DOCSLIB.ORG

Differential Geometry Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential Geometry Notes Differential geometry notes John Kerl November 1, 2005 Abstract The following are notes to help me prepare for the University of Arizona math department’s geometry- topology qualifier in 2006. The content is distilled from Spivak’s A Comprehensive Introduction to Differential Geometry, Spivak’s Calculus on Manifolds, Lang’s Algebra, and other sources. The list of topics is based on, but presented in a different order from, Prof. Doug Pickrell’s fall 2005 differential geometry course (Math 534A). My intention is to gather together precise definitions and examples. This document is a work in progress. Please do not base any important decisions on its contents; for definitive information, please consult the cited references. 1 Contents 1 Categories 3 1.1 Functors . 4 1.2 Hom functors . 4 1.3 Pushouts and pullbacks . 4 2 Linear algebra and tensors 5 2.1 Basis and dual space . 5 2.2 Algebras . 6 2.3 Tensors . 6 2.4 Pushouts and pullbacks . 7 2.5 Tensor algebras . 8 3 Manifolds 9 3.1 Manifolds . 9 3.2 Coordinates and parameterizations . 9 3.3 Chart and atlas . 9 3.4 Differentiable structures . 9 3.5 Maps of manifolds . 9 3.6 Immersion and embedding . 9 4 Tangent spaces 10 4.1 Partial derivatives . 10 4.2 asdfasdfasdf . 11 4.3 asdfasdfasdf . 11 5 Vector bundles 12 6 Flows 13 2 1 Categories Category theory is sometimes ([Lang]) referred to as “abstract nonsense”. However, it is simply an ab- straction of some very concrete things. Namely, we are abstracting the concept of structures (groups, vector spaces, topological spaces, etc.) and mappings (homomorphisms, linear transformations, homeomorphisms, etc.). One oddity is that the definitions below are specifically designed to avoid mentioning the elements of those structures. The primary reason for presenting categories in this paper is as a mnemonic device: for example, category ∗ theory helps us remember the difference between f∗ and f in differential geometry. Definition 1.1. A category C consists of a class of objects and sets of morphisms or arrows between those objects: • For every ordered pair A, B of objects there is a set HomC(A, B) of morphisms from A to B. This is called a hom set. • There is a law of composition: if f ∈ HomC(A, B) and g ∈ HomC(B, C) (which is to say f : A → B and g : B → C), then there is an element g ◦ f of HomC(A, C). • The hom sets are disjoint: If A1 and A2 are different, or if B1 and B2 are different, then HomC(A1,B1) and HomC(A2,B2) have no morphisms in common. • Composition of morphisms is associative: if f : A → B, g : B → C, and h : C → D, then h ◦ (g ◦ f) = (h ◦ g) ◦ f. • Each object A has an identity morphism, written 1A. Since we are defining morphisms without reference to specific elements of an object, we can’t say something like “. such that 1A(x) for all x ∈ A”. Rather, an identity morphism on an object A is characterized by the property that for all other objects B, and for all f : A → B and all g : B → A, f ◦ 1A = f and 1A ◦ g = g. xxx put a picture here Some familiar examples are: • Objects are groups and morphisms are group homomorphisms. • Objects are rings and morphisms are ring homomorphisms. • Objects are vector spaces and morphisms are linear transformations. A perhaps less familiar example is a partially ordered set. Let P be a partially ordered set, with operation 4. Then the poset P is itself the entire category. Objects of P are the elements of P . Morphisms of P are quite literally arrows: there is an arrow from a to b if a 4 b. Composition of arrows is nothing more than the transitivity property of posets. The identity morphisms are provided by the reflexivity property of posets, namely, a 4 a for all a ∈ P . xxx divisibility lattice in Z. Another example is a single group G. There is a single object which is all of G; the hom set HomG(G, G) is all of G. That is, the morphisms are the elements of G. The composition of two morphisms x and y is the usual product xy. The identity morphism is the identity element of G. Lang’s cat1, cat2, cat3. 3 1.1 Functors xxx forgetful functor xxx lang’s fun1, fun2 Definition 1.2. 1.2 Hom functors xxx covt and ctvt Definition 1.3. 1.3 Pushouts and pullbacks Definition 1.4. 4 2 Linear algebra and tensors Everything said in this section is applicable to finite-dimensional vector spaces over an arbitrary field. However, for this paper, the base field is always R. 2.1 Basis and dual space Let V be a finite-dimensional vector space V over R, say of dimension m. Then V has many bases; in particular, it has a standard basis {e1,..., em} with elements of the form e1 = (1, 0, 0,..., 0), e2 = (0, 1, 0,..., 0),..., em = (0, 0, 0,..., 1). Definition 2.1. The dual space of V is the set V ∗ of all linear transformations from V to R. Recall that V ∗ is a vector space over R. It has the same dimension, namely m, as V . Definition 2.2. An element of a dual space is called a linear functional. Definition 2.3. For any basis {v1,..., vm}, there is a basis for the dual space (i.e. a dual basis) ∗ ∗ {v1,..., vm} ∗ where the vi functions are defined to have value ∗ vi (vj) = δij where the δ is Kronecker’s, i.e. δij = 1 when i = j, 0 otherwise. Thus, the standard basis has a standard dual basis ∗ ∗ ∗ {e1,..., em} with ei (ej) = δij. Let λ : V → R and v ∈ V . Given the above basis, we may write m m X ∗ X λ = aiei and v = bjej. i=1 j=1 ∗ ∗ Since the ei ’s are linear functions, and since ei (ej) = δij, we have m m m m m X ∗ X X X ∗ X λ(v) = aiei bjej = aibjei (ej) = aibi. i=1 j=1 i=1 j=1 i=1 Now let f : V → W be a linear transformation, and let λ : W → R be a linear functional, i.e. λ ∈ W ∗. We can map from V to R by going through W first. That is, for v ∈ V , f(v) is in W , and so we can apply λ to it: 5 V H λ ◦ f HHj f R ¨¨* ?¨ λ W Formally, λ ◦ f : V → R. Given λ ∈ W ∗, we have obtained λ ◦ f ∈ V ∗. Post-composing λ by f is said to be a pullback of λ from W ∗ to V ∗. Another way to look at this is that, given f : V → W , we have a map from W ∗ to V ∗: f : V → W f ∗ : W ∗ → V ∗ f ∗(λ) = λ ◦ f (f ∗λ)(v) = λ(fv). In this way, f ∗ takes functionals to functionals: V V ∗ 6 f f ∗ ? W W ∗ xxx have a map from V to V ∗. xxx have a map from W to W ∗. xxx note arrow-reversing; refer to the contravariant hom functor here. ∗ Is it true that f (φ) = φ ◦ f and f∗(φ) = f ◦ φ? Check Lang. 2.2 Algebras xxx write axioms in various ways: ring and subring; vector space w/ multiplication; third way? Examples: extension fields; poly rings; matrix rings 2.3 Tensors (1) Tensor products from abstract algebra. xxx cite chapter and verse from DF, hungerford, grove, etc. (2) Old-fashioned tensors from physics (“transform according to . ”). (3) Tensors as extent-r arrays (4) Tensors as r-linear functions T : V r → R. xxx Discuss the last first. Show all are equivalent. Definition 2.4. Let V be a vector space over R.A tensor is a k-linear map from V k to R. The number k is called the order or degree of the tensor. Often, we call a tensor of order k simply a k-tensor. 6 The term “k-linear” means that a tensor is linear in each slot. For example, for a 2-tensor λ, for all u, v, w ∈ V , λ(u1 + u2, v) = λ(u1, v) + λ(u2, v) λ(u, v1 + v2) = λ(u, v1) + λ(u, v2) λ(au, v) = aλ(u, v) λ(u, bv) = bλ(u, v) As a consequence: λ(au, bv) = abλ(u, v) λ(u1 + u2, v1 + v2) = λ(u1, v1) + λ(u1, v2) + λ(u2, v1) + λ(u2, v2) and so on. xxx linear functional example xxx dot product example xxx write matrix for dotpr xxx 2x2 det example; matrix xxx 3x3 det example; write 3x3x3 array. 2.4 Pushouts and pullbacks xxx ([Spivak1], p. 77) Let f : V → W be a linear transformation, and let λ : W k → R be a k-tensor, i.e. λ ∈ T k(W ). We can map k from V to R by going through W first. That is, for (v1,..., vk) ∈ V ,(f(v1), . , f(vk)) is in W , and so we can apply λ to it: V k H λ ◦ f HHj f R ¨¨* ?¨ λ W k Formally, λ ◦ f : V → R. Given λ ∈ W ∗, we have obtained λ ◦ f ∈ V ∗. Post-composing λ by f is said to be a pullback of λ from W ∗ to V ∗. Another way to look at this is that, given f : V → W , we have a map from W ∗ to V ∗: f : V → W f ∗ : W ∗ → V ∗ f ∗(λ) = λ ◦ f (f ∗λ)(v) = λ(fv). In this way, f ∗ takes functionals to functionals: 7 V T k(V ) 6 f f ∗ ? W T k(W ) 2.5 Tensor algebras 8 3 Manifolds 3.1 Manifolds Definition 3.1.
Load more

Recommended publications
  • Topics in Geometric Group Theory. Part I
    Topics in Geometric Group Theory. Part I Daniele Ettore Otera∗ and Valentin Po´enaru† April 17, 2018 Abstract This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (qsf), geometric simple connec- tivity (gsc), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of discrete groups seen from a topo- logical viewpoint at infinity. Also, we shall outline the main steps of the achievements obtained in the last years around the very general question whether or not all finitely presented groups satisfy these conditions. Keywords: Geometric simple connectivity (gsc), quasi-simple filtration (qsf), inverse- representations, finitely presented groups, universal covers, singularities, Whitehead manifold, handlebody decomposition, almost-convex groups, combings, easy groups. MSC Subject: 57M05; 57M10; 57N35; 57R65; 57S25; 20F65; 20F69. Contents 1 Introduction 2 1.1 Acknowledgments................................... 2 arXiv:1804.05216v1 [math.GT] 14 Apr 2018 2 Topology at infinity, universal covers and discrete groups 2 2.1 The Φ/Ψ-theoryandzippings ............................ 3 2.2 Geometric simple connectivity . 4 2.3 TheWhiteheadmanifold............................... 6 2.4 Dehn-exhaustibility and the QSF property . .... 8 2.5 OnDehn’slemma................................... 10 2.6 Presentations and REPRESENTATIONS of groups . ..... 13 2.7 Good-presentations of finitely presented groups . ........ 16 2.8 Somehistoricalremarks .............................. 17 2.9 Universalcovers................................... 18 2.10 EasyREPRESENTATIONS . 21 ∗Vilnius University, Institute of Data Science and Digital Technologies, Akademijos st. 4, LT-08663, Vilnius, Lithuania. e-mail: [email protected] - [email protected] †Professor Emeritus at Universit´eParis-Sud 11, D´epartement de Math´ematiques, Bˆatiment 425, 91405 Orsay, France.
    [Show full text]
  • Experience Implementing a Performant Category-Theory Library in Coq
    Experience Implementing a Performant Category-Theory Library in Coq Jason Gross, Adam Chlipala, and David I. Spivak Massachusetts Institute of Technology, Cambridge, MA, USA [email protected], [email protected], [email protected] Abstract. We describe our experience implementing a broad category- theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed sub- stantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to repre- sent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such rea- soning. One particular encoding trick to which we draw attention al- lows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful. Keywords: Coq · category theory · homotopy type theory · duality · performance 1 Introduction Category theory [36] is a popular all-encompassing mathematical formalism that casts familiar mathematical ideas from many domains in terms of a few unifying concepts. A category can be described as a directed graph plus algebraic laws stating equivalences between paths through the graph. Because of this spar- tan philosophical grounding, category theory is sometimes referred to in good humor as \formal abstract nonsense." Certainly the popular perception of cat- egory theory is quite far from pragmatic issues of implementation.
    [Show full text]
  • Motivating Abstract Nonsense
    Lecture 1: Motivating abstract nonsense Nilay Kumar May 28, 2014 This summer, the UMS lectures will focus on the basics of category theory. Rather dry and substanceless on its own (at least at first), category theory is difficult to grok without proper motivation. With the proper motivation, however, category theory becomes a powerful language that can concisely express and abstract away the relationships between various mathematical ideas and idioms. This is not to say that it is purely a tool { there are many who study category theory as a field of mathematics with intrinsic interest, but this is beyond the scope of our lectures. Functoriality Let us start with some well-known mathematical examples that are unrelated in content but similar in \form". We will then make this rather vague similarity more precise using the language of categories. Example 1 (Galois theory). Recall from algebra the notion of a finite, Galois field extension K=F and its associated Galois group G = Gal(K=F ). The fundamental theorem of Galois theory tells us that there is a bijection between subfields E ⊂ K containing F and the subgroups H of G. The correspondence sends E to all the elements of G fixing E, and conversely sends H to the fixed field of H. Hence we draw the following diagram: K 0 E H F G I denote the Galois correspondence by squiggly arrows instead of the usual arrows. Unlike most bijections we usually see between say elements of two sets (or groups, vector spaces, etc.), the fundamental theorem gives us a one-to-one correspondence between two very different types of objects: fields and groups.
    [Show full text]
  • Oka Manifolds: from Oka to Stein and Back
    ANNALES DE LA FACULTÉ DES SCIENCES Mathématiques FRANC FORSTNERICˇ Oka manifolds: From Oka to Stein and back Tome XXII, no 4 (2013), p. 747-809. <http://afst.cedram.org/item?id=AFST_2013_6_22_4_747_0> © Université Paul Sabatier, Toulouse, 2013, tous droits réservés. L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales de la Facult´e des Sciences de Toulouse Vol. XXII, n◦ 4, 2013 pp. 747–809 Oka manifolds: From Oka to Stein and back Franc Forstnericˇ(1) ABSTRACT. — Oka theory has its roots in the classical Oka-Grauert prin- ciple whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomor- phic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We de- scribe equivalent characterizations of Oka manifolds, the functorial prop- erties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov’s ellipticity.
    [Show full text]
  • Arxiv:Math/9703207V1 [Math.GT] 20 Mar 1997
    ON INVARIANTS AND HOMOLOGY OF SPACES OF KNOTS IN ARBITRARY MANIFOLDS V. A. VASSILIEV Abstract. The construction of finite-order knot invariants in R3, based on reso- lutions of discriminant sets, can be carried out immediately to the case of knots in an arbitrary three-dimensional manifold M (may be non-orientable) and, moreover, to the calculation of cohomology groups of spaces of knots in arbitrary manifolds of dimension ≥ 3. Obstructions to the integrability of admissible weight systems to well-defined knot invariants in M are identified as 1-dimensional cohomology classes of certain generalized loop spaces of M. Unlike the case M = R3, these obstructions can be non-trivial and provide invariants of the manifold M itself. The corresponding algebraic machinery allows us to obtain on the level of the “abstract nonsense” some of results and problems of the theory, and to extract from others the essential topological (in particular, low-dimensional) part. Introduction Finite-order invariants of knots in R3 appeared in [V2] from a topological study of the discriminant subset of the space of curves in R3 (i.e., the set of all singular curves). In [L], [K], finite-order invariants of knots in 3-manifolds satisfying some conditions were considered: in [L] it was done for manifolds with π1 = π2 = 0, and in [K] for closed irreducible orientable manifolds. In particular, in [L] it is shown that the theory of finite-order invariants of knots in two-connected manifolds is isomorphic to that for R3; the main theorem of [K] asserts that for every closed oriented irreducible 3- manifold M and any connected component of the space C∞(S1, M) the group of finite- arXiv:math/9703207v1 [math.GT] 20 Mar 1997 order invariants of knots from this component contains a subgroup isomorphic to the group of finite-order invariants in R3.
    [Show full text]
  • Perspective the PNAS Way Back Then
    Proc. Natl. Acad. Sci. USA Vol. 94, pp. 5983–5985, June 1997 Perspective The PNAS way back then Saunders Mac Lane, Chairman Proceedings Editorial Board, 1960–1968 Department of Mathematics, University of Chicago, Chicago, IL 60637 Contributed by Saunders Mac Lane, March 27, 1997 This essay is to describe my experience with the Proceedings of ings. Bronk knew that the Proceedings then carried lots of math. the National Academy of Sciences up until 1970. Mathemati- The Council met. Detlev was not a man to put off till to cians and other scientists who were National Academy of tomorrow what might be done today. So he looked about the Science (NAS) members encouraged younger colleagues by table in that splendid Board Room, spotted the only mathe- communicating their research announcements to the Proceed- matician there, and proposed to the Council that I be made ings. For example, I can perhaps cite my own experiences. S. Chairman of the Editorial Board. Then and now the Council Eilenberg and I benefited as follows (communicated, I think, did not often disagree with the President. Probably nobody by M. H. Stone): ‘‘Natural isomorphisms in group theory’’ then knew that I had been on the editorial board of the (1942) Proc. Natl. Acad. Sci. USA 28, 537–543; and “Relations Transactions, then the flagship journal of the American Math- between homology and homotopy groups” (1943) Proc. Natl. ematical Society. Acad. Sci. USA 29, 155–158. Soon after this, the Treasurer of the Academy, concerned The second of these papers was the first introduction of a about costs, requested the introduction of page charges for geometrical object topologists now call an “Eilenberg–Mac papers published in the Proceedings.
    [Show full text]
  • Hodge Theory and Deformations of Maps Compositio Mathematica, Tome 97, No 3 (1995), P
    COMPOSITIO MATHEMATICA ZIV RAN Hodge theory and deformations of maps Compositio Mathematica, tome 97, no 3 (1995), p. 309-328 <http://www.numdam.org/item?id=CM_1995__97_3_309_0> © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 97: 309-328, 1995. 0 1995 Kluwer Academic Publishers. Printed in the Netherlands. 309 Hodge theory and deformations of maps ZIV RAN* Department of Mathematics, University of California, Riverside, CA 92521 Received 23 January 1992; accepted in revised form 30 June 1993; final revision June 1995. The purpose of this paper is to establish and apply a general principle (Theorem 1.1, Section 1) which serves to relate Hodge theory and defor- mation theory. This principle, which we state in a somewhat technical abstract-nonsense framework, takes on two dual forms which say roughly, respectively, that: (0.1) for a functorial homomorphism from a Hodge-type group to an infinitesimal deformation group, im( 1") consists of unobstructed deformations; (0.2) for a functorial homomorphism from an obstruction group to a Hodge-type group, "actual" obstructions lie in ker( n). A familiar example of (0.1) is the case of deformations of manifolds with trivial canonical bundle, treated from a similar viewpoint in [Rl] ; a familiar example of (0.2) is Bloch’s semi-regularity map [B], treated from a similar viewpoint in [R3].
    [Show full text]
  • Relative Cohomology and Projective Twistor Diagrams
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 324, Number 1, March 1991 RELATIVE COHOMOLOGY AND PROJECTIVE TWISTOR DIAGRAMS S. A. HUGGETT AND M. A. SINGER ABSTRACT. The use of relative cohomology in the investigation of functionals on tensor products of twistor cohomology groups is considered and yields a significant reduction in the problem of looking for contours for the evaluation of (projective) twistor diagrams. The method is applied to some simple twistor diagrams and is used to show that the standard twistor kernel for the first order massless scalar ¢ 4 vertex admits a (cohomological) contour for only one of the physical channels. A new kernel is constructed for the ¢4 vertex which admits contours for all channels. 1. INTRODUCTION Since the discovery that the twistor description of massless fields is best un- derstood in terms of the analytic cohomology of twistor space [3, 9, 13], the question of how to interpret twistor diagrams [6, 16] in a cohomologically sat- isfactory way has attracted considerable interest [4, 5, 8]. The introduction of such diagrams is motivated by the desire to extend the twistor description of massless fields so as to include their basic (quantum field theoretic) interactions. A Feynman diagram specifies a kernel which defines, by substitution into an appropriate integral, a functional on some tensor product of Hilbert spaces. Similarly, a twistor diagram specifies a holomorphic kernel h which is to define a functional on some tensor product of analytic cohomology groups. In fact, one generally expects on physical grounds to associate a (finite) family of func- tionals to anyone kernel, since there are generally many physically significant 'channels' for anyone Feynman diagram.
    [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]
  • Homological Algebra
    HOMOLOGICAL ALGEBRA BRIAN TYRRELL Abstract. In this report we will assemble the pieces of homological algebra needed to explore derived functors from their base in exact se- quences of abelian categories to their realisation as a type of δ-functor, first introduced in 1957 by Grothendieck. We also speak briefly on the typical example of a derived functor, the Ext functor, and note some of its properties. Contents 1. Introduction2 2. Background & Opening Definitions3 2.1. Categories3 2.2. Functors5 2.3. Sequences6 3. Leading to Derived Functors7 4. Chain Homotopies 10 5. Derived Functors 14 5.1. Applications: the Ext functor 20 6. Closing remarks 21 References 22 Date: December 16, 2016. 2 BRIAN TYRRELL 1. Introduction We will begin by defining the notion of a category; Definition 1.1. A category is a triple C = (Ob C; Hom C; ◦) where • Ob C is the class of objects of C. • Hom C is the class of morphisms of C. Furthermore, 8X; Y 2 Ob C we associate a set HomC(X; Y ) - the set of morphisms from X to Y - such that (X; Y ) 6= (Z; U) ) HomC(X; Y ) \ HomC(Z; U) = ;. Finally, we require 8X; Y; Z 2 Ob C the operation ◦ : HomC(Y; Z) × HomC(X; Y ) ! HomC(X; Z)(g; f) 7! g ◦ f to be defined, associative and for all objects the identity morphism must ex- ist, that is, 8X 2 Ob C 91X 2 HomC(X; X) such that 8f 2 HomC(X; Y ); g 2 HomC(Z; X), f ◦ 1X = f and 1X ◦ g = g.
    [Show full text]
  • Ghostwhite EL-Algebras: Definition and Applications
    EL1-algebras: Definition and Applications Christian Saemann Maxwell Institute and School of Mathematical and Computer Sciences Heriot–Watt University, Edinburgh Workshop “Homotopy Algebra of Quantum Field Theory and Its Application”, Yukawa Institute, 29.3.2021 Based on: arXiv:2104.????? with Leron Borsten and Hyungrok Kim Christian Saemann EL1-algebras: Definition and Applications Motivation: Five Questions Christian Saemann EL1-algebras: Definition and Applications A first question 4/25 What is the algebraic structure underlying Courant algebroids? Answers in the literature: Roytenberg (2002): An (exact) Courant algebroid is the symplectic dg-manifold ∗ µ µ µ V2 = T [2]T [1]M;! = dx ^ dpµ + dξ ^ dζ ; µ Q = fS; −} ;S = ξ pµ for M some manifold. Dorfman and Courant brackets: 1 [X; Y ]D = fQX; Y g ; [X; Y ]C = 2 fQX; Y g − fQY; Xg cf. Rogers (2011): [−; −]C part of L1-algebra [−; −]D part of dg-Leibniz algebra. Is there more to it? Christian Saemann EL1-algebras: Definition and Applications Relevance of first question 5/25 This may be seem as a niche question, but: Courant algebroids underlie Hitchin’s Generalized Geometry Application in supergravity: Generalized tangent bundle All generalized tangent bundles are symplectic L1-algebroids Dorfman bracket structure relevant in tensor hierarchies Currently relevant: Double and Exceptional Field Theory. In order to further understand supergravity: understand symplectic L1-algebroids! Christian Saemann EL1-algebras: Definition and Applications A second, related question 6/25 What is the alg. structure underlying multisymplectic manifolds? Answers in the literature: Multisymplectic/p-plectic manifold (M; !): p+1 ! 2 Ω (M) ; d! = 0 ; ιX ! = 0 , X = 0 comes with Hamiltonian p − 1-forms α with Xα: ιXα ! = dα This leads to brackets: [α; β]h = LXα β ; [α; β]s = ιXα ιXβ ! cf.
    [Show full text]
  • Foundational Concepts Underlying a Formal Mathematical Basis for Systems Science
    Foundational Concepts Underlying a Formal Mathematical Basis for Systems Science Kenneth A. Lloyd, Jr. Watt Systems Technologies Inc. 10524 Sorrento Drive NW Albuquerque, NM 87114 USA Abstract We propose Category Theory as the formal, mathematical language and theory for studying systems. But, to understand the applicability of Category Theory to Systems Science we must go back to certain prior foundational concepts which, first, must be clearly elucidated and their contexts made explicit. Keywords: science, formal systems, category theory. Science is a Formal System for Developing Valid Knowledge We propose Category Theory as the formal mathematical language for a science of studying systems – all kinds of systems – from the quantum to cosmological, from the social to technological, and beyond. In the same way that Category Theory has been used to study all of mathematics, Category Theory provides a coherently consistent means for studying the systems of science. This approach may lead us to new understandings in the paradigms of science. At present, science is often misunderstood (Pollack, 2003), and there are problems in adequately communicating its concepts without inconsistency, incompleteness or paradoxes. Part of that problem is incompleteness internal to the concepts of science, while another part of the problem manifests externally in communication between elements within the paradigm – specifically, in the way science interacts between our worlds - plural. In communication, Eliezer Yudkowsky of the Machine Intelligence Institute cautions us about expecting short inferential distances (Yudkowsky, 2007), “the legendary difficulty most scientists have in communicating with a lay audience - or even communicating with scientists from other disciplines”. Therefore, inferential distance is the number of necessary transformative steps from a general prior knowledge to a new, adequate understanding of knowledge concepts.
    [Show full text]