DOCSLIB.ORG

Deformation Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Deformation Theory Alexios Terezakis Deformation Theory Master's Thesis University of Athens, Department of Mathe- matics Athens March 14, 2019 Advisor: Aristides Kontogeorgis Master Thesis Committee Ioannis Emmanouil Aristides Kontogeorgis Ioannis Dokas To my parents Contents Eiσαγωγή ix Introduction xiii 1 Basic Definitions 1 1.1 Coefficient-Λ-algebras 1 1.2 Zariski Tangent Space 3 1.3 K¨ahlerDifferentials 5 2 Schlessinger's Representability Theorem 9 2.1 Small Extensions 9 2.2 Functors of Artin rings 11 2.3 Universal Elements 12 2.4 Schlessinger's Theorem 14 3 Examples 23 3.1 The Picard functor 23 3.2 Deformations of curves 26 A Flat Modules 29 Literature 31 Eiσαγωγή 'Enac moduli functor F από mia kathgorÐa C! Sets eÐnai ènac συναρτητής από mia kathgorÐa (schemes, sheaves, morfism¸n metαξύ touc, anaparasτάσεων) sthn kathgorÐa twn συνόlwn, ¸ste o F na stèlnei oikogèneiec antikeimènwn thc C πάνω από mia βάση B se èna stoiqeÐo thc κλάσης isodunamÐac twn antikeimènwn πάνω από to B. 'Ena moduli πρόβλημα lègetai fine όtan o moduli συναρτητής eÐnai representable, δηλαδή όtan υπάρχει èna scheme X kai ènac isomorfiσμός ∼ sunartht¸n hX = F . 'Opou o hX eÐnai o συναρτητής pou stèlnei to antikeÐmeno T sto σύνοlo Hom(T;X), twn morfism¸n T ! X thc kathgorÐac C kai thn απεικόnish f : T1 ! T2 sthn απεικόnish hx(T2) 3 h2 7! h1 = h2 ◦ f 2 hX (T1) mèsw tou διαγράμματoc h1 T1 / X > f h 2 T2 H ύπαρξη ενός tètoiou isomorfiσμού, shmaÐnei όti gia οποιαδήποte antikeÐmena Ti;Tj kai συναρτήσειc fij : Ti ! Tj, υπάρχει mÐa συμβατή oikogèneia isomorfi- sm¸n φi tètoia ¸ste to ακόloujo διάγραμμa na metatÐjetai φj hX (Tj) / F (Tj) hX (fi;j ) F (fi;j ) φi hX (Ti) / F (Ti) 'Ena από ta κλασσικά moduli προβλήματa eÐnai to moduli πρόβλημα twn kampul¸n dedomènou gènouc g. Αυτός o moduli συναρτητής apeikonÐzei κάθε oikogèneia sqetik¸n kampul¸n X ! T πάνω από èna scheme T , sthn κλάση isomorfÐac touc, όπου δύο oikogèneiec X1; X2 eÐnai iσόμορφες όtan υπάρχει ènac isomorfiσμός φ tètoioc ¸ste to παρακάτw διάγραμμα na metatÐjetai φ X1 / X2 ~ T x · Contents Γεωμετρικά, oi καμπύλες πάνω από to Speck, όπου to k eÐnai αλγεβρικά kleistό s¸ma, antistoiχούν sta shmeÐa tou X, αφού h απεικόnish X ! Speck antistoiqeÐ se stoiqeÐo tou συνόlou hX (Speck) = Speck; X), δηλαδή se èna γεωμετρικό shmeÐo tou X. Dustuq¸c, h ύπαρξη automorfism¸n kampul¸n, empodÐzei ton moduli sunarthth από to na eÐnai representable. Gia παράδειgma an C eÐnai mia αλγεβρική καμπύλη me κάποιon mh tetrimmèno automorfiσμό φ, τόte èqoume to ακόloujo διάγραμμα φ C / C id Speck / Speck H σπουδαιόthta tou παραπάνω paradeÐgmatoc ègkeitai sto όti h apeikόnish id : Speck ! Speck den periγράφει thn apεικόnish twn oikogenei¸n. Wc èna ακόμα παράδειgma ja deÐxoume όti o moduli συναρτητής twn elleiptik¸n kampul¸n den eÐnai representable. Mia ελλειπτική καμπύλη πάνω από to C eÐnai mia leÐa probo- λική καμπύλη E, mazÐ me èna stajeropoihmèno shmeÐo e 2 E. Qrhsimopoi¸ntac to Je¸rhma Riemann-Roch se συνδυασμό me thn jewrÐa διαμόρφωσης twn τεσσά- rwn shmeÐwn διακλάδωσης thc διπλής επικάλυψης E ! P1, μπορούμε na deÐxoume όti κάθε ελλειπτική καμπύλη mporeÐ na perigrafeÐ από ta shmeÐa mhdeniσμού tou omoγενούς poλυωνύμου Y 2Z − X(X − Z)(X − λZ); Se αυτό to montèlo to kleisτό shmeÐo e èqei probolikèc suntetagmènec e = [0 : 1 : 0], kai λ 2 A1 − f0; 1g. To polu¸numo αυτό orÐzei mia oikogèneia 1 E! A − f0; 1g; πάνω από thn truphmènh afiνική eujeÐa, sunep¸c to A1−f0; 1g mporeÐ na jewrhjeÐ wc q¸roc paramètrwn gia thn oikogèneia. H αναπαράσtash mia κλάσης isomorfÐac wc Ðna den eÐnai moναδική, υπάρχει mia δράση thc συμμετρικής ομάδας S3 sto A1 −f0; 1g h opoÐa παράγετai από touc automorfiσμούς λ 7! 1/λ, λ 7! 1=(1−λ). An jèloume na παραμετρήσουμε elleiptikèc καμπύλες qwrÐc na κάποια proboλική εμφύτευση prèpei na θεωρήσουμε to phlÐko A1 − f0; 1g proc thn δράση αυτή thc S3. O q¸roc pou ja katαλήξουμε eÐnai o δακτύλιoc twn analloÐwtwn tou C[λ]λ(λ−1) pou eÐnai h j-eujeÐa, me (λ2 − λ + 1)3 j = 28 : λ2(λ − 1)2 Υπάρχει mia antistoiqÐa metαξύ twn κλάσεων isomorfiσμού elleiptik¸n kampul¸n πάνω από to C kai twn migadik¸n arijm¸n j 2 C. Wsτόσο h afiνική eujeÐa A1 den apoteleÐ èna fine moduli q¸ro gia tic elleiptikèc καμπύλεc. Πράγματι, èstw 1 mia oikogèneia elleiptik¸n kampul¸n Et orismènh πάνω από to A − f0g, h opoÐa dÐnetai από thn exÐswsh Y 2Z = X3 − tZ3: Gia κάθε t όlec oi Ðnec èqoun stαθερή j-analloÐwth Ðsh me to 0. 'Estw όti to A1 anaparisτά tic elleiptikèc καμπύλες, τόte h paραπάνω oikogèneia ja prèpei 1 1 na antistoiqeÐ ston stαθερό morfiσμό (A − f0g) ! Aj . 'Omwc h ελλειπτική Contents · xi 2 3 3 καμπύλη E0 : Y Z = X − Z èqei epÐshc j-analloÐwth 0. Sunep¸c h oikogèneia 1 Et ja eÐnai tetrimmènh kai Ðsh me E0 × (A − f0g). Wsτόσο αυτό den eÐnai alhjèc, 1 πάνω από to function field C(t) oi oikogèneiec Et kai (A −f0g) gÐnontai iσόmorfec πάνω από thn epèktash C(t1=6). Υπάρχουν διάφορες teqnikèc pou mac epitrèpoun na katasτήσουμε èna moduli πρόβλημα representable, όπως gia παράδειgma h eiσαγωγή thc ènnoiac twn alge- brik¸n q¸rwn kai twn stacks, ή αλλάζοntac thn ènnoia thc isomorfÐac. An mac epitrèpetai mia υπεραπλούσteush, μπορούμε na pούμε pwc ènac τρόπος na ori- stούν ta stacks ενός moduli q¸rou kampul¸n eÐnai na orÐsoume thn kathgorÐa me antikeÐmena tic proper smooth oikogèneiec X! S, twn opoÐwn oi Ðnec eÐnai sunektikèc καμπύλες dedomènou gènouc. H jewrÐa paramorf¸sewn (Deformation theory) από thn άλλη proèrqetai a- πό thn douleÐa twn Kodaira kai Spencer πάνω se migadikèc poλλαπλόthtec. O Grothendieck metèfere thn jewrÐa αυτή sthn gl¸ssa twn Schemes. Μπορούμε na πούμε όti h jewrÐa paramorf¸sewn eÐnai h diaqeÐrish ενός moduli προβλή- matoc tοπικά, όπου meletούντai oikogèneiec πάνω από to φάσμα topik¸n Artin daktulÐwn. 'Enac daκτύλιoc tou Artin eÐnai ex oriσμού ènac δακτύλιoc ston o- poÐo κάθε fjÐnousa akoloujÐa idewd¸n tou termatÐzei ύσtera από peperasmèna βήματa. 'Ena από ta pio απλά paradeÐgmata (pou den eÐnai s¸ma) eÐnai o δακτύλιoc k[]=h2i, όπου to apoteleÐ èna apeirostό βαθμού 2 me thn ènnoia όti 2 = 0. Sth dhmosÐeush tou o Schlessinger mac parèqei thn gl¸ssa kai ta ergaleÐa na qeiristούμε ta apeirosτά san stoiqeÐa tou ‘εφαπτόμενου χώρου’ kai λύνει thn antÐstoiqh συνήθη διαφορική exÐswsh mèsw tupik¸n dunamoseir¸n daktulÐwn. Sugkekrimèna sto Κεφάλαιo 1 eiσάγουμε tic kathgorÐec pou ja χρησιμοποιή- soume, ton Zariski εφαπτόμενο q¸ro kai pwc na ton orÐsoume gia sunarthtèc kai ta διαφορικά K¨ahler. Sto Κεφάλαιo 2 orÐzoume thn ènnoia thc small extension (μικρής επεκτάσης), thn ènnoia smoothness kai tèloc to κεντρικό apotèlesma, to Je¸rhma tou Schlessinger. Sto teleutaÐo κεφάλαιo αποδεικνύουμε me thn χρή- sh tou Θεωρήματoc tou Schlessinger όti o Picard συναρτητής kai o συναρτητής παραμόρφωσης eÐnai pro-representable. Αθήνα Μάρτιoc 2019. Introduction A moduli functor F from a category C! Sets is a functor from a category (schemes, sheaves,morphisms between them, representations) to the category of sets, so that F it sends families of objects of C over a base B to the element of equivalence class of objects over B. A moduli problem is called fine when the moduli functor is representable, that is there is a scheme X and an isomor- ∼ phism of functors hX = F . The functor hX is the functor sending T to the set Hom(T;X), of morphisms of schemes T ! X, and the map f : T1 ! T2 to the map hX (T2) 3 h2 7! h1 = h2 ◦ f 2 hx(T1) by the diagram h1 T1 / X > f h 2 T2 ∼ The existence of an isomorphism hX = F , means that for every objects Ti;Tj and functions fi;j : Ti ! Tj, there is a compatible set of isomorphisms φi so that the following diagram is commutative φj hX (Tj) / F (Tj) hX (fi;j ) F (fi;j ) φi hX (Ti) / F (Ti) One of the classical moduli problems is the moduli problem of curves of genus g. This moduli functor to any family of relative curves X! T over a scheme T assigns the isomorphy class of it, where two families X1; X2 are isomorphic when there is an isomorphism φ making the following diagram commutative: φ X1 / X2 ~ T Geometrically curves over Speck, where k is an algebraically closed field k, correspond to points of X, since the structure map X ! Speck corresponds to an element in the set hX (Speck) = Hom(Speck; X) i.e. a geometric point of X. xiv · Contents Unfortunately, the existence of automorphisms of curves, prevents the mod- uli functor to be representable. For example if C is an algebraic curve which admits a non-trivial automorphism φ, then we have the diagram φ C / C id Speck / Speck The importance of the above example is that the map id : Speck ! Speck does not describe the map of the families. As an other example we will show that the moduli space of elliptic curves is not representable. An elliptic curve over C is a smooth projective curve E, together with a selected closed point e 2 E. As an application of Riemann-Roch theorem we can show that any elliptic curve can be described as the zero locus of the homogeneous polynomial Y 2Z − X(X − Z)(X − λZ); using also the theory of configuration of the four ramification points of the two cover E ! P1. In this model the closed point e has projective coordinates e = [0 : 1 : 0], and λ 2 A1 − f0; 1g.
Load more

Recommended publications
  • Hodge Decomposition for Higher Order Hochschild Homology
    ANNALES SCIENTIFIQUES DE L’É.N.S. TEIMURAZ PIRASHVILI Hodge decomposition for higher order Hochschild homology Annales scientifiques de l’É.N.S. 4e série, tome 33, no 2 (2000), p. 151-179 <http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 2000, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Sclent. EC. Norm. Sup., 4° serie, t 33, 2000, p. 151 a 179. HODGE DECOMPOSITION FOR HIGHER ORDER HOCHSCHILD HOMOLOGY TEIMURAZ PIRASHVILI ABSTRACT. - Let r be the category of finite pointed sets and F be a functor from r to the category of vector spaces over a characteristic zero field. Loday proved that one has the natural decomposition 7TnF(S1) ^ ©n^ Q^^CF), n ^ 0. We show that for any d ^ 1, there exists a similar decomposition for 7^nF(Sd). Here 3d is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for 7VnF(S1) (respectively 7TnF(S2)) completely determines the decomposition of 7^nF(Sd) for any odd (respectively even) d. These results can be applied to the cohomology of the mapping space Xs , where X is a c^-connected space.
    [Show full text]
  • Derived Smooth Manifolds
    DERIVED SMOOTH MANIFOLDS DAVID I. SPIVAK Abstract. We define a simplicial category called the category of derived man- ifolds. It contains the category of smooth manifolds as a full discrete subcat- egory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C1-rings that is obtained by patching together homotopy zero-sets of smooth functions on Eu- clidean spaces. We show that derived manifolds come equipped with a stable normal bun- dle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived man- ifolds. In particular, the intersection A \ B of submanifolds A; B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ^ [B] = [A \ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory. Contents 1. Introduction 1 2. The axioms 8 3. Main results 14 4. Layout for the construction of dMan 21 5. C1-rings 23 6. Local C1-ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Proofs of technical results 39 9. Derived manifolds are good for doing intersection theory 50 10. Relationship to similar work 52 References 55 1. Introduction Let Ω denote the unoriented cobordism ring (though what we will say applies to other cobordism theories as well, e.g.
    [Show full text]
  • Math 395: Category Theory Northwestern University, Lecture Notes
    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
    [Show full text]
  • Infinite Dimensional Lie Theory from the Point of View of Functional
    Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods.
    [Show full text]
  • On the Smoothness of Functors
    Iranian Journal of Mathematical Sciences and Informatics Vol. 5, No. 1 (2010), pp. 27-39 On the Smoothness of Functors A. Bajravania and A. Rastegarb,∗ aDepartment of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran b Faculty of Mathematics, Sharif University, P. O. Box 11155, Tehran, Iran E-mail: [email protected] E-mail: [email protected] Abstract. In this paper we will try to introduce a good smoothness no- tion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should have. Keywords: Abelian Category, First Order Deformations, Multicategory, Tangent Category, Topologizing Subcategory. 2000 Mathematics subject classification: 14A20, 14A15, 14A22. 1. Introduction Nowadays noncommutative algebraic geometry is in the focus of many basic topics in mathematics and mathematical physics. In these fields, any under consideration space is an abelian category and a morphism between noncom- mutative spaces is a functor between abelian categories. So one may ask to gen- eralize some aspects of morphisms between commutative spaces to morphisms between noncommutative ones. One of the important aspects in commuta- tive case is the notion of smoothness of a morphism which is stated in some languages, for example: by lifting property as a universal language, by projec- tivity of relative cotangent sheaves as an algebraic language and by inducing a surjective morphism on tangent spaces as a geometric language. ∗Corresponding Author Received 25 December 2009; Accepted 10 April 2010 c 2010 Academic Center for Education, Culture and Research TMU 27 28 A.
    [Show full text]
  • Diffeology, Groupoids & Morita Equivalence
    Diffeology, Groupoids & Morita Equivalence M.Sc. Thesis in Mathematics Nesta van der Schaaf 22nd June 2020 Supervisor: Prof. dr. N.P. Landsman Second reader: Dr. I.T. Mărcuț Abstract This thesis consists of two parts. The first (Chapters I and II) is a thorough introduction to the theory of diffeology. We provide a ground-up account of the theory from the viewpoint of plots (in contrast to the sheaf-theoretic treatments). This includes a proof that the category Diffeol of diffeological spaces and smooth maps is complete and cocomplete, (locally) Cartesian closed, and a quasitopos. In addition, we treat many examples, including a detailed recollection of the classification of irrational tori. The second part (Chapters III to VI) is a proposal for a framework of diffeological Morita equivalence. We give definitions of diffeological groupoid actions, -bundles, and -bibundles, general- ising the known theory of Lie groupoids and their corresponding notions. We obtain a bicategory DiffeolBiBund of diffeological groupoids and diffeological bibundles. This has no analogue inthe Lie theory, since we put no ‘principality’ restrictions on these bibundles. We then define a new notion of principality for diffeological (bi)bundles, and subsequently obtain a notion of Morita equivalence by declaring that two diffeological groupoids are equivalent if and only if there existsa biprincipal bibundle between them. Our main new result is the following: two diffeological groupoids are Morita equivalent if and only if they are weakly equivalent in the bicategory DiffeolBiBund. Equivalently, this means that a diffeological bibundle is weakly invertible if and only if it is biprincipal. This significantly generalises the original theorem in the Lie groupoid setting, where an analogous state- ment can only be made if we assume one-sided principality beforehand.
    [Show full text]
  • On the Smoothness of the Quot Functor
    On the Smoothness of the Quot Functor S EBASTIAN W IEGANDT Master of Science Thesis Stockholm, Sweden 2013 On the Smoothness of the Quot Functor S EBASTIAN W IEGANDT Master’s Thesis in Mathematics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2013 Supervisor at KTH was Roy Skjelnes Examiner was Roy Skjelnes TRITA-MAT-E 2013:29 ISRN-KTH/MAT/E--13/29--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract For a commutative ring k, we consider free k-modules E, endowing them with k[x1; : : : ; xm]- module structures through a ring homomorphism k[x1; : : : ; xm] ! EndZ(E). These struc- tures are then inspected by encoding the actions of the unknowns xi in matrices X1;:::;Xm. We further introduce the concepts of lifts and formal smoothness for functors, and define the n QuotF=A=k functor acting on the category of k-algebras, taking some k-algebra B to the set of quotients of the form (F ⊗k B)=N, which are locally free as B-modules. Lastly, we find con- 4 2 crete examples of modules showing that the functors Hilbk[x;y;z]=k and QuotL2 k[x;y]=k[x;y]=k are not formally smooth. Contents 1 Introduction 3 1.1 Background . .3 1.2 Overview ........................................4 1.3 Acknowledgments . .5 2 Preliminaries 6 2.1 Categories . .6 2.1.1 Yoneda's Lemma . .6 2.1.2 Representable Functors . .8 3 Free Quotients 10 3.1 Quotients in Rank One .
    [Show full text]
  • Smooth Functors Vs. Differential Forms
    Hamb. Beitr. Math. Nr. 297 ZMP-HH/08-04 Smooth Functors vs. Differential Forms Urs Schreiber and Konrad Waldorf Department Mathematik Schwerpunkt Algebra und Zahlentheorie Universität Hamburg Bundesstraße 55 D–20146 Hamburg Abstract We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary be- tween fundamental notions from category theory and differential geom- etry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth func- tors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces. arXiv:0802.0663v4 [math.DG] 18 Jul 2011 Table of Contents Introduction 2 1 Review: Smooth Functors and 1-Forms 6 1.1 ThePathGroupoidofaSmoothManifold . 6 1.2 DiffeologicalSpaces. .. .. .. 8 1.3 Equivalence between Functors and Forms . 10 2 Smooth 2-Functors and Differential Forms 14 2.1 ThePath2-GroupoidofaSmoothManifold . 15 2.2 FromFunctorstoForms . 18 2.3 FromFormstoFunctors . 30 2.4 MainTheorem .......................... 42 3 Examples of Smooth 2-Functors 44 3.1 Connectionson(non-abelian)Gerbes . 44 3.2 DerivativesofSmoothFunctors . 47 3.3 Classical Solutions in BF-Theory . 48 4 Transgression to Loop Spaces 49 4.1 Generalization to Diffeological Spaces . 49 4.2 InducedStructureontheLoopSpace . 54 Appendix 61 A.1 Basic2-CategoryTheory . 61 A.2 Lie 2-Groups and Smooth Crossed Modules . 65 A.3 ProofofLemma2.16 ....................... 69 Table of Notations 72 References 73 Introduction The present article is the second of three articles aiming at a general and sys- tematical approach to connections on (non-abelian) gerbes and their surface holonomy.
    [Show full text]
  • On the Smoothness of Functors
    On the Smoothness of Functors A. Bajravani Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran P. O. Box 14115-134 A. Rastegar∗ Faculty of Mathematics, Sharif University, Tehran, Iran P. O. Box 11155 Abstract In this paper we will try to introduce a good smoothness notion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should have. Keywords: Abelian Category, First Order Deformations, Multicategory, Tangent Category, Topologizing Subcategory. Mathematics Subject Classification: 14A20, 14A15, 14A22. 1 Introduction Nowadays noncommutative algebraic geometry is in the focus of many basic topics in mathematics and mathematical physics. In these fields, any under consideration space is an abelian category and a morphism between noncommutative spaces is a functor between abelian categories. So arXiv:1007.5314v1 [math.AG] 29 Jul 2010 one may ask to generalize some aspects of morphisms between commutative spaces to morphisms between noncommutative ones. One of the important aspects in commutative case is the notion of smoothness of a morphism which is stated in some languages, for example: by lifting property E-mail Addresses: [email protected] [email protected] (*)Corresponding Author 1 as a universal language, by projectivity of relative cotangent sheaves as an algebraic language and by inducing a surjective morphism on tangent spaces as a geometric language. In this paper, in order to generalize the notion of smooth morphism to a functor we propose three different approaches. A glance description for the first one is as follows: linear approxima- tions of a space are important and powerful tools.
    [Show full text]
  • Journal of Pure and Applied Algebra Tannaka Duality for Proper
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 214 (2010) 750–768 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa Tannaka duality for proper Lie groupoidsI Giorgio Trentinaglia ∗ Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands article info a b s t r a c t Article history: By replacing the category of smooth vector bundles of finite rank over a manifold with Received 13 March 2009 the category of what we call smooth Euclidean fields, which is a proper enlargement of the Received in revised form 18 July 2009 former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we Available online 16 September 2009 are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Communicated by C.A. Weibel Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field. MSC: ' 2009 Elsevier B.V. All rights reserved. 58H05 22D35 0. Introduction The classical duality theory of Tannaka leads to the result that a compact group can be reconstructed from a purely discrete, algebraic object, namely the ring of its continuous, finite dimensional representations or, more precisely, the algebra of its representative functions. Compare [1,2]. The same theory can be efficiently recast in categorical terms. This alternative point of view on Tannaka duality stems from Grothendieck's theory of motives in algebraic geometry [3–5].
    [Show full text]
  • 3.5 Associated Vector Bundles
    3.5 Associated vector bundles Recall that the tangent bundle (T M; πM ;M) is a bundle of vector spaces TM = tpTpM −1 k which has local trivializations Φ : πM (U) −! U × R which preserve the projections to U, and which are linear maps for fixed p. By the smoothness and linearity of these local trivializations, we note that, for charts (Ui ;'i ), we have −1 −1 k gij = (Φj ◦ Φi ) = T ('j ◦ 'i ): Ui \ Uj −! GL(R ) is a collection of smooth matrix-valued functions, called the \transition functions" of the bundle. These obviously satisfy the gluing conditions or \cocycle condition" gij gjk gki = 1k×k on Ui \ Uj \ Uk . We will now use the tangent bundle to create other bundles, and for this we will use functors from the category of vector spaces VectR to itself obtained from linear algebra. Example 3.29 (Cotangent bundle). Consider the duality functor V 7! V ∗ = Hom(V; R) which is contravariant, i.e. if A : V −! W then A∗ : W ∗ −! V ∗. Also, it is a smooth functor in the sense that the map A 7! A∗ is smooth map of vector spaces (in this case it is the identity map, essentially). The idea is to apply this functor to the bundle fibrewise, to apply it to the trivializations fiberwise, and to use the smoothness of the functor to obtain the manifold structure on the result. Therefore we can form the set ∗ ∗ T M = tp(TpM) ; −1 k which also has a projection map pM to M. And, for each trivialization Φ: πM (U) −! U × R , we obtain ∗ −1 −1 k ∗ bijections F = (Φ ) : pM (U) −! U × R .
    [Show full text]
  • Super Symmetric Field Theories and Integral Modular Functions
    Super symmetric field theories and integral modular functions Stephan Stolz and Peter Teichner Preliminary Version of March 9, 2007 Contents 1 Introduction 2 2 Field theories a la Atiyah-Segal 6 2.1 The categories TV and TV± ................... 7 2.2 The Riemannian bordism category RBd ............. 10 2.3 Quantum field theories, preliminary definition . 17 2.4 Consequencesoftheaxioms . 18 3 QFT’s as smooth functors 24 3.1 Smoothcategoriesandfunctors . 24 3.2 The smooth category of locally convex vector spaces . .. 26 3.3 The smoothRiemannianbordismcategory . 27 3.4 QFT’s of dimension d ....................... 32 4 Super symmetric quantum field theories 32 4.1 Supermanifolds.......................... 32 4.2 Super Riemannian structures on 1 1-manifolds . 36 4.3 Super Riemannian structures on 2|1-manifolds . 40 d|1| 4.4 The super bordism category RB ................ 46 4.5 ExtendingTVtoasupercategory. 49 4.6 Super symmetric quantum field theories . 49 1 5 Partition functions of susy QFT’s 51 5.1 Partition functions of QFT’s of dimension 1 1 ......... 51 | 5.2 Partition functions of QFT’s of dimension 2 1 ......... 54 | 1 Introduction thm:main The main result of this paper, Theorem 1, is that the partition function of a 2-dimensional super symmetric quantum field theory is an integral mod- ular function. The bulk of this work consists of incorporating a suitable geometric notion of super symmetry into the axiomatic description of quan- tum field theories going back to Atiyah and Segal. Roughly speaking, we replace the usual functors from Riemannian manifolds to (locally convex) vector spaces by functors from Riemannian super manifolds to super vector spaces and work in families parametrized by complex super manifolds.
    [Show full text]