamated coproduct" or just "amalgamation" refers to [[pushouts]] (seen, if you will, as [[coproducts]] in a [[coslice category]]). Beware though, while the terminology "amalgamation" indeed neatly captures the nature of [[pushouts]] in general [[categories]], traditionally it is not widely used in this generality. Moreover, but only vaguely related: * in [[model theory]] there is a notion of [[amalgamation property]] of [[structures in model theory]]. Finally, the term [[amalgamation theory]] is used here and there... category: disambiguation [[!redirects amalgamations]] [[!redirects amalgamated coproduct]] [[!redirects amalgamated coproducts]] [[!redirects amalgam]] [[!redirects amalgams]]
amalgamation property	https://ncatlab.org/nlab/source/amalgamation+property	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Model theory +-- {: .hide} [[!include model theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea A class of [[structures in model theory]] has the __amalgamation property__ if for any three structures $A,B,C$ and embeddings $f_B: A\hookrightarrow B$, $f_C: A\hookrightarrow C$, there exist embeddings $g_B:B\hookrightarrow D$ and $g_C: C\hookrightarrow D$ such that $g_B\circ f_B = g_C\circ f_C$. One of the simplest cases is when the free amalgam of structures $B\oplus_A C$ exists. ## Related entries * [[amalgamation]] * [[Galois type]] * [[Ore condition]] ## References The amalgamation method for generating strongly minimal theories is introduced in * [[Ehud Hrushovski]], _A new strongly minimal set_. Ann. Pure Appl. Logic, 62(2):147--166, 1993. Stability in model theory, III (Trento, 1991). Recent references include * [[Ehud Hrushovski]], _Groupoids, imaginaries and internal covers_, [arxiv/math.LO/0603413](http://arxiv.org/abs/math/0603413) * Uri Andrews, _Amalgamation constructions and recursive model theory_, thesis, [pdf](http://www.math.wisc.edu/~andrews/andrewsthesis.pdf) * John T. Baldwin, Alexei Kolesnikov, [[Saharon Shelah]], _The amalgamation spectrum_, J. Symbolic Logic __74__:3 (2009) 914-928, [MR2548468](http://www.ams.org/mathscinet-getitem?mr=2548468), [doi](http:/dx.doi.org/10.2178/jsl/1245158091[euclid](http://projecteuclid.org/euclid.jsl/1245158091) * J D Brody, _Model theory of graphs_, thesis, [pdf](http://tarski.fandm.edu/brody/mainthesis.pdf) category: model theory [[!redirects amalgamation properties]]
amalgamation theory	https://ncatlab.org/nlab/source/amalgamation+theory	## Related concepts * [[amalgamation]] * [[amalgamated free product of groups]] * [[amalgamation property]] [[!redirects amalgamation theoies]]
Amanda Peet	https://ncatlab.org/nlab/source/Amanda+Peet	* [webpage](http://ap.io/home/about-prof-peet/) ## Selected writings On [[S-branes]]: * {#KruczenskiMyersPeet02} [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], _Supergravity S-Branes_, JHEP 0205 (2002) 039 ([arXiv:hep-th/0204144](https://arxiv.org/abs/hep-th/0204144)) On [[D-brane polarization]] into [[supertubes]]: * [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], David J. Winters, _Aspects of supertubes_, JHEP 0205:017, 2002 ([arXiv:hep-th/0204103](https://arxiv.org/abs/hep-th/0204103)) category: people
Amar Hadzihasanovic	https://ncatlab.org/nlab/source/Amar+Hadzihasanovic	* [website](https://www.irif.fr/~ahadziha/) ## Selected writings On [[string diagrams]] and [[quantum entanglement]]: * [[Amar Hadzihasanovic]], _The algebra of entanglement and the geometry of composition_, ([arXiv:1709.08086](https://arxiv.org/abs/1709.08086)) On [[rewriting]] in [[higher categories]] * [[Amar Hadzihasanovic]], _Diagrammatic sets and rewriting in weak higher categories_, ([arXiv:2007.14505](https://arxiv.org/abs/2007.14505)) On the [[tensor product]] of [[PROs]] as the [[smash product]] of [[pointed spaces]]: * [[Amar Hadzihasanovic]], _The smash product of monoidal theories_, ([arXiv:2101.10361](https://arxiv.org/abs/2101.10361)) category: people
Amaria Javed	https://ncatlab.org/nlab/source/Amaria+Javed	member of [[CQTS]] * [GoogleScholar page](https://scholar.google.ae/citations?user=NTzY1lUAAAAJ&hl=en) ## Selected writings on [[solitons]] in [[photonic crystals]]: * [[Amaria Javed]], Alaa Shaheen U. Al Khawaja, Amplifying optical signals with discrete solitons in waveguide arrays, Physics Letters A 384 26 (2020) 126654 [[doi:10.1016/j.physleta.2020.126654](https://doi.org/10.1016/j.physleta.2020.126654)] * Alaa Shaheen, [[Amaria Javed]], U. Al Khawaja, Adding binary numbers with discrete solitons in waveguide arrays, Phys. Scr. 95 (2020) 085107 [[arXiv:2108.01406](https://arxiv.org/abs/2108.01406), [doi:10.1088/1402-4896/aba2b2](https://doi.org/10.1088/1402-4896/aba2b2)] category: people
amazing right adjoint	https://ncatlab.org/nlab/source/amazing+right+adjoint	[[!redirects fractional exponential]] +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Differential geometry +--{: .hide} [[!include synthetic differential geometry - contents]] =-- #### Compact objects +--{: .hide} [[!include compact object - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition# [[William Lawvere]]'s definition of an _atomic_ [[infinitesimal space]] is as an [[object]] $\Delta$ in a [[topos]] $\mathcal{T}$ such that the [[inner hom]] [[functor]] $(-)^\Delta : \mathcal{T} \to \mathcal{T}$ has a [[right adjoint]] (is an [[atomic object]]). Notice that by definition of [[inner hom]], $(-)^\Delta$ always has a [[left adjoint]]. A [[right adjoint]] can only exist for very particular objects. Therefore the term amazing right adjoint. ## Right adjoints to representable exponentials Assume $\mathcal{T} = Sh(C)$ is a [[Grothendieck topos]], that the [[Grothendieck topology]] on the [[site]] $C$ is [[subcanonical coverage\|subcanonical]]. Let $\Delta \in C \hookrightarrow Sh(C)$ be a [[representable functor\|representable object]]. Then, $(-)^\Delta: Sh(C)\to Sh(C)$ has a [[right adjoint]], hence $\Delta$ is an atomic [[infinitesimal space]], precisely if $(-)^\Delta: C\to C$ preserves [[colimit]]s. This is a special case of the general [[adjoint functor theorem#in_toposes]]. For if $(-)^\Delta$ preserves colimits, its [[right adjoint]] is $$ (-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y)) \,. $$ The $Y_\Delta$ defined this way is indeed a sheaf, due to the assumption that $(-)^\Delta$ preserves colimits. So this is indeed a [[right adjoint]]. ## Related concepts * A [[topos]] $\mathcal{X}$ is a [[local topos]] (over [[Set]]) if its [[global section]] functor $\Gamma = Hom(1_{\mathcal{X}}, -)$ admits a [[right adjoint]]. This is hence an "external" version of the amazing right adjoint, exhibiting $1_{\mathcal{X}}$ as "atomic". * The topic of amazing right adjoints appears to have been studied mostly for toposes, but a related definition has been given for any category (with products): if $O$ is an [[exponentiable object]] of a category $\mathcal{C}$, then $O$ being a [[tiny object]] in the sense of [Definition 0.1](#Yetter1987) means that the [[internal hom\|endofunctor]] $(-)^O$ has a right adjoint. (This adjoint is sometimes denoted $(-)^{1/O}$, c.f. [Lawvere, p.269](#Lawvere2002), and called 'fractional exponential') In this situation, one then has, symbolically $$ (-)\times O \quad \dashv\quad (-)^O \quad \dashv\quad (-)^{1/O} $$ ## References# The ubiquity of right adjoints to exponential functors in the context of [[synthetic differential geometry]] was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment [[lambda calculus]] with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997). * [[William Lawvere]], _Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body_, Cah.Top.Géom.Diff.Cat. 21 no.4 (1980) pp.377-392. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_4/CTGDC_1980__21_4_377_0/CTGDC_1980__21_4_377_0.pdf)) * [[William Lawvere]], _[[Toposes of laws of motion]]_, 1997. * {#Law04} [[William Lawvere]], _Left and right adjoint operations on spaces and data types_ , Theor. Comp. Sci. 316 (2004) pp.105-111. * [[Ieke Moerdijk]], [[Gonzalo Reyes]], [[Models for Smooth Infinitesimal Analysis]] , Springer Heidelberg 1991. (appendix 4) * [[Anders Kock]], [[Gonzalo E. Reyes]], _Aspects of Fractional Exponent Functors_ , TAC 5 (1999) pp.251-265. ([pdf](http://www.tac.mta.ca/tac/volumes/1999/n10/n10.pdf)) * {#Yetter1987} [[David Yetter]], _On Right Adjoints of Exponential Functors_ , JPAA 45 (1987) pp.287-304. (Corrections in JPAA 58 (1989) pp.103-105) * {#Lawvere2002} [[William Lawvere]], Categorical algebra for continuum micro physics, Journal of Pure and Applied Algebra 175 (2002) 267–287
Ambidexterity in K(n)-Local Stable Homotopy Theory	https://ncatlab.org/nlab/source/Ambidexterity+in+K%28n%29-Local+Stable+Homotopy+Theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Stable Homotopy theory +--{: .hide} [[!include stable homotopy theory - contents]] =-- =-- =-- This page collects some links related to * [[Michael Hopkins]], [[Jacob Lurie]]: Ambidexterity in $K(n)$-Local Stable Homotopy Theory (2013) [pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf) also: talk at _Notre Dame Graduate Summer School on Topology and Field Theories_ and _Harvard lecture_ 2012 ([video part 1](http://www.youtube.com/watch?v=eQayYLDw1VA), [part 2](http://www.youtube.com/watch?v=OEShrQyvmS4), [part 3](http://www.youtube.com/watch?v=nOIcdn1iUR4) [part 4](http://www.youtube.com/watch?v=ZwnClYedaYM), [pdf lecture notes](http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf) by Chris Elliott) on [[biproduct\|bilimits]] in [[K(n)-local stable homotopy theory]] and generally on [[ambidextrous adjunctions]] and their un-twisted [[Wirthmüller isomorphisms]] for [[(∞,1)-module bundles\|∞-module bundles]] in [[semiadditive (∞,1)-categories]]. (The untwisted [[Wirthmüller isomorphism]] is the map $\mu$ in Construction 4.0.7 and then the [[norm map]] in Remark 4.1.12. The induced [[integration]] map considered in Construction 4.0.7, Notation 4.1.6 there is also discussed (for the general twisted case) in the article _[[schreiber:Quantization via Linear homotopy types]]_, see section 4.3 there for details.) The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of [[quantization]] in sections 3 and 8 of * [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ in P. R. Kotiuga (ed.) A celebration of the mathematical legacy of Raoul Bott AMS (2010) ([arXiv:0905.0731](http://arxiv.org/abs/0905.0731)) For the 2014 installment of UOregon's Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on [[spectral algebraic geometry]], one of which is * [[Jacob Lurie]], [Ambidexterity](http://media.uoregon.edu/channel/2014/05/23/moursund-mathematics-lecture-series-dr-jacob-lurie-lecture-2-ambidexterity/) #Contents# * table of contents {:toc} ## 1. Multiplicative aspects of Dieudonne Theory ## 2. The Morava $K$-theory of Eilenberg-MacLane Spaces * [[Morava K-theory]] ## 3. Alternating powers of $p$-Divisible groups ## 4. Ambidexterity ### 4.1 Beck-Chevalley fibrations and Norm maps * [[Beck-Chevalley condition]] * [[pointwise (∞,1)-Kan extension]] ### 4.2 Properties of the norm ### 4.3 Local systems * [[local system]] * [[ambidextrous space]] ### 4.4 Examples * [[semiadditive (∞,1)-category]] ## 5. Ambidexterity of $K(n)$-Local stable homotopy theory ### 5.1 Ambidexterity and duality ### 5.2 The main theorem * [[K(n)-local stable homotopy theory]] ### 5.3 Cartier duality * [[Cartier duality]] ### 5.4 The global sections functor category: reference
ambidextrous adjunction	https://ncatlab.org/nlab/source/ambidextrous+adjunction	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### 2-Category theory +--{: .hide} [[!include 2-category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition An [[adjoint triple]] $F \dashv G \dashv H$ is called an _ambidextrous adjunction_ (or sometimes _ambiadjunction_ or _ambijunction_, for short) if the [[left adjoint]] $F$ and the [[right adjoint]] $H$ of $G$ are [[equivalence\|equivalent]] $F \simeq H$, or more precisely: equipped with a specified equivalence. In fact, often $F$ is identified with $H$, which is the situation of a strongly adjoint pair $(F \dashv G \dashv F)$ originally considered by [Morita 1965](#Morita65). Some authors refer to this situation by saying that $G$ is a [[Frobenius functor]] (ie. a functor which has a [[left adjoint]] that is also a [[right adjoint]]). Sometimes $F$ is said to be biadjoint to $G$ (not to be confused with [[biadjoint]] in the sense of [[biadjunction]]). Functor $G$ which has a left and right adjoint which are equivalent is said to be [[Frobenius functor]]. In the special case that $G$ is a [[fully faithful functor]] with an ambidextrous adjoint one also speaks of an [[essential localization]] (cf. [[bireflective subcategory]]). ## Properties ### Frobenius algebra structure The [[monad]] induced by an ambidextrous adjunction is a [[Frobenius monoid]] object in [[endofunctors]]. (e.g. [Lauda 05, theorem 17](#Lauda05)), hence a [[Frobenius monad]]. ### Fiberwise characterization of ambidextrous Kan extension {#FiberwiseCharacterization} Let $\mathcal{D} \in Cat_\infty$ be an [[(∞,1)-category]] with small [[(∞,1)-colimits]]. For $f \;\colon\; X \longrightarrow Y$ a morphism of [[∞-groupoids]], write $$ f^\ast \;\colon\; [Y,\mathcal{D}] \longrightarrow [X,\mathcal{D}] $$ for the induced pullback of [[(∞,1)-functor (∞,1)-categories]] (which one may think of as the categories of $\mathcal{D}$-valued [[local systems]] over $X$ and $Y$, respectively). The [[left adjoint]] and [[right adjoint]] (if it exists) of this are left and right [[(∞,1)-Kan extension]]. +-- {: .num_defn #AmbidextrousKanExtension} ###### Definition Say that a morphism $f$ is $\mathcal{D}$-ambidextrous if $(f_! \dashv f^\ast)$ is an ambidextrous adjunction $(f_! \simeq f_\ast)$ and in addtion all pullbacks of $f$ satisfy some property (...). Say that an [[∞-groupoid]] $A \in Grpd_\infty$ is _$\mathcal{D}$-ambidextrous_ if its [[terminal object in an (∞,1)-category\|terminal]] map is. =-- ([Hopkins-Lurie 14, def. 4.1.11](#{#HopkinsLurie14})) +-- {: .num_prop} ###### Proposition A morphism $f \colon X \to Y$ between [[∞-groupoids]], is $\mathcal{D}$-ambidextrous, def. \ref{AmbidextrousKanExtension}, precisely if each [[homotopy fiber]] $X_y$ of $f$ is. =-- ([Hopkins-Lurie 14, prop. 4.3.5](#HopkinsLurie14)) ## Examples +-- {: .num_example} ###### Example (coincident limits and colimits) Let $\mathcal{C}$ be a [[small category]] and $\mathcal{D}$ any category and consider the functor $const \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}]$ that sends objects to constant [[presheaves]] with this value. Then the [[right adjoint]] of this functor is, if it exists, the [[limit]] construction, and the [[left adjoint]] is, if it exists, the [[colimit]] construction. (See also at _[[Kan extension]]_.) Therefore if both exist as an ambidextrous adjunction, then this means that limits in $\mathcal{D}$ over [[diagrams]] of shape $\mathcal{C}$ coincide with the [[colimits]] over these diagrams. If $\mathcal{C}$ is a [[finite set]], then this situation is traditionally referred to as _[[biproducts]]_. Generally therefore this is sometimes called _[[bilimits]]_ (but see the discussion of the terminology there). In ([Hopkins-Lurie 14, section 4.3](#HopkinsLuire14)) such $\mathcal{C}$ is called _$\mathcal{D}$-ambidextrous_ (or rather, they consider $\mathcal{C}$ an [[∞-groupoid]] and hence call it a _$\mathcal{D}$-ambidextrous space_). Concrete examples of this include those discussed at _[[K(n)-local stable homotopy theory]]_. =-- +-- {: .num_example} ###### Example (Yoga of six functors) A [[Wirthmüller context]] in the presence of an un-twisted [[Wirthmüller isomorphism]] is an ambidextrous adjunction. ###### Example Every [[self-adjoint functor]] forms an ambidextrous adjunction. =-- ## Related concepts * [[Wirthmüller context]] * [[infinitesimal cohesion]] ## References Ambidextrous adjunctions were maybe first considered under the name strongly adjoint pairs (of functors), in: * {#Morita65} [[Kiiti Morita]], Adjoint pairs of functors and Frobenius extensions, Science Reports of the Tokyo Kyoiku Daigaku, Section A 9 202/208 (1965) 40-71 [[jstor:43698658](https://www.jstor.org/stable/43698658)] The terminology "[[Frobenius functors]]" for "strongly adjoint pairs" is due to * {#CaenepeelMilitaruZhu97} [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997) 4311-4342 [[1997-349-11/S0002-9947-97-02004-7](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7), [pdf](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7/S0002-9947-97-02004-7.pdf)] * F. Castaño Iglesias, [[José Gómez Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10 (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)] The case of [[bireflective subcategories]]: * [[Peter Freyd]], [[Peter O’Hearn]], [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], Bireflectivity, Theoretical Computer Science 228 1–2 (1999) 49-76 [<a href="https://doi.org/10.1016/S0304-3975(98)00354-5">doi:10.1016/S0304-3975(98)00354-5</a>] On the [[Frobenius monads]] induced by ambidextrous adjuntions: * {#Street04} [[Ross Street]], Frobenius monads and pseudomonoids, J. Math. Phys. 45 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)] * {#Lauda05} [[Aaron Lauda]], Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories 16 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)] See also: * {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], [[Ambidexterity in K(n)-Local Stable Homotopy Theory]] (2014) with some review in: * [[Peter Haine]], Ambidexterity (2018) [[pdf](https://math.berkeley.edu/~phaine/files/Ambidexterity_4.pdf), [[Haine-Ambidexterity.pdf:file]]] On the issue of equipping an ambidextrous adjunction $F \dashv G \dashv H$ with a specific equivalence between $F$ and $H$: * [[Qiaochu Yuan]], [MO:377104](https://mathoverflow.net/a/377104) Connection to [[Hopf adjunction]]s * Harshit Yadav, _Frobenius monoidal functors from (co)Hopf adjunctions_, [arXiv:2209.15606](https://arxiv.org/abs/2209.15606) [[!redirects ambidextrous adjunctions]] [[!redirects ambidextrous adjoint]] [[!redirects ambidextrous adjoints]] [[!redirects ambidextrous space]] [[!redirects ambidextrous spaces]] [[!redirects strongly adjoint pair]] [[!redirects strongly adjoint pairs]] [[!redirects biadjoint pair]] [[!redirects ambiadjunction]] [[!redirects ambiadjunctions]] [[!redirects ambijunction]] [[!redirects ambijunctions]]
ambient category	https://ncatlab.org/nlab/source/ambient+category	An ambient category $C$ is just a category from which [[internalization\|internal structures]] take their objects and structure morphisms. If $C$ has extra structure, like being ([[semi-abelian category\|semi]]-)[[abelian category\|abelian]] or a [[site]], then one can do extra things; this is made precise through the concept of [[doctrine]]. More generally an ambient category could be seen as a 'universe of discourse', as when $C$ is a [[topos]]; see [[foundations]].
ambimorphic object > history	https://ncatlab.org/nlab/source/ambimorphic+object+%3E+history	< [[ambimorphic object]] [[!redirects ambimorphic object -- history]]
Ambrus Kaposi	https://ncatlab.org/nlab/source/Ambrus+Kaposi	* [website](https://akaposi.github.io/) ## Selected writings * [[Thorsten Altenkirch]], [[Ambrus Kaposi]], Type Theory in Type Theory using Quotient Inductive Types, POPL17 (2017) [[pdf](http://www.cs.nott.ac.uk/~psztxa/publ/tt-in-tt.pdf)] On [[higher inductive-inductive types]]: * {#KaposiKovacs} [[Ambrus Kaposi]], [[András Kovács]], A Syntax for Higher Inductive-Inductive Types, at 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018) (2018) 20:1-20:18 [[pdf](http://drops.dagstuhl.de/opus/volltexte/2018/9190/), [doi:10.4230/LIPIcs.FSCD.2018.20](https://doi.org/10.4230/LIPIcs.FSCD.2018.20)] * [[Ambrus Kaposi]], [[András Kovács]], Signatures and Induction Principles for Higher Inductive-Inductive Types, Logical Methods in Computer Science 16 1 (2020) lmcs:6100 [[arXiv:1902.00297](https://arxiv.org/abs/1902.00297), <a href="https://doi.org/10.23638/LMCS-16(1:10)2020">doi:10.23638/LMCS-16(1:10)2020</a>] Exposition: * [[Ambrus Kaposi]], Quotient inductive-inductive types and higher friends, [[Homotopy Type Theory Electronic Seminar Talks]], 22 October 2020 ([video](https://www.youtube.com/watch?v=a9_KjX1WM84), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kaposi-2020-10-22-HoTTEST.pdf)) category: people
amenable category	https://ncatlab.org/nlab/source/amenable+category	An amenable category is an [[additive category]] in which all [[idempotent]]s split.
amenable topological groupoid	https://ncatlab.org/nlab/source/amenable+topological+groupoid	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Higher geometry +--{: .hide} [[!include higher geometry - contents]] =-- #### Topology +--{: .hide} [[!include topology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition ([AD-Renault 00](#AD-Renault00)) Recalled for instance in ([Sims-Williams 12, p. 4](#SimsWilliams)). ([ANS 05, def. 1.3](#ANS05)) ## Properties ### Groupoid convolution algebra +-- {: .num_prop} ###### Proposition The [[groupoid convolution algebra]] of an amenable topological groupoid is in the [[bootstrap category]]. =-- ([Tu 99, prop. 10.7](#Tu99)), recalled as ([Uuye 11, example 3.6](#Uuye11)). +-- {: .num_prop} ###### Proposition For an amenable [[Lie groupoid]] $\mathcal{G}$, the full [[groupoid convolution algebra]] and the reduced one are [[natural isomorphism\|naturally isomorphic]]. =-- This is due to ([AD-Renault 00](#AD-Renault00)), recalled for instance as ([ANS 05, prop. 1.9](#ANS05)) ## References * C. Anantharaman-Delaroche, [[Jean Renault]], _Amenable Groupoids_ Monogr. Enseign. Math., vol. 36 Enseignement Math., Geneva (2000) ([webpage, including extensive review](http://www.unige.ch/math/EnsMath/EM_MONO/m36.html)) {#AD-Renault00} * Aidan Sims, Dana P. Williams, _Amenability for Fell bundles over groupoids_ ([arXiv:1201.0792](http://arxiv.org/abs/1201.0792)) {#SimsWilliams12} * Johannes Aastrup, Ryszard Nest, Elmar Schrohe, _A Continuous Field of C-algebras and the Tangent Groupoid for Manifolds with Boundary_ ([arXiv:math/0507317](http://arxiv.org/abs/math/0507317)) {#ANS05} [[Jean-Louis Tu]], _La conjecture de Baum-Connes pour les feuilletages moyennables_, K-Theory 17 (1999), no. 3, 215–264. MR 1703305 (2000g:19004) ([Portico](http://www.portico.org/Portico/article/access/viewHtml.por?journalId=ISSN_09203036&issueId=ISSN_09203036v17i3&articleId=pgg1zfpwkq7&fileType=html:Header&fileValid=true), subscription needed) {#Tu99} * [[Otgonbayar Uuye]], _A note on the Künneth theorem for nonnuclear C*-algebras_ ([arXiv:1111.7228](http://arxiv.org/abs/1111.7228)) {#Uuye11} [[!redirects amenable topological groupoids]] [[!redirects amenable topological group]] [[!redirects amenable topological groups]] [[!redirects amenable groupoid]] [[!redirects amenable groupoids]] [[!redirects amenable group]] [[!redirects amenable groups]] [[!redirects amenable Lie groupoid]] [[!redirects amenable Lie groupoids]]
Amer Iqbal	https://ncatlab.org/nlab/source/Amer+Iqbal	* [Wikipedia entry](https://en.wikipedia.org/wiki/Amer_Iqbal) ## Selected writings On [[string junctions]] in relation to [[Lie algebras]] and [[Lie algebra representations]]: * Oliver DeWolfe, Tamas Hauer, [[Amer Iqbal]], [[Barton Zwiebach]], _Uncovering the Symmetries on $[p,q]$ 7-branes: Beyond the Kodaira Classification_, Adv. Theor. Math. Phys. 3 (1999) 1785-1833 ([arXiv:hep-th/9812028](https://arxiv.org/abs/hep-th/9812028)) On a [[mysterious duality]]: * {#INV01} [[Amer Iqbal]], [[Andrew Neitzke]], [[Cumrun Vafa]], _A mysterious duality_, ([arXiv:hep-th/0111068](https://arxiv.org/abs/hep-th/0111068)) On [[M-strings]]/[[self-dual strings]]: * [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1)) and the [[M-string elliptic genus]]: * [[Stefan Hohenegger]], [[Amer Iqbal]], _M-strings, Elliptic Genera and $\mathcal{N}=4$ String Amplitudes_, Fortschritte der PhysikVolume 62, Issue 3 ([arXiv:1310.1325](http://arxiv.org/abs/1310.1325)) * [[Stefan Hohenegger]], [[Amer Iqbal]], Soo-Jong Rey, _M String, Monopole String and Modular Forms_, Phys. Rev. D 92, 066005 (2015) ([arXiv:1503.06983](https://arxiv.org/abs/1503.06983)) On [[E-strings]]: * [[Amer Iqbal]], _A note on E-strings_, Adv. Theor. Math. Phys. 7 (2003) 1-23 ([arXiv:hep-th/0206064](https://arxiv.org/abs/hep-th/0206064)) ## Related $n$Lab entries * [[mysterious duality]] category: people
Amihay Hanany	https://ncatlab.org/nlab/source/Amihay+Hanany	Amihay Hanany is professor for theoretical [[physics]] at Imperial College London. * [webpage](http://www.imperial.ac.uk/people/a.hanany) ## Selected writings On [[small instantons]] as [[NS5-branes]]/[[M5-branes]] in [[E8]]-[[heterotic string theory]]/[[heterotic M-theory]]: * {#GanorHanany96} [[Ori Ganor]], [[Amihay Hanany]], _Small $E_8$ Instantons and Tensionless Non-critical Strings_, Nucl. Phys. B474 (1996) 122-140 ([arXiv:hep-th/9602120](https://arxiv.org/abs/hep-th/9602120)) * [[Amihay Hanany]], Noppadol Mekareeya, _The Small $E_8$ Instanton and the Kraft Procesi Transition_, JHEP07 (2018) 098 ([arXiv:1801.01129](https://arxiv.org/abs/1801.01129)) On [[geometric engineering of quantum field theories]] involving [[NS5-branes]] and introducing the [[Hanany-Witten effect]] and other effects of [[NS5-brane]]/[[D-brane]] [[brane intersection\|intersections]]: * {#HanayWitten97} [[Amihay Hanany]], [[Edward Witten]], Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics, Nucl. Phys.B 492 (1997) 152-190 [[arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230), <a href="https://doi.org/10.1016/S0550-3213(97)80030-2">doi:10.1016/S0550-3213(97)80030-2</a>] On [[D-branes]] [[brane intersection\|intersecting]] [[NS5-branes]]: * {#BrodieHanany97} [[John Brodie]], [[Amihay Hanany]], _Type IIA Superstrings, Chiral Symmetry, and N=1 4D Gauge Theory Dualities_, Nucl.Phys. B506 (1997) 157-182 ([arXiv:hep-th/9704043](https://arxiv.org/abs/hep-th/9704043)) On [[(p,q)5-brane webs]]: * [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116)) * [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170)) and their intersection with [[orientifolds]]: * [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242)) Discussion of a [[BFSS matrix model\|BFSS-like]] [[matrix model]] for [[MK6-branes]]: * [[Amihay Hanany]], Gilad Lifschytz, _M(atrix) Theory on $T^6$ and a m(atrix) Theory Description of KK Monopoles_, Nucl. Phys. B519:195-213, 1998 ([arXiv:hep-th/9708037](https://arxiv.org/abs/hep-th/9708037)) On [[toric duality]]: * [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085)) * [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259)) * [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063)) * [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228)) On [[discrete torsion]]: * [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023)) On [[vortex strings]]: * [[Amihay Hanany]], [[David Tong]], _Vortices, Instantons and Branes_, JHEP 0307 (2003) 037 ([arXiv:hep-th/0306150](https://arxiv.org/abs/hep-th/0306150)) * [[Amihay Hanany]], [[David Tong]], _Vortex Strings and Four-Dimensional Gauge Dynamics_, JHEP 0404 (2004) 066 ([arXiv:hep-th/0403158](https://arxiv.org/abs/hep-th/0403158)) * {#Tong09} [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060)) * [[David Tong]], _Vortices, Strings, and Vortex Strings_ ([[TongVortexStrings.pdf:file]]) On [[heterotic M-theory on ADE-orbifolds]]: * Santiago Cabrera, [[Amihay Hanany]], [[Marcus Sperling]], _Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories_, J. High Energ. Phys. (2019) 2019: 71 ([arXiv:1904.12293](https://arxiv.org/abs/1904.12293)) On [[D=3 N=4 super Yang-Mills theory]]: * [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657)) ## Related entries * [[super Yang-Mills theory]] * [[Seiberg duality]] * [[AdS-CFT]] category: people
Amir Dembo	https://ncatlab.org/nlab/source/Amir+Dembo	* [webpage](http://statweb.stanford.edu/~adembo/) ## related $n$Lab entries * [[probability theory]] category: people
Amir-Kian Kashani-Poor	https://ncatlab.org/nlab/source/Amir-Kian+Kashani-Poor	* [webpage](www.phys.ens.fr/~kashani/index.html) ## related $n$Lab entries * [[topological recursion]] * [[topological string]] * [[matrix model]] category: people
Amirhossein Tajdini	https://ncatlab.org/nlab/source/Amirhossein+Tajdini	* [InSpire page](https://inspirehep.net/authors/1889144) * [ResearchGate page](https://www.researchgate.net/scientific-contributions/Amirhossein-Tajdini-2116760582) ## Selected writings Review of the claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]: * [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. 2020 13 (2020) \[<a href="https://doi.org/10.1007/JHEP05(2020)013">doi:10.1007/JHEP05(2020)013</a>, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\] Review: * [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. 93 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)] category: people
Amit Giveon	https://ncatlab.org/nlab/source/Amit+Giveon	* [webpage](http://old.phys.huji.ac.il/~hep/amit.giveon.html) ## Selected writings On [[geometric engineering of QFT]] in [[intersecting D-brane models]] subject to the [[s-rule]]: * [[Amit Giveon]], [[David Kutasov]], _Brane Dynamics and Gauge Theory_, Rev. Mod. Phys. 71:983-1084, 1999 ([arXiv:hep-th/9802067](https://arxiv.org/abs/hep-th/9802067)) On [[NS5-brane]]-[[brane intersections\|intersections]] with [[D-branes]]: * {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], D-Branes in the Background of NS Fivebranes, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)] category: people
Amit Patel	https://ncatlab.org/nlab/source/Amit+Patel	* [institute page](https://www.math.colostate.edu/~akp/) * [personal page](http://akpatel.org/) ## Selected writings Introducing [[well groups]]: * {#EMP11} [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], Quantifying Transversality by Measuring the Robustness of Intersections, Foundations of Computational Mathematics, 11 3 (2011) 345–361 $[$[arXiv:0911.2142](https://arxiv.org/abs/0911.2142)$]$ * [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], The Robustness of Level Sets, In: M. de Berg, U. Meyer (eds.) _Algorithms – ESA 2010_. ESA 2010. Lecture Notes in Computer Science 6346 Springer (2010) $[$[doi:10.1007/978-3-642-15775-2_1](https://doi.org/10.1007/978-3-642-15775-2_1)$]$ * [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], Homology and Robustness of Level and Interlevel Sets, Homology, Homotopy and Applications, 15 (2013) 51-72 $[$[euclid:1383943667](https://projecteuclid.org/euclid.hha/1383943667)$]$ category: people
Amit Sever	https://ncatlab.org/nlab/source/Amit+Sever	* [Institute page](https://en-exact-sciences.tau.ac.il/profile/asever) ## Selected writings On [[Wilson line]]-[[quantum observables]] and [[bosonization]] in [[Chern-Simons theory\|Chern-Simons]]/matter theory (such as the [[ABJM model]]): * [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, talk at [[Strings 2022]] [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)] category: people
Amit Sharma	https://ncatlab.org/nlab/source/Amit+Sharma	* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=214520) ## Selected writings On [[compact closed category\|compact closure]] in [[homotopical algebra]] and relating to the [Barrat-Priddy theorem](stable+cohomotopy#AsAlgebraicKTheoryOverTheFieldWithOneElement): * [[Amit Sharma]], Compact closed categories and Γ-categories (with an appendix by [[André Joyal]]), Theory and Applications of Categories 37 37 (2021) 1222-1261 [[arXiv:2010.09216](https://arxiv.org/abs/2010.09216), [tac:37-37](http://www.tac.mta.ca/tac/volumes/37/37/37-37abs.html)] On [[Picard groupoids]]: * [[Amit Sharma]], Picard groupoids and $\Gamma$-categories, [[Cahiers]] LXIV 3 (2023) [[arXiv:2002.05811](https://arxiv.org/abs/2002.05811), [pdf](http://cahierstgdc.com/wp-content/uploads/2023/07/SHARMA_LXIV-3.pdf)] category: people
Amitai Regev	https://ncatlab.org/nlab/source/Amitai+Regev	* [Wikipedia entry](https://en.wikipedia.org/wiki/Amitai_Regev) ## Selected writings Expressing the number of [[standard Young tableaux]] with at most 3 rows through [[Catalan numbers]] and [[Motzkin numbers]]: * [[Amitai Regev]], Asymptotic values for degrees associated with strips of young diagrams, Advances in Mathematics Volume 41, Issue 2, August 1981, Pages 115-136 (<a href="https://doi.org/10.1016/0001-8708(81)90012-8">doi:10.1016/0001-8708(81)90012-8</a>) category: people
Amitsur complex	https://ncatlab.org/nlab/source/Amitsur+complex	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebra +--{: .hide} [[!include higher algebra - contents]] =-- #### Homological algebra +--{: .hide} [[!include homological algebra - contents]] =-- #### Locality and descent +--{: .hide} [[!include descent and locality - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition Given a [[commutative ring]] $R$ and an $R$-[[associative algebra]] $A$, hence a [[ring]] [[homomorphism]] $R \longrightarrow A$, the _Amitsur complex_ is the [[Moore complex]] of the dual [[Cech nerve]] of $Spec(A) \to Spec(R)$, hence the [[chain complex]] of the form $$ R \to A \to A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots $$ with [[differentials]] given by the alternating sum of the coface-maps. (See also at _[[Sweedler coring]]_, at _[[commutative Hopf algebroid]]_ and at _[[Adams spectral sequence]]_ for the same or similar constructions.) ## Properties ### Descent theorem +-- {: .num_theorem } ###### Theorem (descent theorem) If $A \to B$ is [[faithfully flat]] then its Amitsur complex is [[exact sequence\|exact]]. =-- This is due to ([[Grothendieck]], [[FGA]]1) The following reproduces the proof in low degree from [Milne, prop. 6.8](#Milne) +-- {: .proof} ###### Proof We show that $$ 0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B $$ is an [[exact sequence]] if $f \colon A \longrightarrow B$ is [[faithfully flat]]. First observe that the statement follows if $A \to B$ admits a [[section]] $s \colon B \to A$. Because then we can define a map $$ k \colon B \otimes_A B \longrightarrow B $$ $$ k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2)) \,. $$ This is such that applied to a coboundary it yields $$ k(1 \otimes b - b \otimes 1) = f(s(b)) - b $$ and hence it exhibits every cocycle $b$ as a coboundary $b = f(s(b))$. So the statement is true for the special morphism $$ B \to B \otimes_A B $$ $$ b \mapsto b \otimes 1 $$ because that has a section given by the multiplication map. But now observe that the morphism $B \to B \otimes_A B$ is the [[tensor product]] of the morphism $f$ with $B$ over $A$. That $A \to B$ is [[faithfully flat]] by assumption, hence that it exhibits $B$ as a [[faithfully flat module]] over $A$ means by definition that the Amitsur complex for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact. =-- ### As a bar construction For $\phi \colon B \longrightarrow A$ a [[homomorphism]] of suitable [[monoids]], there is the corresponding pull-push [[adjunction]] ([[extension of scalars]] $\dashv$ [[restriction of scalars]]) on [[categories of modules]] $$ ((- )\otimes_B A \dashv \phi^\ast ) \;\colon\; Mod_A \stackrel{\overset{(-)\otimes_B A}{\leftarrow}}{\underset{\phi^\ast}{\longrightarrow}} Mod_B \,. $$ The [[bar construction]] of the corresponding [[monad]] -- the [[higher monadic descent\|higher]] [[monadic descent]] objects -- is the corresponding Amitsur complex. (e.g. [Hess 10, section 6](#Hess10)) ## Related concepts * [[Adams-Novikov spectral sequence]] ## References The Amitsur complex was introduced in * [[Shimshon Amitsur]], _Simple algebras and cohomology groups of arbitrary fields_, Transactions of the American Mathematical Society Vol. 90, No. 1 (Jan., 1959), pp. 73-112 ([JSTOR](http://www.jstor.org/stable/1993268)) His results were simplified in * Alex Rosenberg and Daniel Zelinsky, _On Amitsur's complex_, Transactions of the American Mathematical Society Vol. 97, No. 2 (Nov., 1960), pp. 327-356 The statement of proof the descent theorem for the Amitsur complex is due to * [[Alexander Grothendieck]], [[FGA]]1 A review of the proof in low degree is in * {#Milne} [[James Milne]], prop. 6.8 of _[[Lectures on Étale Cohomology]]_ Discussion from the point of view of [[Sweedler corings]] and a full proof of the descent theorem is in * [[Tomasz Brzezinski]], [[Robert Wisbauer]], section 29 of _Corings and Comodules_, Cambridge University Press, London Math. Soc. LN 309 (2003), ([errata pdf](http://www.math.uni-duesseldorf.de/~wisbauer/corinerr.pdf)) Disucssion from the point of view of [[higher monadic descent]] is in * {#Hess10} [[Kathryn Hess]], section 6 of _A general framework for homotopic descent and codescent_ ([arXiv:1001.1556](http://arxiv.org/abs/1001.1556)) Discussion of [[formal completion]] of [[(infinity,1)-modules]] in terms of [[totalization]] of Amitsur complexes is in * {#Carlsson07} [[Gunnar Carlsson]], _Derived completions in stable homotopy theory_ ([arXiv:0707.2585](http://arxiv.org/abs/0707.2585)) [[!redirects Amitsur complexes]] [[!redirects descent theorem]] [[!redirects descent theorems]]
amnestic functor	https://ncatlab.org/nlab/source/amnestic+functor	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- # Amnestic functors * table of contents {: toc} ## Idea A [[strict functor]] is _amnestic_ if its [[domain]] has no more duplication of [[isomorphic]] [[objects]] than its [[codomain]]. As ‘amnestic’ is basically a fancy synonym of ‘forgetful’, the idea is to identify a property that one would like in a [[forgetful functor]]. For example, the notion of amnestic functors formalizes the sense in which the [[concrete category]] $Met_cont$ of [[metric spaces]] and [[continuous maps]] is often better thought of as the category $Met Top$ of [[metrizable topological spaces]] (and [[continuous maps]]), by identifying a property that the [[forgetful functor]] $Met Top \to Set$ has but $Met_cont \to Set$ does not. There is a corresponding notion of _amnesticization_ of a functor (called the _amnestic modification_ in _[[The Joy of Cats]]_), which replaces the domain with an [[equivalent category]], relative to which the functor becomes amnestic. Applying this to $Met_cont \to Set$ produces a category [[isomorphic category\|isomorphic]] to $Met Top$. Although not needed in this case, we need the [[axiom of choice]] (AC) in general to prove that every functor has an amnesticization. Without AC, we can use amnestic [[anafunctors]] to make everything work out, although much of the convenience is lost. Amnesticity is really a property of [[strict functors]] (or anafunctors) between [[strict category\|strict]] [[groupoids]]. Groupoids, because the non-isic [[morphisms]] play no role in the definition; only the categories\' [[cores]] matter, and a functor is amnestic iff its core is amnestic. Strict, because the definition requires us to state (in two places) that some isomorphic objects are equal; weakening the definition to follow the [[principle of equivalence]] leads to a trivial property that every functor satisfies. (That is, up to [[equivalence of categories\|equivalence]], every functor is amnestic, which is because every functor is equivalent to its amnesticization.) ## Definitions +-- {: .num_defn} ###### Definition Let $C$ and $D$ be two [[strict categories]], and let $U$ be a [[strict functor]] from $D$ to $C$. We say that $U$ is __amnestic__ if its [[groupoid core]] reflects [[identity morphisms]]. Explicitly, $U$ is amnestic iff, for every [[isomorphism]] $f \colon a \to B$ in $D$ that $U$ takes to an [[identity morphism]] $U(f) = id_{U(a)} = id_{U(b)}$, then already $f$ itself is an [[identity morphism]]. In other words, a functor is amnestic if its strict [[fibers]] are [[gaunt category\|gaunt]]. =-- Observe that this is similar to a [[conservative functor]], which reflects isomorphisms rather than identities. If we follow the [[principle of equivalence]] and refuse to state [[equalities]] between objects, then we must modify the hypothesis to say that $U(a)$ and $U(b)$ are [[isomorphic]] in $C$ (say via $g\colon U(a) \to U b$) and $U(f)$ is the identity relative to this isomorphism (so $U(f) = \id_{U(b)} \circ g \circ \id_{U(a)}$; since we can simply let $g$ be $U(f)$, this is trivial (beyond the initial isomorphism $f\colon a \to b$). Similarly, we must modify the conclusion to say that $a$ and $b$ are isomorphic (say via $h\colon a \to b$) and $f$ is the identity relative to this isomorphism (so $f = \id_b \circ h \circ \id_a$); since we can simply let $h$ be $f$, this is also trivial. Thus up to [[equivalence of categories\|equivalence]], this property is trivial; on the other hand, it is preserved by [[isomorphism of categories\|isomorphism]]. +-- {: .num_defn #anafunctor} ###### Definition Now let $C$ and $D$ be two [[strict categories]], and let $U$ be an [[anafunctor]] from $D$ to $C$. We again say that $U$ is __amnestic__ if its [[core]] reflects [[identity morphisms]]. Explicitly, now $U$ is amnestic iff, whenever $a$ and $b$ are [[objects]] of $D$, $f$ is an [[isomorphism]] in $D$ from $a$ to $b$, $\alpha$ is a specification of $a$ for the anafunctor $U$, $\beta$ is a specification of $b$ for the anafunctor $U$, $U_\alpha(a)$ and $U_\beta(b)$ are equal objects in $C$, and $U_{\alpha,\beta}(f)$ is the [[identity morphism]] on this object in $C$, then $a$ and $b$ are equal objects in $D$, and $f$ is the identity morphism on this object in $D$. =-- We might also demand that $\alpha = \beta$; this is automatic if $U$ is saturated. ## Properties * An amnestic [[full and faithful functor]] is automatically an [[isocofibration]], i.e. injective on objects: if $U D' = U D$, then there is some _isomorphism_ $f : D' \to D$ in $\mathcal{D}$ such that $U f = id_{U D}$, but then we must have $f = id_{U D'} = id_{U D}$, so $D' = D$. * An amnestic [[isofibration]] has the following lifting property: for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a _unique_ isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$. Indeed, if $\tilde{f}' : \tilde{C}' \to D$ were any other isomorphism such that $U \tilde{f}' = f$, then $U (\tilde{f}^{-1} \circ \tilde{f}') = id_C$, so we must have $\tilde{f} = \tilde{f}'$. Amnestic isofibrations are occasionally called discrete isofibrations, but this term may be misleading, because they are not [[isofibrations]] with discrete fibres. * If the [[composite]] $U \circ K$ is an amnestic functor, then $K$ is also amnestic. ## Examples * Most famous [[forgetful functors]] are amnestic, such as $Grp \to Set$, $Top \to Set$, $Top Grp \to Grp$, and $Top Grp \to Top$. Even $Met \to Set$ is amnestic, since the morphisms in [[Met]] are [[short maps]]. However, $Met_cont \to Set$, where the morphisms are [[continuous maps]], is not amnestic, as is shown by any set with two different but topologically equivalent metrics (such as $\mathbb{R}^2$ with the $l^1$ and $l^\infty$ metrics). * The forgetful functor from a groupoid of [[structured sets]] is amnestic. The examples above may all be defined by starting from such a groupoid and specifying which functions preserve the structure (and so are morphisms in the category of structured sets). So long as this produces no additional isomorphisms, the forgetful functor will be amnestic. But in the case of $Met_cont$, any non-isometric homeomorphism will be an isomorphism that was not in the original groupoid (consisting only of surjective isometries), and so this forgetful functor is not amnestic. * Any strictly [[monadic functor]] is amnestic. Conversely, any monadic functor that is also an amnestic isofibration is necessarily strictly monadic. ## Related concepts * [[conservative functor]] * [[concrete category]] * [[structure]] * [[stuff, structure, property]] ## References * J. Adámek, H. Herrlich and G.E. Strecker: [[The Joy of Cats]], Chapter I, Definition 3.27. * G. Preuß: Theory of Topological Structures: An Approach to Categorical Topology. D. Reidel Publishing Company. Mathematics and Its Applications, Dordrecht, Holland 1988. p. 178, footnote 31 [[!redirects amnestic functor]] [[!redirects amnestic functors]] [[!redirects amnestic anafunctor]] [[!redirects amnestic anafunctors]] [[!redirects amnesticization]] [[!redirects amnesticizations]] [[!redirects amnesticisation]] [[!redirects amnesticisations]] [[!redirects amnestic modification]] [[!redirects amnestic modifications]] [[!redirects amnestic isofibration]] [[!redirects amnestic isofibrations]] [[!redirects discrete isofibration]] [[!redirects discrete isofibrations]]
Amnon Neeman	https://ncatlab.org/nlab/source/Amnon+Neeman	# Amnon Neeman * [web site](http://wwwmaths.anu.edu.au/~neeman/) ## Selected writing On [[triangulated categories]]: * {#Neeman01} [[Amnon Neeman]], Triangulated Categories, Annals of Mathematics Studies 213, Princeton University Press (2001) [[ISBN:9780691086866](https://press.princeton.edu/titles/7102.html), [pdf](http://www.mi-ras.ru/~akuznet/homalg/Neeman%20Triangulated%20categories.pdf)] The [[theorem of the heart]]: * [[Amnon Neeman]], K-Theory for Triangulated Categories III(A): The Theorem of the Heart, Asian Journal of Mathematics 2 3 (1998) 495-589 [[arXiv:math/9901091](https://arxiv.org/abs/math/9901091), [doi:10.4310/AJM.1998.v2.n3.a4](https://doi.org/10.4310/AJM.1998.v2.n3.a4)] * [[Amnon Neeman]], K-theory for triangulated categories III(B): The theorem of the heart, Asian Journal of Mathematics 3 3 (1999) 557–608 [[doi:10.4310/AJM.1999.v3.n3.a2](https://doi.org/10.4310/AJM.1999.v3.n3.a2), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0003/AJM-1999-0003-0003-a002.pdf)] * [[Amnon Neeman]], K-Theory for Triangulated Categories $3\frac{3}{4}$: A Direct Proof of the Theorem of the Heart, K-Theory 22 1-2 (2001) 1-144 [[doi:10.1023/A:1011172502978](http://dx.doi.org/10.1023/A:1011172502978)] category: people
Amnon Yekutieli	https://ncatlab.org/nlab/source/Amnon+Yekutieli	Here are abstracts of a few papers of mine that connect to higher category theory, with links to the full texts. I will be glad to discuss this material. ### 1. Twisted Deformation Quantization of Algebraic Varieties ### Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties. [full paper - eprint](http://arxiv.org/abs/0905.0488), [survey](http://arxiv.org/abs/0801.3233) [lecture notes](http://www.math.bgu.ac.il/%7Eamyekut/lectures/twisted-defs/notes.pdf) ### 2. Nonabelian Multiplicative Integration on Surfaces ### We construct a 2-dimensional twisted nonabelian multiplicative integral. This is done in the context of a Lie crossed module (an object composed of two Lie groups interacting), and a pointed manifold. The integrand is a connection-curvature pair, that consists of a Lie algebra valued 1-form and a Lie algebra valued 2-form, satisfying a certain differential equation. The geometric cycle of the integration is a kite in the pointed manifold. A kite is made up of a 2-dimensional simplex in the manifold, together with a path connecting this simplex to the base point of the manifold. The multiplicative integral is an element of the second Lie group in the crossed module. We prove several properties of the multiplicative integral. Among them is the 2-dimensional nonabelian Stokes Theorem, which is a generalization of Schlesinger's Theorem. Our main result is the 3-dimensional nonabelian StokesTheorem. This is a totally new result. The methods we used are: the CBH Theorem for the nonabelian exponential map; piecewise smooth geometry of polyhedra; and some basic algebraic topology. The motivation for this work comes from twisted deformation quantization. In the paper (no. 1 above) we encountered a problem of gluing nonabelian gerbes, where the input was certain data in differential graded algebras. The 2-dimensional multiplicative integral gives rise, in that situation, to a nonabelian 2-cochain; and the 3-dimensional Stokes Theorem shows that this cochain is a twisted 2-cocycle. (This was superseded by a simpler approach; see no. 3 below.) [eprint](http://arxiv.org/abs/1007.1250) [lecture notes](http://www.math.bgu.ac.il/~amyekut/lectures/multi-integ/notes.pdf) [book](http://www.worldscientific.com/worldscibooks/10.1142/9537) ### 3. Combinatorial Descent Data for Gerbes ### We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data. This result is used to construct the twisted quantization in paper no. 1 above (replacing the earlier approach with surface integration). [eprint](http://arxiv.org/abs/1109.1919) [lecture notes](http://www.math.bgu.ac.il/~amyekut/lectures/higher-descent/notes.pdf) Contact: email to <amyekut@math.bgu.ac.il> * [webpage](http://www.math.bgu.ac.il/~amyekut/) [[!redirects Amnon Yekutieli]] [[!redirects amnon yekutieli]] [[!redirects A. Yekutieli]] [[!redirects Amnon+Yekutieli]] category: people
ample sheaf	https://ncatlab.org/nlab/source/ample+sheaf	## Definition ### General (...) ### In complex geometry In [[complex analytic geometry]] a [[line bundle]] is ample precisely if its [[Chern class]] is represented by a [[Kähler form]]. See also at _[[positive line bundle]]_. ## References * Wikipedia, _[Ample line bundle](http://en.wikipedia.org/wiki/Ample_line_bundle)_ [[!redirects ample sheaves]] [[!redirects ample line bundle]] [[!redirects ample line bundles]] [[!redirects ample vector bundle]] [[!redirects ample vector bundles]] [[!redirects ample module]] [[!redirects ample modules]]
amplimorphism	https://ncatlab.org/nlab/source/amplimorphism	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +-- {: .hide} [[!include higher algebra - contents]] =-- #### Operator algebra +-- {: .hide} [[!include AQFT and operator algebra contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The concept of _amplimorphism_ is a way to present [[bimodules]] in terms of linear maps. For $A$ an [[associative algebra]] (not necessarily unital) over some [[ring]] $R$ (possibly with extra structure, in applications often a [[C-algebra]]) then an _amplimorphism_ from $A$ to $A$ is an algebra homomorphism of the form $$ \alpha \;\colon\; A \longrightarrow A \otimes_R Mat_n(R) \simeq Mat_n(A) $$ for $n \in \mathbb{N}$ and $Mat_n(R)$ the ring of [[matrices]] with [[coefficients]] in $R$ under [[matrix multiplication]]. This map induces the $R$-[[bimodules]] $N_\alpha \subset A \otimes R^n$ on elements $N_\alpha = \{\psi \;\|\; \alpha(1)\psi = \psi\}$ with left $A$-action given by $\alpha$ and right $A$ action given by componentwise multiplication with $A$ from the right. (If $R = \mathbb{C}$ and $A$ is a [[C-algebra]] then this is canonically equipped with the structure of a [[Hilbert bimodule]]). ## References At least in the context of [[AQFT]] amplimorphisms were introduced * K. Szlachányi, K. Vecsernyes, _Quantum symmetry and braid group statistics in $G$-spin models_, Commun. Math. Phys. __156__:1 (1993), 127-168, [euclid](http://projecteuclid.org/euclid.cmp/1104253519) [doi](http://dx.doi.org/10.1007/BF02096735) The concept is recalled for instance in * Ezio Vasselli, page 6 of _The $C^\ast$-algebra of a vector bundle and fields of Cuntz algebras_, Journal of Functional Analysis 222(2) (2005), 491-502, [arXiv:math/0404166](http://arxiv.org/abs/math/0404166) * Fernando Lledó, Ezio Vasselli, section 3.3 of _Realization of minimal $C^\ast$-dynamical systems in terms of Cuntz-Pimsner algebras_ ([arXiv:math.OA/0702775](http://arxiv.org/abs/math.OA/0702775)) [[!redirects amplimorphisms]]
amplitude	https://ncatlab.org/nlab/source/amplitude	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Harmonic analysis +-- {: .hide} [[!include harmonic analysis - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The _amplitude_ of a [[wave]] is a measure for its maximal change with [[space]] and [[time]]. ## Definition For $n \in \mathbb{N}$ and $\mathbb{R}^n$ the corresponding [[Cartesian space]], a [[plane wave]] is a function $\mathbb{R}^n \to \mathbb{C}$ of the form $$ \vec x \mapsto A e^{-2 \pi i \vec k \cdot \vec x} \,. $$ Here $a \in \mathbb{C}$ is the _amplitude_ of the [[plane wave]] (while $\vec k$ s its [[wave vector]]). ## Related concepts * [[probability amplitude]] * [[absolute value]] [[!redirects amplitudes]]
amplituhedron	https://ncatlab.org/nlab/source/amplituhedron	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Physics +-- {: .hide} [[!include physicscontents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The [[scattering amplitudes]] in [[Yang-Mills theory]] are traditionally expressed as sums of typically very many [[Feynman diagrams]], which in turn are [[integrals]] over certain [[algebraic functions]]. At least in highly [[supersymmetry\|supersymmetric]] versions of [[Yang-Mills theory]] such as [[N=4 D=4 super Yang-Mills theory]] this can be expressed more efficiently (see [Dixon 13](#Dixon13) for a review of the general method and see at _[[string theory results applied elsewhere]]_) as fewer [[integrals]] of some integrand over some [[domain]] of certain convex [[polyhedra]] inside a [[positive Grassmanian]] (which is an infinite dimensional space related to the study of the phenomenon of [[total positivity]]). Therefore this has been called the _amplituhedron_ ([Arkani-Hamed & Trnka13](#ArkaniHameTrnka13)). In slightly more detail, the scattering amplitudes in [[N=4 D=4 super Yang-Mills theory]] can be given as functions of $4n$ real variables, the [[momenta]] of $n$ scattering [[particles]], and at loop number $\ell$ they are given by $4 \ell$-fold integrals of [[algebraic functions]] of $4n+4 \ell$ real variables. (When $\ell=0$, then ([ABCGPT 12, sections 16.4](#ABCGPT12)) claims that all the [[Galois conjugates]] of the [[algebraic function]] occur symmetrically; so that one indeed has [[rational functions]].) This formulation is typically a drastic improvement of computational complexity for fixed $k$ ([[helicity]]) and fixed $\ell$ (loop number) but variable $n$. The [[Feynman diagram]] description is exponential in $n$, while the complexity of the amplituhedron description grows much more mildly. In particular for $k = 0$ and $k = 1$ the resulting amplitude is just 0 and the amplituhedron description gives this immediately, but for $k = 1$ there are still many nontrivial Feynman integrals. That the sum of all these indeed vanish was maybe known earlier, though, due to a result by Parke and Taylor. For $k = 2$ again there is the "[[Parke-Taylor formula]]" efficiently expressing the amplitudes [[MHV amplitudes]]. See also at [[motives in physics]] ## References Review: * {#Dixon13} [[Lance Dixon]], _Calculating Amplitudes_, December 2013 ([web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/)) * [[Henriette Elvang]], [[Yu-tin Huang]], Scattering Amplitudes, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)] * Livia Ferro, Tomasz Lukowski, _Amplituhedra, and Beyond_, Topical Review invited by Journal of Physics A: Mathematical and Theoretical ([arXiv:2007.04342 ](https://arxiv.org/abs/2007.04342)) For scattering amplitudes via the "amplituhedron" the integrand is discussed in * {#ABCGPT12} [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, [[Alexander Goncharov\|Alexander B. Goncharov]], [[Alexander Postnikov]], Jaroslav Trnka, _Scattering amplitudes and the positive Grassmannian_, [arxiv/1212.5605](http://arxiv.org/abs/1212.5605) and the integration domain in * {#ArkaniHameTrnka13} [[Nima Arkani-Hamed]], Jaroslav Trnka, _The Amplituhedron_, [arxiv/1312.2007](http://arxiv.org/abs/1312.2007) Simple aspects of four particle scattering are treated in * [[Nima Arkani-Hamed]], Jaroslav Trnka, _Into the Amplituhedron_, [arxiv/1312.7878](http://arxiv.org/abs/1312.7878) {#ArkaniHameTrnka13b}. Earlier lecture notes and announcements include * Jaroslav Trnka, _The amplituhedron_, [pdf](http://www.staff.science.uu.nl/~tonge105/igst13/Trnka.pdf) * Nima Arkani-Hamed, _Grassmannian polytopes and scattering amplitudes_, lecture at Perimeter Institute, [video](http://pirsa.org/11080047) Informed online discussion includes * Logan Maingi on MathOverflow [here](http://mathoverflow.net/questions/142841/the-amplituhedron-minus-the-physics/143421#143421) * David Speyer [here](http://www.scottaaronson.com/blog/?p=1537#comment-88216) Journalistic coverage includes * Natalie Wolchover, _A Jewel at the Heart of Quantum Physics_, Quanta Magazine, Sep 17, 2013 (hosted at Simons Foundation) [html](https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics) See also * Wikipedia: _[Amplituhedron](http://en.wikipedia.org/wiki/Amplituhedron)_ * Thomas Lam, _Amplituhedron cells and Stanley symmetric functions_, [arxiv/1408.5531](http://arxiv.org/abs/1408.5531)
Amélia Liao	https://ncatlab.org/nlab/source/Am%C3%A9lia+Liao	* [Website](https://amelia.how/) ## Talks On [[univalent categories]] in [[homotopy type theory]]: * [[Amélia Liao]], Univalent Category Theory, [[Homotopy Type Theory Electronic Seminar Talks]], 6 October 2022, ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Liao-2022-10-06-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=asY6dXkR2yg)) On [[displayed categories]]: * [[Amélia Liao]], [Displayed Categories as Building Blocks](https://www.fmf.uni-lj.si/en/news/event/698/amelia-liao-displayed-categories-as-building-blocks/), [Seminar for foundations of mathematics and theoretical computer science](https://www.fmf.uni-lj.si/en/research/seminar-for-foundations-of-mathematics/), 22 September 2022, [Faculty of Mathematics and Physics](https://www.fmf.uni-lj.si/en/), University of Ljubljana ## Related entries * [[1lab]] category: people [[!redirects Amelia Liao]]
An Descheemaeker	https://ncatlab.org/nlab/source/An+Descheemaeker	* [Institute page I](https://www.kuleuven.be/wieiswie/nl/person/00011206) * [Institute page II (publications)](https://kuleuven.academia.edu/AnDescheemaeker) ## Selected writing On relative [[completions of groups]]: * [[Carles Casacuberta]], [[An Descheemaeker]], Relative group completions, Journal of Algebra, 285 2 (2005) 451-469 [[doi:10.1016/j.jalgebra.2004.10.023](https://doi.org/10.1016/j.jalgebra.2004.10.023)] category: people
an elementary treatment of Hilbert spaces	https://ncatlab.org/nlab/source/an+elementary+treatment+of+Hilbert+spaces	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Functional analysis +--{: .hide} [[!include functional analysis - contents]] =-- =-- =-- #Contents# * automatic table of contents goes here {:toc} ## Purpose of this page The purpose of this page is to examine how much of the theory of [[Hilbert space]]s can be done in an "elementary" fashion. More specifically, to develop as much as possible of the standard theory without using the fact that Hilbert spaces are special normed [[vector space]]s, or special [[metric space]]s. The reason for trying this is in the spirit of [[centipede mathematics]]. One can view the theory of [[locally convex space\|locally convex topological vector spaces]] as the result of pulling off most of the legs of Hilbert spaces, though perhaps one should go one step further and say that Hilbert spaces themselves are the result of pulling the "finite dimensional" leg off the Euclidean centipede. Indeed, this is almost the standard treatment of LCTVSs except that the usual starting point is [[normed vector space\|normed vector spaces]] rather than Hilbert spaces. In that view, Hilbert spaces are special normed vector spaces rather than normed vector spaces being Hilbert spaces without a leg or two. Although the intention is that the treatment be elementary, we shall remark on the relationship to the standard theory and thus the commentary will not necessarily be elementary. For example, concepts that are traditionally defined by using the metric space structure of a Hilbert space will need recasting and we shall need to reassure the reader that the new definition is equivalent to the original. We shall work over $\mathbb{C}$ throughout. ## Basic Definitions We start with the basic definition of an inner product space. +-- {: .num_defn #ipsp} ###### Definition An inner product space is a vector space, say $V$, equipped with a function $\|V\| \times \|V\| \to \mathbb{C}$, written $\langle u, v \rangle$, satisfying: 1. $\langle v, u\rangle = \overline{\langle u, v\rangle}$, 2. $\langle v + \lambda w, u \rangle = \langle v, u \rangle + \lambda \langle w, u\rangle$, 3. $\langle v, v\rangle \in [0,\infty)$ with $\langle v, v\rangle = 0$ if and only if $v = 0$. =-- Ideally, we want to deal solely with Hilbert spaces but first we need to figure out how to deal with completeness without recourse to metric space theory. We do this by using orthonormal families. +-- {: .num_defn #ofam} ###### Definition Let $(V,\langle -, - \rangle)$ be an inner product space. An orthogonal family in $V$ is a subset $B \subseteq V$ with the property that $\langle b, b'\rangle = 0$ whenever $b,b' \in B$ are distinct. The family is said to be orthonormal if, in addition, $\langle b, b\rangle = 1$ for all $b \in B$. Two vectors, say $u$ and $v$, are said to be orthogonal if the family $\{u,v\}$ is an orthogonal family. =-- Using orthogonal families, we can express the notion of completeness as follows. +-- {: .num_defn #hilb} ###### Definition A Hilbert space is an inner product space, $(H, \langle -, -\rangle)$ in which the following property holds. Let $(b_n)$ be an orthonormal sequence and $(\lambda_n)$ a sequence of positive real numbers such that $\sum \lambda_n^2$ is bounded. Then there is some $v \in H$ such that for all $u \in H$, $$ \sum \lambda_n \langle b_n, u \rangle = \langle v, u \rangle. $$ =-- +-- {: .un_remark} ###### Remark We need to justify this notion of completeness. One direction is simple: the sequence of partial sums of the series $\sum \lambda_n b_n$ is Cauchy and so if the space is complete, it has a limit and this limit satisfies the criterion. The other direction takes a little more effort. The quickest (but dirtiest) route is simply to observe that if that condition is satisfied then whenever we have an isometry from $\ell^0$ (with its standard inner product) into our space then it extends to $\ell^2$. A slightly more concrete route is as follows. Start with a Cauchy sequence in $H$, say $(x_n)$, and then apply the [[Gram–Schmidt process]]. This results in an orthonormal sequence, say $(b_n)$. For each $k$, we define $\lambda_k$ as the limit (in $\mathbb{C}$) of $(\langle x_n, b_k\rangle)$. We can make $\lambda_k$ a positive real number by taking the phase factor in to $b_k$. Let $(s_n)$ be the sequence of partial sums of the series $\sum \lambda_k b_k$. Then $(s_n)$ is Cauchy and the interpolation of $(s_n)$ and $(x_n)$ is also Cauchy. By assumption, $(s_n)$ has a weak limit. As it is Cauchy, the existence of a weak limit is enough to show that it has a strong limit. Thus $(x_n)$ also converges and so $H$ is complete. The tenor of the new definition and its equivalence to the standard one is a theme that will run throughout this page. In our elementary treatment, weak definitions are to be preferred to the standard strong ones. Their equivalence exposes some of the deep results of Hilbert space theory. =-- As this is intended as an elementary treatment, it is likely that at some point we will want to assume that our Hilbert space is "small", by which (of course) we mean "[[separable space\|separable]]". Fortunately, it is not hard to formulate separability without recourse to metric spaces. +-- {: .num_defn #sep} ###### Definition An inner product space is separable if it contains a sequence, say $(x_n)$ with the property that $\langle x, x_n\rangle = 0$ for all $n$ implies that $x = 0$. =-- ## Cauchy--Schwarz The Cauchy--Schwarz inequality is one of the basic results of Hilbert space theory. As we wish to avoid any mention of a norm, we state it as follows. +-- {: .num_proposition #csineq} ###### Proposition Let $H$ be a Hilbert space, $u,v \in H$. Then $$ \|\langle u, v \rangle\|^2 \le \langle u, u \rangle \langle v, v \rangle $$ with equality if and only if $u$ and $v$ are collinear. =-- +-- {: .query} I removed the square roots here, since there doesn\'t seem to be much point to them if you\'re not going to connect with a metric. (On the other hand, surely even an elementary treatment of Hilbert spaces may deign to mention the geometric concept of distance? It\'s one thing to not assume a knowledge of metric spaces, but it\'s another thing to refuse to even mention norms.) ---Toby As you can (probably) tell, I'm making this up as I go along! I've not decided yet whether to allow distances or not, so at the moment I'm avoiding them if possible. Part of my goal is to avoid overwhelming the "reader" with notation just for the sake of it. So I may well implicitly use the norm as $\langle v, v\rangle$ but without introducing the $\\|v\\|$ notation. But although I expect I'll be the main contributor here, I don't particularly want to be so I'm more than happy for others to weigh in with their opinions on what an "elementary treatment" would look like. If I disagree then it'll force me to think carefully _why_ I disagree and that can only improve things. Pondering a little more, I think that I'm avoiding distance not because I don't think it belongs in an elementary treatment - as you imply, what could be more simple to understand than distance? - but because it's a way of keeping metric spaces at bay: once I start talking about distances then it'll be easy to talk about metrics and the like. I think you're right about the square roots, by the way. ---Andrew Since accidentally using metric space theory is always a danger, I understand not wanting to mention distances and norms right away. On the other hand, I do think that any introductory treatment ought to mention them at some point, and even to prove that they satisfy the triangle inequality and Cauchy completeness. But those should be theorems specifically about the concept of distance in a Hilbert space, not anything fundamental to the development of the general theory of Hilbert spaces. So sure, keep them out for now. ---Toby =-- +-- {: .proof} ###### Proof The "if and only if" part gives us the key to the most direct proof. It will simplify matters a little if we assume that $u$ and $v$ are non-zero, the case where one of them is zero being easy to establish. Now if $u$ and $v$ are collinear then there is some $\lambda \in \mathbb{C}$ such that $u = \lambda v$. There is only one possibility for $\lambda$ which can be found by taking inner products with $v$: if $\lambda$ exists such that $u = \lambda v$ then $\lambda = \langle u, v\rangle/\langle v, v\rangle$. So we consider the question: is $\langle v, v\rangle u = \langle u, v\rangle v$? Or, equivalently, is $\langle v, v\rangle u - \langle u, v\rangle v = 0$? We have a test for when a vector is non-zero using the inner product. One of the axioms for an inner product says that a vector $w$ is zero if and only if $\langle w, w\rangle = 0$. Moreover, we also know that in any case $\langle w, w\rangle \ge 0$. Thus we have \[ \label{csineq} \left\langle \langle v, v\rangle u - \langle u, v\rangle v, \langle v, v\rangle u - \langle u, v\rangle v \right\rangle \ge 0 \] with equality if and only if $u$ and $v$ are collinear. Before expanding out the left hand side of this, we note that $$ \left\langle \langle v, v\rangle u - \langle u, v\rangle v , v \right\rangle = 0 $$ whence \eqref{csineq} simplifies to $$ \left\langle \langle v, v\rangle u - \langle u, v\rangle v, u \right\rangle \ge 0 $$ with equality if and only if $u$ and $v$ are collinear. Expanding this out yields $$ \langle v,v\rangle \langle u,u\rangle - \langle u, v \rangle \langle v, u\rangle \ge 0 $$ with equality if and only if $u$ and $v$ are collinear. Rearranging and square-rooting produces the traditional statement of the Cauchy--Schwarz inequality. =-- ## Subspaces and Complements
An Essay on the Foundations of Geometry	https://ncatlab.org/nlab/source/An+Essay+on+the+Foundations+of+Geometry	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Philosophy +-- {: .hide} [[!include philosophy - contents]] =-- =-- =-- * [[Bertrand Russell]], _An Essay on the Foundations of Geometry_, Cambridge 1897, [online version](http://bertrandrussellsocietylibrary.org/br-efg/br-efg.html) In his early philosophical work Russell concerned himself with German [[idealism]], which was at that time propounded in Britain by [[J.M. Ellis McTaggart]], while he studied as well the mathematical innovations of the 19th century. The latter questioned the [[Kant\|Kantian]] view of [[space]] -- especially in [[geometry]] where a developing plethora of new concepts of space, e.g. [[metric space]] or [[Riemannian manifold]], were hardly compatible with the Kantian Transcendental Aesthetic in which there exists a perception ["Anschauung"] of a space such that every space occurs only as a subspace of this space ([[Critique of Pure Reason]] [A24](https://de.wikisource.org/wiki/Seite:Kant_Critik_der_reinen_Vernunft_024.png)/B39): > ... when we talk of divers spaces, we mean only parts of one and the same space. In the essay under consideration Russell proposes an alternative version of Kant's argument that continues the idea of a space given _a priori_, but only as a "form of externality" and not a concrete perception of a space. About this space three properties can be deduced _a priori_ (constant [[curvature]], [[finite number\|finite]] [[dimension\|dimensionality]], and existence of straight [[lines]]). At the end of his essay Russell recapitulates: > ... > In the second chapter, we endeavoured, by a criticism of some geometrical philosophies, to prepare the ground for a constructive theory of Geometry. We saw that Kant, in applying the argument of the Transcendental Aesthetic to space, had gone too far, since its logical scope extended only to some form of externality in general. We saw that Riemann, Helmholtz and Erdmann, misled by the quantitative bias, overlooked the qualitative substratum required by all judgments of quantity, and thus mistook the direction in which the necessary axioms of Geometry are to found. We also rejected Helmholtz's view that Geometry depends on Physics, because we found that Physics must assume a knowledge of Geometry before it can become possible. ... > Proceeding, in the third chapter, to a constructive theory of Geometry, we saw that projective Geometry, which has no reference to quantity, is necessarily true for any form of externality. Its three axioms---homogeneity, dimension, and the straight line---were all deduced from the conception of a form of externality, and, since some such form is necessary to experience, were all declared _a priori_. In metrical Geometry, on the contrary, we found an empirical element arising out of the alternatives of Euclidean and non-Euclidean space. ... > In the present chapter, we completed our proof of the apriority of the projective and equivalent metrical axioms by showing the necessity, for experience, of some form of externality, given by sensation or intuition, and not merely inferred from other data. Without this, we said, a knowledge of diverse but interrelated things, the corner-stone of all experience, would be impossible. ... ## Related entries * [[space form]] \| [[synthetic geometry]] \| \|-------------------------\| \| [[Euclidean geometry]] \| \| [[hyperbolic geometry]] \| \| [[elliptic geometry]] \| ## References * [[Kant]], _[[Critique of Pure Reason]]_, second edition (referenced to as "B") 1787 category: reference
An Exercise in Kantization	https://ncatlab.org/nlab/source/An+Exercise+in+Kantization	## Motivation Once, [[Johan Alm]] wrote some very neat notes: * [Quantization as a Kan Extension](http://ncatlab.org/nlab/files/Kantization09May27.pdf) A discussion is available at the [$n$Café](https://golem.ph.utexas.edu/category/2009/05/alm_on_quantization_as_a_kan_e.html). The paper proves an aspect of the "[$n$-Café Quantum Conjecture](http://golem.ph.utexas.edu/category/2007/06/the_ncaf_quantum_conjecture.html)", suggested by [[Urs Schreiber]]. There is some old discussion at previous versions of the page [[Quantization as a Kan Extension]], and some on-page discussion that is now stored [at the nForum](https://nforum.ncatlab.org/discussion/12529/an-exercise-in-kantization/?Focus=90941#Comment_90941). ## Further work Discussion of quantization in terms of Kan extension and [[pushforward in generalized cohomology]] is now at _[[motivic quantization]]_, with notes being at _[[schreiber:master thesis Nuiten]]_ and _[[schreiber:Quantization via Linear homotopy types]]_ . category: reference
An Huang	https://ncatlab.org/nlab/source/An+Huang	* [webpage](http://people.brandeis.edu/~anhuang/) * [institute page](http://www.brandeis.edu/facultyguide/person.html?emplid=e225bcb771bc4cefeed959ca4b802ac0055303e5) ## selected writings * {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013)) (on [[p-adic string theory]] and the [[Riemann zeta function]]) ## related $n$Lab entries * [[p-adic string theory]] * [[Riemann zeta function]] category: people
An Introduction to Homological Algebra	https://ncatlab.org/nlab/source/An+Introduction+to+Homological+Algebra	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Homological algebra +--{: .hide} [[!include homological algebra - contents]] =-- =-- =-- This entry provides a hyperlinked index for the textbook * [[Charles Weibel]], An Introduction to Homological Algebra Cambridge University Press (1994) [doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136) [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf) which gives a first exposition to central concepts in _[[homological algebra]]_. For a more comprehensive account of the theory see also chapters 8 and 12-18 of * [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006) and see the $n$Lab lecture notes * _[[schreiber:Introduction to Homological Algebra]]_ ([[IntroductionToHomologicalAlgebra-170509.pdf:file]]) #Contents# * table of contents {:toc} ## 1 Chain complexes ### 1.1 Complexes of $R$-modules * [[abelian group]], [[commutative ring]], [[module]] * [[exact sequence]] Definition 1.1.1 [[chain complex]] * [[chain map]], [[chain homotopy]] * [[category of chain complexes]] Exercise 1.1.2 [homology is functorial](chain+map#OnHomology) Exercise 1.1.3 [exact sequences of chain complexes are split](split+exact+sequence#OfVectorSpaces) Exercise 1.1.4 [[internal hom of chain complexes]] Definition 1.1.2 [[quasi-isomorphism]] [[cochain complex]], [[bounded chain complex]] Exercise 1.1.5 [exactness and weak nullity](exact+sequence#ExactnessAndQuasiIsomorphisms) Application 1.1.3 [[chain on a simplicial set]], [[simplicial homology]] Exercise 1.17 [simplicial homology of the tetrahedron](simplicial+homology#OfTetrahedon) Application 1.1.4 [[singular homology]] ### 1.2 Operations on chain complexes * [[additive and abelian categories]] * [[Ab-enriched category]] * [[pre-additive category]] * [[additive category]], * [[additive functor]] Exercise 1.2.1 [homology respects direct product](chain+homology+and+cohomology#ChainHomologyRespectsDirectProduct) Definition 1.2.1 [[kernel]], [[cokernel]] * [[quotient]] Exercise 1.2.2 [in an abelian category kernels/cokernels are the monos/epis](Mod#RModIsAbelian) Exercise 1.2.3 [(co)kernels of chain maps are degreewise (co)kernels](category+of+chain+complexes#KernelsOfChainComplexes) Definition 1.2.2 [[abelian category]], [[abelian subcategory]] Theorem 1.2.3 [a category of chain complexes is itself abelian](category+of+chain+complexes#IsAbelian) Exercise 1.2.4 [exact sequence of chain complexes is degreewise exact](category+of+chain+complexes#ShortExactSequencesDegreewise) $R$[[Mod]] Example 1.2.4 [[double complex]] Sing trick 1.2.5 [double complex with commuting/anti-commuting differentials](double+complex#EquivalenceOfTheTwoDefinitions) Total complex 1.2.6 [[total complex]] Exercise 1.2.5 [total complex of a bounded degreewise exact double complex is itself exact](total+complex#TotOfBoundedDegreewiseExactIsExact) Example 1.2.4 [[double complex]] Truncations 1.2.7 [[truncation of a chain complex]] Translation 1.2.8 [[suspension of a chain complex]] Exercise 1.2.8 [[mapping cone]] ### 1.3 Long exact sequences Theorem 1.3.1 [[connecting homomorphism]], [[long exact sequences in homology]] Exercise 1.3.1 [[3x3 lemma]], Snake lemma 1.3.2 [[snake lemma]] Exercise 1.3.3 [[5 lemma]] Remark 1.3.5 [[exact triangle]] ### 1.4 Chain homotopies * [[homotopy theory]] Definition 1.4.1 [[split exact sequence]] Exercise 1.4.1 [splitness of exact sequences of free modules](split+exact+sequence#OfVectorSpaces) Definition 1.4.3 [[null homotopy]] Exercise 1.4.3 [split exact means identity is null homotopic](split+exact+sequence#RelationToChainHomotopy) Definition 1.4.4 [[chain homotopy]] Lemma 1.4.5 [chain homotopy respects homology](chain%20map#OnHomology) Exercise 1.4.5 [[homotopy category of chain complexes]] ### 1.5 Mapping cones and cyclinders 1.5.1 [[mapping cone]] 1.5.5 [[mapping cylinder]] 1.5.8 [[fiber sequence]] ### 1.6 More on abelian categories Theorem 1.6.1 [[Freyd-Mitchell embedding theorem]] Functor categories 1.6.4 [[functor category]] [[presheaf]] Definition 1.6.5 [[abelian sheaf]] Definition 1.6.6 left/right [[exact functor]] Yoneda embedding 1.6.10 [[Yoneda embedding]] Yoneda lemma 1.6.11 [[Yoneda lemma]] [proof of the Freyd-Mitchell embedding theorem](http://ncatlab.org/nlab/show/Freyd-Mitchell+embedding+theorem#Proof) ## 2 Derived functors [[derived functor in homological algebra]] ### 2.1 $\delta$-Functor Definition 2.1.1 [[delta-functor]] ### 2.2 Projective resolutions [[projective module]] ([[cofibrant object]] in the [[model structure on chain complexes]]) Definition 2.2.4 [[projective resolution]] ([[cofibrant replacement]]) Horseshoe lemma 2.2.8 [[horseshoe lemma]] ### 2.3 Injective resolutions [[injective module]] ([[fibrant object]] in the other [[model structure on chain complexes]]) Baer's criterion 2.3.1 [[Baer's criterion]] Definition 2.3.5 [[injective resolution]] ([[fibrant replacement]]) Definition 2.3.9 [[adjoint functor]] ### 2.4 Left derived functors [[left derived functor]] ### 2.5 Right derived functors [[right derived functor]] Application 2.5.4 [[global section functor]], [[abelian sheaf cohomology]] ### 2.6 Adjoint functors and left/right exactness [[adjoint functor]] Definition 2.6.4 [[Tor]] Application 2.6.5 [[sheafification]] Application 2.6.6 [[direct image]], [[inverse image]] Application 2.6.7 [[colimit]] Variation 2.6.9 [[limit]] Definition 2.6.13 [[filtered category]], [[filtered colimit]] ### 2.7 Balancing $Tor$ and $Ext$ Tensor product of complexes 2.7.1 [[tensor product of chain complexes]] Lemma 2.7.3 [[acyclic assembly lemma]] ## 3 Tor and Ext [[Tor]] and [[Ext]] ### 3.1 $Tor$ for abelian groups Proposition 3.1.2-3.1.3 [relation to torsion subgroups](Tor#RelationToTorsionGroups) ### 3.2 $Tor$ and flatness Definition 3.2.1 [[flat module]] Definition 3.2.3 [[Pontrjagin duality]] Flat resolution lemma 3.2.8 [[flat resolution lemma]] Corollary 3.2.13 [Localization for Tor](Tor#Localization) ### 3.3 $Ext$ for nice rings Corollary 3.3.11 [Localization for Ext](Ext#Localization) ### 3.4 $Ext$ and extensions [[extension]] [[group extension]] Vista 3.4.6 [[Yoneda extension group]] ### 3.5 Derived functors of the inverse limit [[tower]] [[additive and abelian categories\|(AB4)-category]] [[directed limit]] Definition 3.5.1 [[lim^1]] Definition 3.5.6 [[Mittag-Leffler condition]] Exercise 3.5.5 [[pullback]] ### 3.6 Universal coefficient theorem Theorem 2.6.1 [[Künneth formula]] Universal cofficient theorem for homology 3.6.2 [universal coefficient theorem in homology](universal+coefficient+theorem#InHomology) Theorem 3.6.3 [[Künneth formula for complexes]] Application 3.6.4 [universal coefficient theorem in topology](universal+coefficient+theorem#InTopology) Universal coefficient theorem in cohomology 3.6.5 [universal coefficient theorem in cohomology](universal+coefficient+theorem#InCohomology) [[Eilenberg-Zilber theorem]] Exercise 3.6.2 [[hereditary ring]] ## 4 Homological dimension ### 4.1 Dimensions * [[dimension]] * [[global dimension theorem]] * [[homological dimension]] ### 4.2 Rings of Small Dimension ### 4.3 Change of Rings Theorem ### 4.4 Local rings * [[local ring]] ### 4.5 Koszul Complexes * [[Koszul complex]] ### 4.6 Local Cohomology ## 5 Spectral sequences ### 5.1 Introduction ### 5.2 Terminology * [[spectral sequence]] ### 5.3 Leray-Serre Spectral Sequence * [[Leray-Serre spectral sequence]] ### 5.4 Spectral sequence of a filtration * [[spectral sequence of a filtration]] ### 5.5 Convergence ### 5.6 Spectral sequence of a double complex * [[spectral sequence of a double complex]] ### 5.7 Hypercohomology * [[hypercohomology]] ### 5.8 Grothendieck spectral sequence * [[Grothendieck spectral sequence]] ### 5.9 Exact couples * [[exact couple]] * [[Bockstein spectral sequence]] ## 6 Group homology and cohomology * [[group cohomology]] ## 7 Lie algebra homology and cohomology * [[Lie algebra homology]], [[Lie algebra cohomology]] ## 8 Simplicial methods in homological algebra ### 8.1 Simplicial object * [[simplicial object]] ### 8.2 Operations on simplicial objects ### 8.3 Simplicial homotopy groups * [[simplicial homotopy group]] ### 8.4 The Dold-Kan correspondence * [[Dold-Kan correspondence]] ### 8.5 The Eilenberg-Zilber theorem * [[Eilenberg-Zilber theorem]] ### 8.6 Canonical resolutions * [[monad]](=triple) [[comonad]] [[bar construction]], [[classifying space]] ### 8.7 Cotriple homology * [[bar construction]] * [[monadic descent]] ### 8.8 Andre-Quillen Homology and Cohomology * [[Kähler differential]] * [[cotangent complex]] ## 9 Hochschild and cyclic homology ### 9.1 Hochschild Homology and Cohomology of Algebras * [[Hochschild cohomology]] ### 9.2 Derivations, Differentials and Separable Algebra * [[derivation]] * [[differential]] * [[separable algebra]] ### 9.3 $H^2$, Extensions, Smooth Algebras * [[smooth algebra]] ### 9.4 Hochschild products * [[Hochschild-Kostant-Rosenberg theorem]] ### 9.5 Morita Invariance ### 9.6 Cyclic Homology * [[cyclic cohomology]] ### 9.7 Group Rings ### 9.8 Mixed Complexes ### 9.9 Graded Algebras ### 9.10 Lie Algebras of Matrices ## 10 The derived category * [[derived category]] ## A Category Theory Language * [[category theory]] ### A.1 Categories * [[category]] ### A.2 Functors * [[functor]] ### A.3 Natural transformations * [[natural transformation]] ### A.4 Abelian categories * [[abelian category]] ### A.5 Limits and colimits * [[limit]], [[colimit]] ### A.6 Adjoint functors * [[adjoint functor]] category: reference [[!redirects An introduction to homological algebra]]
Ana Canas da Silva	https://ncatlab.org/nlab/source/Ana+Canas+da+Silva	* [webpage](https://people.math.ethz.ch/~acannas/) ## related $n$Lab entries * [[geometric quantization]] * [[geometric quantization by push-forward]] category: people
Ana Cannas da Silva	https://ncatlab.org/nlab/source/Ana+Cannas+da+Silva	* [webpage](http://www.math.ethz.ch/~acannas/) * [Wikipedia entry](https://en.wikipedia.org/wiki/Ana_Cannas_da_Silva) ## Selected writings On [[geometric quantization]]: * [[Ana Cannas da Silva]], [[Yael Karshon]], [[Susan Tolman]], Quantization of Presymplectic Manifolds and Circle Actions, Transactions of the AMS 352 2 (2000) 525-552 [[arXiv:dg-ga/9705008](http://arxiv.org/abs/dg-ga/9705008), [jstor:118052](https://www.jstor.org/stable/118052)] On [[symplectic geometry]]: * [[Ana Cannas da Silva]], Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7)] ## Related entries * [[Kähler manifold]] * [[tubular neighbourhood]] * [[tubular neighbourhood theorem]] * [[Guillemin-Sternberg geometric quantization conjecture]] category: people [[!redirects Ana Cannas]]
Ana-Maria Raclariu	https://ncatlab.org/nlab/source/Ana-Maria+Raclariu	* [website](https://perimeterinstitute.ca/people/ana-maria-raclariu) ## Selected publications On [[celestial holography]]: * [[Nima Arkani-Hamed]], Monica Pate, [[Ana-Maria Raclariu]], [[Andrew Strominger]], Celestial Amplitudes from UV to IR, JHEP 08 (2021) 062 ([arXiv:2012.04208](https://arxiv.org/abs/2012.04208)) * [[Ana-Maria Raclariu]], Lectures on Celestial Holography ([arXiv:2107.02075](https://arxiv.org/abs/2107.02075)) category:people
anabelian geometry	https://ncatlab.org/nlab/source/anabelian+geometry	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In _anabelian geometry_ one studies how much information about a [[space]] $X$ (specifically: an [[algebraic variety]]) is contained already in its first [[étale homotopy group]] $\pi^{et}_1(X,x)$ (specifically: the [[algebraic fundamental group]]). The term "anabelian" is supposed to be alluding to the fact that "the _less_ [[abelian group\|abelian]] $\pi^{et}_1(X,x)$ is, the more information it carries about $X$." Precisely: an [[anabelian group]] is a non-[[trivial group]] for which every [[finite index subgroup]] has trivial [[center]]. Accordingly, an algebraic variety whose [[isomorphism class]] is entirely determined by $\pi^{et}_1(X,x)$ is called an anabelian variety. An early conjecture motivating the theory (in [Grothendieck 84](#Grothendieck)) was that all [[hyperbolic curves]] over [[number fields]] are anabelian varieties. This was eventually proven by various authors in various cases. In ([Uchida](#Uchida)) and ([Neukirch](#Neukirch)) it was shown that an isomorphism between [[Galois groups]] of [[number fields]] implies the existence of an isomorphism between those number fields. For algebraic curves over [[finite fields]], over [[number fields]] and over [[p-adic field]] the statement was eventually completed by ([Mochizuki 96](#Mochizuki96)). Grothendieck also conjectured the existence of higher-dimensional anabelian varieties, but these are still very mysterious. ## Étale homotopy types * [[Alexander Schmidt]], [[Jakob Stix]], _Anabelian geometry with étale homotopy types_, ([pdf](https://www.mathi.uni-heidelberg.de/~schmidt/papers/anab-hotype.pdf)). ##Related entries * [[algebraic fundamental group]] also called the 'geometric fundamental group' by Grothendieck. * [[child's drawing]]/ Dessins d'enfant. * [[inter-universal Teichmüller theory]] * [[profinite completion]] * [[Grothendieck-Teichmüller group]] * [[section conjecture]] * [[arithmetic geometry]] ## References The notion of anabelian geometry originates in * [[Alexandre Grothendieck]], letter to Faltings (June 27, 1983) ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf)), written in response to [[Faltings]]' work on the [[Mordell Conjecture]], and the note of the [[Long March]]: * [[Alexandre Grothendieck]], La Longue Marche, Notes ([link](https://agrothendieck.github.io/galoispoincaregrothendieck/galois.pdf)). There is some discussion of the area in * {#Grothendieck} [[Alexander Grothendieck]], _[[Esquisse d'un programme]]_ (1984). A relation with the theory of [[motive]]s is in * [[Alexander Schmidt]], _Motivic aspects of Anabelian geometry_, Advanced studies in pure mathematics 63, 2012. Galois-Teichmüller theory and Arithmetic geometry. pp 503 - 517 [pdf](https://www.mathi.uni-heidelberg.de/~schmidt/papers/63117.pdf) Surveys: * {#Tschinkel14} [[Yuri Tschinkel]], _Introduction to anabelian geometry_, talk at _[Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html)_, Oxford 2014 ([slides pdf](http://www.cims.nyu.edu/~tschinke/.talks/oxford14/oxford14.pdf)) * [[Leila Schneps]], page 60 (2) of _Grothendieck's "Long march through Galois theory"_ ([pdf](http://www.math.jussieu.fr/~leila/SchnepsLM.pdf)) A comprehensive introduction is in * [[Fedor Bogomolov]], [[Yuri Tschinkel]], Introduction to birational anabelian geometry [[pdf](http://www.math.nyu.edu/~tschinke/papers/yuri/10msri/msri7.pdf), [arXiv:1011.0883](https://arxiv.org/abs/1011.0883)] Examples are discussed in * Yasutaka Ihara, Hiroaki Nakamura, _Some illustrative examples for anabelian geometry in high dimensions_, in [[Leila Schneps]], P. Lochak (eds) _Geometric Galois Actions I_, London Math. Soc. Lect. Note Series 242 ([pdf](http://www.math.okayama-u.ac.jp/~h-naka/zoo/lion/INanabel.pdf)) The classification of anabelian varieties for [[number fields]] was shown in * J. Neukirch, _Kennzeichnung der $p$-adischen und der endlichen algebraischen Zahlkörper_, Invent. Math. 6 (1969), p. 296–314. {#Neukirch} * J. Neukirch, _Über die absoluten Galoisgruppen algebraischer Zahlkörper_, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris (1977) * {#Uchida} K. Uchida. _Isomorphisms of Galois groups_, J. Math. Soc. Japan 28 (1976), no. 4, 617–620. * K. Uchida, _Isomorphisms of Galois groups of algebraic function fields_, Ann. Math. (2) 106 (1977), no. 3, p. 589–598. and for [[algebraic curves]] in * [[Shinichi Mochizuki]], _The profinite Grothendieck conjecture for hyperbolic curves over number fields_, J. Math. Sci. Univ. Tokyo 3 (1996), 571–627. {#Mochizuki96} * [[Shinichi Mochizuki]], _The absolute anabelian geometry of canonical curves_ (2002) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Canonical%20Liftings.pdf)) See also * Frans Oort, [Lecture notes](http://www.staff.science.uu.nl/~oort0109/IC.AnabelianWorkshop.ps). Informal notes (not for publication) made available for the Lorentz Center workshop 'Anabelian number theory and geometry', December 3-5, 2001 * [[Nikolai Durov\|N. V. Durov]], _Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties)_, preprint POMI 13/2012 (in Russian) [abstract](http://www.pdmi.ras.ru/preprint/2012/12-13.html), [pdf.gz](ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2012/13-12_rus.pdf.gz) * [[Taylor Duypuy]], _[Anabelian geometry](https://www.youtube.com/playlist?list=PLJmfLfPx1OedXBno8vlpGd89_3wsdfCEc)_ * {#Morrow17} [[Jackson Morrow]], _Kummer classes and Anabelian geometry_, notes from Super [QVNTS](http://www.dms.umontreal.ca/%7Eqvnts/): [Kummer Classes and Anabelian Geometry](http://www.uvm.edu/%7Etdupuy/anabelian.html) 2017 ([pdf](https://www.dropbox.com/s/kiszgkrrsjrqu4p/VermontNotes_Final.pdf?dl=0)) [[!redirects anabelian variety]] [[!redirects anabelian varieties]]
anabelian group	https://ncatlab.org/nlab/source/anabelian+group	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Group Theory +-- {: .hide} [[!include group theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea An _anabelian group_ is a group that is "far from being an [[abelian group]]" in a precise sense: ## Definition It is a non-[[trivial group]] for which every [[finite index subgroup]] has trivial [[center]]. ## Related concepts * [[anabelian geometry]]. [[!redirects anabelian groups]]
anabelioid	https://ncatlab.org/nlab/source/anabelioid	\tableofcontents \section{Introduction} An anabelioid is a category intended to play the role of a 'generalised geometric object' in algebraic/arithmetic geometry. Its definition is simple: a finite product of [[Galois category\|Galois categories]], or in other words of [[classifying topos\|classifying topoi]] of [[profinite group\|profinite groups]]. The significance comes from the fact that in [[anabelian geometry]], an [[algebraic variety]] is essentially determined by its [[étale fundamental group\|algebraic fundamental group]], which arises from a Galois category associated to the algebraic variety. The idea, due to [[Shinichi Mochizuki]], is that one can develop the geometry of these Galois categories themselves, and products of Galois categories in general; thus, develop a form of categorical algebraic geometry. To quote from Remark 1.1.4.1 of [Mochizuki2004](#Mochizuki2004): > >_The introduction of anabelioids allows us to work with both "algebro-geometric anabelioids" (i.e., anabelioids arising from (anabelian) varieties) and "abstract anabelioids" (i.e., those which do not necessarily arise from an (anabelian) variety) as geometric objects on an equal footing. _ > >The reason that it is important to deal with "geometric objects" as opposed to groups, is that: > > > _We wish to study what happens as one varies the basepoint of one of these geometric objects._ \section{Details} The following definitions follow [Mochizuki2004](#Mochizuki2004). \begin{defn} A _connected anabelioid_ is exactly a [[Galois category]]. \end{defn} \begin{defn} An _anabelioid_ is a category equivalent to a [[finite product]] of connected anabelioids, that is, to a finite product of Galois categories. \end{defn} \begin{rmk} An anabelioid is also known as a _multi-Galois category_. \end{rmk} \section{Associated notions} * [[finite étale morphism of anabelioids]] [[!redirects multi-Galois category]] \section{References} * {#Mochizuki2004} _The geometry of anabelioids_, [[Shinichi Mochizuki]], 2004, Publ. Res. Inst. Math. Sci., 40, No. 3, 819-881. [paper](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Anabelioids.pdf) [Zentralblatt review](https://zbmath.org/?q=an%3A1113.14021)
anabicategory	https://ncatlab.org/nlab/source/anabicategory	An anabicategory is a particular notion of _weak [[2-category]]_ appropriate in the absence of the [[axiom of choice]] (including in many [[internal category\|internal]] contexts). It is derived from the notion of [[bicategory]] by replacing the composition functors $\circ: B(y,z) \times B(x,y) \to B(x,z)$ with [[anafunctor]]s (and therefore the [[associator]]s and [[unitor]]s with [[ananatural transformation\|ananatural transformations]]). +-- {.query} [[Zoran Škoda]]: I understand that there is a version of $Cat$ using anafunctors, but it is not clear to me what are the axioms for a variant of 2-category which this example belongs to. I do not understand what you mean by replacing hom-functor with anafunctor in that definition: I mean which definition of bicategory is phrased in terms of hom-functors; standard definition talks associators and so on...Please be more explicit. [[Mike Shulman\|Mike]]: Is this version clearer? _Toby_: Or look at the complete definition in the reference that I just added. =-- If [[Cat]] is defined as consisting of ([[small category\|small]]) [[category\|categories]], anafunctors, and [[ananatural transformation\|ananatural transformations]] (as is most appropriate in the absence of choice), then $Cat$ is more naturally an anabicategory rather than any stricter notion. +--{: .query} [[Mike Shulman\|Mike]]: Is that only because of the non-canonicity of pullbacks in $Set$? If our version of $Set$ has canonical chosen pullbacks, do we get an ordinary bicategory? _Toby_: H\'m ... as I recall, you get an ordinary bicategory using canonical pullbacks and general anafunctors, but if you move to saturated anafunctors, then you still only get an anabicategory. I should check this and then rewrite (here and at [[Cat]]) to say it correctly. (Of course, if you really use anafunctors, then you should only want an anabicategory.) =-- # Explicit definition # An anabicategory $\mathcal{C}$ consists of a set $\mathcal{C}_0$ of objects, a set $\mathcal{C}_1$ of 1-cells, which are relations $\mathcal{C}_0 \times \mathcal{C}_0$, and a set $\mathcal{C}_2$ of 2-cells, which are relations $\mathcal{C}_1 \times \mathcal{C}_1$. (Note that only $\mathcal{C}_2$ needs a defined equality relation, so the other two sets only need to be [[preset]]s.) On these sets are defined several relations; for each relation $R$ if we have $R(x, y)$ $y$ is said to be a _value_ of $R$ at $x$. Some pairs $x, y$ with $R(x, y)$ are also said to be _specified values_; generally if $R$ has some value at $x$, at least one such value will be specified. We omit the word "value" when it is clear from context. 1. There are _source_ and _target_ relations $\mathcal{C}_1 \times \mathcal{C}_0$ and $\mathcal{C}_2 \times \mathcal{C}_1$, and every 1- or 2- cell $f$ has some specified value for its source and target. 1. There are _2-source_ and _2-target_ relations $\mathcal{C}_2 \times \mathcal{C}_0$, and every 2- cell $f$ has some specified value for its 2-source and 2-target. 1. There are _identity_ relations $Id:\mathcal{C}_0 \times \mathcal{C}_1$ and $\mathcal{C}_1 \times \mathcal{C}_2$, and every object or 1-cell has some specified value for its identity. 1. There is a _composition_ relation $\circ:(\mathcal{C}_1 \times \mathcal{C}_1) \times \mathcal{C}_1$; whenever $f_1$ and $f_2$ are such that some specified target of $f_1$ is also a specified source of $f_2$, there is some specified value of the composite of $f_1$ and $f_2$. 1. There is a _1-composition_ relation $\circ_1:(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$; whenever $f_1$ and $f_2$ are such that some specified 2-target at $f_1$ is also a specified 2-source at $f_2$, there is some specified value of the 1-composite of $f_1$ and $f_2$. 1. There is a _2-composition_ relation $\circ_2:(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$; whenever $f_1$ and $f_2$ are such that some specified target at $f_1$ is also a specified source at $f_2$, there is some specified value of the 2-composite of $f_1$ and $f_2$. The relations satisfy the following conditions: 1. For any 2-cell $f$, and any values $x$ and $y$ of its source and target (respectively), the set of targets of $x$, of targets of $y$, and of 2-targets of $f$ are all equal (as subsets of $\mathcal{C}_1$), and likewise for sources/2-sources. If $x$ or $y$ is a specified value of either relation, each specified value of its source is a specified value of the 2-source, and likewise for targets/2-targets. 1. Any source of an identity $x$ is also a target of $x$, and vice versa. (However, we do not require the sets of specified values to match.) For any specified identity $y$ of $x$, $x$ should be a specified source and target of $y$. 1. If $h$ is composition of $f \circ g$, the set of targets of $f$ is equal to the set of targets of $h$, and the set of sources of $g$ is equal to the set of sources of $h$. If $h$ is a specified composition, any specified target of $f$ is a specified target of $h$, and any specified source of $g$ is a specified source of $h$. (In general the converses may fail.) Moreover, if the composition $f \circ g$ has any value, the set of targets of $g$ is equal to the set of sources of $f$. 1. If $h$ is a value of the 2-composition $f \circ_2 g$, the set of targets of $f$ is equal to the set of targets of $h$, and the set of sources of $g$ is equal to the set of sources of $h$. If $h$ is a specified value of the 2-composition, any specified target of $f$ is a specified target of $h$, and any specified source of $g$ is a specified source of $h$. Moreover, if the 2-composition $f \circ_2 g$ has any value, the set of targets of $g$ is equal to the set of sources of $f$. 1. If $h$ is a value of the 1-composition $f \circ_1 g$, each source of $h$ is a composition of some source of $f$ and some source of $g$, and each such composition is a source of $h$. For any specified sources of $f$ and $g$, each specified value of their composition is a specified source of $h$. The same results hold with source replaced by target. Moreover, if the 1-composition $f \circ_1 g$ has any value, the set of 2-targets of $g$ is equal to the set of 2-sources of $f$. There are also conditions defining equivalence relations on $\mathcal{C}_0$ and $\mathcal{C}_1$. (The equivalence relation for $\mathcal{C}_2$ is simply equality.) All the relations defined above are required to be invariant under the appropriate equivalence relations (note, however, that their specified values do not need to be invariant). So in particular, the identity relation $\mathcal{C}_1\times \mathcal{C}_2$ and the 1- and 2- composition relations $(\mathcal{C}_2 \times \mathcal{C}_2) \times \mathcal{C}_2$ are functional: any two values of the identity on some 1-cell $x$, or any two values of one of the compositions on some 2-cells $f$ and $g$, must be equal (as elements of $\mathcal{C}_2$.) * A 2-cell $f$ is said to be an _isomorphism_ (between any of its sources and any of its targets) if there are some 2-cell $g$ and 1-cell $x$ such that the 2-composition of $f$ and $g$ is the identity on $x$. We say that $g$ is an _inverse_ of $f$. * Two 1-cells $f$ and $g$ are _isomorphic_ if there is a 2-cell isomorphism $h$ with $f$ as a source and $g$ as a target (or equivalently, vice versa). This is the chosen equivalence relation for $\mathcal{C}_1$. * A 1-cell $f$ is said to be an _equivalence_ (between any of its sources and any of its targets) if there are some 1-cell $g$ and object $x$ such that some composition of $f$ and $g$ in either direction is isomorphic to an identity on $x$. * Two objects $x$ and $y$ are _equivalent_ if there is a 1-cell equivalence $g$ with $x$ as a source and $y$ as a target (or equivalently, vice versa). This is the chosen equivalence relation for $\mathcal{C}_0$. Finally there are coherence relations: 1. Let $f$ be an identity 2-cell (in other words, a value of the identity relation for some 1-cell), and $g$ be any 2-cell. Then, if there is a 2-composition $f \circ_2 g$ or $g \circ_2 f$, any such value is equal to $g$. It follows (by considering their 2-composition) that identities on isomorphic 1-cells are equal, and that when two 1-cells are isomorphic, this fact will be witnessed by the identity on either object. 1. Let $f$ and $g$ be identity 2-cells on, say, $x$ and $y$ respectively, and suppose that $f \circ_1 g$ exists. Then it is an identity 2-cell on any value of $x \circ y$. Equivalently: the 1-composition of isomorphisms is an isomorphism. 1. Let $f$ be an identity 1-cell, and let $g$ be any 1-cell. Then, if there is a composition $f \circ g$ or $g \circ f$, any such value is isomorphic to $g$. It follows (by considering their composition) that identities on equivalent objects are isomorphic, and that when two objects are equivalent, this fact will be witnessed by the identity on either object. 1. Let $f$, $g$, and $h$ be 2-cells. If $F = f \circ_2 g$, and $G = g \circ_2 h$, then $F \circ_2 h = f \circ_2 G$. (The latter values exist because the set of targets of $G$ and $g$ are equal, as are the set of sources of $g$ and $F$.) 1. Let $f$, $g$, and $h$ be 2-cells. If $F = f \circ_1 g$, and $G = g \circ_1 h$, then $F \circ_1 h = f \circ_1 G$. (The latter values exist because the set of 2-targets of $G$ and $g$ are equal, as are the set of 2-sources of $g$ and $F$.) It follows that composition of 1-cells is also associative, up to isomorphism. 1. Let $f_1$, $f_2$, $g_1$, and $g_2$ be 2-cells. If $F = f_1 \circ_1 f_2$, $G = g_1 \circ_1 g_2$, $H_1 = f_1 \circ_2 g_1$, and $H_2 = f_2 \circ_2 g_2$, then $F \circ_1 G = H_1 \circ_2 H_2$. (It follows from the preceding relations that these compositions exist.) So every anabicategory is "ana[[skeletal]]" (every equivalence is an identity), and "[[strict category\|anastrict]]:" its associators and unitors are also identities. Every [[bicategory]] can be made into an anabicategory; we simply make the source, target, identity, and composition maps into relations with the original functions as specified values, then define the full values by isomorphism and equivalence. This will satisfy the coherence relations because the original bicategory satisfied the bicategory coherence relations. On the other hand, we can also assume that every value of the defining relations is specified; then the anabicategory is said to be _saturated_. Any such anabicategory is clearly [[anafunctors\|anaequivalent]] to a strict 2-category (just map each object or 1-cell to its equivalence class, and vice versa). For there to be a weak equivalence between the anabicategory and the 2-category, however, we would need the [[Axiom of Choice]] (to pick out a single element of each equivalence class to serve as its image). # Reference # * [[Michael Makkai]]; Avoiding the axiom of choice in general category theory; section 3 (which is part 4 [here](http://www.math.mcgill.ca/makkai/anafun/)).
anafunction	https://ncatlab.org/nlab/source/anafunction	# Anafunctions * table of contents {: toc} ## Idea An anafunction is a [[relation]] between two [[sets]] $A,B$ such that to each [[element]] of $A$ there corresponds exactly one element of $B$. It can (usually) equivalently be defined as a [[span]] $A \leftarrow F \to B$ of functions whose first leg $F\to A$ is both [[injective function\|injective]] and [[surjective function\|surjective]]. Every [[function]] yields an anafunction, namely its [[graph of a function\|graph]], and this embeds the set of functions from $A$ to $B$ into the set of anafunctions. Conversely, in ordinary mathematics every anafunction is the graph of some function, so these two sets are isomorphic (and indeed, in some foundations such as [[material set theory]], equal). Thus in such cases the notion of anafunction is unneeded. However, in more exotic contexts where the [[function comprehension\|function comprehension principle]] (a.k.a. the "axiom of unique choice") fails, such as the [[internal logic]] of a [[quasitopos]] or a [[tripos]], it may be necessary to distinguish anafunctions from functions. Indeed, the fact that every anafunction is a function, or equivalently that every injective and surjective function is an [[isomorphism]], is essentially the definition of the function comprehension principle. Anafunctions are a [[decategorification]] of the notion of [[anafunctor]], and take their name from the latter. Traditionally they would be called "total functional relations". ## In homotopy type theory In [[dependent type theory]] ([[homotopy type theory]]), an anafunction is a [[type family]] $R$ indexed by types $A$ and $B$ such that for each [[term\|element]] $a \colon A$, the [[dependent sum type]] $\sum_{b \colon B} R(a, b)$ is a [[contractible type]]: $$ \mathrm{isAnafunction}(R) \;\coloneqq\; \prod_{a:A} \mathrm{isContr} \left( \sum_{b \colon B} R(a, b) \right) \,. $$ Since the [[principle of unique choice]] holds in dependent type theory, it follows that given any anafunction $a\colon A, b \colon B \vdash R(a, b) \; \mathrm{type}$, there is a function $a \colon A \vdash f(a) \colon B$. The [[categorical semantics]] of an anafunction in [[homotopy type theory]] is an [[infinity-anafunctor\|$\infty$-anafunctor]], since the [[identity types]] between two elements of a type are not required to be [[mere propositions]]. Given any [[type family]] $R$ indexed by types $A$ and $B$, there is a type family $R^\op$ indexed by $B$ and $A$, defined by $R^\op(b, a) \coloneqq R(a, b)$ for all $a \con A$ and $b \colon B$. An anafunction $R$ is an [[equivalence of types]] if both $R$ and $R^\op$ are anafunctions. ## See also * [[anafunctor]] * [[infinity-anafunctor]] * [[one-to-one correspondence]] * [[equivalence of types]] * [[dependent anafunction]] ## References For anafunctions in [[foundations of mathematics]] without the [[principle of unique choice]] see: * [[Mike Shulman]], Mathematics without the principle of unique choice. [[MathOverflow]]. June 5, 2018. [Web](https://mathoverflow.net/questions/302037/mathematics-without-the-principle-of-unique-choice). [[!redirects anafunctions]]
anafunctor	https://ncatlab.org/nlab/source/anafunctor	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Homotopy theory +-- {: .hide} [[!include homotopy - contents]] =-- =-- =-- # Anafunctors * automatic table of contents goes here {:toc} ## Idea An _anafunctor_ $F\colon C \to D$ is a generalized [[functor]]. A basic fact in ordinary [[category theory]] is that a [[functor]] $f\colon C \to D$ is an [[equivalence of categories]] -- in that there is a functor $g\colon D \to C$ and natural isomorphisms $f \circ g \simeq Id_D$ and $g \circ f \simeq Id_C$ -- if and only if it is [[essentially surjective functor\|essentially surjective]] and [[full and faithful functor\|fully faithful]]. However, the "if" part of this statement depends crucially on the [[axiom of choice]]: the functor $g$ is obtained by _choosing_ for each [[object]] $d \in D$ an object $c \in C$ such that $f(c) \simeq d$. In fact, the statement that "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom of choice. The notion of anafunctor is a generalization of the usual notion of functor, which enables us to recover a version of this statement without the axiom of choice. It was first studied in detail ([Makkai](#Makkai)) with [[foundations\|foundational concerns]] in mind, although it also appears unnamed in ([Kelly](#Kelly)). Later, they were applied by [[Toby Bartels]] to [[internal category\|internal categories]], where the axiom of choice is simply not an option. These "internal anafunctors" actually turned out to be known already (at least up to equivalence) in some contexts, in particular as [[Hilsum-Skandalis morphism]]s between [[Lie groupoid]]s. We present three motivations and applications of anafunctors: * [Anafunctors as a tool for handling foundational issues](#FoundationalMotivation) * [Anafunctors as morphisms in internal categories](#InternalCatMotivation) * [Anafunctors as morphisms in a homotopy category](#HomotopicalMotivation) ### Foundational motivation {#FoundationalMotivation} There is a sense in which the construction of inverse equivalences is not a "real" use of the axiom of choice, because the choice of $d$ is determined up to unique isomorphism. Note that in ordinary set theory, the axiom of choice is not necessary to make choices that are uniquely defined; this is sometimes called the "axiom of non-choice" or the "function comprehension principle." Since in category theory, an object can only be expected to be determined up to unique isomorphism, it is natural to regard the above statement as really being a "[[functor comprehension principle]]" or an "axiom of non-choice for categories." The fact that the full axiom of choice is required to make such choices is then an artifact of the usual [[foundation]]al choice to define categories as having a [[set]] of objects. In fact, however, we can recover the functor comprehension principle while maintaining the definition of categories in set theory if we modify the notion of functor. This results in the notion of _anafunctor_, which is essentially "a functor which determines its values on objects only up to isomorphism." In particular, an anafunctor is an [[equivalence of categories]] (in the sense of having an inverse anafunctor) precisely if it is essentially surjective and full and faithful. From this point of view, an anafunctor is not necessarily a fundamental notion, but rather an artifact that makes it possible to approximate the "natural" theory of categories, which doesn't need choice but has a functor comprehension principle, while still working in a set-theoretic foundation lacking choice. Every functor may be interpreted as an anafunctor; that every anafunctor is equivalent to a functor is equivalent to the [[axiom of choice]], in which case the inclusion of functors into anafunctors is in fact an [[equivalence of categories]]. But if you ignore functors and deal only with anafunctors (or saturated anafunctors), then the theory becomes entirely [[constructive mathematics\|constructive]] (without using the axiom of choice or even [[excluded middle]]). Thus, anafunctors (or even saturated anafunctors) are the correct notion to use if you are doing [[constructive mathematics]] and you still want to [[foundations\|found]] mathematics on some sort of [[set theory]]. ### Motivation from internal categories {#InternalCatMotivation} Since questions concerning the axiom of choice tend to look a bit esoteric to those not actively interested in questions of [[foundations]], it is helpful (and useful!) to think of this more generally in terms of [[internal category]] theory, where the concept is of independent use, and in fact well known by other names than "anafunctor". Consider some ambient category $\mathcal{E}$ [[internalization\|internal]] to which we want to do category theory. A good example to keep in mind is the category [[Top]] of [[topological space]]s. We observe that "the axiom of choice fails in [[Top]]", but that this is a very non-esoteric and obvious statement: it just means that not every continuous [[epimorphism]] $P \to X$ between topological spaces has a _continuous_ [[section]]. Then it may easily happen that an internal functor $f\colon C \to D$ between [[internal category\|internal categories]] in $\mathcal{E}$ (for instance between topological categories) is fully faithful, in the "internal" sense that $$\array{C_1 & \overset{}{\to} & D_1\\ \downarrow && \downarrow\\ C_0 \times C_0 & \underset{}{\to} & D_0\times D_0}$$ is a [[pullback]], and essentially surjective, in the "internal" sense that $C_0 \times_{D_0} Iso(D) \to D_0$ is surjective (or even a quotient map, i.e. a [[regular epimorphism]]), but for which there still does not exist a weak inverse in $Cat(\mathcal{E})$. For example, let $X$ be a topological space and consider the functor $C(U) \to X$, where $X$ is regarded as an internal category in $Top$ which is [[discrete category\|discrete]] in the categorical sense, and $C(U)$ is the [[Cech nerve\|Cech groupoid]] associated to an open [[cover]] $X = \bigcup_i U_i$ of $X$. That means that the space of objects of $C(U)$ is $\coprod_i U_i$, and its space of morphisms is $\coprod_{i,j} (U_i\cap U_j)$. Then the functor $C(U)\to X$ is fully faithful (essentially by definition of the morphisms in $C(U)$) and essentially surjective (since the $U_i$ are a cover of $X$) in the senses above. However, in general it does not admit a weak inverse, since the continuous function $\coprod_i U_i \to X$ does not have a continuous section if $X$ is connected unless one of the $U_i$ is already equal to $X$. Note that the foundational point of view also fits in this picture; we can simply take $\mathcal{E} = Set_{\not AC}$ to be the category [[Set]] of [[set]]s in a model of set theory that need not satisfy the [[axiom of choice]]. Here the same thing may happen: not every fully faithful and essentially surjective functor has a weak inverse. ### Homotopical motivation {#HomotopicalMotivation} There is a standard way to deal with such situations where we are faced with a category -- here the category $Cat(\mathcal{E})$ of categories internal to $\mathcal{E}$ -- some of whose morphisms look like they ought to have inverses, but do not: we call these would-be invertible morphisms _weak equivalences_ such that our category becomes a [[category with weak equivalences]] or a [[homotopical category]]. Then we pass to the corresponding [[homotopy category]]: the universal "improvement" of our category such that all the would-be invertible morphisms do become invertible. Here we take the weak equivalences in $Cat(\mathcal{E})$ to be the internal functors that are internally fully faithful and essentially surjective. It turns out that this choice of weak equivalences is particularly well-behaved in that it actually forms a [[calculus of fractions]]. Due to the early work on abstract [[homotopy theory]] by Gabriel and Zisman, there is simple explicit construction of the corresponding [[homotopy category]] $Ho(Cat(\mathcal{E}))$ in this case: the objects are the same as those of $Cat(E)$ -- hence [[internal category\|categories internal to]] $\mathcal{E}$ for us -- and the morphisms $f\colon C \to D$ are [[span]]s of morphism in $Cat(\mathcal{E})$ $$ \array{ \hat C &\stackrel{\hat f}{\to}& D \\ \downarrow^{\mathrlap{\in W}} \\ C } \,, $$ where the left leg is a weak equivalence, hence for us: where the left leg is an internal functor that is $k$-surjective for all $k$. (This is the beginning of the construction of the [[Dwyer-Kan localization]] at our chosen weak equivalences.) For the case $\mathcal{E} = Top$ such a span is a morphism out of a _Cech cover_. For instance for $C = X$ a topological space regarded as a topological category, for $G$ a [[topological group]] and $D = \mathbf{B}G$ its [[delooping]] one-object topological groupoid, such a span is a [[Cech cohomology\|Cech cocycle]] on $X$ with values in $G$. And finally: for the case that $\mathcal{E} = Set_{\not C}$ is the category of sets without the axiom of choice, such a span is an anafunctor: a functor $\hat C \to C$ that is surjective on objects and [[full and faithful functor\|full and faithful]], together with a functor $\hat C \to D$ out of the "resolution" of $C$. So one can understand ordinary anafunctors as follows: 1. first we consider that the [[axiom of choice]] may fail, which makes previously invertible functors non-invertible; 1. then we universally _force_ the now non-invertible functors to become invertible after all, by throwing in formal inverses for them. More generally, in any [[category of fibrant objects]] the morphisms in the [[homotopy category]] are represented by [[span]]s of the form $$ \array{ \hat X &\to & Y \\ \downarrow^{\mathrlap{\simeq}} \\ X } $$ with the left leg being an acyclic fibration. (This is a special case of the general statements of [[simplicial localization]]). ## Definitions Given categories $C$ and $D$, an __anafunctor__ $F\colon C \to D$ may be rather slickly defined as a [[span]] of ordinary ([[strict functor\|strict]]) [[functors]] $C \overset{\sigma}\leftarrow \overline{F} \overset{\tau}\rightarrow D$ (where $\overline{F}$ is some category), with the property that the functor $\sigma\colon {\overline{F}} \to C$ is both [[faithful functor\|faithful]] and (strictly!) surjective on both objects and morphisms (therefore both [[full functor\|full]] and [[essentially surjective functor\|essentially surjective on objects]]). It is also possible to define an anafunctor as a span in which $\sigma$ is merely a [[equivalence of categories\|weak equivalence]] (that is, faithful, full, and essentially surjective on objects), although that is slightly more complicated to work with. ### Explicit set-theoretic definition {#SetDef} In more explicit detail, an __anafunctor__ $F\colon C \to D$ consists of: * a set ${\|F\|}$ of __specifications__ of $F$ (which corresponds to the set of objects of $\overline{F}$); * maps $\sigma\colon {\|F\|} \to C$ and $\tau\colon {\|F\|} \to D$ (taking values in objects). Given $x\colon C$ and $y\colon D$, we say that $y$ is a __specified value__ of $F$ at $x$ if, for some $s\colon {\|F\|}$, $x = \sigma(s)$ and $y = \tau(s)$; in this case, $s$ __specifies__ $y$ as a value of $F$ at $x$, and we write $F_s(x) = y$. That is, $$ F_s(x) \coloneqq \tau(s) .$$ We say that $y$ is a __value__ of $F$ at $x$ if $y$ is isomorphic (in $D$) to some specified value of $F$ at $x$; we write $F(x) \cong y$. (There is no notion of the value of $F$ at $x$, except in the up-to-isomorphism sense of the [[generalised the]], and $F(x) = y$ is a meaningless statement.); * for each $s, t\colon {\|F\|}$ and morphism $f\colon \sigma(s) \to \sigma(t)$ in $C$, a morphism $$ F_{s,t}(f)\colon F_s(x) \to F_t(y) $$ in $D$, where $x \coloneqq \sigma(s)$ and $y \coloneqq \sigma(t)$. Similarly to the above, we can define whether a given morphism $g$ in $D$ is a __specified value__ of $F$ at a given morphism $f$ in $C$ or whether $g$ is (merely) a __value__ of $F$ at $f$. (Again, there is no notion of the value of $F$ at $f$.); * $\sigma$ is a [[surjective function]]. Thus, $F$ has some value at any given object or morphism of $C$. (In the [[internalization\|internalized]] case, this requirement can become quite complicated; for example, internal to [[Diff]], one requires a [[surjective submersion]].); * $F$ preserves [[identity morphism\|identities]]. That is, given $s\colon {\|F\|}$, the value of $F$ specified by $s$ and $s$ at the identity of $\sigma(s)$ is the identity of $\tau(s)$, or (in symbols) $F_{s,s}(\id_{\sigma(s)}) = \id_{\tau(s)}$, or (whenever this makes sense) $$ F_{s,s}(\id_x) = \id_{F_s(x)} ;$$ * $F$ preserves [[composition]]. That is, given $s, t, u\colon {\|F\|}$, $f\colon \sigma(s) \to \sigma(t)$, and $g\colon \sigma(t) \to \sigma(u)$, $$ F_{s,u}(f;g) = F_{s,t}(f);F_{t,u}(g) .$$ (Here the semicolon indicates composition in the anti-Leibniz order.). From the above explicit data, the category $\overline{F}$ is constructed as follows: the objects of $\overline{F}$ are the elements of ${\|F\|}$, while a morphism $s \to t$ in $\overline{F}$ is simply a morphism $\sigma(s) \to \sigma(t)$ in $C$. Then $\sigma$ extends to a surjective faithful functor from $\overline{F}$ to $C$ (acting as the identity on morphisms), and $\tau$ extends to a functor from $\overline{F}$ to $D$ (mapping the morphism $f\colon s \to t$ in $\overline{F}$ to $F_{s,t}(f)\colon \tau(s) \to \tau(t)$ in $D$). An anafunctor $F$ is __saturated__ if, whenever $F(x) \cong y$, $F_s(x) = y$ for some unique specification $s$, where the unicity of $s$ depends not only on $x$ and $y$ but also on how $y$ is a value of $F$ at $x$. To be precise: if $g\colon y' \to y$ is an isomorphism in $D$ and $F_{s'}(x) = y'$ for some specification $s'$, then there is a unique specification $s$ such that $F_{s',s}(\id_x) = g$ (where in particular, $\sigma(s) = x$ and $F_s(x) = y$). Every anafunctor $F\colon C \to D$ has a _saturation_ $\overline{F}$; $\overline{F}$ is a saturated anafunctor and $F \cong \overline{F}$ in the category of anafunctors from $C$ to $D$. In fact, the inclusion of the saturated anafunctors into the anafunctors (as a full subcategory) is an equivalence of categories (given fixed $C$ and $D$). The usual notions of [[full functors]] and [[faithful functors]] can be generalized to anafunctors. An anafunctor $F$ is __full__ if the maps $F_{s, t}: Hom(\sigma(s), \sigma(t)) \to Hom(\tau(s), \tau(t))$ are all surjective, and it is __faithful__ if the maps are injective. Anafunctors can be composed via pullback. Given anafunctors $F: C \to D$, $G: D \to E$, we can form the pullback $$ \array{ \|GF\| & \to & \|G\| & \overset{\tau_G}\to E\\ \downarrow & & \downarrow\mathrlap{\sigma_G}\\ \|F\| & \underset{\tau_F}\to & D\\ \mathllap{\sigma_F}\downarrow\\ C } $$ More explicitly, the specifications are given by pairs $(s, t) \in \|F\| \times \|G\|$ such that $\tau_F(s) = \sigma_G(t)$, and $\sigma_{G F}(s, t) = \sigma_F(s)$, $\tau_{G F}(s, t) = \tau_G(t)$. For $(s, t), (s', t') \in \|GF\|$ and an arrow $f: \sigma_F(s) \to \sigma_F(s')$, we obtain an arrow $F_{s, s'}(f): \tau(s) \to \tau(s')$. This is also an arrow $\sigma_G(t) \to \sigma_G(t')$, so we can lift this map to $G_{t, t'}F_{s, s'}(f): \tau(t) \to \tau(t')$, and this completes the description of the anafunctor. The other axioms can be verified straightforwardly. Categories, anafunctors, and a suitably defined notion of [[ananatural transformation]] between them form a [[bicategory]] $Cat_{ana}$; an internal [[equivalence]] in this 2-category is called an anaequivalence. Every functor may be interpreted as an anafunctor, with ${\|F\|}$ always taken to be (the set of objects in) $C$ itself and $\sigma$ the [[identity functor]]. Indeed, there is a [[2-functor]] to $Cat_{ana}$ from the [[strict 2-category]] $Str Cat$ of categories, functors and natural transformations; this functor is an [[equivalence of categories\|equivalence]] if and only if the [[axiom of choice]] holds. Thus, most mathematicians will identify $Cat_{ana}$ and $Str Cat$ as simply [[Cat]], the $2$-category of categories; however, mathematicians who doubt the axiom of choice will distinguish them. While anafunctors exist in any case, there is an ideological statement that may be implied by their use: that $Cat$ is really $Cat_{ana}$ rather than $Str Cat$. In any case, (modulo "size issues" which one may want to impose) the inclusion of $Str Cat$ into $Cat_{ana}$ has a right adjoint, described using [[clique]]s. Accordingly, we can instead define anafunctors by means of clique categories, taking an anafunctor from $C$ into $D$ to be a genuine functor from $C$ into $Clique(D)$ (and the 2-category of anafunctors as the [[Kleisli category]] for the $Clique(-)$ 2-monad (in particular, natural transformations between anafunctors into $D$ are simply natural transformations of the corresponding genuine functors into $Clique(D)$)). ### Internal definition using covers We generalise the slick definition of anafunctors as spans rather than the detailed definition involving specified values. Let $S$ be a category containing a collection of morphisms called "covers" such that * every [[isomorphism]] is a cover, * covers are closed under composition, * any [[pullback]] of a cover exists and is a cover ([[pullback stability]]), * every cover is the [[quotient object]] of its [[kernel pair]], i.e. is an [[effective epimorphism]]. (Since all pullbacks of covers exist, this is equivalent to saying that every cover is a [[regular epimorphism]].) Note that these are precisely the axioms saying that the singleton families $\{p\colon V\to U\}$ where $p$ is a cover form a [[subcanonical coverage\|subcanonical]] [[Grothendieck pretopology]]. One important class of examples is when $S$ is a [[regular category]] and the covers are the [[regular epimorphisms]]. Another is when $S$ is the category of smooth manifolds and the covers are the surjective submersions. In such a situation, if $C$ and $D$ are [[internal category\|internal categories]] in $S$, we define an __anafunctor__ $C\to D$ to consist of a span $C\leftarrow F \to D$ of internal functors such that: 1. $F_0\to C_0$ (the map of objects) is a cover. 2. $F\to C$ is [[ff morphism\|fully-faithful]], in the internal sense that the following is a pullback square: $$\array{F_1 & \to & C_1 \\ \downarrow && \downarrow \\ F_0\times F_0& \to & C_0\times C_0}$$ Note that assuming $F_0\to C_0$ is a cover, so is $F_0\times F_0\to C_0\times C_0$ (it is a composition of pullbacks of $F_0\to C_0$); thus the above pullback always exists. By the remarks above, if $S$ is [[Set]] and "cover" means "[[surjection]]" (an example where the covers are the regular epimorphisms), then we recover the original external notion of ([[small category\|small]]) anafunctor. An anafunctor, defined in this way, is saturated just when the map $core(F) \to core(C\times D)$ of [[cores]] is an [[isofibration]], so we need an internal notion of core to define saturated anafunctors internally. An anafunctor is an anaequivalence when $F\to D$ is fully faithful and [[essentially surjective functor\|essentially surjective]], meaning the canonical map $F_0\times_{C_0}C_1 \to C_0$ has splits over a cover of $C_0$. For [[Lie groupoids]], these are the [[Morita equivalences]]. If $C\leftarrow F \to D$ and $C\leftarrow G \to D$ are internal anafunctors, we define an __[[ananatural transformation]]__ between them (or simply a _natural transformation_, given the context) to be a [[natural transformation]] between the two induced internal functors $F\times_C G \to D$. We can then prove that internal categories, anafunctors, and natural transformations form a [[bicategory]]. (Interestingly, you may need the axiom of choice in the [[metalogic]] to conclude this, depending on whether there is a natural way to choose the necessary pullbacks; else you get an [[anabicategory]], in which the composition functors are anafunctors.) The role of the assumptions about covers is: * Identity maps must be covers in order to have identity anafunctors (and more generally, for every functor to give rise to an anafunctor). * To compose anafunctors by pullback, the pullbacks of covers must exist and be covers, and covers must be closed under composition. * To define $F \times_C G$ and obtain a notion of natural transformation, we again need covers to have pullbacks. * To define composition of natural transformations between anafunctors, we need covers to be effective; see diagram (118) in [HGT1](#HGT1). Note: in Section 1.1.5 of [HGT1](#HGT1), the following additional axiom was assumed on the class of covers: * every [[congruence]] involving a cover has a quotient object which is a cover. This is not needed for anafunctors but is used to relate descent to bundles (and then to $2$-bundles). ### As an operation on 2-categories _This section is work in progress by [[David Roberts\|me]]_ While internal anafunctors seem to require a lot of baggage, they can be defined very elegantly by working with the 2-category $Cat(S)$ (or some sub-2-category thereof) as a 2-category. One first makes the observation: * functors internal to $S$ which are fully faithful and whose object component belongs to a singleton Grothendieck pretopology themselves form a (strict) singleton pretopology on $Cat(S)$. Thus one can consider anafunctors as spans in a 2-category where the source leg belongs to a strict, subcanonical singleton Grothendieck pretopology, all of whose covers are [[ff morphism\|ff]]. ### Homotopy-theoretic interpretation Observe that the surjective-on-objects equivalences are precisely the [[model category\|acyclic fibrations]] for the [[canonical model structure]] on [[Cat]]. Therefore, anafunctors can be identified with the "one-step generalized morphisms" in $Cat$ whose first leg is not just a [[weak equivalence]] but an acyclic fibration. However, it appears that the canonical model structure on Cat only exists (with its weak equivalences being the fully faithful and essentially surjective maps) under the assumption of some choice---though full AC is not needed, [[COSHEP]] suffices. More generally, it is proven in [EKV](#EKV) that if $S$ has a Grothendieck coverage, then under suitable additional conditions on $S$ (and, of course, the axiom of choice assumed external to $S$), there is a [[model category\|model structure]] on the category $Cat(S)$ of internal categories in $S$ relative to that coverage. The internal anafunctors relative to the given coverage, as defined above, can then once again be identified with the spans whose first leg is an acyclic fibration. Since all objects in the canonical model structure on Cat are fibrant, according to Kenneth Brown's theorem in [[homotopical cohomology theory]] it follows that one-step generalized morphisms already realize the full localization, i.e. they represent all morphisms in the [[homotopy category]] $Ho(Cat)$. If we specialize to groupoids, with their canonical model structure by [[Ronnie Brown\|Brown]]--[[Marek Golasiński\|Golasiński]], then by the general idea of [[homotopical cohomology theory]] this means that anafunctors between groupoids represent [[nonabelian cocycle]]s on groupoids with values in groupoids. By the notion of [[descent\|codescent]] such homotopical cocycles are related to [[descent\|descent data]] that enters the definition of [[sheaf\|sheaves]] and [[stack]]s. ## Examples We will use the [explicit set-theoretic definition](/nlab/show/anafunctor#SetDef) in this section. ### Product anafunctor Given a category $C$ with [[cartesian product\|binary products]], we can form a product functor $P: C \times C \to C$ that sends a pair of objects $(A, B)$ to a product of them. This requires picking a product for each pair, and hence requires the [[axiom of choice]]. However, we can form the product anafunctor without using choice. The specifications are given by $\|P\| = \text{product diagrams in }\;C$, and the maps $\sigma: \|P\| \to C \times C$ and $\tau: \|P\| \to C$ are given by $$ \sigma(A \leftarrow D \rightarrow B) = (A, B),\;\; \tau(A \leftarrow D \rightarrow B) = D. $$ Given product diagrams $A \leftarrow D \rightarrow B$ and $A' \leftarrow D' \rightarrow B'$, and a map $(f, g): (A, B) \to (A', C')$ in $C \times C$, we obtain $P(f, g): D \to D'$ by the universal property of the product. The compatibility conditions are easy to check. ### Anafunctor from functor Suppose we have a usual functor $F: C \to D$. Then we can obtain an anafunctor as the span $$ \array{ C & \overset{F}\to & D\\ \mathllap{id_C} \downarrow \\ C } $$ The composition of anafunctors agree with the composition of functors. ### Inverses of anafunctors Given an anafunctor $F: C \to D$ with $\sigma: \|F\| \to C$ and $\tau: \|F\| \to D$, if $F$ (ie. $\tau$) is essentially surjective, then its saturation is strictly surjective. Then given a saturated full and faithful essentially surjective anafunctor, we can obtain an inverse anafunctor $F^{-1}: D \to C$ by swapping $\tau$ and $\sigma$ around. The conditions of being full and faithful and essentially surjective guarantees the axioms are still satisfied. ## Questions of size {#SizeQuestions} Even if $C$ and $D$ are [[small categories]], then the category $Ana(C,D)$ of anafunctors from $C$ to $D$ is not necessarily even essentially small, and thus the 2-category $Cat_{ana}$ of categories and anafunctors is not [[cartesian closed category\|cartesian closed]]. Some models in which this fails to be true are sketched in [this MO discussion](#HenryMO). This is true, however, under the assumption of [[COSHEP]], since in that case (as above) anafunctors represent maps in $Ho(Cat)$, which is locally small by general model category theory. More specifically, under COSHEP every anafunctor $C\leftarrow F \to D$ is equivalent to one where the set of objects of $F$ is $C_0'$, where $C_0'\to C_0$ is a projective cover of the set $C_0$ of objects of $C$. COSHEP is actually stronger than necessary for this; all that is really needed is [[WISC]], i.e. for any set $X$, the full subcategory of $Set/X$ consisting of surjections has a [[weakly initial set]]. For in that case, any anafunctor $C\leftarrow F \to D$ is equivalent to one where the set of objects of $F$, equipped with its surjection to $C_0$, belongs to the weakly initial set. Note that COSHEP implies WISC, as do the [[axiom of multiple choice]] and the axiom of [[small violations of choice]] (SVC). In [Makkai's paper](#Makkai), he proves that $Ana(C,D)$ is essentially small under the assumption of his [[small cardinality selection axiom]] (SCSA), which also follows from SVC. Although SCSA and WISC carry the same feel that "choice is violated only in a small way," Makkai's proof from SCSA is an "injective" approach, in that the set of possibilities for the objects of $F$ is constructed mainly from $D$, rather than purely from $C$ as in the "projective" approach above using COSHEP or WISC. Another axiom ensuring that $Ana(C,D)$ is essentially small is the [[axiom of stack completions]] (ASC), since if $D\to \hat{D}$ is an intrinsic stack completion we have $Ana(C,D) \simeq Fun(C,\hat{D})$. In particular, the bicategory of categories and anafunctors is locally essentially small and cartesian closed in the internal logic of any [[Grothendieck topos]], because the latter satisfies WISC, SCSA, and ASC. ##Anafunctors in homotopy type theory Anafunctors are unnecessary when using "saturated/univalent" categories in [[homotopy type theory]] (see Def. 9.1.3 of [[the HoTT book]], and Chap. 9 notes), because of their [[functor comprehension principle]]. An anafunctor is a span whose first leg is a surjective and fully faithful functor, but for saturated categories any such functor is an equivalence (in the strong sense of having an inverse), so any anafunctor is equivalent to a functor. ## Related concepts ### Anafunctors versus representable profunctors A different way of describing "a functor whose values are determined only up to isomorphism" is with a representable [[profunctor]] (a.k.a. distributor). A profunctor $A ⇸ B$ is a functor $F\colon A \to [B^{op},Set]$, and we call it representable if each [[presheaf]] $F(a)$ is [[representable functor\|representable]]. In the presence of [[axiom of choice\|AC]], one can then choose a representing object $F_a\in B$ for each $a$ and thereby define a functor $A\to B$, but without choice this is generally not possible. However, one can define an anafunctor from $A$ to $B$, whose specifications at $a$ are "representations" of $F(a)$. Conversely, given an anafunctor represented by a span $A \xleftarrow{g} P \xrightarrow{f} B$, one can define a profunctor $A ⇸ B$ as the composite $f_* \circ g^$, where $f_(p)(b) = B(b,f(p))$ and $g_(a)(p) = A(g(p),a)$, and this profunctor will be representable. This defines a bijective-on-objects [[equivalence of 2-categories]] between $Cat_{ana}$ and $Prof_{rep}$, the [[locally full sub-2-category]] of [[Prof]] determined by the representable profunctors. (This appears to have been written down first [here](http://permalink.gmane.org/gmane.science.mathematics.categories/6485) by [[Jean Benabou]]). Anafunctors and representable profunctors each have advantages. For purposes which require only the 2-category $Cat_{ana}\simeq Prof_{rep}$, either one is of course sufficient. For instance, a non-[[cleavage\|cloven]] [[Grothendieck fibration]] can equally well be turned into a [[pseudofunctor]] valued in $Cat_{ana}$ or in $Prof_{rep}$. However, in this case $Prof_{rep}$ is arguably more natural, since an arbitrary functor (not necessarily a fibration) $A \to B$ can be turned into a normal [[lax 2-functor]] $B^{op}\to Prof$, which is a pseudofunctor landing in $Prof_{rep}$ exactly when the given functor was a fibration. (It is a pseudofunctor landing in $Prof_{corep}$ iff the functor was an opfibration, and it is a pseudofunctor iff the functor was [[exponentiable functor\|exponentiable]].) More generally, one good point about using representable profunctors is that they fit in immediately with the general notion of profunctor. On the other hand, sometimes it requires a little contortion to put something in the form of a representable distributor. For instance, if $A$ has binary [[products]], then there is obviously a product-assigning representable distributor $P\colon A \times A ⇸ A$ defined by $P(a_1,a_2)(a) = Hom_A(a,a_1) \times Hom_A(a,a_2)$. But if A has binary [[coproducts]], then in order to define a coproduct-assigning representable distributor $C\colon A \times A ⇸ A$, one needs to say something like $C(a_1,a_2)(a)=$ the set of triples $(a_3,p_1,p_2,f)$ where $p_i\colon a_i \to a_3$ are the injections into a coproduct and $f$ is a morphism $a \to a_3$, modulo an equivalence relation $(a_3,p_1,p_2,f) \sim (a_3',p_1',p_2',f')$ if there exists a (necessarily unique iso)morphism $g\colon a_3 \to a_3'$ commuting with all the structure maps. (This is essentially making explicit the functor $Cat_{ana} \to Prof_{rep}$ defined above.) Of course, $C$ is more easily defined as a [[corepresentable functor\|corepresentable]] distributor. But if we want to define a functor that involves both limits and colimits, like $(a,b,c) \mapsto a \times (b + c)$, then it is not "naturally" represented as either a representable or a corepresentable profunctor. However, with anafunctors, all of these functors can be represented "naturally" in analogous ways. In the first case, we consider the span $A\times A \leftarrow P \to A$, where $P$ is the category of binary product diagrams in $A$. In the second case, we consider the span $A\times A \leftarrow C \to A$, where $C$ is the category of binary coproduct diagrams. And in the third case, we consider the span $A\times A\times A \leftarrow D \to A$, where $D$ is the category of binary coproduct diagrams together with a product diagram one of whose factors is the vertex of the coproduct diagram. Roughly speaking, anafunctors are formulated exactly in order to describe "functors defined up to isomorphism," while representable distributors describe "functors valued in representable presheaves." "Objects defined up to isomorphism" and "representable presheaves" are formally equivalent (without invoking AC), but not every "naturally occurring" object-defined-up-to-isomorphism is "given in nature" by the presheaf it represents. Some are given by the [[copresheaf]] they corepresent; others aren't given directly in either of those ways. One can also think of an anafunctor as a particularly convenient "presentation" of a representable distributor. A further reason that the notion of anafunctor is useful is that when working with [[internal categories]], the quotienting operations necessary to define the composite of internal profunctors may not exist, whereas internal anafunctors can always be composed. Thus, when working with (for instance) [[smooth manifold\|smooth]] categories or groupoids, profunctors are not so much an option, but anafunctors are well-behaved (see the papers by Bartels and Roberts referenced below). ## Generalizations ### Higher versions see [[infinity-anafunctor]] ### Lower version see [[anafunction]] ### Additive version The notion of abelian [[butterfly]] introduced by Behrang Noohi [Weak maps of 2-groups](http://arxiv.org/abs/math.CT/0506313) is the additive version of the notion of (saturated) anafunctor: the equivalence between, on the one hand, internal groupoids and internal functors and, on the other hand, arrows and commutative squares in an abelian category extends to an equivalence between saturated anafunctors and butterflies. ## References The term "anafunctor" was introduced by Michael Makkai in * {#Makkai} [[Michael Makkai]], _Avoiding the axiom of choice in general category theory_, Journal of Pure and Applied Algebra 108 isse 2 (1996) pp 109-173, doi:[10.1016/0022-4049(95)00029-1](https://doi.org/10.1016/0022-4049%2895%2900029-1), ([author's page](http://www.math.mcgill.ca/makkai/anafun/)) motivated in part to complete the analogy prophase:anaphase::profunctor:??. The concept also appears, unnamed, in the article [[Max Kelly]], _Complete functors in homology I_ (1963) {#Kelly} on [[homological algebra]]. The popularity of the term was notably pushed by [[Toby Bartels]], who considered [[internalization]]s of Makkai's definition in * [[Toby Bartels]], _Higher Gauge Theory I: 2-Bundles_ ([arXiv:math.CT/0410328](http://arxiv.org/abs/math.CT/0410328)) {#HGT1} A development and exposition of the general setup taking Makkai's and Bartels' motivations and the theory of [[homotopical category\|homotopical categories]] into account is * {#Roberts12} [[David Roberts]], _Internal categories, anafunctors and localisations_, [[Theory and Applications of Categories]], Vol. 26, 2012, No. 29, pp 788-829, [journal version](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html), [arXiv:1101.2363](http://arxiv.org/abs/1101.2363) and a purely formal construction using 2-categories appears in the notes * {#Roberts18} [[David Roberts]], _The construction of formal anafunctors_ (2018) arXiv:[1808.04552](https://arxiv.org/abs/1808.04552), doi:[10.25909/5b6cfd1a73e55](https://doi.org/10.25909/5b6cfd1a73e55) See also * [[Erik Palmgren]], _Locally cartesian closed categories without chosen constructions_, [TAC](http://www.tac.mta.ca/tac/volumes/20/1/20-01abs.html). Since anafunctors are a special case of a more general concept, they, or the general theory applying to them, has been considered under different terms elsewhere. The general question of [[model category]] structures on categories of [[internal category\|internal categories]] is discussed in * T. Everaert, R.W. Kieboom and T. Van der Linden , _Model structures for homotopy of internal categories_ TAC, Vol. 15, CT2004, No. 3, pp 66-94. ([web](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html) ([pdf](http://www.tac.mta.ca/tac/volumes/15/3/15-03.pdf))) {#EKV} Closely related, still a bit more general, are the considerations in * Jardine, _Cocycle categories_ K-theory 0782 ([web](http://www.math.uiuc.edu/K-theory/0782/)) ([pdf](http://www.math.uiuc.edu/K-theory/0782/coc-cat3.pdf)) Some models of set theory in which the bicategory of anafunctors fails to be small are sketched in the answers to * {#HenryMO} Simon Henry, _Non smallness of the set of anafunctors without AC?_, URL (version: 2017-03-16): <https://mathoverflow.net/q/264585> [[!redirects anafunctor]] [[!redirects anafunctors]]
anafunctor category	https://ncatlab.org/nlab/source/anafunctor+category	Given [[category\|categories]] $C$ and $D$, the _anafunctor category_ $D^C$ has [[anafunctor\|anafunctors]] $F: C \to D$ as [[object\|objects]] and [[ananatural transformation\|ananatural transformations]] between these as [[morphism\|morphisms]]. This is the appropriate notion of [[functor category]] to use in the absence of the [[axiom of choice]] (including many [[internal category\|internal]] situations). Functor categories serve as the [[hom-category\|hom-categories]] in the [[anabicategory]] [[Cat]].
analogy	https://ncatlab.org/nlab/source/analogy	#Contents# * table of contents {:toc} ## Examples * [[function field analogy]] * [[MKR analogy]] * [[rubber-sheet analogy of gravity]] ## Related concepts [[!include mathematical statements --- contents]] ## References * Wikipedia, _[Analogy](http://en.wikipedia.org/wiki/Analogy)_ [[!redirects analogies]] [[!redirects analog]] [[!redirects analogs]]
analyser	https://ncatlab.org/nlab/source/analyser	Before the modern theory of [[operad]]s was (re)invented there was a variant introduced earlier in * M. Lazard, _Lois de groupes et analyseurs_, Ann. École Norm. Sup. __72__ (1955), pp. 299–400. under the name __analyser__ which is often even nowdays referred by its original French version analyseur. There is a generalization motivated by some questions in noncommutative algebra, the pseudoanalyser: * Ralf Holtkamp, _A pseudo-analyzer approach to formal group laws not of operad type_, Journal of Algebra __237__, 1, 2001, p. 382-405, [doi](http://dx.doi.org/10.1006/jabr.2000.8566) [[!redirects analysers]] [[!redirects analyseur]]
analysis	https://ncatlab.org/nlab/source/analysis	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Functional analysis +-- {: .hide} [[!include functional analysis - contents]] =-- #### Topology +-- {: .hide} [[!include topology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea {#Idea} In [[mathematics]], _analysis_ usually refers to any of a broad family of fields that deals with a general theory of _[[limit of a sequence\|limits]]_ in the sense of [[convergence]] of [[sequences]] (or more generally of [[nets]]), particularly those fields that pursue developments that originated in "the [[calculus]]", i.e., the theory of [[differentiation]] ([[differential calculus]]) and [[integration]] ([[integral calculus]]) of [[real numbers\|real]] and [[complex numbers\|complex]]-valued [[functions]]. The classical foundation of this general subject is usually based on the idea that the [[real numbers\|real number system]] is describable as a [[complete space\|complete]] [[ordered field]], or more generally on the concept of [[metric spaces]]. Their [[distance]] functions allow to formalize concepts like [[continuous functions\|continuity]] and [[convergence]] in terms of existence of sufficiently small [[open balls]]. Many concepts of this "[[epsilontic analysis]]" have equivalent formulations in terms of simple [[combinatorics]] of [[open subsets]] with respect to the [[metric topology]] of metric spaces, and this way the field of analysis has a large overlap with the field of _[[topology]]_, this is particularly true for [[functional analysis]] and the theory of [[topological vector spaces]]. Analysis can also refer to other responses to the problem of founding these developments, especially "[[infinitesimal analysis]]" which admits [[infinitesimal quantities]] not found in the classical real number system and which takes various forms, for example the [[nonstandard analysis]] first introduced by [[Abraham Robinson]], or "[[synthetic differential geometry\|synthetic differential analysis]]" whose rigorous foundations were largely introduced by [[William Lawvere]] and other [[category theory\|category theorists]] who, following the example of [[Alexander Grothendieck]], consider [[nilpotent infinitesimals]] (instead of invertible ones à la Robinson) as a basis for understanding [[differentiation]]. ## Entries related to analysis ### On mainstream analysis Some of the $n$lab entries related to __mathematical analysis__ include [[metric space]], [[normed vector space]], [[metric topology]], [[sequence]], [[net]], [[convergence]], [[functional analysis]], [[harmonic analysis]], [[complex analysis]], [[Weierstrass preparation theorem]], [[several complex variables]], [[Fourier transform]], [[Pontrjagin dual]], [[differential geometry]], [[Legendre polynomial]], [[dilogarithm]], [[Hilbert space]], [[Banach space]], [[Banach algebra]], [[topological vector space]], [[locally convex space]], [[operator algebras]], [[Gelfand spectrum]], [[measure space]], [[measurable function]], [[Lebesgue space]], [[Sobolev space]], [[bounded operator]], [[compact operator]], [[Fredholm operator]], [[distribution]] (generalized function), [[hyperfunction]].[[spectral theory]], [[integral]], [[integration]]...and a book entry [[Handbook of analysis and its foundations]]. Many of the basic notions used in analysis courses are described in $n$lab in the more general [[topology\|topological]] context if they belong there, e.g. [[compact space]], [[continuous map]], [[compact-open topology]] and so on. Many of the aspects of [[analytic geometry]] are treated in terms of Riemann surfaces, [[monodromy]], [[local system]]s and so on. ### On foundations Alternative [[foundations]], especially [[constructive mathematics\|constructive]] and those using [[topos theory]], are of traditional interest to the [[category theory]] community. For example the [[synthetic differential geometry]] of Lawvere and Kock (more in next paragraph) and the [[nonstandard analysis]] of Robinson, and its variant, [[internal set]] theory of Nelson are some of the principal examples. See also [[Fermat theory]], [[natural numbers object]], [[infinitesimal number]] etc. Many statements are about the versions without the [[axiom of choice]] and so on; we like to state clean and minimal conditions when possible. ### On smoothness and generalized Lie theory Various smoothness concepts in geometry, rarely studied in standard courses of analysis, but sometimes relevant, were studied to fair extent (and sometimes with innovations) in the $n$lab. These smoothness concepts are built using some primitive notions in rather generalized (often categorical) setups: [[Kähler differential]], [[differential form]], [[tangent space]], [[jet bundles]], resolution of diagonal, [[infinitesimal object]], [[microlinear space]], [[generalized smooth algebra]], [[tangent category]], [[cotangent complex]] as defining ingredients of various notions of smoothness and smooth spaces. Main framework to systematize in geometry similar notion studied in $n$lab is [[synthetic differential geometry]] but many other examples are also represented. Let us mention [[generalized smooth space]], [[stratifold]], [[Frölicher space]], and some graded and super analogues ([[supermanifold]], [[NQ-supermanifold]], [[integration over supermanifolds]]); some concepts of smoothness are rather algebraic, e.g. [[formal smoothness]] of [[Grothendieck]]; see also [[algebraic approaches to differential calculus]]. Special attention in $n$lab has been paid to smooth group like objects like [[Lie group]], [[Lie groupoid]] and their superanalogues and [[categorification]]s, as well as to their tangent structures like [[Lie algebroids]] and their interrelations ([[Lie theory]]: [[integration]], [[Lie integration]]). ### On geometric function theory and quantization Some other entries are related to the conceptual and categorical understanding of Feynman [[path integral]], however so far from physical, conceptual and formal point of view only (and not of analytic theory). This is closely related to understanding various higher categorical spaces of [[sections]] in geometry and in study of sigma-models in physics. This is here called [[geometric function theory]] (cf. [[space and quantity]], [[geometric quantization]]...). ### On quantization and the geometry of differential operators Very relevant for [[quantization]] is also the geometric study of differential operators (see [[D-geometry]], [[diffiety]]) and distributions (cf. [[microlocal analysis]]), by analysis of oscillating integrals ([[semiclassical approximation]]), [[symplectic geometry]] (esp. the geometry of [[lagrangian submanifold]]s which could often be viewed as quantum points) etc. Some of the topological properties of differential operators are studied in [[index theory]], where special role have so called [[Dirac operator]]s. Sometimes it is possible or even useful to avoid fine analysis by using the [[algebraic approaches to differential calculus]] and [[regular differential operator\|differential operators]], what also makes possible some noncommutative analogues. ### On contructivism and computable analysis * [[constructive analysis]], [[computable analysis]] ## Related concepts * [[geometric analysis]] * [[epsilontic analysis]] * [[infinitesimal analysis]] * [[matrix analysis]] ## References {#References} ### General {#ReferencesGeneral} Textbooks accounts: * [[Tom Apostol]], Mathematical Analysis (1973) [ISBN:0201002884, [pdf](http://webpages.iust.ac.ir/amtehrani/files/Addison%20Wesley%20-%20Mathematical%20Analysis%20_%20Apostol%20%285Th%20Ed%29%20%281981%29.pdf)] * {#Rudin64} [[Walter Rudin]], _Principles of Mathematical Analysis_, McGraw-Hill (1964, 1976) [[pdf](https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf)] * [[Eric Schechter]], _[[Handbook of Analysis and its Foundations]]_, Academic Press (1996) ([web](http://www.math.vanderbilt.edu/~schectex/ccc/)) Discussion of the history, amplifying its roots all the way back in [[Zeno's paradoxes of motion]] is in * {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949 See also * Wikipedia, _[Mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis)_ See also the references at _[[calculus]]_. ### Constructive analysis {#ReferencesConstructiveAnalysis} The formulation of [[analysis]] in [[constructive mathematics]], hence [[constructive analysis]], was maybe initiated in * {#Bishop} [[Errett Bishop]], _Foundations of constructive analysis._ McGraw-Hill, (1967) together with the basic notion of [[Bishop set]]/[[setoid]]. Implementations of constructive [[real number]] analysis in [[type theory]] implemented in [[Coq]] are discussed in * R. O'Connor, _A Monadic, Functional Implementation of Real Numbers_. MSCS, 17(1):129-159, 2007 ([arXiv:0605058](http://arxiv.org/abs/cs/0605058)) * R. O'Connor, _Certified exact transcendental real number computation in Coq_, In TPHOLs 2008, LNCS 5170, pages 246--261, 2008. * R. O'Connor, _Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory_, PhD thesis, Radboud University Nijmegen, 2009. * Robbert Krebbers, [[Bas Spitters]], _Type classes for efficient exact real arithmetic in Coq_ ([arXiv:1106.3448](http://arxiv.org/abs/1106.3448/)) On [[constructive analysis]] such as in [[univalent foundations]] ([[homotopy type theory]]), see the references [there](https://ncatlab.org/nlab/show/constructive+analysis#References), such as: * {#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), [[Booij-AnalysisInUF.pdf:file]]] [[!redirects mathematical analysis]] [[!redirects real analysis]]
analysis - contents	https://ncatlab.org/nlab/source/analysis+-+contents	[[analysis]] ([[differential calculus\|differential]]/[[integral calculus\|integral]] [[calculus]], [[functional analysis]], [[topology]]) [[epsilontic analysis]] [[infinitesimal analysis]] [[computable analysis]] _[[Introduction to Topology -- 1\|Introduction]]_ ## Basic concepts [[triangle inequality]] [[metric space]], [[normed vector space]] [[open ball]], [[open subset]], [[neighbourhood]] [[metric topology]] [[sequence]], [[net]] [[convergence]], [[limit of a sequence]] [[compact space\|compactness]], [[sequentially compact space\|sequential compactness]] [[differentiation]], [[integration]] [[topological vector space]] ## Basic facts [[continuous metric space valued function on compact metric space is uniformly continuous]] ... ## Theorems [[intermediate value theorem]] [[extreme value theorem]] [[Heine-Borel theorem]] ...
Analysis Situs	https://ncatlab.org/nlab/source/Analysis+Situs	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Topology +--{: .hide} [[!include topology - contents]] =-- #### Algebraic topology +--{: .hide} [[!include algebraic topology - contents]] =-- #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- =-- =-- This page is about the article * [[Henri Poincaré]]: Analysis Situs Journal de l'École Polytechnique. (2). 1: 1–123 (1895) original: [gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7) English translation by [[John Stillwell]]: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]] which laid modern foundations of [[topology]] (such as introducing the notion and terminology of [[homeomorphisms]]) and introduced basic concepts of [[algebraic topology]] (such as [[singular homology]]) and of [[homotopy theory]] (such as the [[fundamental group]]). See also: * Wikipedia, <a href="https://en.wikipedia.org/wiki/Analysis_Situs_(paper)">Analysis Situs_(paper)</a> * [[Peter Hilton]], Subjective History of Homology and Homotopy Theory, Mathematics Magazine 61 5 (1988) 282-291 $[$[doi:10.2307/2689545](https://doi.org/10.2307/2689545)$]$ category: reference
analytic (∞,1)-functor	https://ncatlab.org/nlab/source/analytic+%28%E2%88%9E%2C1%29-functor	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Goodwillie calculus +--{: .hide} [[!include Goodwillie calculus - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In the context of [[Goodwillie calculus]] an [[(∞,1)-functor]] is called _analytic_ if it behaves in [[analogy]] with an [[analytic function]] in that it its [[Goodwillie-Taylor tower]] converges to it. ## Definition ### (Co-)Cartesian cubical diagrams Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]]. +-- {: .num_defn #StronglyCoCartesian} ###### Definition An $n$-[[cube]] in $\mathcal{C}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{C}$, is called _strongly homotopy co-cartesian_ or just _strongly co-cartesian_, if all its 2-dimensional square faces are [[homotopy pushout]] [[diagrams]] in $\mathcal{C}$. =-- +-- {: .num_defn #Cartesian} ###### Definition An $n$-[[cube]] in $\mathcal{D}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{D}$, is called _homotopy cartesian_ or just _cartesian_, if its "first" object exhibits a [[homotopy limit]]-[[cone]] over the remaining objects. =-- ### Analytic functors +-- {: .num_defn #rhoAnalyticFunctor} ###### Definition An [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is _stably $n$-excisive_ with constants $c$ and $\kappa$_ -- or _satisfies "condition $E_n(c,\kappa)$"_ -- if for every strongly co-Cartesian $(n)+1$-cube $X$ in $\mathcal{C}$, def. \ref{StronglyCoCartesian}, such that $X(\emptyset) \to X(s)$ is $k_s$-[[n-connected object of an (infinity,1)-topos\|connective]] for $k_s \geq \kappa$ for all $s\in \{1,\cdots, n+1\}$, then $F(X)$ is an $(n+1)$-cube in $\mathcal{D}$ such that the comparison map $$ F(\emptyset) \longrightarrow \underset{{K\,subcube\,of\,F(X)}\atop {K \neq \emptyset}}{\lim} F(K) $$ (to the indicated [[homotopy limit]]) is $(-c + \sum_s k_s)$-[[n-connected object of an (infinity,1)-topos\|connective]]. The functor $F$ is called _$\rho$-analytic_ if there is $q$ such that it satisfies the condition $E_n(n\rho - q,\rho + 1)$ for all $n$. =-- (e.g. [Johnson 95, def. 1.1, def. 1.3](#Johnson95)) ## Properties ### Convergence of the Goodwillie-Taylor tower For $\rho$-analytic functors their [[Goodwillie-Taylor tower]] converges to them on $\rho$-[[n-connected object of an (infinity,1)-topos\|connective objects]]. See there. ## Examples ### The identity functor on homotopy types The identity $(\infty,1)$-functor on [[∞Grpd]] is 1-analytic, def. \ref{rhoAnalyticFunctor}. For $n = 2$ this is the statement of the [[Blakers-Massey theorem]], for $n \gt 2$ this is the statment of the [higher cubical BM-theorems](Blakers-Massey+theorem#HigherCubical). (see e.g. [Munson-Volic 15, example 10.1.18](#MunsonVolic15)) ## Related concepts * [[excisive (∞,1)-functor]] * [[polynomial (∞,1)-functor]] ## References The concept is due to * [[Tom Goodwillie]], _Calculus II: Analytic functors_, K-Theory 01/1991; 5(4):295-332. DOI: 10.1007/BF00535644 Review includes * [[Tom Goodwillie]], section 3 of _The differential calculus of homotopy functors_, Proceedings of the International Congress of Mathematicians in Kyoto 1990, Vol. I, Math. Soc. Japan, 1991, pp. 621–6 ([article pdf](https://math.mit.edu/juvitop/old/notes/2009_Fall/goodwillie-icm.pdf), [full proceedings Vol I pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.2/ICM1990.2.ocr.pdf), [[GoodwillieICM1990.pdf:file]]) * {#Johnson95} Brenda Johnson, _The derivatives of homotopy theory_, Transactions of the AMS, Volume 347, 1995 ([pdf](http://www.ams.org/journals/tran/1995-347-04/S0002-9947-1995-1297532-6/S0002-9947-1995-1297532-6.pdf)) A textbook account is in * {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], _Cubical homotopy theory_, Cambridge University Press, 2015 [pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf) [[!redirects analytic (∞,1)-functors]] [[!redirects analytic (infinity,1)-functor]] [[!redirects analytic (infinity,1)-functors]] [[!redirects analytic functor]] [[!redirects analytic functors]]
analytic affine line	https://ncatlab.org/nlab/source/analytic+affine+line	#Contents# * table of contents {:toc} ## Idea The analytic affine line is a [[Berkovich analytic space]] that is the analog of the [[affine line]] in [[analytic geometry]]. It is the [[analytic spectrum]] of the [[polynomial]] ring over the given base [[field]]. ## References * [[Jérôme Poineau]], _An introduction to the Berkovich line over $\mathbb{Z}$_ ([pdf](http://www-irma.u-strasbg.fr/~poineau/Textes/LineoverZ.pdf)) For general references see at _[[analytic space]]_. [[!redirects analytic affine lines]] [[!redirects Berkovich line]] [[!redirects Berkovich lines]]
analytic completion of a ring	https://ncatlab.org/nlab/source/analytic+completion+of+a+ring	[[!redirects analytic completion]] #Contents# * table of contents {:toc} ## Idea For $A$ a [[commutative ring]] and $p$ a [[prime number]], then there is the [[formal completion\|topological completion]] at $(p)$, the _[[p-completion]]_ $A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_n A/(p^n A)$. The _analytic completion_ is instead the [[quotient]] $$ A[ [ x ] ]/(x-p)A[ [ x ] ] \,. $$ For $A= \mathbb{Z}$ the [[integers]] both constructions agree, up to [[isomorphism]], and yield the [[p-adic integers]]. In general though they are different. ## References * {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, Vol 304, Springer 1972 * [[Charles Rezk]], _Analytic completion_ ([pdf](https://rezk.web.illinois.edu/analytic-paper.pdf)) [[!redirects analytic completions]]
analytic continuation	https://ncatlab.org/nlab/source/analytic+continuation	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Complex geometry +--{: .hide} [[!include complex geometry - contents]] =-- #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In [[analysis]], _analytic continuation_ refers to the extension of the [[domain]] of an [[analytic function]]. In particular it often refers to the extension of an analytic function on the [[real line]] to a [[holomorphic function]] on the [[complex plane]]. ## Related concepts * [[analytically continued Chern-Simons theory]] * [[zeta function]] * [[zeta function regularization]] * [[Wick rotation]], [[thermal quantum field theory]] ## References See also * Wikipedia, _[Analytic continuation](http://en.wikipedia.org/wiki/Analytic_continuation)_ In [[p-adic geometry]] the need for analytic continuation motivates the [[G-topology]], see the introduction of * {#BoschGuntzerRemmert84} [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf)) [[!redirects analytic continuations]]
analytic function	https://ncatlab.org/nlab/source/analytic+function	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analysis +-- {: .hide} [[!include analysis - contents]] =-- #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- # Analytic functions * table of contents {: toc} ## Idea An _analytic function_ is a [[function]] that is locally given by a converging [[power series]]. ## Definitions Let $V$ and $W$ be [[complete space\|complete Hausdorff]] [[topological vector spaces]], let $W$ be [[locally convex space\|locally convex]], let $c$ be an [[element]] of $V$, and let $(a_0,a_1,a_2,\ldots)$ be an [[infinite sequence]] of [[homogeneous operator]]s from $V$ to $W$ with each $a_k$ of degree $k$. Given an element $c$ of $V$, consider the [[infinite series]] $$ \sum_k a_k(x - c) $$ (a [[power series]]). Let $U$ be the [[interior]] of the set of $x$ such that this series converges in $W$; we call $U$ the __domain of convergence__ of the power series. This series defines a [[function]] from $U$ to $W$; we are really interested in the case where $U$ is [[inhabited set\|inhabited]], in which case it is a [[balanced neighbourhood]] of $c$ in $V$ (which is Proposition 5.3 of [Bochnak--Siciak](#BS)). Let $D$ be any [[subset]] of $V$ and $f$ any [[continuous function]] from $D$ to $W$. This function $f$ is __analytic__ if, for every $c \in D$, there is a power series as above with inhabited domain of convergence $U$ such that $$ f(x) = \sum_k a_k(x - c) $$ for every $x$ in both $D$ and $U$. (That $f$ is continuous follows automatically in many cases, including of course the finite-dimensional case.) ## Generalisation The vector spaces $V$ and $W$ may be generalised to [[analytic manifold]]s and (more generally) [[analytic space]]s. However, these are [[manifolds]] and [[varieties]] modelled on vector spaces using analytic [[transition functions]], so the notion of analytic function between vector spaces is most fundamental. ## Complex-analytic functions of one variable If $W$ is a vector space over the [[complex numbers]], then we have this very nice theorem, due essentially to [[Édouard Goursat]]: +-- {: .un_theorem} ###### Theorem A function from $D \subseteq \mathbb{C}$ to $W$ is [[differentiable function\|differentiable]] if and only if it is analytic. =-- (Differentiability here is in the usual sense, that the difference quotient converges in $W$.) See [[holomorphic function]] and [[Goursat theorem]]. ## Related concepts * [[analytic geometry]] * [[holomorphic function]], [[meromorphic function]] * [[smooth function]] * [[analytic (∞,1)-functor]] * [[HoTT book real numbers]] ## References The theory of analytic function was constructed to some extent by * M. Krasner (1940) and in full generality by * [[John Tate]] (1961) Textbook accounts: * {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs (1965) * {#BS} Jacek Bochnak and Józef Siciak, _Analytic functions in topological vector spaces_; Studia Mathematica 39 (1971); ([pdf](http://matwbn.icm.edu.pl/ksiazki/sm/sm39/sm3916.pdf)). * {#Schanuel} [[Stephen Schanuel]], _Continuous extrapolation to triangular matrices characterizes smooth functions_, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. [web](http://www.sciencedirect.com/science/journal/00224049/24/1) [[!redirects analytic function]] [[!redirects analytic functions]] [[!redirects analytic map]] [[!redirects analytic maps]] [[!redirects complex analytic]]
analytic geometry	https://ncatlab.org/nlab/source/analytic+geometry	> This entry is about [[geometry]] based on the study of _[[analytic functions]]_, hence about [[analytic varieties]]. > This is unrelated to "analytic geometry" in the sense of methods in the geometry of $n$-dimensional [[Euclidean space]] involving [[coordinate]] calculations (as opposed to [[synthetic geometry]]); which is usually combined with linear algebra taught in a geometric way. For this latter meaning see at _[[coordinate system]]_. +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In research mathematics, when one says _analytic geometry_, then "analytic" refers to [[analytic functions]] in the sense of [[Taylor expansion]] and by __analytic geometry__ one usually means the study of geometry of [[complex manifolds]]/[[complex analytic spaces]], as well as their analytic subsets, [[Stein domains]] and related notions. More generally one may replace the complex numbers by [[non-archimedean fields]] in which case one speaks of _[[rigid analytic geometry]]_. Similarly to an [[algebraic variety]], an [[analytic variety]] is locally given as a [[zero locus]] of a [[finite set]] of [[analytic functions]], i.e. of [[holomorphic functions]] in complex analytic geometry. A short survey can be found in a chapter of Dieudonne's _Panorama of pure mathematics_. In addition to analytic geometry over complex numbers, there is also another formalism which allows for [[archimedean field\|nonarchimedean]] [[ground fields]]. This is the subject of [[rigid analytic geometry]] or [[global analytic geometry]]. Similarly to [[schemes]], rigid analytic varieties are glued from [[Bercovich spectrum\|Bercovich spectra]] of certain commutative Banach algebras, so-called [[affinoid]]s, in a certain [[Grothendieck topology]]. (See _[[analytic space]]_.) There are several variants of the formalism (e.g. due Huber). The subject is closely related to [[formal geometry]] and has its main applications in [[arithmetic geometry]] and [[representation theory]]. It is an open problem to find an appropriate analogue of rigid analytic geometry in [[noncommutative geometry]], which is supposed to play an important role in [[mirror symmetry]]. Local properties of analytic manifolds and spaces are studied in [[local analytic geometry]]. ## Theorems (...) * [[Ostrowski's theorem]] (...) ## Holomorphic functions of several complex variables This section is about certain aspects of [[holomorphic functions]] $\mathbb{C}^n \to \mathbb{C}$. Currently it concentrates on aspects of relevance in the application to [[AQFT]], such as a version of the [edge-of-the-wedge] (http://en.wikipedia.org/wiki/Edge-of-the-wedge_theorem) theorem. From the viewpoint of complex [[manifold]]s this is the _local_ theory that describes the situation in coordinate patches. * [Wikipedia] (http://en.wikipedia.org/wiki/Several_complex_variables) When we talk about holomorphic functions in the following and do not specify the domain, we will always assume that the domain is an open, simply connected subset of $\mathbb{C}^n$. ### relevance to quantum field theory (QFT) In the [[AQFT]] formulation (actually the following description is the [[Heisenberg picture]] of quantum mechanics in a nutshell) [[selfadjoint operator]]s $A$ on a [[Hilbert space]] $\mathcal{H}$ are the [[observables]] of a physical system, while normed vectors $x, y \in \mathcal{H}$ represent the [[states]] the system can be in. The [[real number]] $\langle y, A x \rangle$ represents the [[probability]] that a system starting in state $x$ will be in state $y$ after a [[measurement]] of $A$. In [[AQFT]] we often encounter a set of operators indexed by several complex variables $z = (z_1, z_2, ...)$ and try to deduce properties of the theory from the function $f(z) := \langle y, A(z)x \rangle$. In this way, the theory of [[holomorphic functions]] of several variables is promoted to an irreplaceable tool in quantum field theory. ### Hartogs' theorem One striking difference of functions of several _real_ variables and several _complex_ variables is described by Hartogs' theorem on separate analyticity: * Theorem (Hartogs): Let f be a $\mathbb{C}$-valued function defined in an open set $U \subset \mathbb{C}^n$. Suppose that f is analytic in each variable $z_j$ when the other coordinates $z_k$ are fixed. Then f is analytic as a function of all n coordinates. _remark_: There are no other assumptions about f necessary, it needs not to be continous or even measurable. Note that in the real case the property of being partially differentiable alone encodes nearly no information about a function. Reference: * Paul Garrett: [Hartogs theorem] (http://www.math.umn.edu/~garrett/m/complex/hartogs.pdf) _relevance_: When reading [[AQFT]] literature you will often encounter the claim that given functions are holomorphic, Hartogs' theorem simplifies the task of checking these claims considerably, because you have to check the holomorphy in every single variable only. ### analogues from the one-dimensional theory Some results remain true in the multi dimensional case. * Identity theorem: If a holomorphic function is zero in a neighbourhood of a point, it is the zero function. _remark_: As usual the domain is supposed to be an open, simply connected (not necessarily proper) subset of $\mathbb{C}^n$, which implies that the point of the precondition of the theorem is an interior point of the domain. ### domains of holomorphy One of the most notably difference of the theory of _one_ complex variable and of _several_ complex variables is that the [riemann mapping theorem] (http://en.wikipedia.org/wiki/Riemann_mapping_theorem) fails in several complex variables, which is in a certain sense the reason why in several complex variables there are domains which can be enlarged such that _all_ holomorphic functions extend to the larger domain. _handwaving_ why this is not possible in one dimension: According to the riemann mapping theorem every domain (open, simply connected proper subset of $\mathbb{C}$) is biholomorph equivalent to the open disk $E: = \{ z: \|z\| \lt 1 \}$, which means that the rings of holomorphic functions are isomorph, too. But the ring of holomorphic functions on E has to every point in the boundary of E a function that has a pole in this point, so that E cannot be enlarged in a way that all holomorphic functions are extentable. Therefore this applies to every domain. Some domains in $\mathbb{C}^n$ _do_ have the property that they cannot be enlarged, and since this is an interesting property, the name domain of holomorphy was coined for these, and the question how they could be described was promoted to an interesting research topic in the beginning of the 20th century. * [Wikipedia] (http://en.wikipedia.org/wiki/Domain_of_holomorphy) #### edge of the wedge theorems These theorems desribe situations where holomorphic functions defined on specific domains (the wedges) can be continued to holomorphic functions of larger domains. * [Wikipedia] (http://en.wikipedia.org/wiki/Edge-of-the-wedge_theorem) They are a valuable tool in [[AQFT]] (and were in fact discovered by one of the fathers of the theory, [Nikolay Bogolyubov] (http://en.wikipedia.org/wiki/Nikolay_Bogoliubov)). We state here one version that will be of use to the nLab: * theorem (edge of the wedge): Let $K := \{ z \in \mathbb{C}^n: \|z\| \le r \}$ be a ball in $\mathbb{C}^n$ and let $\mathcal{C} \subset \mathbb{R}^n$ be an open convex cone such that $\mathcal{C} \cap (- \mathcal{C}) \neq \emptyset $. We put $z = x + iy$ with $x, y \in \mathbb{R}^n$ and define an open domain G by $G:= \{ z = x + iy: z \in K, y \in \mathcal{C} \}$. Let $f$ be a holomorphic function in G and assume that $$ \lim_{y \to 0, y \in \mathcal{C}} f(x + iy) $$ exists for all $x \in B_r \subset \mathbb{R}^n$, where $B_r$ is an open ball with radius r. (The limit may not depend on the specific sequence chosen). Then $f$ is holomorph extendable into an open region $G \cup G_0$ with $$ G_0 := \bigcup_{x \in B_r} \{ z: \|z-x\| \le \theta \cdot dist(x, \partial U) \} $$ with $0 \lt \theta \lt 1$ a constant that is independent from $x, B_r$, and $f$. _proof_: V.S.Vladimirov, "theory of functions of several complex variables" ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0125.31904&format=complete)) ## Related concepts * [[several complex variables]], * [[rigid analytic geometry]], [[analytic spectrum]], [[analytic space]], [[Stein space]], [[Berkovich space]], [[additive analytic geometry]] * [[G-topology]] * [[Oka principle]] * [[p-adic geometry]] * [[B1-homotopy theory]] * [[global analytic geometry]] * [[overconvergent global analytic geometry]] ## References {#References} Course notes on ([[global analytic geometry\|global]]) analytic geometry are in * [[Frédéric Paugam]], _Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf)) and for [[rigid analytic geometry]] in * Kiran Sridhara Kedlaya, _Introduction to Rigid Analytic Geometry_ ([web](http://www-math.mit.edu/~kedlaya/18.727/notes.html)) * [[Brian Conrad]], _Several approaches to non-archimedean geometry_ ([pdf](http://math.stanford.edu/~conrad/papers/aws.pdf)) A gentle and modern introduction to [[complex manifolds]] that starts with an extensive exposition of the local theory is this: * [[Daniel Huybrechts]], _Complex geometry. An introduction._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1055.14001&format=complete)) * [[Hans Grauert]], Reinhold Remmert, _Theory of Stein spaces_, Grundlehren der Math. Wissenschaften __236__, Springer 1979, xxi+249 pp.; _Coherent analytic sheaves_, Grundlehren der Math. Wissenschaften __265__, Springer 1984. xviii+249 pp.; _Komplexe Räume_, Math. Ann. __136__, 1958, 245–318, [DOI](http://dx.doi.org/10.1007/BF01362011) Discussion of [[Berkovich space]] analytic geometry as [[algebraic geometry]] in the general sense of [[Bertrand Toën]] and [[Gabriele Vezzosi]] is in * [[Oren Ben-Bassat]], [[Kobi Kremnizer]], _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338)) For more see the references at [[rigid analytic geometry]] and at [[analytic space]]. category: analysis, geometry
analytic geometry -- contents	https://ncatlab.org/nlab/source/analytic+geometry+--+contents	[[analytic geometry]] ([[complex analytic geometry\|complex]], [[rigid analytic geometry\|rigid]], [[global analytic geometry\|global]]) [[geometry]]+[[analysis]]/[[analytic number theory]] ## Basic concepts [[analytic function]] [[analytic space]], [[analytic variety]], [[Berkovich space]] * [[polydisc]] * [[affinoid algebra]], [[analytic spectrum]] * [[complex analytic ∞-groupoid\|analytic ∞-groupoid]] [[analytification]] ## Theorems [[GAGA]]
analytic geometry ingredients -- table	https://ncatlab.org/nlab/source/analytic+geometry+ingredients+--+table	\| [[algebra\|algebraic structure]] \| [[group]] \| [[ring]] \| [[field]] \| [[vector space]] \| [[associative algebra\|algebra]] \| \|---\|-----------\|-----------\|-------------------\|--------------\|---\| \| (submultiplicative) [[norm]] \| [[normed group]] \| [[normed ring]] \| [[normed field]] \| [[normed vector space]] \| [[normed algebra]] \| \| multiplicative [[norm]] ([[absolute value]]/[[valuation]]) \| \| \| [[valued field]] \| \| \| \| [[complete space\|completeness]] \| [[complete normed group]] \| [[Banach ring]] \| [[complete field]] \| [[Banach vector space]] \| [[Banach algebra]] \|
analytic Langlands program	https://ncatlab.org/nlab/source/analytic+Langlands+program	[[!redirects analytic langlands program]] #Contents# * table of contents {:toc} ## Idea The aim of analytic version of the [[Langlands program]] is to find a theory of analytic motives that gives a full correspondence between all (not only algebraic) automorphic representations and all analytic motives. There are many interesting automorphic representation that have nothing very clear to do with algebraic varieties (Mass forms, etc...). They should however be related to natural analytic motivic coefficients. ## References * [[Pavel Etingof]], [[Edward Frenkel]], [[David Kazhdan]], A general framework for the analytic Langlands correspondence [[arXiv:2311.03743](https://arxiv.org/abs/2311.03743)] [[!redirects analytuc Langlands correspondence]]
analytic manifold	https://ncatlab.org/nlab/source/analytic+manifold	#Contents# * table of contents {:toc} ## Idea An analytic manifold is a [[manifold]] with [[analytic function\|analytic]] transition functions over some [[field]]. The most widely studied are real-analytic and complex analytic manifolds. Analytic manifolds are studied in [[analytic geometry]]. ## References * L. Schwarz, _Lectures on complex analytic manifolds_ (1955) ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr04.pdf)) [[!redirects analytic manifolds]] [[!redirects complex analytic manifold]]
analytic Markov's principle	https://ncatlab.org/nlab/source/analytic+Markov%27s+principle	#Contents# * table of contents {:toc} ## Definition ## The analytic Markov's principle states that the [[pseudo-order]] on the [[Dedekind real numbers]] is a [[stable relation]]: for all [[real numbers]] $r \in \mathbb{R}$ and $s \in \mathbb{R}$, $\neg \neg (r \lt s)$ implies $r \lt s$. This is equivalent to the usual formulation of the analytic Markov's principle, which says that for all real numbers $x \in \mathbb{R}$, $\neg (x \leq 0)$ implies $0 \lt x$. For if we take $x = s - r$, this becomes $\neg (s - r \leq 0)$ implies $0 \lt s - r$, and by the order and arithmetic properties of the real numbers, this is equivalent to $\neg (s \leq r)$ implies $r \lt s$, which is the same as $\neg \neg (r \lt s)$ implies $r \lt s$. Other equivalent statements include that the [[tight apartness relation]] on the Dedekind real numbers is a [[stable relation]]. The analytic Markov's principle makes sense for any [[ordered local ring\|ordered]] [[local Artinian algebra\|local Artinian $\mathbb{R}$-algebra]] as well, where the relation $\lt$ is in general only a [[strict weak order]] instead of a [[pseudo-order]], the [[preorder]] $\geq$ is not a [[partial order]], and the equivalence relation $a \approx b$ derived from the preorder holds if and only if $a - b$ is [[nilpotent]]. The quotient of the Weil $\mathbb{R}$-algebra by its [[nilradical]] is the Dedekind real numbers satisfying the analytic Markov's principle. ## See also ## * [[Markov's principle]] * [[axiom of real cohesion]] * [[intermediate value theorem]] ## References ## * [[Mike Shulman]], Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
analytic monad	https://ncatlab.org/nlab/source/analytic+monad	## Idea Analytic monads are [[monads]] on [[Set]] that correspond to [[operads]] in [[Set]]. ## Definition More precisely, an [[operad]] $O$ in [[Set]] induces a [[monad]] $T$ on [[Set]]: $$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$$ Such a monad $T$ is equipped with a canonical weakly [[cartesian natural transformation]] to the [[monad]] $Sym$ arising from the [[commutative operad]]. [[finitary monad\|Finitary monads]] correspond to [[algebraic theories]], and the analytic monads correspond to algebraic theories that are "linear regular", that is to say, they can be presented using only equations where the same variables appear on both sides and exactly once on each side (see e.g. [SzawielZawadowski](#SzawielZawadowski)). ## Properties A theorem of [Joyal](#Joyal) states that there is a [[monoidal equivalence]] between the [[monoidal category]] of [[endofunctors]] $Set\to Set$ that admits a weakly [[cartesian natural transformation]] to $Sym$ and the [[monoidal category]] of [[species]], i.e., symmetric sequences in [[Set]] with the [[substitution product]]. In particular, the category of analytic [[monads]] on [[Set]] is equivalent to the category of [[operads]] in [[Set]]. ## The colored case The correspondence carries over to [[colored operads]] (with a [[set]] of colors $C$) if we use the [[slice category]] $Set/C$ instead of [[Set]]. ## The nonsymmetric case A similar correspondence can be established for [[nonsymmetric operads]], except that we must include the data of a _cartesian_ (not weakly cartesian) transformation to the [[monad]] of the [[associative operad]], which is no longer unique. See Example 4.2.14 in [Leinster's book](#Leinster). ## The homotopical case The correspondence generalizes to [[(∞,1)-categories]], with some statements becoming more elegant. See [Gepner–Haugseng–Kock](#GHK). ## Related concepts * [[species]] * [[operad]] * [[monad]] ## References * {#Joyal} [[André Joyal]], _Foncteurs analytiques et espèces de structures_, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159. [doi](http://dx.doi.org/10.1007/bfb0072514). * [[Mark Weber]], _Generic morphisms, parametric representations and weakly Cartesian monads_, Theory Appl. Categ. 13 (2004), 191–234. * [[Tom Leinster]], _[[Higher Operads, Higher Categories]]_, London Mathematical Society Lecture Note Series 298 (2004), [doi](http://dx.doi.org/10.1017/cbo9780511525896). * {#GHK} [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, [arXiv:1712.06469](https://arxiv.org/abs/1712.06469). * {#SzawielZawadowski} [[Stanislaw Szawiel]] and [[Marek Zawadowski]], _Theories of analytic monads_, Math. Struct. Comp. Sci. 24 (2014). [arxiv:1204.2703](https://arxiv.org/pdf/1204.2703.pdf)
analytic motives	https://ncatlab.org/nlab/source/analytic+motives	Analytic motives give a common generalization of classical homotopy theory (Ayoub's analytic motives over C), algebraic motivic homotopy theory (strict analytic motives over Z with the trivial non-archimedean norm), and Ayoub's non-archimedean motives. The basic idea is to extend motivic homotopy theory to a general base Banach ring, by replacing the affine line by the overconvergent unit disc. Analytic stable homotopy theory gives a natural setting to define homotopy invariant cohomologies, that are particularly interesting when one works over an extension of the field of rational numbers. However, they seem to be less well adapted to the study of torsion phenomena in de Rham type cohomology theories over an integral base, because homotopy invariance is not available in this situation, and imposing it artificially kills the p-torsion information in caracteristic p. References: Joseph Ayoub: Betti realization of motives and motives for rigid analytic varieties. [[Frederic Paugam]]: Overconvergent global analytic geometry (in preparation).
analytic motivic homotopy theory	https://ncatlab.org/nlab/source/analytic+motivic+homotopy+theory	#Contents# * table of contents {:toc} ## Idea Analytic motivic homotopy theory is an analytic generalization of [[motivic homotopy theory]] that gives back classical homotopy theory when one works with analytic varieties over $\mathbb{C}$ and motivic homotopy theory when one works with strict analytic varieties over a ring equipped with its trivial norm. The proper setting for analytic motivic homotopy theory of quasi-projective varieties seems to be the setting of [[logarithmic motivic homotopy theory]]. ## Related subjects [[motivic homotopy theory]] [[logarithmic motivic homotopy theory]] ## References [[Joseph Ayoub]] _La realisation de Betti et les six opérations_ (defines complex analytic motives and shows that they are equivalent to classical homotopy types) [[Joseph Ayoub]] _Motives of rigid analytic varieties_ (defines rigid motives and uses them to study the functor of vanishing cycles on a characteristic $0$ "trait"). [[Frédéric Paugam]] _Overconvergent global analytic geometry_ (defines motives over an arbitrary base Banach ring)
analytic number theory	https://ncatlab.org/nlab/source/analytic+number+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The branch of [[number theory]] that uses [[analysis]]. Related to [[global analytic geometry]] as [[number theory]] is to [[arithmetic geometry]]. ## References * Wikipedia, _[Analytic number theory](http://en.wikipedia.org/wiki/Analytic_number_theory)_
analytic philosophy	https://ncatlab.org/nlab/source/analytic+philosophy	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Philosophy +-- {: .hide} [[!include philosophy - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea _Analytic philosophy_ is a school of [[philosophy]] emphasizing clarity of argument, [[formal logic]], and aiming for strong connections with natural [[sciences]] such as [[physics]]. > Analytic philosophy is characterized above all by the goal of clarity, the insistence on explicit argumentation in philosophy, and the demand that any view expressed be exposed to the rigours of critical evaluation and discussion by peers. (European Society for Analytic Philosophy, homepage of website <http://www.dif.unige.it/esap>; accessed 18 October 2011) Analytic philosophy to some extent defined itself, via people like [[Bertrand Russell]] in a movement known as the "revolt again idealism", in opposition to forms of German idealism, notably the [[objective idealism]] of [[Georg Hegel]] as expressed in his _[[Science of Logic]]_. See at _[Perception of Hegel's Naturphilosophie](Georg+Hegel#PerceptionOfHegelsPhilosophy)_ for more on this. Specifically, analytic philosophy aims to detect the underlying logical form of propositions and analyse the concepts they rely upon. Since the surface grammar of natural language is taken to be often misleading, this work of analysis is often done in terms of formal languages such as [[first-order logic]] or [[modal logic]]. Examples: * Where it might appear that holding the belief that 'Unpunctuality is reprehensible' commits one to the existence of something denoted by 'unpunctuality', rewording the sentence, as Gilbert Ryle did, as 'Whoever is unpunctual deserves that other people should reprove him for being unpunctual' avoids this commitment. As a further step, the meaning of 'X deserves Y' would now be a candidate for analysis. * [[Martin Heidegger\|Heidegger]] in _What is Metaphysics?_ passes from his claims "What is to be investigated is being only and — nothing else; being alone and further — nothing; solely being, and beyond being — nothing" to a discussion of what he takes to be their subject by asking "What about this Nothing?". For Carnap this was just to be have misled by the grammar which appear to make 'Nothing' a subject, but is to be properly analyzed in terms of negation and universal quantification. So convinced was Russell of the power of the then new first-order logic that he wrote: >The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds are beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion. (Bertrand Russell, '[[Logic as the Essence of Philosophy]]', 1914) See also [SEP: Conceptions of Analysis in Analytic Philosophy](http://plato.stanford.edu/entries/analysis/#6). Remark. While indeed [[first-order logic]] has little to no resemblance to [[Science of Logic\|Hegel's logic]], for its refinement by [[type theory]] in the guise of [[modal type theory]] the situation is quite different. [[William Lawvere]] has argued that key aspects of [[Science of Logic\|Hegel's logic]] do have a useful formalization in [[modal type theory]] (Lawvere mostly considered this in the corresponding [[categorical semantics]]). Lawvere's formalization of Hegel's concepts of _[[unity of opposites]]_, _[[Aufhebung]]_, _[[category of being]]_ etc. lead to non-trivial [[theorems]] in the [[foundations of mathematics]] and for the [[geometry of physics]]. ## References * Wikipedia, _[Analytic philosophy](http://en.wikipedia.org/wiki/Analytic_philosophy)_
analytic ring	https://ncatlab.org/nlab/source/analytic+ring	#Contents# * table of contents {:toc} ## Definition \begin{definition}([ScholzeLCM](#ScholzeLCM), Def. 7.1) In [[condensed mathematics]], a _pre-analytic ring_ is a [[condensed ring]] $\underline{A}$ together with a [[functor]] $S\mapsto\mathcal{A}[S]$ from [[extremally disconnected topological spaces]] to $\underline{\mathcal{A}}$-[[modules]] in [[condensed abelian groups]] taking finite [[disjoint unions]] to [[products]], and a [[natural transformation]] $S \to\mathcal{A}[S]$. \end{definition} \begin{definition}([ScholzeLCM](#ScholzeLCM), Def. 7.4) An _analytic ring_ is a pre-analytic ring $\mathcal{A}$ such that for any [[chain complex]] $C$ of $\underline{\mathcal{A}}$-modules in [[condensed abelian groups]] such that all $C_{i}$ are [[direct sums]] of objects of the form $\mathcal{A}[T]$ for varying [[extremally disconnected topological space\|extremally disconnected]] $T$, the map $$R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)\to R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)$$ of [[condensed abelian groups]] is an [[isomorphism]] for all extremally disconnected $S$. \end{definition} ## Examples The following examples come from [ScholzeLCM](#ScholzeLCM), Examples 7.3, as examples of pre-analytic rings. They are shown to be analytic rings as well later on in the same reference. * The analytic ring $\mathbb{Z}_{\square}$ is given by $\underline{\mathcal{A}}=\mathbb{Z}$ and $S\mapsto \mathbb{Z}[S]^{\square}$ (in the notation of [[solid abelian group]]). * For $A$ a discrete ring, the analytic ring $(A,\mathbb{Z})_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto \mathbb{Z}_{\square}[S]\otimes_{\mathbb{Z}} A$. * For $A$ a finitely generated $\mathbb{Z}$-algebra, the condensed ring $A_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto A_{\square}[S] \;\coloneqq\; \underset {\underset{i}{\leftarrow}} {\lim} \; A\big[S_{i}\big] \, $. ## Related concepts * [[condensed mathematics]] * [[solid module]] ## References * {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)
analytic space	https://ncatlab.org/nlab/source/analytic+space	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- #### Complex geometry +--{: .hide} [[!include complex geometry - contents]] =-- =-- =-- # Analytic spaces * table of contents {: toc} ## Idea Analytic spaces are [[spaces]] that are locally modeled on [[Isbell duality\|formal duals]] of sub-[[associative algebra\|algebras]] of [[power series]] algebras on elements with certain [[convergence]] properties with respect to given [[seminorms]]. This is in contrast to ([[formal geometry\|formal]]) [[algebraic spaces]] (([[formal scheme\|formal]]) [[schemes]]) where no [[convergence]] properties are considered. In [[complex number\|complex]] [[analytic geometry]] analytic spaces -- [[complex analytic space]] -- are a vast generalization of complex [[analytic manifolds]] and are usually treated in the formalism of [[locally ringed spaces]]. In this case the _[[GAGA]]-principle_ closely relates [[complex analytic geometry]] with [[algebraic geometry]] over the [[complex numbers]]. In the case of non-archimedean ground field, the topology of the affine space is totally disconnected what requires different approach than, say, over complex numbers. This leads to several variants like [[rigid analytic geometry]], [[Berkovich space]]s. [[Huber's adic spaces]] and so on. ## Related entries * [[analytic geometry]], * [[complex analytic space]], [[analytic variety]], [[Berkovich space]] * [[Banach analytic space]] * [[B1-homotopy theory]] * [[perfectoid space]] ## References * Wikipedia, _[Analytic space](http://en.wikipedia.org/wiki/Analytic_space)_ Discussion for [[complex analytic spaces]] and [[Stein spaces]] is in * [[Hans Grauert]], [[Reinhold Remmert]], _Theory of Stein spaces_, Grundlehren der Math. Wissenschaften __236__, Springer 1979, xxi+249 pp.; _Coherent analytic sheaves_, Grundlehren der Math. Wissenschaften __265__, Springer 1984. xviii+249 pp.; _Komplexe Räume_, Math. Ann. __136__, 1958, 245–318, [DOI](http://dx.doi.org/10.1007/BF01362011) [[!redirects analytic space]] [[!redirects analytic spaces]]
analytic spectrum	https://ncatlab.org/nlab/source/analytic+spectrum	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The concept of _analytic spectrum_ is a realization of the concept of [[spectrum (geometry)]] in the context of [[non-archimedean analytic geometry]]. Given an [[affinoid algebra]] $A$ over a [[non-archimedean field]], then the concept of [[spectrum of a commutative ring]] whose points are [[prime ideals]]/[[maximal ideals]] does not produce a sensible space that admits [[analytic geometry]]. Rather, instead of regarding points of the spectrum as ring homomorphisms to $A \to \mathbb{R}$, the analytic spectrum instead takes points to be multiplicative [[seminorms]] ${\vert -\vert} \colon A \to \mathbb{R}_{\geq 0}$ bounded by the norm on the given field. If $Spec_{an}(A)$ is the set of all such multiplicative seminorms, then for $x \in Spec_{an}(A)$ a point one writes the corresponding seminorm as ${\vert - \vert}_x$ and thinks of it as being the norm on the function algebra on $Spec(A)$ which is given by "evaluating functions at $x$ and then applying the field norm to that". One turns this $Spec_{an}(A)$ into a [[topological space]] in the usual way by choosing the weakest topology such that under this assignment the original elements of $A$ become [[continuous function]] on $Spec_{an}(A)$. Globalizing this analytic spectrum construction leads to the concept of [[Berkovich analytic space]]. ## Definition For $A$ a [[normed ring]], its _analytic spectrum_ or _Berkovich spectrum_ $Spec_an A$ is the set of all non-zero multiplicative [[seminorms]] on $A$, regarded as a [[topological space]] when equipped with the weakest topology such that all [[functions]] $$ Spec_{an} A \to \mathbb{R}_+ $$ of the form $$ x \mapsto {\vert x(a)\vert} $$ for $a \in A$ are [[continuous function\|continuous]]. If $A$ is equipped with the structure of a [[Banach ring]], one takes the _[[bounded map\|bounded]]_ multiplicative seminorms. So a point in the analytic spectrum of $A$ corresponds to a non-zero [[function]] $$ {\Vert -\Vert} : A \to \mathbb{R} $$ to the [[real numbers]], such that for all $x, y \in A$ 1. ${\Vert x \Vert} \geq 0$; 1. ${\Vert x y \Vert} = {\Vert x \Vert} {\Vert y \Vert}$; 1. ${\Vert x + y \Vert} \leq {\Vert x \Vert} + {\Vert y \Vert}$ and boundedness means that there exists $C \gt 0$ such that for all $x \in A$ $$ {\Vert x\Vert} \leq C {\vert x \vert}_A \,. $$ (e.g. [Berkovich 09, def. 1.2.3](#Berkovich09)) ## Examples ### Affine line For $k$ a [[field]] and $k[T]$ the [[polynomial ring]] over $k$ in one generator, $$ \mathbb{A}_k := Spec_{an} k[T] $$ is the [[analytic affine line]] over $k$. If $k = \mathbb{C}$, then $\mathbb{A}_k = \mathbb{C}$ is the ordinary [[complex plane]]. ## Related concepts * [[spectrum (geometry)]] * [[analytic geometry]], [[analytic space]] * [[p-adic geometry]] ## References ### Original The notion originates in * {#Berkovich90} [[Vladimir Berkovich]], _Spectral theory and analytic geometry over non-Archimedean fields_, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp. ### Expositions and Lecture notes Introductory exposition of the Berkovich analytic spectrum includes * [[Sarah Brodsky]], _Non-archimedean geometry_, brief lecture notes, 2012 ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf)) * [[Scott Carnahan]], _Berkovich spaces I_ ([web](http://sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i/)) * {#Poineau07} [[Jérôme Poineau]], _Global analytic geometry_, pages 20-23 in EMS newsletter September 2007 ([pdf](http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf)) * [[Frédéric Paugam]], section 2.1.4 of _Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf)) * {#Berkovich09} [[Vladimir Berkovich]], section 1 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf)) [[!redirects analytic spectra]] [[!redirects Berkovich analytic spectrum]] [[!redirects Berkovich analytic spectra]] [[!redirects Berkovich spectrum]] [[!redirects Berkovich spectra]] [[!redirects Bercovich spectrum]] [[!redirects Bercovich spectra]]
analytic torsion	https://ncatlab.org/nlab/source/analytic+torsion	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Riemannian geometry +--{: .hide} [[!include Riemannian geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea What is called _analytic torsion_ or _Ray-Singer torsion_ ([Ray-Singer 73](#RaySinger73)) is the invariant $T(X,g)$ of a [[Riemannian manifold]] $(X,g)$ given by a product of powers of the [[functional determinants]] $det_{reg} \Delta\|_{\Omega^p}$ of the [[Laplace operators]] $\Delta\|_{\Omega^p}$ of the manifold acting on the space of [[differential p-forms]]: $$ T(X,g) \coloneqq \underset{p}{\prod} \left(det_{reg} \Delta\|_{\Omega^p}\right)^{-(-1)^p \frac{p}{2}} \,. $$ This analytic torsion is an [[analogy\|analogue]] in [[analysis]] of the invariant of [[topological manifolds]] called _[[Reidemeister torsion]]_. The two agree for compact Riemannian manifolds ([Cheeger 77](#Cheeger77)). ## Properties ### Relation to Iwasawa theory According to ([Morishita 09](arithmetic+topology#Morishita09)) the relation between Reidemeister torsion and analytic torsion is analogous to that between [[Iwasawa polynomials]] and [[zeta functions]] obtained by [[adelic integration]]. (...) ### Relation to Selberg and Ruelle zeta function {#RelationToSelbergZeta} The [[special value of L-functions\|special value]] of a [[Ruelle zeta function]] at $s= 0$ is expressed by [[Reidemeister torsion]] ([Fried 86](#Fried86)) ### Relation to Chern-Simons theory Analytic torsion appears as one factor in the [[perturbation theory\|perturbative]] [[path integral]] [[quantization]] of [[Chern-Simons field theory]]. See there at _[Quantization -- Perturbative -- Path integral quantization](http://ncatlab.org/nlab/show/Chern-Simons+theory#PerturbativePathIntegralQuantization)_. ## References * {#RaySinger73} D. Ray, [[Isadore Singer]], _Analytic torsion for complex manifolds_, Ann. Math. __98__, 1 (1973), 154--177. * {#Cheeger77} [[Jeff Cheeger]], _Analytic torsion and Reidemeister torsion_, Proc. Natl. Acad. Sci. USA __74__, No. 7, pp. 2651-2654 (1977), [pdf](http://www.pnas.org/content/74/7/2651.full.pdf) * Wikipedia, _[Analytic torsion](http://en.wikipedia.org/wiki/Analytic_torsion)_ * A.A. Bytsenko, A.E. Goncalves, W. da Cruz, _Analytic Torsion on Hyperbolic Manifolds and the Semiclassical Approximation for Chern-Simons Theory_ ([arXiv:hep-th/9805187](http://arxiv.org/abs/hep-th/9805187)) Review of the role played in the perturbative [[quantization of Chern-Simons theory]] includes * {#Young} M. B. Young, section 2 of _Chern-Simons theory, knots and moduli spaces of connections_ ([pdf](http://www.math.sunysb.edu/~myoung/CS.pdf)) Discussion for [[hyperbolic manifolds]] in terms of the [[Selberg zeta function]]/[[Ruelle zeta function]] is due to * {#Fried86} [[David Fried]], _Analytic torsion and closed geodesics on hyperbolic manifolds_, Invent. math. 84, 523-540 (1986) ([pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0084&divID=LOG_0026&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0084%7C&targetFileName=PPN356556735_0084_LOG_0026.pdf&)) with further developments including * {#BunkeOlbrich94a} [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407012](http://arxiv.org/abs/dg-ga/9407012)) * {#Park09} Jinsung Park, _Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps_, Journal of Functional Analysis Volume 257, Issue 6, 15 September 2009, Pages 1713–1758 In relation to the [[topological string]] and [[black hole entropy]]: * [[Cumrun Vafa]], Ray-Singer Torsion, Topological Strings and Black Holes [[arXiv:2401.12816](https://arxiv.org/abs/2401.12816)] [[!redirects Ray-Singer torsion]]
analytic variety	https://ncatlab.org/nlab/source/analytic+variety	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- #### Complex geometry +--{: .hide} [[!include complex geometry - contents]] =-- =-- =-- # Analytic varieties * table of contents {: toc} ## Idea Analytic varieties form an analogue of [[algebraic varieties]] in analytic context; they are more general than [[analytic manifolds]] in allowing [[singularity\|singularities]]. While an algebraic variety is the loci of zeros of some set of [[polynomials]], an analytic varieties is the loci of zeros of some set of [[analytic functions]]. By [[Chow's theorem]] every [[complex number\|complex]] [[projective variety\|projective]] analytic variety is algebraic; this is based on the machinery of Weierstrass (the [[Weierstrass preparation theorem]] etc.). ## Related concepts * [[analytic space]] * [[complex analytic variety]] ## Literature * P. Griffiths, J. Harris, _Principles of algebraic geometry_ [[!redirects analytic variety]] [[!redirects analytic varieties]]
analytic versus synthetic	https://ncatlab.org/nlab/source/analytic+versus+synthetic	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Philosophy +-- {: .hide} [[!include philosophy - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The analytic-synthetic distinction has a long history stretching back to the ancient Greeks. It has come to mean different things according to the discipline in which it is employed, but each use can trace its origins to the classical version. In the classical world, thinkers such as Aristotle, Euclid, Pappus and Proclus, used these terms to distinguish between methods of enquiry. A synthetic solution to a problem relies on reasoning from first principles, the kind of reasoning we see displayed in Euclid's _Elements_. The solution is thought to be _put together_ (συντίθημι). The kinds of first principle allowed are definitions, common notions and postulates, the latter being concerned with the specific subject matter at hand. By contrast, an analytic solution operates by working backwards from the problem to see what needs to be the case to be able to resolve it. Thus it analyses, or _unravels_ (ἀναλύω), the problem. This exercise might then make contact with things already known from first principles, or lead to new such principles. Often analytic discovery was written up in synthetic fashion. In the seventeenth century, [[Descartes]] understood the distinction in the same way. When asked by Mersennes why he did not present his philosophical arguments in the synthetic fashion, he replied that he considered presentation according to the analytic method as more persuasive. This allowed the reader to see the necessity of the first principles reached, for instance, famously the _Cogito_, 'I think therefore I am'. Descartes' approach to geometry via coordinates allowed him to resolve open questions bequeathed by Pappus and others from the ancient world (see [Domski](#Domski)). Since it could be seen as operating according to an analytic method, it was named _analytic geometry_. ## Analytic-synthetic distinction in philosophy Later in the seventeenth century, we find [[Leibniz]] arguing that for any true statement, universal or singular, necessary or contingent, its subject contains within it the predicate stated to hold of it. For some of these propositions, such as identity statements, this is obvious, but others require considerable work to reveal this to be so: >Implicit containment (or exclusion) was to be revealed by the sort of "analysis of notions" that Leibniz had already emphasized as a crucial philosophical method in his influential paper "Meditations on Knowledge, Truth, and Ideas", and this role accounts both for the general importance of analysis within German rationalism and for Kant’s choice of the term 'analytic' to describe such containment truths. ([Anderson 15, p. 9](#Anderson15)) [[Kant]] famously disagreed with this claim. For him the truth of some propositions relies unavoidably on intuition or empirical sensation along with conceptual understanding. Thus, an _analytic_ proposition for Kant distinguishes a proposition whose predicate concept is wholly contained in its subject concept. A famous example is 'All bachelors are unmarried.' This is sometimes glossed today as _true by virtue of definition_. By contrast, in a _synthetic_ proposition the predicate concept is not wholly contained in the subject content. Kant gives 'All bodies are heavy' as an example of a synthetic statement, whereas 'All bodies are extended' is analytic. Ascertaining that bodies are heavy unavoidably requires empirical sensation. Note, all the same, that Kant continues to use 'analytic' and 'synthetic' in their original methodological sense: > I have adopted in this work the method that is, I believe, most suitable if one wants to proceed analytically from common cognition to the determination of its supreme principle, and in turn synthetically from the examination of this principle and its sources back to the common cognition in which we find it used. (Groundwork of the Metaphysics of Morals, Preface) With the introduction of his new logic, [[Frege]] defines analyticity in terms of a proposition's logical form. Where Kant had taken contentful mathematical statements as synthetic yet knowable _a priori_ (i.e., not relying on empirical data), Frege now considered arithmetic statements as analytic by virtue of his logicist analysis of number as a class of equinumerous concepts. So where Kant could argue that knowledge of $7+ 5=12$ relied upon intuitive synthesis >no matter how long I analyze my concept of such a possible sum [of seven and five] I will still not find twelve in it, for Frege, such a statement may be established purely by logical means. In the context of his [[Martin-Löf dependent type theory\|dependent type theory]], [[Per Martin-Löf]] ([ML94](#ML94)) draws on Kant to relate the analytic-synthetic distinction to the distinction between judgmental and propositional [[equality]]. Wherever you must construct an element to establish a proposition, that proposition is synthetic. ## Analytic and synthetic geometry A distinction between _analytic_ and _synthetic_ methods is often made in geometry, leading on from the description of Descartes' geometry as analytic. In [Elementary Mathematics from an Advanced Standpoint: Geometry](http://books.google.co.uk/books?id=fj-ryrSBuxAC), Felix Klein wrote in 1908 > _Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates._ Rightly understood, there exists only a _difference of gradation_ between these two kinds of geometry, according as one gives _more prominence to the figures or to the formulas._ Analytic geometry which dispenses entirely with geometric representation can hardly be called geometry; synthetic geometry does not get very far unless it makes use of a suitable language of formulas to give precise expression to its results. (p. 55) He continues > In mathematics, however, as everywhere else, men are inclined to form parties, so that there arose _schools of pure synthesists_ and _schools of pure analysts_, who placed chief emphasis upon absolute "purity of method," and who were thus more one-sided than the nature of the subject demanded. Thus the analytic geometricians often lost themselves in blind calculations, devoid of any geometric representation, The synthesists, on the other hand, saw salvation in an artificial avoidance of all formulas, and thus they accomplished nothing more, finally, than to develop their own peculiar language formulas, different from ordinary formulas. (pp. 55-56) #Related entries# * [[synthetic mathematics]] * [[synthetic differential geometry]] ## References * {#Domski} Mary Domski, _Descartes’ Mathematics_, ([SEP](https://plato.stanford.edu/entries/descartes-mathematics)) * {#ML94} [[Per Martin-Löf]], _Analytic and Synthetic Judgements in Type Theory_, ([article](http://archive-pml.github.io/martin-lof/pdfs/Martin-Lof-Analytic-and-Synthetic-Judgements-in-Type-Theory.pdf)) * {#Anderson15} R. Lanier Anderson, _The Poverty of Conceptual Truth: Kant's Analytic/Synthetic Distinction and the Limits of Metaphysics_, Oxford University Press, 2015.
analytical index	https://ncatlab.org/nlab/source/analytical+index	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Index theory +-- {: .hide} [[!include index theory - contents]] =-- #### Integration theory +--{: .hide} [[!include integration theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea By [[pseudo-differential operator\|pseudo-differential analysis]] an [[elliptic operator]] acting on [[sections]] of two [[vector bundles]] on a [[manifold]] is a [[Fredholm operator]] and hence has closed [[kernel]] and [[cokernel]] of finite [[dimension]]. The difference of these two dimensions is the _analytical index_ of the operator. More generally, for $(E_p, D_p)$ an [[elliptic complex]], its analytical index is the alternating sum $$ ind_{an}(E_p, D_p) = \sum_p (-1)^p dim (ker (D_p)) \,. $$ ## Properties This index does not the depend of the [[Sobolev space]] used to get a [[bounded operator]] (by elliptic regularity the kernel is made up of smooth sections and the same for the cokernel as it is the kernel of the adjoint). By [[topological K-theory]] one can associate to it also a [[topological index]]. The [[Atiyah-Singer index theorem]] say that this two indexes coincide. ## Related concepts * [[index theory]] * [[topological index]] * analytical index * [[Atiyah-Singer index theorem]] [[!redirects analytical indices]] [[!redirects analytic index]] [[!redirects analytic indices]] [[!redirects index of an elliptic complex]] [[!redirects index of elliptic complexes]]
analytically continued Chern-Simons theory	https://ncatlab.org/nlab/source/analytically+continued+Chern-Simons+theory	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### $\infty$-Chern-Simons theory +--{: .hide} [[!include infinity-Chern-Simons theory - contents]] =-- #### Quantum field theory +--{: .hide} [[!include functorial quantum field theory - contents]] =-- #### Physics +--{: .hide} [[!include physicscontents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Where ordinary 3d [[Chern-Simons theory]] is given by an [[action functional]] with values in the [[circle group]] $\mathbb{R}/\mathbb{Z}$ on a space of [[special unitary group]]-[[principal connections]], its "[[analytic continuation]]"([Gukov 03](#Gukov03), [Witten 10](#Witten10)) instead is defined on complex [[special linear group]]-principal connections and its values are elements in $\mathbb{C}/\mathbb{Z}$ (see also at _[[Chern-Simons theory with complex gauge group]]_). The [[Wilson line]] [[quantum observables]] of analytically continued Chern-Simons theory are accordingly [[analytic continuations]] of [[knot invariants]] ([Garoufalidis 07](#Garoufalidis07)). ## Properties Discussion of the [[phase space]] with its complex symplectic form is in [Gukov 03, section 2.2](#Gukov03) ## Related concepts * [[Chern-Simons theory with complex gauge group]] * [[complex volume]], [[Cheeger-Simons classes]] * [[volume conjecture]] * [[holomorphic Chern-Simons theory]] * [[Spin Chern-Simons theory]] * [[analytically continued Wess-Zumino-Witten theory]] ## References * {#Gukov03} [[Sergei Gukov]], _Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial_, Commun.Math.Phys. 255 (2005) 577-627 ([arXiv:hep-th/0306165](http://xxx.lanl.gov/abs/hep-th/0306165)) * {#Garoufalidis07} [[Stavros Garoufalidis]], _Chern-Simons theory, analytic continuation and arithmetic_ ([arXiv:0711.1716](http://arxiv.org/abs/0711.1716)) * {#Witten10} [[Edward Witten]], _Analytic Continuation Of Chern-Simons Theory_, Chern–Simons Gauge Theory, 20, 347-446. ([arXiv:1001.2933](http://arxiv.org/abs/1001.2933)) * {#Dimofte11} [[Tudor Dimofte]], _Quantum Riemann Surfaces in Chern-Simons Theory_ ([arXiv:1102.4847](http://arxiv.org/abs/1102.4847)) * {#Witten14} [[Edward Witten]], _Two Lectures On The Jones Polynomial And Khovanov Homology_ ([arXiv:1401.6996](http://arxiv.org/abs/1401.6996)) [[!redirects analytic Chern-Simons theory]]
analytically continued Wess-Zumino-Witten theory	https://ncatlab.org/nlab/source/analytically+continued+Wess-Zumino-Witten+theory	## Idea [[WZW model]] for non-compact [[complex Lie groups]] such as $SL(2,\mathbb{C})$ ## Related concepts * [[analytically continued Chern-Simons theory]] * [[Liouville theory]] ## References * Nobuyuki Ishibashi, _Extra Observables in Gauged WZW Models_, Nucl.Phys. B379 (1992) 199-219 ([arXiv:hep-th/9110071](http://arxiv.org/abs/hep-th/9110071)) * [[Dmitri Sorokin]], Francesco Toppan, _Hamiltonian Reduction of Supersymmetric WZNW Models on Bosonic Groups and Superstrings_, Nucl.Phys. B480 (1996) 457-484 ([arXiv:hep-th/9603187](http://arxiv.org/abs/hep-th/9603187)) * Jian-Feng Wu, Yang Zhou, _From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality_ ([arXiv:0911.1922](http://arxiv.org/abs/0911.1922))
analytification	https://ncatlab.org/nlab/source/analytification	> under construction +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Analytic geometry +--{: .hide} [[!include analytic geometry -- contents]] =-- #### Complex geometry +--{: .hide} [[!include complex geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Analytification is the process of universally turning an [[algebraic space]] into an [[analytic space]]. ## Definition Let $X \to Spec(\mathbb{C})$ be a [[scheme]] of [[locally finite type]] over the [[complex numbers]]. Its set $X(\mathbb{C})$ of "complex points" is the set of [[maximal ideals]], since $\mathbb{C}$ is an [[algebraically closed field]], e.g. [Neeman 07, prop. 4.2.4](#Neeman07)). This set $X(\mathbb{C})$ canonically carries the [[complex analytic topology]]. As such it is a [[topological space]] written $X^{an}$. Equipped with the canonical [[structure sheaf]] $\mathcal{O}_{X^{an}}$ this is a [[complex analytic space]]. This $(X^{an}, \mathcal{O}_{X^{an}})$ is called the _analytification_ of $X$. This construction extends to a [[functor]] from the [[category]] of [[schemes]] over $\mathbb{C}$ to that of [[complex analytic spaces]]. See e.g. ([Neeman 07, section 4, p.71](#Neeman07), [Danilov 91, chapter 3, paragraph 1, section 1.1 (p.61)](#Danilov91) Generalization to [[structured (infinity,1)-toposes]] is in ([Lurie 08, remark 4.4.13](#Lurie08)). ## Examples The analytification of the [[projective space]] $\mathbb{P}^1$ is the [[complex projective space]] $(\mathbb{P}^1)^{an} \simeq \mathbb{C}\mathbb{P}^1$, hence the [[Riemann sphere]]. The analytification of an [[elliptic curve]] is the [[complex torus]]. see e.g. ([Danilov 91, example in chapter 3, paragraph 1, section 1.1. (p. 61)](#Danilov91)) ## Properties ### Existence and fully faithfulness (GAGA) {#Existence} The analytification of an [[algebraic space]] over the [[complex numbers]] which is 1. [[locally of finite type]] 1. [[locally separated]] is a [[complex analytic space]]. Moreover, under suitable conditions analytification is a [[fully faithful functor]]. This is a classical result due to ([Artin 70, theorem 7.3](#Artin70)). A textbook account of the proof is in ([Neeman 07, section 10](#Neeman07)). Discussion in more general [[analytic geometry]] is in ([Conrad-Temkin 09, section 2.2](#ConradTemkin09)). Generalization to [[algebraic stacks]]/[[Deligne-Mumford stacks]]/[[geometric stacks]] is in ([Lurie 04](#Lurie04), [Hall 11](#Hall11), [Geraschenko & Zureick-Brown 12](#GeraschenkoZureickBrown12)). ### As geometric realization in $\mathbb{A}^1$-homotopy theory {#GeometricRealizationInA1Homotopy} For $k \hookrightarrow \mathbb{C}$ a field, then the functor that takes a smooth complex scheme to the the [[homotopy type]] underlying its analytification induces [[geometric realization]] $$ Sh_\infty(Sch^{sm}_k) \to Sh_\infty(Sch^{sm}_k)^{\mathbb{A}^1} \to \infty Grpd $$ ([Isaksen 01](#Isaksen01), [Dugger-Isaksen 05, theorem 5.2](#DuggerIsaksen05)) ## Related concepts * [[GAGA]] ## References ### Complex analytification Original articles include * {#Artin70} [[Michael Artin]], _Algebraization of formal moduli: II. Existence of modifications_, Annals of Math., 91 no. 1 (1970), pp. 88–135. * [[Alexander Grothendieck]], [[SGA]] I, Exposé XII A review of that is in * Yan Zhao, _Géométrie algébrique et géométrie analytique_, 2013 ([pdf](http://pub.math.leidenuniv.nl/~jinj/2013/efg/gaga.pdf)) Textbook accounts include * {#Neeman07} [[Amnon Neeman]], _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry)) * {#Danilov91} [[Vladimir Danilov]], chapter 3 of _Cohomology of algebraic varieties_, in I. Shafarevich (ed.), _Algebraic Geometry II_, volume 35 of _Encyclopedia of mathematical sciences_, Springer 1991 ([GoogleBooks](http://books.google.de/books?id=ZhzXJHUgcRUC&lpg=PA67&ots=aVQoeMkBwc&dq=analytification&pg=PA61&redir_esc=y#v=onepage&q=analytification&f=false))) Discussion for [[real analytic spaces]] includes * {#Huisman02} [[Johannes Huisman]], section 2 of _The exponential sequence in real algebraic geometry and Harnack's Inequality for proper reduced real schemes_, Communications in Algebra, Volume 30, Issue 10, 2002 ([pdf](http://pageperso.univ-brest.fr/~huisman/rech/publications/exphi.pdf)) Generalizations to [[higher geometry]] are in * {#ConradTemkin09} [[Brian Conrad]], M. Temkin, _Non-Archimedean analytification of algebraic spaces_, J. Algebraic Geom. 18 (2009), no. 4, 731–788 ([arXiv:0706.3441](http://arxiv.org/abs/0706.3441)) * {#Lurie04} [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_, ([arXiv:math.AG/0412266](http://arxiv.org/abs/math/0412266)) * {#Lurie08} [[Jacob Lurie]], _[[Structured Spaces]]_, 2008 * {#Hall11} [[Jack Hall]], _Generalizing the GAGA Principle_ ([arXiv:1101.5123](http://arxiv.org/abs/1101.5123)) * {#GeraschenkoZureickBrown12} [[Anton Geraschenko]], David Zureick-Brown, _Formal GAGA for good moduli spaces_ ([arXiv:1208.2882](http://arxiv.org/abs/1208.2882)) See also * [[Walter Gubler]]. _Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros)_ ([arXiv:1303.7364](http://arxiv.org/abs/1303.7364)) Discussion in the context of [[hypercovers]] and [[A1-homotopy theory]] is in * {#Isaksen01} [[Daniel Isaksen]], _Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes_, 2001 ([K-theory archive](http://www.math.uiuc.edu/K-theory/0495/)) * {#DuggerIsaksen05} [[Daniel Dugger]] and [[Daniel Isaksen]], _Hypercovers in topology_, 2005 ([pdf](http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf), [K-Theory archive](http://www.math.uiuc.edu/K-theory/0528/)) ### Non-archimedean analytification {#ReferencesNonArchimedeanAnalytification} Discussion in more general [[rigid analytic geometry]] is in * [[Brian Conrad]], Michael Temkin, _Non-archimedean analytification of algebraic spaces_ ([arXiv:0706.3441](http://arxiv.org/pdf/0706.3441)) [[!redirects analytifications]]
ananatural transformation	https://ncatlab.org/nlab/source/ananatural+transformation	# Ananatural transformations * table of contents {: toc} ## Idea Just as [[natural transformation\|natural transformations]] go between [[functor\|functors]], ananatural transformations go between [[anafunctor\|anafunctors]]. Given two functors interpreted as anafunctors, the natural transformations and ananatural transformations between them are the same, so the term 'ananatural' is overkill; one only needs it to emphasise the ana-context and otherwise can just say 'natural'. That is, a 'natural transformation' between anafunctors unambigously means an ananatural transformation. ## Definitions Given two [[categories]] $C$ and $D$ and two anafunctors $F, G\colon C \to D$, let us interpret $F,G$ as [[spans]] $C \leftarrow \overline{F} \rightarrow D$ and $C \leftarrow \overline{G} \rightarrow D$ of [[strict functors]] (where each backwards-pointing arrow is strictly surjective and faithful; see the definition of [[anafunctor]]). Form the [[strict 2-limit\|strict]] $2$-[[pullback]] $P \coloneqq \overline{F} \times_C \overline{G}$ and consider the strict functors $P \to \overline{F} \to D$ and $P \to \overline{G} \to D$. Then an ananatural transformation from $F$ to $G$ is simply a natural transformation between these two strict functors. More explicitly, if $F,G$ are given by sets ${\|F\|}, {\|G\|}$ of specifications and additional maps (see the other definition of [[anafunctor]]), then an __ananatural transformation__ from $F$ to $G$ consists of a coherent family of morphisms of $D$ indexed by the elements of $\|F\|$ and $\|G\|$ with common values in $C$. That is: * for each object $x$ of $C$, each $F$-specification $s$ over $x$, and each $G$-specification $t$ over $x$, we have a morphism $$ \eta_{s,t}(x)\colon F_s(x) \to G_t(x) $$ in $D$; * for each morphism $f\colon x \to y$ in $C$, each pair of $F$-specifications $s,t$ over $x,y$, and each pair of $G$-specifications $u,v$ over $x,y$, the diagram $$ \array { F_s(x) & \overset{\eta_{s,u}(x)}\rightarrow & G_u(x) \\ F_{s,t}(f) \downarrow & & \downarrow F_{u,v}(f) \\ F_t(y) & \underset{\eta_{t,v}(y)}\rightarrow & G_v(y) } $$ is a [[commutative square]]. Of course, an __ananatural isomorphism__ is an [[invertible morphism\|invertible]] ananatural transformation. ## Composition Just as natural transformations can be composed vertically to form the morphisms of a [[functor category]], so ananatural transformations can be composed vertically to form an anafunctor category. Just as natural transformations can also be whiskered by functors and composed horizontally to make a [[strict 2-category]] $Str Cat$ of (strict) categories, (strict) functors and natural transformations, so ananatural transformations can also be whiskered by anafunctors and composed horizontally to make a [[bicategory]] $Cat_{ana}$ of (strict) categories, anafunctors and (ana)natural transformations. Assuming the [[axiom of choice]], $Cat_{ana}$ is equivalent to $Str Cat$; without choice (and [[internalisation\|internally]]), $Cat_{ana}$ has better properties than $Str Cat$ and we will usually identify the former with [[Cat]]. [[!redirects ananatural transformation]] [[!redirects ananatural transformations]] [[!redirects ananatural isomorphism]] [[!redirects ananatural isomorphisms]]
Anand Dessai	https://ncatlab.org/nlab/source/Anand+Dessai	* [webpage](https://homeweb.unifr.ch/dessaia/pub/) ## Selected writings On the [[Witten genus]] and the [[Stolz conjecture]]: * [[Anand Dessai]], _Some geometric properties of the Witten genus_, in: [[Christian Ausoni]], [[Kathryn Hess]], [[Jérôme Scherer]] (eds.) _Alpine Perspectives on Algebraic Topology_, Contemporary Mathematics 504 (2009) ([pdf](http://homeweb2.unifr.ch/dessaia/pub/papers/Arolla/DessaiArollaFinalRevised30June09.pdf), [[DessaiEllipticGenus.pdf:file]], [doi:10.1090/conm/504](http://dx.doi.org/10.1090/conm/504)) On the [[Witten genus]] on manifolds with [[SU(2)]]-[[action]]: * [[Anand Dessai]], _The Witten genus and $S^3$-actions on manifolds_, 1994 ([pdf](https://homeweb.unifr.ch/dessaia/pub/papers/MZpreprint6_Witten_S3.pdf), [[DessaiWittenGenusS3.pdf:file]]) On the [[rigidity theorem for elliptic genera]]: * [[Anand Dessai]], Rainer Jung, _On the Rigidity Theorem for Elliptic Genera_, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) ([jstor:117694](https://www.jstor.org/stable/117694)) category: people
Anand Pillay	https://ncatlab.org/nlab/source/Anand+Pillay	__Anand Pillay__ is a model theorist, now at University of Notre Dame. * [homepage](https://www3.nd.edu/~apillay) at Notre Dame Related items: [[geometric stability theory]], [[definable set]], [[differential Galois theory]] * _Remarks on Galois cohomology and definability_, The Journal of Symbolic Logic __62__:2 (1997) 487-492 [doi](https://doi.org/10.2307/2275542) * Moshe Kamensky, Anand Pillay, _Interpretations and differential Galois extensions_, IMRN 24 (2016) 7390–7413 [doi](https://doi.org/10.1093/imrn/rnw019) * Z. Chatzidakis, Anand Pillay, _Generalized Picard-Vessiot extensions and differential Galois cohomology_, 2017, [pdf](https://www3.nd.edu/~apillay/generalizedPV-PHS.zoe.pdf)
Ananda Roy	https://ncatlab.org/nlab/source/Ananda+Roy	* [personal page](https://sites.rutgers.edu/ananda-roy/) * [institute page](https://physics.rutgers.edu/people/faculty-list/faculty-profile/roy-ananda) ## Selected writings On [[topological quantum computation]]: * [[Ananda Roy]], [[David P. DiVincenzo]], Topological Quantum Computing, Lecture notes of the [48th IFF Spring School](https://www.fz-juelich.de/en/pgi/expertise/schools-and-courses/iff-spring-school/the-iff-spring-school-topics-and-history/springschool2017) (2017) $[$[arXiv:1701.05052](https://arxiv.org/abs/1701.05052)$]$ category: people
Anastasia Volovich	https://ncatlab.org/nlab/source/Anastasia+Volovich	* [Wikipedia entry](https://en.wikipedia.org/wiki/Anastasia_Volovich) * [Institute page](https://vivo.brown.edu/display/avolovic) ## Selected writings On [[twistor string theory]]: * {#RoibanVolovich04} [[Radu Roiban]], [[Anastasia Volovich]], All Googly Amplitudes from the B-model in Twistor Space, Phys. Rev. Lett. 93 (2004) 131602 ([arXiv:hep-th/0402121](https://arxiv.org/abs/hep-th/0402121)) * {#RoibanSpradlinVolovich04} [[Radu Roiban]], [[Marcus Spradlin]], [[Anastasia Volovich]], On the Tree-Level S-Matrix of Yang-Mills Theory, Phys. Rev. D70 : 026009, 2004 ([arXiv:hep-th/0403190](https://arxiv.org/abs/hep-th/0403190)) category: people
Anatoly Libgober	https://ncatlab.org/nlab/source/Anatoly+Libgober	* [personal page](http://homepages.math.uic.edu/~libgober/) * [Wikipedia entry](https://en.wikipedia.org/wiki/Anatoly_Libgober) ## Selected writings On [[braid groups]], [[braid representations]] and some [[knot theory]]: * [[Joan S. Birman]], [[Anatoly Libgober]] (eds.) Braids, Contemporary Mathematics 78 (1988) [[doi:10.1090/conm/078](http://dx.doi.org/10.1090/conm/078)] On [[local systems]] and their [[twisted de Rham cohomology]]/[[Dolbeault cohomology]]: * [[Anatoly Libgober]], [[Sergey Yuzvinsky]], Cohomology of local systems, Advanced Studies in Pure Mathematics 27, Mathematics Society of Japan (2000) 169-184 [[pdf](http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf), [doi:10.2969/aspm/02710169](https://doi.org/10.2969/aspm/02710169)] category: people
Anatoly Malcev	https://ncatlab.org/nlab/source/Anatoly+Malcev	[[!redirects A. I. Mal'cev]] Anatoly Malcev (Анато́лий Ива́нович Ма́льцев; [ˈmɐlʲt͡sef]) was a Russian mathematician. ## Related entries * [[Malcev completion]] * [[Malcev variety]] * [[Malcev category]] * [[Kan object]] * [[maximal compact subgroup]] ## References [List of publications at Math-Net.Ru](http://www.mathnet.ru/php/person.phtml?&personid=26552&option_lang=eng). [[!redirects А. И. Мальцев]] [[!redirects А. Мальцев]] [[!redirects Анатолий Иванович Мальцев]] [[!redirects Мальцев]] [[!redirects Malcev]] [[!redirects Mal'cev]] [[!redirects Maltsev]] [[!redirects Mal'tsev]] [[!redirects A. I. Malcev]] [[!redirects A. Malcev]] [[!redirects A. I. Mal'cev]] [[!redirects A. Mal'cev]] [[!redirects A. I. Maltsev]] [[!redirects A. Maltsev]] [[!redirects A. I. Mal'tsev]] [[!redirects A. Mal'tsev]] [[!redirects Anatoly I. Malcev]] [[!redirects Anatoly Malcev]] [[!redirects Anatoly I. Mal'cev]] [[!redirects Anatoly Mal'cev]] [[!redirects Anatoly I. Maltsev]] [[!redirects Anatoly Maltsev]] [[!redirects Anatoly I. Mal'tsev]] [[!redirects Anatoly Mal'tsev]] category: people
Anatoly Shirshov	https://ncatlab.org/nlab/source/Anatoly+Shirshov	__Anatolij Illarionovič Širšov__ (Anatoly/Anatolii Illarinovich Shirshov, orig. Анатолий Илларионович Ширшов) was an [[algebra\|algebraist]] working in Novosibirsk (in former Soviet Union). * Анатолий Илларионович Ширшов (к 80-летию со дня рождения), Алгебра и логика, 40:4 (2001), I–IV, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=al&paperid=333&what=fullt&option_lang=rus) * wikipedia [ru](http://ru.wikipedia.org/wiki/%D0%A8%D0%B8%D1%80%D1%88%D0%BE%D0%B2,_%D0%90%D0%BD%D0%B0%D1%82%D0%BE%D0%BB%D0%B8%D0%B9_%D0%98%D0%BB%D0%BB%D0%B0%D1%80%D0%B8%D0%BE%D0%BD%D0%BE%D0%B2%D0%B8%D1%87) * L. A. Bokutʹ, I. P. Shestakov, _Some results by A. I. Shirshov and his school_, Second International Conference on Algebra (Barnaul, 1991), 1--12, Contemp. Math. __184__, Amer. Math. Soc. 1995. * A. I. Širšov, _Some algorithm problems for Lie algebras_ (Russian) Sibirsk. Mat. Ž. __3__ 1962, 292--296, [MR0183753](http://www.ams.org/mathscinet-getitem?mr=0183753) category: people [[!redirects Anatoli Shirshov]] [[!redirects Anatoly Shirshov]] [[!redirects Anatoliy Shirshov]] [[!redirects Anatolii Shirshov]] [[!redirects Anatolij Shirshov]] [[!redirects Anatoli Širšov]] [[!redirects Anatoly Širšov]] [[!redirects Anatoliy Širšov]] [[!redirects Anatolii Širšov]] [[!redirects Anatolij Širšov]] [[!redirects Anatolij Illarionovič Širšov]] [[!redirects Анатолий Ширшов]] [[!redirects Анатолий Илларионович Ширшов]]
Anatoly Vershik	https://ncatlab.org/nlab/source/Anatoly+Vershik	* [Wikipedia entry](https://en.wikipedia.org/wiki/Anatoly_Vershik) ## Selected writings On the asymptotic number of [[standard Young tableaux]] (dimension of [[Specht modules]]) with respect to the [[Plancherel measure]]: * {#VershikKerov85} [[Anatoly Vershik]], [[Sergei Kerov]], Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group, Functional Analysis and Its Applications volume 19, pages 21–31 (1985) ([doi:10.1007/BF01086021](https://doi.org/10.1007/BF01086021)) On the [[representation theory of the symmetric group]] via the [[Gelfand-Tsetlin basis]]/[[seminormal representation]]: * {#VershikOkounkov04} [[Anatoly Vershik]], [[Andrei Okounkov]], A New Approach to the Representation Theory of the Symmetric Groups, Part I: Selecta Mathematica, New Series 2, 581-605 ([arXiv:math/0503040](https://arxiv.org/abs/math/0503040), [doi:10.1007/BF02433451](https://doi.org/10.1007/BF02433451)); Part II (incorporates Part I in revised and improved form): Russian version: [Записки научных семинаров ПОМИ 307 (2004), 57–98](ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v307/p057.ps.gz) (Zapiski nauchnyh seminarov POMI 307 (2004), 57–98); English version: Journal of Mathematical Sciences 131 (2005), 5471–5494 ([doi:10.1007/s10958-005-0421-7](https://doi.org/10.1007/s10958-005-0421-7)) * [[Anatoly Vershik]], Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis, pp. 619 in: [[Pavel Etingof]], [[Vladimir S. Retakh]], [[Isadore Singer]] (eds.) The Unity of Mathematics In Honor of the Ninetieth Birthday of [[I. M. Gelfand]], Birkhäuser 2006 ([arXiv:math/0503140](https://arxiv.org/abs/math/0503140), [ISBN:978-0-8176-4467-3](https://www.springer.com/gp/book/9780817640767)) category: people
Anaxagoras	https://ncatlab.org/nlab/source/Anaxagoras	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Philosophy +-- {: .hide} [[!include philosophy - contents]] =-- =-- =-- ## Hegel on Anaxagoras In the _[[Science of Logic]]_, [[Hegel]] writes > [WdL §54](Science+of+Logic#54) Anaxagoras wird als derjenige gepriesen, der zuerst den Gedanken ausgesprochen habe, daß der [[nous]], der Gedanke, das Princip der Welt, daß das Wesen der Welt als der Gedanke bestimmt ist. Er hat damit den Grund zu einer Intellektualansicht des Universums gelegt, deren reine Gestalt die Logik seyn muß. Es ist in ihr nicht um ein Denken über etwas, das für sich außer dem Denken zu Grunde läge, zu thun, um Formen, welche bloße Merkmale der Wahrheit abgeben sollten; sondern die nothwendigen Formen und eigenen Bestimmungen des Denkens sind der Inhalt und die höchste Wahrheit selbst. > WdL §54 Anaxagoras is praised as the man who first declared that [[Nous]], thought, is the principle of the world, that the essence of the world is to be defined as thought. In so doing he laid the foundation for an intellectual view of the universe, the pure form of which must be logic. What we are dealing with in logic is not a thinking about something which exists independently as a base for our thinking and apart from it, nor forms which are supposed to provide mere signs or distinguishing marks of truth; on the contrary, the necessary forms and self-consciousness of thought are the content and the ultimate truth itself. See also the paragraph from the [[Lectures on the History of Philosophy]] starting with "[Das Prinzip des Anaxagoras war ...](#DasPrinzipDesAnaxagorasWar)". The following is Hegel on Anaxagoras in his _[[Lectures on the History of Philosophy]]_. ### Introduction > Hier fängt erst an, ein Licht aufzugehen (es ist zwar noch schwach): Der Verstand wird als das Prinzip anerkannt. Von Anaxagoras sagt schon Aristoteles: »Der aber, der sagte, daß die Vernunft wie in dem Lebenden, so auch in der Natur die Ursache ist der Welt und aller Ordnung, ist wie ein Nüchterner erschienen gegen die [, die] vorher ins Blinde (eikê) sprachen.« Die Philosophen vor Anaxagoras, sagt Aristoteles, »sind den Fechtern, die wir Naturalisten nennen, zu vergleichen. Wie diese oft sich in ihrem Herumtummeln gute Stöße tun, aber nicht nach der Kunst, so scheinen auch diese Philosophen kein Bewußtsein über das zu haben, was sie sagen.« Dies Bewußtsein hat zuerst Anaxagoras gehabt, indem er sagt, der Gedanke ist das an und für sich seiende Allgemeine, der reine Gedanke ist das Wahre. Anaxagoras ist wie ein Nüchterner unter Trunkenen erschienen; aber auch sein Stoß geht noch ziemlich ins Blaue hinein. Als Prinzipien haben wir gesehen das Sein, das Werden, das Eins; es sind Gedanken, allgemeine, nichts Sinnliches, auch nicht Vorstellungen der Phantasie; der Inhalt, die Teile desselben aber sind genommen aus dem Sinnlichen, es sind Gedanken in irgendeiner Bestimmung. Anaxagoras nun sagt, das Allgemeine – nicht Götter, sinnliche Prinzipien, Elemente, noch Gedanken, die wesentlich als bestimmte sind (Reflexionsbestimmungen) –, sondern der Gedanke selbst, an und für sich, das Allgemeine ohne Gegensatz, alles in sich befassend, ist die Substanz. Hierbei müssen wir uns nicht den subjektiven Gedanken vorstellen; wir denken beim Denken sogleich an unser Denken, wie es im Bewußtsein ist. Hier ist dagegen der ganz objektive Gedanke gemeint, das Allgemeine, der tätige Verstand; wie wir sagen, es ist Verstand, Vernunft in der Welt, auch in der Natur, – oder wie[369] wir von Gattungen in der Natur sprechen: sie sind das Allgemeine. Ein Hund ist ein Tier, dies ist seine Gattung, sein Substantielles; er selbst ist dies. Dies Gesetz, dieser Verstand, diese Vernunft ist selbst immanent in der Natur, ist das Wesen der Natur; sie ist nicht von außen formiert, wie die Menschen einen Stuhl machen. Der Tisch ist auch vernünftig gemacht, aber es ist ein äußerlicher Verstand diesem Holze. Aber diese äußere Form, die der Verstand sein soll, fällt uns beim Sprechen vom Verstande sogleich ein. Hier aber ist der Verstand, das Allgemeine gemeint, was die immanente Natur des Gegenstandes selbst ist. Dies ist das Prinzip. Bisher hatten wir Gedanken, jetzt ist es der Gedanke selbst, der zum Prinzip gemacht ist. > Der [[nous]] ist nichtdenkendes Wesen draußen, das die Welt eingerichtet; so ist der Gedanke des Anaxagoras ganz verdorben, ihm alles philosophische Interesse benommen. Denn ein Individuelles, Einzelnes draußen ist ganz in die Vorstellung herabgefallen und deren Dualismus; ein denkendes sogenanntes Wesen ist kein Gedanke mehr, ist ein Subjekt. Das wahrhaft Allgemeine ist nicht abstrakt, gerade das Allgemeine (Gute, Schöne, Zweck) [ist] dies, in und aus sich selbst das Besondere an und für sich zu bestimmen, – nicht äußerlicher Zweck. ### Life > Vor seiner Philosophie haben wir seine Lebensumstände zu betrachten. Mit ihm sehen wir die Philosophie in das eigentliche Griechenland, das bisher noch keine hatte, und zwar nach Athen wandern; bisher war Kleinasien oder Italien der Sitz der Philosophie gewesen. Anaxagoras, selbst ein Kleinasiate, lebte vorzüglich in Athen, das, wie das Haupt der griechischen Macht, so jetzt auch der Sitz und Mittelpunkt der Künste und der Wissenschaften war. Anaxagoras lebte in der großen Zeit zwischen den medischen Kriegen und dem Zeitalter des Perikles. Er fällt in die schönste Zeit des griechischen athenischen Lebens und berührt noch den Untergang oder vielmehr den Übergang in den Untergang, in das Sterben des schönen athenischen Lebens. Die Schlacht bei[370] Marathon ist in der 72., die bei Salamis in der 75. Olympiade; in der 81. (456 v. Chr.) kam Anaxagoras nach Athen. Anaxagoras, Ol. 70 (500 v. Chr.) geboren, ist früher als Demokrit und dem Alter nach ebenso als Empedokles, doch mit diesen überhaupt wie mit Parmenides gleichzeitig; er ist so alt als Zenon. Seine Vaterstadt ist Klazomenai in Lydien, nicht sehr weit von Kolophon und dann Ephesos, auf einer Landenge, die eine große Halbinsel mit dem festen Lande zusammenhängt. > Anaxagoras schließt diese Periode, nach ihm beginnt eine neue. Er wird, nach der beliebten Ansicht vom genealogischen Übergehen der Prinzipien von Lehrer auf Schüler, weil er aus Ionien war, oft als ein Fortsetzer der ionischen Schule, als ionischer Philosoph vorgestellt; denn Hermotimos von Klazomenai war sein Lehrer. Auch wird er zu diesem Behufe zu einem Schüler des Anaximenes gemacht, dessen Geburt aber in Ol. 55-58 gesetzt wird, also fünfzehn Olympiaden (d.h. 60 Jahre) früher. Sein Leben besteht kurz darin, daß er sich auf das Studium der Wissenschaften legte, von den öffentlichen Angelegenheiten zurückzog, viele Reisen machte und zuletzt nach einigen im dreißigsten, wahrscheinlicher im fünfundvierzigsten Jahre seines Alters nach Athen kam. Er kam am günstigsten, in der blühendsten Zeit der Stadt dahin; Perikles stand an der Spitze des Staats und erhob ihn zum höchsten Glanze, das ist der Silberblick des atheniensischen Lebens. Perikles suchte den Anaxagoras auf und lebte mit ihm in sehr vertrautem Umgang. Athen hatte damals den Kulminationspunkt seiner schönen Größe erreicht; besonders ist der Gegensatz von Athen und Lakedämon in dieser Zeit interessant. Athen und Lakedämon waren die beiden griechischen Nationen, die um die erste Stelle in Griechenland miteinander[371] wetteiferten. Athens ist im Gegensatze Lakedämons zu erwähnen, – der Prinzipien dieser berühmten Staaten. Keine Kunst und Wissenschaft war bei den Lakedämoniern. Daß Athen der Sitz der Wissenschaften und schönen Künste war, hatte es der Eigentümlichkeit seiner Verfassung und seines ganzen Geistes zu danken. > Lakedämon ist ebenso seiner Verfassung nach hochzuachten. Die Lakedämonier hatten den strengen dorischen Geist durch ihre konsequente Verfassung geordnet – eine Verfassung, worin der Hauptzug ist, daß die Individualität, alle persönliche Besonderheit dem Allgemeinen, dem Zwecke des Staats, dem Leben des Staats untergeordnet ist oder vielmehr aufgeopfert war, – daß das Individuum das Bewußtsein seiner Ehre, Gültigkeit usf. nur in dem Bewußtsein der Tätigkeit, des Lebens, des Handelns für den Staat hat. Ein Volk von solcher gediegenen Einheit, worin Wille des Einzelnen eigentlich ganz verschwunden ist, machte einen unüberwindlichen Zusammenhang aus; und Lakedämon wurde deswegen an die Spitze der Griechen gestellt und erhielt die Hegemonie, wie wir sie in den trojanischen Zeiten bei den Argivern sahen. > Dies ist ein großes Prinzip, was in jedem wahrhaften Staate sein muß, was aber bei den Lakedämoniern in seiner Einseitigkeit geblieben ist; diese Einseitigkeit ist von den Atheniensern gemieden, und dadurch sind sie größer geworden. In Lakedämon war die Eigentümlichkeit, Persönlichkeit, Individualität nachgesetzt, und so, daß das Individuum nicht für sich seine freie Ausbildung, Äußerung haben konnte; sie war nicht anerkannt, daher nicht in Übereinstimmung gebracht, nicht in Einheit gesetzt mit dem allgemeinen Zweck des Staats. Dieses allgemeine Leben, dies Aufheben des Rechts der Besonderheit, der Subjektivität, ging bei den Lakedämoniern sehr weit; und wir finden dasselbe Prinzip in Platons Republik auf seine Weise. > Aber das Allgemeine ist nur lebendiger Geist, insofern das einzelne Bewußtsein sich als solches in ihm findet, – das[372] Allgemeine nicht das unmittelbare Leben und Sein des Individuums, die Substanz nur ausmacht, sondern das bewußte Leben. Wie die Einzelheit, die von dem Allgemeinen sich trennt, ohnmächtig ist und zugrunde geht, ebenso kann die einseitige allgemeine, die seiende Sitte der Individualität nicht widerstehen. Der lakedämonische Geist, der auf die Freiheit des Bewußtseins nicht gerechnet und dessen Allgemeines sich von ihr isoliert hatte, mußte sie deswegen als dem Allgemeinen entgegengesetzt hervorbrechen sehen. Und wenn wir zuerst als Befreier Griechenlands von seinen Tyrannen die Spartaner auftreten sehen, denen selbst Athen die Verjagung der Nachkommen des Peisistratos verdankt, so geht ihr Verhältnis zu ihren Bundesgenossen bald in gemeine, niederträchtige Gewalt über und im Inneren, im Staate ebenso in eine harte Aristokratie sowie die festgesetzte Gleichheit des Eigentums (oder Bestimmtheit des Eigentums, daß bei jeder Familie ihr Erbgut bliebe und durch Verbannung eigentlichen Geldes und Handels und Wandels die Möglichkeit der Ungleichheit des Reichtums verbannt würde) in eine Habsucht über, die dem Allgemeinen entgegen brutal und niederträchtig wurde. > Dies wesentliche Moment der Besonderheit, nicht in den Staat aufgenommen, damit nicht gesetzlich, sittlich (moralisch zunächst) gemacht, erscheint als Laster. Alle Momente der Idee sind vorhanden in einer vernünftigen Organisation; ist Leber isoliert als Galle, so würde sie darum nicht mehr und nicht weniger tätig sein, -aber als feindlich, sie zeigte sich als vom Körper, der leiblichen Ökonomie sich isolierend. > Den Atheniensern hatte hingegen Solon nicht nur Gleichheit der Rechte, Einheit des Geistes zu ihrer Verfassung gemacht, sondern auch der Individualität ihren Spielraum gegeben, dem Volke (nicht Ephoren) die Staatsgewalt anvertraut, die es nach Verjagung seiner Tyrannen an sich nahm und so in Wahrheit ein freies Volk wurde. Der Einzelne hatte selbst das Ganze in ihm, sein Bewußtsein und Tun im Ganzen; Ausbildung des freien Bewußtseins muß sich darin finden.[373] > Bei den Atheniensern war auch Demokratie und reinere Demokratie als in Sparta. Jeder Bürger hatte sein substantielles Bewußtsein in der Einheit mit den Gesetzen, mit dem Staate; aber zugleich war der Individualität, dem Geiste, dem Gedanken des Individuums freigelassen, sich zu gewähren, zu äußern, zu ergehen. So sehen wir in diesem Prinzip die Freiheit der Individualität in ihrer Größe auftreten. Das Prinzip der subjektiven Freiheit erscheint zunächst noch verbunden, in Einigkeit mit der allgemeinen Grundlage der griechischen Sittlichkeit, des Gesetzlichen, selbst mit der Mythologie; und so brachte es in seinem Ergehen, indem der Geist, das Genie seine Konzeptionen frei ausgebären konnte, diese großen Kunstwerke der bildenden schönen Künste und die unsterblichen Werke der Poesie und Geschichte hervor. Das Prinzip der Subjektivität hatte insofern noch nicht die Form angenommen, daß die Besonderheit als solche freigelassen, auch der Inhalt ein subjektiv besonderer, wenigstens im Unterschiede von der allgemeinen Grundlage, der allgemeinen Sittlichkeit, der allgemeinen Religion, den allgemeinen Gesetzen sein sollte. Wir sehen also nicht besonders modifizierte Einfälle haben, sondern den großen, sittlichen, gediegenen göttlichen Inhalt in diesen Werken für das Bewußtsein zum Gegenstande gemacht, vor das Bewußtsein überhaupt gebracht. Wir werden später sehen die Form der Subjektivität für sich freiwerden und in den Gegensatz treten gegen das Substantielle, die Sitte, die Religion, das Gesetz. > Die Grundlage von diesem Prinzip der Subjektivität, aber die noch ganz allgemeine Grundlage, sehen wir im Anaxagoras. Er lebte etwas früher als Sokrates, aber sie kannten sich noch. Er kam in dieser Zeit, deren Prinzip eben angegeben ist, nach Athen. > Athen, nach den persischen Kriegen, unterwarf sich den größten Teil der griechischen Inseln sowie eine Menge Seestädte in Thrakien und sonst weiter hinein ins Schwarze Meer. In diesem edlen, freien, gebildeten Volke der Erste[374] des Staats zu sein, – dies Glück wurde Perikles, und dieser Umstand erhebt ihn in der Schätzung der Individualität so hoch, wie wenige Menschen gesetzt werden können. Von allem, was groß unter den Menschen ist, ist die Herrschaft über den Willen der Menschen, die einen Willen haben, das Größte, denn diese herrschende Individualität muß wie die allgemeinste, so die lebendigste sein, – ein Los für Sterbliche, wie es wenige oder keins mehr gibt. Die Größe seiner Individualität war ebenso tief als durchgebildet, ebenso ernst (er hat nie gelacht) als energisch und ruhig; Athen hatte ihn den ganzen Tag. Von Perikles sind uns bei Thukydides einige Reden an das Volk erhalten, denen es wohl wenige Werke an die Seite zu setzen gibt. Unter Perikles findet sich die höchste Ausbildung des sittlichen Gemeinwesens, der Schwebepunkt, wo die Individualität noch unter und im Allgemeinen gehalten ist. Gleich darauf wird die Individualität übermächtig, indem ihre Lebendigkeit in die Extreme gefallen, da der Staat noch nicht als Staat selbständig in sich organisiert ist. Indem das Wesen des athenischen Staats der allgemeine Geist, der Religionsglaube an dies ihr Wesen war, so verschwindet mit dem Verschwinden dieses Glaubens das innere Wesen des Volks, da der Geist nicht als Begriff wie in unseren Staaten ist. Der rasche Übergang hierzu ist der nous die Subjektivität, als Wesen, Reflexionin-sich, – nicht Abstraktion. Athen war der Sitz, ein Kranz von Sternen der Kunst und Wissenschaft. Wie sich die größten Künstler in Athen sammelten, ebenso haben sich die berühmtesten Philosophen und Sophisten dort aufgehalten: Aischylos, Sophokles, Aristophanes, Thukydides, Diogenes von Apollonia, Protagoras, Anaxagoras und andere Kleinasiaten. Kleinasien selbst fiel unter die Herrschaft der Perser, und mit dem Verluste ihrer Freiheit starb auch die Philosophie bei ihnen aus. > Anaxagoras lebte in dieser Zeit in Athen, ein Freund des[375] Perikles, ehe dieser mit Staatsgeschäften sich beschäftigte. Aber es wird auch gesagt, er sei in Dürftigkeit gekommen, weil Perikles ihn vernachlässigt habe, – die Lampe nicht mit 01 versehen, die ihn erleuchtet. > Wichtiger ist, daß Anaxagoras, wie nachher Sokrates und viele andere Philosophen, angeklagt wurde wegen Verachtung der Götter, welche das Volk dafür nähme. Es tritt Gegensatz der Prosa des Verstandes gegen poetisch religiöse Ansicht ein. Bestimmt wird erzählt, Anaxagoras habe die Sonne, die Sterne für glühende Steine gehalten (es wird ihm auch, nach anderen, Schuld gegeben, etwas, das die Propheten für ein Wunder – Omen – ausgegeben, auf natürliche Weise erklärt zu haben); es steht damit in Verbindung, daß er es vorausgesagt haben soll, daß am Tage von Aigos Potamos, wo die Athener gegen Lysander ihre letzte Flotte verloren, ein Stein vom Himmel fiel. Überhaupt konnte schon bei Thales, Anaximander usf. die Bemerkung gemacht werden, daß sie Sonne, Mond, Erde und Gestirne zu Dingen machten und sie auf verschiedene Weise vorstellten, – Vorstellungen, die übrigens keine weitere Beachtung verdienen; denn diese Seite gehört eigentlich der Bildung. Alle ihre Vorstellungen von solchen Gegenständen enthalten das Gemeinschaftliche, daß die Natur durch sie entgöttert worden. Sie vertilgten die poetische Ansicht der Natur, die allem, was sonst jetzt für leblos gilt, ein eigentliches Leben, etwa auch Empfinden und, wenn man will, ein Sein überhaupt nach Weise des Bewußtseins zuschrieb. Diese poetische Ansicht zogen sie in die prosaische herab. Die Sonne wird als materielles Ding genommen, wie wir sie dafür halten, ist nicht mehr lebendiger Gott; diese Gegenstände sind uns bloße Dinge, dem Geiste äußerliche, geistlose Gegenstände. Dinge kann man herleiten von Denken. Dies[376] Denken tut wesentlich, daß es solche Gegenstände, Vorstellungen davon, die man göttlich, poetisch nennen kann, mit dem ganzen Umfange des Aberglaubens, verjagt, sie herabsetzt zu dem, was man natürliche Dinge nennt. Denn im Denken weiß der Geist sich als das wahrhaft Seiende, Wirkliche. Denken ist die Identität seiner und des Seins; für den Geist setzt sich im Denken das Ungeistige, Äußerliche zu Dingen herab, zum Negativen des Geistes. > Der Verlust jener Ansicht ist nicht zu beklagen, als ob damit die Einheit mit der Natur, schöner Glaube, unschuldige Reinheit und Kindlichkeit des Geistes verlorengegangen wäre. Unschuldig und kindlich mag sie wohl sein; aber die Vernunft ist eben das Herausgehen aus solcher Unschuld und Einheit mit der Natur. Sobald der Geist sich selbst erfaßt, für sich ist, muß er eben darum das Andere seiner sich als ein Negatives des Bewußtseins entgegensetzen, d.h. zu einem Geistlosen, zu bewußt- und leblosen Dingen bestimmen – und erst aus diesem Gegenstande zu sich kommen. Es ist dies diese Befestigung der sich bewegenden Dinge, die wir in den Mythen der Alten antreffen, die z.B. erzählen, daß die Argonauten die Felsen an der Meerenge des Hellesponts, die vorher wie Scheren sich bewegt hätten, festgestellt haben. Ebenso befestigte die fortschreitende Bildung das, was vorher eigene Bewegung und Leben in sich selbst zu haben gemeint wurde, und machte es zu ruhenden Dingen. Dieser Übergang solcher mythischen Ansicht zur prosaischen kommt hier zum Bewußtsein der Athenienser. Solche prosaische Ansicht setzt voraus, daß dem Menschen innerlich ganz andere Forderungen aufgehen, als er sonst gehabt hat. Darin liegen also die Spuren der wichtigen, notwendigen Konversion, die in den Vorstellungen der Menschen durch das Erstarken des Denkens, durch das Bewußtsein seiner selbst, durch die Philosophie gemacht ist. > Die Erscheinung solcher Anklage des Atheismus, die wir noch ausführlicher bei Sokrates berühren werden, ist bei Anaxagoras äußerlich aus dem näheren Grunde begreiflich,[377] daß die Athener auf Perikles eifersüchtig waren; daß die mit Perikles um den ersten Platz in Athen wetteiferten und ihn nicht unmittelbar (öffentlich) anzutasten wagten, seine Lieblinge gerichtlich angriffen und der Neid ihn durch die Anklage seines Freundes zu kränken suchte. So hatte man auch seine Freundin Aspasia zur Anklage gebracht; und der edle Perikles mußte, um sie von der Verdammung zu retten, die einzelnen Bürger Athens mit Tränen um ihre Lossprechung bitten. Das athenische Volk forderte in seiner Freiheit an seine Machthaber, denen es ein Übergewicht zuließ solche Akte, durch welche sie sich ebenso das Bewußtsein ihrer Demütigung vor dem Volke gaben; das Volk revanchierte sich so für das Übergewicht, welches die großen Männer hatten, übte selbst die Nemesis und setzte sich in Gleichgewicht mit ihnen, so wie sie wiederum das Gefühl ihrer Abhängigkeit, Unterwürfigkeit und Machtlosigkeit vor ihm dartaten. > Die Nachrichten über den Erfolg dieser Anklage des Anaxagoras sind ganz widersprechend und zweifelhaft; wenigstens befreite ihn Perikles von der Verurteilung zum Tode. Und entweder wurde er, nach einigen, nur zur Verbannung verurteilt, nachdem Perikles ihn vor das Volk geführt und für ihn sprach und bat, der schon durch sein Alter und Abzehrung und Schwäche das Mitleid des Volkes erregte. Oder andere sagen, er sei mit Hilfe des Perikles aus Athen geflohen und wurde abwesend zum Tode verurteilt und das Urteil nicht an ihm vollzogen. Oder andere sagen, er sei freigesprochen worden; aber aus Verdruß über diese Anklage und Besorgnis ihrer Wiederholung habe er Athen freiwillig verlassen, und im etlichen und 60. oder 70. Jahre sei er in Lampsakos in der 88. Olympiade (428 v. Chr.) gestorben. ### The general principle of thought > Der Zusammenhang seiner Philosophie mit den vorhergehenden ist: In Heraklits Idee als Bewegung sind alle Momente absolut verschwindende; Empedokles ist Zusammenfassen dieser Bewegung in die Einheit, aber eine synthetische, ebenso Leukipp und Demokrit, – aber so, daß bei Empedokles die Momente dieser Einheit die seienden Elemente des Feuers, Wassers usf. sind, bei diesen aber reine Abstraktionen, an sich seiende Wesen, Gedanken; hierdurch aber ist unmittelbar die Allgemeinheit gesetzt, denn die Entgegengesetzten haben keinen sinnlichen Halt mehr. Die Einheit kehrt als allgemeine in sich zurück aus der Entgegensetzung (in dem Synthesieren ist das Entgegengesetzte noch getrennt von ihr für sich, nicht der Gedanke selbst das Sein), – der Gedanke, als reiner freier Prozeß in sich selbst, das sich selbst bestimmende Allgemeine, nicht unterschieden vom bewußten Gedanken. Im Anaxagoras tut sich ein ganz anderes Reich auf. > Aristoteles sagt: »Anaxagoras hat erst diese Bestimmungen angefangen«, – ist also der erste, der das absolute Wesen als Verstand oder Allgemeines aussprach, als Denken (nicht Vernunft). Aristoteles und dann andere nach ihm führen das trockene Faktum an, daß ein Hermotimos auch aus Klazomenai dazu Veranlassung gegeben; aber deutlich, bestimmt habe dies Anaxagoras getan. Damit ist dann wenig gedient, da wir weiter nichts erfahren von Hermotimos' Philosophie; viel mag es nicht gewesen sein. Andere haben viel historische Untersuchungen über diesen Hermotimos angestellt. Dieser Name kommt noch einmal vor: α) Wir haben ihn schon angeführt in der Liste derer, von denen erzählt wird, daß Pythagoras, vor seinem Leben als Pythagoras, sie gewesen sei. β) Wir haben eine Geschichte von[379] Hermotimos: er habe nämlich die eigene Gabe besessen, als Seele seinen Leib zu verlassen. Dies sei ihm aber am Ende schlecht bekommen, denn seine Frau, mit der er Händel hatte und die sonst wohl wußte, wie es damit war, zeigte diesen von seiner Seele verlassenen Leib ihren Bekannten als tot, und er wurde verbrannt, ehe die Seele sich eingestellt hatte, die sich freilich wird verwundert haben. Es ist nicht der Mühe wert, zu untersuchen, was dieser alten Geschichte zugrunde liegt, d.h. wie wir die Sache ansehen wollen; man könnte an Verzückung dabei denken. Wir haben eine Menge solcher Geschichten von alten Philosophen, wie von Pherekydes, Epimenides usf.; daß dieser z.B. (eine Schlafmütze) siebenundfünfzig Jahre geschlafen habe. > {#DasPrinzipDesAnaxagorasWar} Das Prinzip des Anaxagoras war, daß er den [[nous]] Gedanken oder Verstand überhaupt, als das einfache Wesen der Welt, für das Absolute erkannt hat. Die Einfachheit des nous ist nicht ein Sein, sondern Allgemeinheit (Einheit). Allgemeines ist einfach und von sich unterschieden, – aber so, daß der Unterschied unmittelbar aufgehoben wird und diese Identität gesetzt, für sich ist, das Wesen nicht ein Scheinen in sich, Einzelheit – Reflexion an und für sich bestimmt – ist. Dies Allgemeine für sich, abgetrennt, existiert rein nur als Denken. Es existiert auch als Natur, gegenständliches Wesen, aber dann nicht mehr rein für sich, sondern die Besonderheit als ein Unmittelbares an ihm habend; so Raum und Zeit z.B., das Ideellste, Allgemeinste der Natur als solcher. Aber es gibt keinen reinen Raum und Zeit und Bewegung, sondern dies Allgemeine hat die Besonderheit unmittelbar an ihm, – bestimmter Raum, Luft, Erde man kann keinen reinen Raum zeigen, sowenig als die Materie. Denken ist also dies Allgemeine, aber rein für sich: Ich bin Ich, Ich = Ich. Ich unterscheide eins von mir, aber dieselbe reine Einheit bleibt, – nicht Bewegung, ein Unterschied,[380] der nicht unterschieden, Fürmichsein. Und in allem, was ich denke, wenn das Denken einen bestimmten Inhalt hat, so ist es mein Gedanke, – ich bin mir in diesem Gegenstande ebenso bewußt. > {#DiesAllgemeineSoFuerSichSeiende} Dies Allgemeine, so für sich Seiende, tritt aber ebenso dem Einzelnen – oder der Gedanke dem Seienden – bestimmt gegenüber. Hier wäre nun die spekulative Einheit dieses Allgemeinen mit dem Einzelnen zu betrachten, wie diese als absolute Einheit gesetzt ist; aber dies wird freilich bei den Alten nicht angetroffen, – den Begriff selbst zu begreifen. Den sich zu einem System realisierenden, als Universum organisierten Verstand, diesen reinen Begriff haben wir nicht zu erwarten. Wie Anaxagoras den [[nous]] erklärt, den Begriff desselben gegeben, gibt Aristoteles näher an: Das Allgemeine hat die zwei Seiten, α) reine Bewegung zu sein, und β) das Allgemeine, Ruhende, Einfache. Es ist darum zu tun, das Prinzip der Bewegung aufzuzeigen, – daß dies das Sichselbstbewegende und dies das Denken (als für sich existierend) ist. So Aristoteles. »Nous ist ihm (Anaxagoras) dasselbe mit Seele.« So unterscheiden wir Seele als das Sichselbstbewegende, unmittelbar Einzelne; aber als einfach ist der nous das Allgemeine. Der Gedanke bewegt um etwas willen, der Zweck ist das erste Einfache (die Gattung ist Zweck), er ist das Erste, welches sich zum Resultate macht, – bei den Alten Gutes und Böses, d.h. eben Zweck als Positives und als Negatives. > Diese Bestimmung ist eine sehr wichtige; sie hatte auch bei Anaxagoras noch keine große Ausführung. Während die bisherigen Prinzipien (Aristoteles unterscheidet zuerst Qualität, poion dann Materie und Stoff) stoffartig sind, außer dem Heraklitischen Prozeß, der drittens Prinzip der Bewegung ist, so tritt viertens das Umweswillen, die Zweckbestimmung mit dem nous ein. Dies ist das in sich Konkrete. Aristoteles fügt nach der oben (S. 217) angeführten Stelle[381] hinzu: »Nach diesen« (Ioniern und anderen) »und nach solchen Ursachen« (Wasser, Feuer usf.), »da sie nicht hinreichend sind, die Natur der Dinge zu erzeugen (gennêsai), sind die Philosophen von der Wahrheit selbst, wie schon gesagt, gezwungen worden, weiter zu gehen nach dem damit verbundenen Prinzip (tên echomenên archên). Denn daß auf einer Seite alles sich gut und schön verhalte, anderes aber erzeugt werde, – dazu ist weder die Erde noch sonst ein Prinzip hinreichend, auch scheinen jene dies nicht gemeint zu haben, noch macht es sich gut (kalôs echei), dem Selbstbewegen und dem Zufall (automatô kai tychê) ein solches Werk zu überlassen.« Gut und schön drückt den einfachen, ruhenden Begriff aus, Veränderung den Begriff in seiner Bewegung. > Mit diesem Prinzip kommen nun folgende Bestimmungen herein: a) Verstand überhaupt ist die sich selbst bestimmende Tätigkeit; dies fehlte bisher. Das Werden des Heraklit, was nur Prozeß ist (eimarenê), ist noch nicht das sich selbständig, unabhängig Bestimmende. In der sich selbst bestimmenden Tätigkeit ist zugleich enthalten, daß die Tätigkeit, indem sie den Prozeß macht, sich erhält als das Allgemeine, Sichselbstgleiche. Das Feuer (der Prozeß nach Heraklit) erstirbt; es ist Übergang ins Andere, keine Selbständigkeit. Es ist auch Kreis, Rückkehr zum Feuer; aber das Prinzip erhält sich in seinen Bestimmungen nicht. Es ist nur Übergehen ins Entgegengesetzte gesetzt, – nicht das Allgemeine, welches sich in beiden Formen erhält. ß) Bestimmung der Allgemeinheit liegt darin, wenn sie auch noch nicht förmlich ausgedrückt ist; es bleibt in der Beziehung auf sich in der Bestimmung. Darin liegt 7) Zweck, das Gute. > Ich habe schon neulich (S. 348 f.) auf den Begriff des Zwecks aufmerksam gemacht. Wir dürfen dabei nicht bloß an die Form des Zwecks denken, wie er in uns, in den Bewußten ist. Wir haben einen Zweck; er ist meine Vorstellung, sie ist[382] für sich, kann sich realisieren oder auch nicht. Es liegt im Zweck die Tätigkeit des Realisierens, wir vollführen diese Bestimmung; und das Produzierte muß dem Zwecke gemäß sein, – wenn man nicht ungeschickt ist, muß das Objekt nichts anderes enthalten als der Zweck. Es ist ein Übergang von der Subjektivität zur Objektivität: ich bin unzufrieden mit meinem Zweck, daß er nur subjektiv ist; meine Tätigkeit ist, ihm diesen Mangel zu benehmen und ihn objektiv zu machen. In der Objektivität hat sich der Zweck erhalten. Ich habe z.B. den Zweck, ein Haus zu bauen, ich bin deshalb tätig; es kommt heraus das Haus, der Zweck ist darin realisiert. > Wir müssen aber nicht bei der Vorstellung von diesem subjektiven Zweck stehenbleiben, wo beide, ich und der Zweck, selbständig existieren, wie wir dies gewöhnlich tun. Z.B. Gott, als weise, regiert nach Zwecken; da ist die Vorstellung, daß der Zweck für sich ist, in einem vorstellenden, weisen Wesen. > {DasAllgemeineDesZwecks} Das Allgemeine des Zwecks ist aber, daß er eine für sich feste Bestimmung ist und daß dann diese Bestimmung, die durch die Bestimmung der Tätigkeit gesetzt ist, weiter tätig ist, den Zweck zu realisieren, ihm Dasein zu geben; aber dies Dasein ist beherrscht durch den Zweck, und er ist darin erhalten. Dies ist, daß der Zweck das Wahrhafte, die Seele einer Sache ist. Das Gute gibt sich selbst Inhalt, indem es mit diesem Inhalt tätig ist, dieser Inhalt sich an Anderes wendet, so erhält sich in der Realität die erste Bestimmung, und es kommt kein anderer Inhalt heraus. Was vorher schon vorhanden war, und was nachher ist, nachdem der Inhalt in die Äußerlichkeit getreten ist, beides ist dasselbe; und das ist der Zweck. > {#DasGroessteBeispielBietetDasLebendige} Das größte Beispiel hiervon bietet das Lebendige dar; es erhält sich so, weil es an sich Zweck ist. Das Lebendige existiert, arbeitet, hat Triebe, diese Triebe sind seine Zwecke; es weiß nichts von diesen Zwecken, es ist bloß lebendig, – es sind erste Bestimmungen, die fest sind. Das Tier arbeitet, diese Triebe zu befriedigen, d.h. den Zweck zu erreichen; es verhält sich zu äußeren Dingen, teils mechanisch, teils chemisch. Aber das Verhältnis seiner Tätigkeit bleibt nicht mechanisch, chemisch. Das Produkt, das Resultat ist vielmehr das Tier selbst, es ist Selbstzweck, es bringt in seiner Tätigkeit nur sich selbst hervor; jene mechanischen usf. Verhältnisse werden darin vernichtet und verkehrt. Im mechanischen und chemischen Verhältnis ist dagegen das Resultat ein Anderes; das Chemische erhält sich nicht. Im Zwecke aber ist das Resultat der Anfang, – Anfang und Ende sind gleich. Selbsterhaltung ist fortdauerndes Produzieren, wodurch nichts Neues entsteht – Zurücknahme der Tätigkeit zum Hervorbringen seiner selbst –, immer nur das Alte. ## related entries * [[Nous]] ## References * Wikipedia, _[Anaxagoras](http://en.wikipedia.org/wiki/Anaxagoras)_ * [[Georg Hegel]], _[[Science of Logic]]_ * [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Anaxagoras](Lectures+on+the+History+of+Philosophy#Anaxagoras)_ category: people
Anayeli Ramirez	https://ncatlab.org/nlab/source/Anayeli+Ramirez	* [webpage](https://uniovi.academia.edu/AnayeliRamirez) ## Selected writings On [[black brane\|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT\|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] as [[defect field theory]] "inside" [[AdS7-CFT6]]: * [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636)) * [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510)) * [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669)) On [[AdS2/CFT1]] on [[D1-D3 brane intersections]]: * [[Giuseppe Dibitetto]], [[Yolanda Lozano]], [[Nicolò Petri]], [[Anayeli Ramirez]], _Holographic Description of M-branes via $AdS_2$_ ([arXiv:1912.09932](https://arxiv.org/abs/1912.09932)) * [[Yolanda Lozano]], [[Carlos Nunez]], [[Anayeli Ramirez]], Stefano Speziali, _New $AdS_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics_ ([arXiv:2011.00005](https://arxiv.org/abs/2011.00005)) category: people
Anders Kock	https://ncatlab.org/nlab/source/Anders+Kock	Anders Kock is a mathematician at Aarhus University, Denmark. He has proved important results in [[category theory]] and particularly in [[synthetic differential geometry]]. [Homepage](https://users-math.au.dk/kock) ## Selected writings On the [[2-category]] of [[complete categories]] being [[monadic functor\|monadic]] over [[Cat]]: * [[Anders Kock]], _Limit monads in categories_, PhD thesis (advisor [[W. Lawvere]]), University of Chicago (1967) [[proquest](https://www.proquest.com/openview/61883e900f1fd8585fe4dfc4c3f271dc)] Introducing the notion of [[strong monads]] and relating to [[monoidal monads]] and [[commutative monads]]: * [[Anders Kock]], Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970) 1-10 [[doi:10.1007/BF01220868](https://doi.org/10.1007/BF01220868)] * [[Anders Kock]], Closed categories generated by commutative monads, Journal of the Australian Mathematical Society 12 4 (Nov 1971) 405-424 [[doi:10.1017/S1446788700010272](https://doi.org/10.1017/S1446788700010272), [[KockMonoidalMonads.pdf:file]]] * [[Anders Kock]], Strong functors and monoidal monads, Arch. Math 23 (1972) 113–120 [[doi:10.1007/BF01304852](https://doi.org/10.1007/BF01304852), [pdf](http://home.imf.au.dk/kock/SFMM.pdf)] On [[fiber bundles]] [[internalization\|internal]] to [[finitely complete categories]]: * [[Anders Kock]], Fibre bundles in general categories, Journal of Pure and Applied Algebra 56 3 (1989) 233-245 [<a href="https://doi.org/10.1016/0022-4049(89)90059-5">doi:10.1016/0022-4049(89)90059-5</a>] * [[Anders Kock]], Generalized fibre bundles, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics 1348 (2006) 194-207 [[doi:10.1007/BFb0081359](https://doi.org/10.1007/BFb0081359)] On [[synthetic differential geometry]]: * {#Kock81} [[Anders Kock]], _Synthetic Differential Geometry_, Cambridge University Press (1981, 2006) [[pdf](http://home.imf.au.dk/kock/sdg99.pdf), [doi:10.1017/CBO9780511550812](https://doi.org/10.1017/CBO9780511550812)] * {#Kock10} [[Anders Kock]], _Synthetic geometry of manifolds_, Cambridge Tracts in Mathematics 180 (2010) [[pdf](http://home.imf.au.dk/kock/SGM-final.pdf), [doi:10.1017/CBO9780511691690](https://doi.org/10.1017/CBO9780511691690)] See also: * A. Kock, _The algebraic theory of moving frames_, Cahiers Top. Géom. Diff. Catégoriques 23 (1982) 347--362 * A. Kock. _Closed categories generated by commutative monads_, J. Austral. Math. Soc. 12(04):405--424, 1971 * A. Kock, [[I. Moerdijk]], _Presentations of étendues_ , Cah. Top. Géom. Diff. Cat. __XXXII__ 2 (1991) 145--164. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_2/CTGDC_1991__32_2_145_0/CTGDC_1991__32_2_145_0.pdf)) * A. Kock, [[I. Moerdijk]], _Every étendue comes from a local equivalence relation_ , JPAA __82_ (1992) 155--174. * [[E. J. Dubuc]], A. Kock, _On 1-form classifiers_, Commun. Alg. 12 (1984) 1471--1531 [doi](https://doi.org/10.1080/00927878408823064) * E. J. Dubuc, [[G. M. Kelly]], _A presentation of topoi as algebraic relative to categories or graphs_, J. Algebra __81__ (1983) 420--433 See * [[hyperdoctrine]] * [[Kock-Lawvere axiom]] * [[synthetic differential geometry]] ## Students * [[Maren Justesen]] category: people [[!redirects A. Kock]]
Anders Mörtberg	https://ncatlab.org/nlab/source/Anders+M%C3%B6rtberg	* [webpage](https://staff.math.su.se/anders.mortberg/) ## Selected writings On [[affine schemes]] in [[cubical type theory]]: * [[Anders Mörtberg]], [[Max Zeuner]], A Univalent Formalization of Affine Schemes ([arXiv:2212.02902](https://arxiv.org/abs/2212.02902)) Introducing the [[programming language]] [[Cubical Agda]] implementing [[univalence\|univalent]] [[cubical type theory\|cubical]] [[homotopy type theory]] with [[higher inductive types]]: * [[Anders Mörtberg]], Cubical Agda (2018) [[blog post](https://homotopytypetheory.org/2018/12/06/cubical-agda)] * {#VMA19} [[Andrea Vezzosi]], [[Anders Mörtberg]], [[Andreas Abel]], Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types, Proceedings of the ACM on Programming Languages 3 ICFP 87 (2019) 1–29 [[doi:10.1145/3341691](https://doi.org/10.1145/3341691), [pdf](https://www.cse.chalmers.se/~abela/icfp19.pdf)] On [[cubical type theory\|cubical]] [[homotopy type theory]] implemented in the [[proof assistant]] [[Cubical Agda]]: * {#Mortberg19} [[Anders Mörtberg]], _Cubical methods in HoTT/UF_, 2019 ([pdf](http://staff.math.su.se/anders.mortberg/papers/cubicalmethods.pdf)) Exposition in view of [[synthetic homotopy theory]]: * [[Anders Mörtberg]], [[Loïc Pujet]], Cubical synthetic homotopy theory, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [[doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), [pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf)] On [[ordinary cohomology in homotopy type theory]]: * [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], Synthetic Integral Cohomology in Cubical Agda, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) 216 (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$ > (in [[cubical type theory\|cubical]] [[Agda]]) ## Related $n$Lab entries * [[cubical type theory]] category: people [[!redirects Anders Moertberg]] [[!redirects Anders Mortberg]]
Anders S. Buch	https://ncatlab.org/nlab/source/Anders+S.+Buch	* [personal page](https://sites.math.rutgers.edu/~asbuch/) ## Selected writings On [[quantum K-theory rings]]: * [[Anders S. Buch]], [[Leonardo C. Mihalcea]], Quantum K-theory of Grassmannians, Duke Math. J. 156 3 (2011) 501-538 [[arXiv:0810.0981](https://arxiv.org/abs/0810.0981), [doi:10.1215/00127094-2010-218](https://doi.org/10.1215/00127094-2010-218)] category: people
Anderson duality	https://ncatlab.org/nlab/source/Anderson+duality	+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Stable Homotopy theory +--{: .hide} [[!include stable homotopy theory - contents]] =-- #### Duality +-- {: .hide} [[!include duality - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The [[stable (infinity,1)-category of spectra]] has a [[dualizing object in a closed category\|dualizing object]] (dualizing module) on a suitable subcategory of finite spectra. It is called the _[[Anderson spectrum]]_ $I_{\mathbb{Z}}$ ([[Representability Theorems\|Lurie, Example 4.3.9]]). The [[duality]] that this induces is called _Anderson duality_. ## Examples The Anderson dual of the [[sphere spectrum]] is discussed in * ([Hopkins-Singer 05, appendix B](#HopkinsSinger05)) in the context of constructing a [[quadratic refinement]] of the [[intersection pairing]] on [[ordinary differential cohomology]] * in ([Freed 14, section 5.1.1](#Freed14)) in the context of invertible [[extended topological field theories]]. The Anderson dual of [[KU]] is (complex conjugation-equivariantly) the 4-fold [[suspension spectrum]] $\Sigma^4 KU$ ([Heard-Stojanoska 14, theorem 8.2](#HeardStojanoska14)). This implies that, nonequivariantly $KU$ is Anderson self-dual and the Anderson dual of $KO$ is $\Sigma^4KO$, which were both first proven by [Anderson](#Anderson69). Similarly [[Tmf]]$[1/2]$ is Anderson dual to its 21-fold suspension ([Stojanoska 12](#Stojanoska12)). ## Related concepts * [[Brown-Comenetz duality]] * [[Spanier-Whitehead duality]] ## References ### General Original articles include * {#Anderson69} [[Donald W. Anderson]], _Universal coefficient theorems for K-theory_, mimeographed notes, Univ. California, Berkeley, Calif., 1969 ([pdf](https://faculty.tcu.edu/gfriedman/notes/Anderson-UCT.pdf)) * Zen-ichi Yosimura, Universal coefficient sequences for cohomology theories of CW-spectra, Osaka J. Math. 12 (1975), no. 2, 305–323. MR 52 #9212 See also * [[Jacob Lurie]], section 4.2 of _[[Representability Theorems]]_ ### Examples The Anderson dual of the [[sphere spectrum]] is discussed (in a context of [[extended TQFTs]]) in * {#HopkinsSinger05} [[Michael Hopkins]], [[Isadore Singer]], appendix B of, _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_, 2005 * {#Freed14} [[Dan Freed]], section 5.1.1 of _Short-range entanglement and invertible field theories_ ([arXiv:1406.7278](http://arxiv.org/abs/1406.7278)) * [[Daniel Freed]], [[Michael Hopkins]], section 5.3 of _Reflection positivity and invertible topological phases_ ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527)) The Anderson duals of [[KU]] and of [[tmf]] are discussed in * {#Stojanoska12} [[Vesna Stojanoska]], _Duality for Topological Modular Forms_, Doc. Math. 17 (2012) 271-311 ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968)) * {#HeardStojanoska14} [[Drew Heard]], [[Vesna Stojanoska]], _K-theory, reality, and duality_ ([arXiv:1401.2581](http://arxiv.org/abs/1401.2581)) In the context of [[heterotic string theory]]: * [[Yuji Tachikawa]], [[Mayuko Yamashita]], Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)] ### Equivariant duality Anderson duality in [[equivariant stable homotopy theory]] is discussed in * {#Ricka15} [[Nicolas Ricka]], _Equivariant Anderson duality and Mackey functor duality_ ([arXiv:1408.1581](http://arxiv.org/abs/1408.1581)) [[!redirects Anderson spectrum]] [[!redirects Anderson dual]] [[!redirects Anderson duals]]
Andor Lukacs	https://ncatlab.org/nlab/source/Andor+Lukacs	* [website](http://www.staff.science.uu.nl/~lukac101/) category: people
Andre Geim	https://ncatlab.org/nlab/source/Andre+Geim	* [Wikipedia entry](https://en.wikipedia.org/wiki/Andre_Geim) * [institute page](http://www.condmat.physics.manchester.ac.uk/people/academic/geim/) ## Selected writings Announcing the isolation of [[graphene]]: * [[Konstantin Novoselov]], [[Andre Geim]], S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, _Electric field effect in atomically thin carbon films_, Science __306__, no. 5696, pp. 666-669 (2004) ([doi:10.1126/science.1102896](http://dx.doi.org/10.1126/science.1102896), [arXiv:cond-mat/0410550](https://arxiv.org/abs/cond-mat/0410550)) and discussion of its [[electron band structure]]: * A. H. Castro Neto, F. Guinea, N. M. R. Peres, [[Konstantin S. Novoselov]], and [[Andre K. Geim]], The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109 $[$[doi:10.1103/RevModPhys.81.109](https://doi.org/10.1103/RevModPhys.81.109)$]$ category: people [[!redirects Andre K. Geim]]
Andre Hirschowitz	https://ncatlab.org/nlab/source/Andre+Hirschowitz	* [website](http://math.unice.fr/~ah/) ## related $n$Lab entries * [[∞-stack]] * [[schematic homotopy type]] category: people [[!redirects Andre Hirschowicz]]
Andre Kornell	https://ncatlab.org/nlab/source/Andre+Kornell	* [Institute page](https://math.berkeley.edu/people/grad/andre-kornell) * [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=220518) ## Selected writings On [[quantum sets]] with [[quantum relations]]: * [[Andre Kornell]], Quantum Sets, J. Math. Phys. 61 102202 (2020) [[doi:10.1063/1.5054128](https://doi.org/10.1063/1.5054128)] * [[Andre Kornell]], Discrete quantum structures [[arXiv:2004.04377](https://arxiv.org/abs/2004.04377)] * [[Andre Kornell]], Discrete quantum structures I: Quantum predicate logic, J. Noncommut. Geom. (2023) [[doi:10.4171/jncg/531](https://doi.org/10.4171/jncg/531)] Introducing a notion of [[quantum CPOs]] for [[quantum computation]] (via [[quantum sets]] carrying [[quantum relations]]): * [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], Quantum CPOs, EPTCS 340 (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)] On an [[axiom\|axiomatic]] characterization of the [[compact closed dagger category]] [[Hilb]] of [[Hilbert spaces]]: * [[Chris Heunen]], [[Andre Kornell]], Axioms for the category of Hilbert spaces ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418)) category: people