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as Groups Categories Questions by: John C. Baez, 2004 September 30 Answers by: Toby Bartels1 , 2004 October 7 Here s a little homework just to make sure you understand the concepts of category, functor and natural transformation, as de ned in the handout Some De nitions Everyone Should Know . Recall that a set with an associative binary product and an element serving as the unit for this product is called a monoid: examples include (N, +, 0) and (N, , 1). A monoid where every element has a two-sided inverse is called a group. A category C with only one object (say ) is the same thing as a monoid, since all C has is a set of morphisms f : that can be composed associatively, together with a morphism 1 : serving as the unit for composition. Similarly, a category with only one object and all morphisms invertible is the same as a group! So, among other things, category theory is a massive generalization of group theory. This means that whenever you encounter a de nition in category theory, you should gure out what it amounts to in the case of groups. In what follows, you can either do problems 1 5 or problem 6. I greatly prefer answers in LaTeX. 1. Suppose that G and H are groups, and regard them as one-object categories with all morphisms invertible. Figure out what a functor F : G H amounts to. What are such functors usually called? F consists of a function F : G H, such that F (1G ) = 1H and for any pair of elements f, g of G, F (f g) = F (f )F (g). This is usually called a group homomorphism from G to H. 2. Suppose G and H are groups regarded as categories, and let F, F : G H be a pair of functors. Figure out what a natural transformation : F F amounts to. consists of an element of H, such that for any element f of G, F (f ) = F (f ). 3. Suppose G is a group regarded as a category and let 1G : G G be the identity functor. Figure out what a natural transformation : 1G 1G amounts to. What is the set of all such natural transformations usually called? consists of an element of G, such that for any element f of G, f = f . The set of such is usually called the centre (or center) of G. 4. Let Vect be the category of vector spaces over your favorite eld, where the morphisms are linear transformations. Suppose G is a group as regarded a category. Figure out what a functor F : G Vect amounts to. What is such a functor usually called? F consists of an object F of Vect and a function F : G hom(F, F ), such that F (1G ) = 1F and for any pair of elements f, g of G, F (f g) = F (f )F (g). This is usually called a linear representation of G. 5. Suppose G is a group regarded as a category and let F, F : G Vect be functors. Figure out what a natural transformation : F F amounts to. What is such a natural transformation usually 1I reserve no legal rights whatsoever to any of my creative work; see http://toby.bartels.name/copyright/. called? consists of a linear operator : F F , such that for any element f of G, this diagram commutes: F F (f ) /F F F (f ) /F is usually called an intertwining operator from F to F . 6. Suppose G is a Lie group, regarded as a one-object category where the morphisms form a manifold. Let Aut(G) be the category whose objects are smooth invertible functors F : G G and whose morphisms are smooth invertible natural transformations : F F . The objects of Aut(G) form a Lie group. Any object F in Aut(G) gives a subset [F ] consisting all objects that are isomorphic to it. What do these subsets look like for G = SO(3)? How about for G = SU(2)? First, let me state some facts about automorphisms of Lie groups. To begin with, every element of G de nes an automorphism of G by conjugation. This de nes a group homomorphism G Ob(Aut(G)); let Inn(G) be the image of this homomorphism. Then Inn(G) is normal; let Out(G) be the quotient group. If G is a connected compact real form of a simple Lie group, then Out(G) is isomorphic to the symmetry group of the Dynkin diagram of G. Also, Inn(G) is isomorphic to G/Z, where Z is the centre of G. Now, both SO(3) and SU(2) are connected compact real forms of the Lie group A 1 : The symmetry group of this Dynkin diagram is trivial, so Ob(Aut(G)) = Inn(G) in both cases. Since Z(SO(3)) is trivial, I have Ob(Aut(G)) = Inn(G) SO(3) in both cases. = So, suppose F and F are elements of G, and let be F (F ) 1 . Then F (f ) = F f F 1 F (F ) 1 = F f (F ) 1 = F (F ) 1 F f (F ) 1 = F (f ). Thus, F F in Aut(G). Therefore, there is a unique isomorphism class [F ], which is all of = Ob(Aut(G)) SO(3). =
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weinstein.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Groupoids: Unifying Internal and External Symmetry A Tour through Some Examples Alan Weinstein Introduction Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps larg...
connection_prasad.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Connections as Functors Prasad Senesi November 8, 2004 1. Dene F : K[a, b] V ect as follows: If c is an object in K[a, b], i.e., c [a, b], F : c Rn If f : t0 t1 is a morphism in K[a, b], F (f ) : v (t1 ), where : [a, b] Rn is the unique solut...
geometryampl.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: On the geometry of 2-categories and their classifying spaces M. BULLEJOS (bullejos@ugr.es) and A.M. CEGARRA (cegarra@ugr.es) Departamento de Algebra, Universidad de Granada Abstract. In this paper we prove that realizations of geometric nerves are ...
spring_garett.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: A Spring in Imaginary Time Math 241 Homework John Baez Answers by Garett Leskowitz One of the stranger aspects of Lagrangian dynamics is how it turns into statics when we replace the time coordinate t by it or in the jargon of physicists, when we W...
s04week03.pdf
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4.30.02.pdf
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ACStreetfest.pdf
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Description: Loop Groups and Lie 2-Algebras Alissa S. Crans Joint work with: John Baez Urs Schreiber 7 ; 77 ; 7 ? ? ? ? ? ? ? ? 5 { BB C C ; t...
church_rosser.pdf
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10039A.pdf
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Prosser-Poster-2006-3.pdf
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Description: The Reese T. Prosser Mathematics Lecture Series Presents Tales of the Dodecahedron: from Pythagoras through Plato to Poincar John Baez University of California - Riverside Friday, November 10, 2006 - 7:00 pm Kemeny Hall Room 008 (Free & open to the...
schmo1.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: 1 Introduction These notes are derived from the Photons, Shmotons thread that ran on sci. physics.research in the summer of 1997 later corrected to Photons, Schmotons. As editor, Ive wielded a pretty free hand, correcting minor errors, imposing a ...
derek.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Cartan Geometry and MacDowellMansouri Gravity: The Work of Derek Wise John C. Baez Groupe de Thorie de lAstroparticule et Cosmologie e July 10, 2007 For more, see: http:/math.ucr.edu/derek Cartan Geometry Euclidean Geometry generalize symmetry gr...
Phys._Rev._Lett._92_105502_(2004).pdf
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Description: VOLUME 92, NUMBER 10 PHYSICA L R EVIEW LET T ERS week ending 12 MARCH 2004 Anomalous Pressure-Induced Transition(s) in Ice XI Koichiro Umemoto,1,2 Renata M. Wentzcovitch,1,2 Stefano Baroni,1 and Stefano de Gironcoli1 SISSAScuola Internazionale Sup...
history.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: A PREHISTORY OF n-CATEGORICAL PHYSICS JOHN C. BAEZ AARON LAUDA January 3, 2008 Abstract We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophistica...
runge.pdf
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Description: The Kepler Problem Revisited: The LaplaceRungeLenz Vector March 6, 2008 John Baez Whenever we have two particles interacting by a central force in 3d Euclidean space, we have conservation of energy, momentum, and angular momentum. However, when the ...
cyclic_miguel.pdf
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Description: Linear orders, cyclic orders and permutations Miguel Carrin Alvarez o 1, 2. Generating functions. The empty set is vacuously ordered in one way; a 1-element set is trivially ordered in 1 way; and an n-element set can be linearly ordered by choosing a...
cm05week08.pdf
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classical7.pdf
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Description: Classical Mechanics, Lecture 7 January 31, 2008 lecture by John Baez notes by Alex Honung 1 The Poisson Bracket If we have one particle in Rn , we call Rn the conguration space - the space of possible positions of the whole system. If we have n pa...
inverse_square_rolle.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics Homework January 24, 2008 John Baez homework by Brian Rolle The Kepler Problem 1 1. Show that the energy of a particle is given by E = m(r2 2 + r2 ) + V (r) and the angular 2 momentum J has z coordinate j = mr2 and vanishing ...
s04week09.pdf
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galilei_rolle.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics Homework February 6, 2008 John Baez homework by Brian Rolle The Euclidean Group 4. Suppose that f : Rn Rn is a map that preserves distances: |f (x) f (y)| = |x y| for all x, y Rn . Then f (x) = Rx + u for some (R, u) E(n). |...
spring_erin.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: A Spring in Imaginary Time Erin P. J. Pearse 1. Suppose you have a spring in Rn with xed endpoints, tracing out a curve q : [s0 , s1 ] Rn , n q(s0 ) = a, q(s1 ) = b. If the spring is in a potential V R R, what curve will the spring trace out when...
action_toby.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Action as a Functor Questions by: John C. Baez, October 4, 2004 Answers by: Toby Bartels1 , 2004 October 14 In class I said some mysterious things about how the electromagnetic eld is a U(1) connection on spacetime, while the Feynman path integral i...
gomez-et-al-2004.pdf
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Description: INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 17 (2004) 15711606 NONLINEARITY PII: S0951-7715(04)67794-2 Connecting orbits and invariant manifolds in the spatial restricted three-body problem G G mez1 , W S Koon2,5 , M W Lo3 , J E Marsden2 , J Masde...
timeline.pdf
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Description: A Brief History of Time John C. Baez, September 23, 2004 Note: all gures are approximate! Working Backwards from Now 60 years ago Invention of the computer. 130 years ago Invention of the telephone. 180 years ago Fossil fuel revolution: coal, tra...
cm05week01.pdf
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action_miguel.pdf
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Description: Action as a functor Miguel Carrin Alvarez o 2004 October 14 1. Connections as functors. The functor S must map : p q to S(): S(p) S(q), where S(p) and S(q) are objects in R. But R is a category with one object only, so S(p) = S(q) = and the functo...
spring_mike.pdf
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Description: m ! ! 1 Ua#8Xg@1 {0 p pv1 E31 USo v\' v1 a |s# \'1 o#d8rpv{dr \' Q y\' pp p# cUSv j m y 1 ! 1 1 1 ! 1 l eAc`bex2#ede2Q032|X2edcaY@8i3D2c2t0xF cQDsUhS|caQ33D8Y3tYth t ez b ~ b A H b A F H T` f b ` CG H...
w07week05b.pdf
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5.2.02.pdf
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runge_childress.pdf
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Description: Classical Mechanics March 11, 2008 John Baez Homework by: Scot Childress The LaplaceLungeRenz Vector Let q denote the dierence in position of two particles in R3 , |q| the magnitude of q, and m the reduced mass. Supposing that the particles exert an...
hw7.pdf
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w07week07a.pdf
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f04week00.pdf
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fqxi_narrative.pdf
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Description: Categorifying Fundamental Physics John Baez Despite the incredible progress over most of the 20th century, and a continuing ow of new observational discoveries neutrino oscillations, dark matter, dark energy, and evidence for ination theoretical re...
pbaez.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Stanford Department of Mathematics Colloquium HIGHER GAUGE THEORY JOHN BAEZ University of California at Riverside Abstract Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both strin...
groupoidification.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Groupoidication Made Easy John C. Baez, Alexander E. Honung, and Christopher D. Walker Department of Mathematics, University of California Riverside, CA 92521 USA December 30, 2008 Abstract Groupoidication is a form of categorication in which vector ...
thesis_acrans.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: UNIVERSITY OF CALIFORNIA RIVERSIDE Lie 2-Algebras A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Alissa Susan Crans August 2004 Dissertation Committee: Professor John B...
nam2.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Higher Categories, Higher Gauge Theory II John C. Baez joint work with: Toby Bartels, Alissa Crans, James Dolan, Aaron Lauda, Urs Schreiber, Danny Stevenson. Unni Namboodiri Lectures April 10th, 2006 x hol() & 8 y hol() hol() $ : Not...
octopus.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Being an Octopus1 John C. Baez, April 27, 2004 In their book Combinatorial Species and Tree-Like Structures, Bergeron, Labelle and Leroux discuss a structure type Oct, called being an octopus. Instead of dening it, Id like you to guess it from some ...
f03week10.pdf
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f03week02.pdf
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res.1405.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: UNIVERSITY OF CALIFORNIA, SANTA CRUZ AS/SCP/1405-1 COMMITTEE ON THE LIBRARY Resolution on Ties with Elsevier Journals To the Academic Senate, Santa Cruz Division: Facing a challenge to scholarly communication The University of California system fac...
f04week02.pdf
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hw5.pdf
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angular.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics Homework February 21, 2008 John Baez Angular Momentum and Rotations In this problem we will see that angular momentum generates rotations for a particle in R n . We begin by recalling a bit about rotations. Let O(n) be the orthog...
hazewinkel_et_al.pdf
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Description: ...
categories1_derek.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Derek Wise October 7, 2004 Quantum Gravity Seminar HW #1 Groups as Categories In what follows, by group I always mean a group in the categorical sense: A group is a groupoid with one object. I use the traditional backward order of composition: f g m...
classical13.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics, Lecture 13 February 21, 2008 lecture by John Baez notes by Alex Honung 1 Lie Algebras Lie groups are groups that are also manifolds. These let us describe continuous symmetries in physics. A classic example is SO(n) - the grou...
galilei.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics Homework February 7, 2008 John Baez This homework gives you some choice in which problems you do, depending on how ambitious you feel. In the section on the Euclidean group you can either do problems 13, or if youre feeling more ...
cm05week10.pdf
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Description: ...
spring_alex.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: A Spring in Imaginary Time QG Homework 2 Fall 2006 Alex Honung October 18, 2006 One of the stranger aspects of Lagrangian dynamics is how it turns into statics when we replace the time coordinate t by it - or in the jargon of physicists, when we Wick...
quantum.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Quantum Quandaries: A Category-Theoretic Perspective John C. Baez Department of Mathematics, University of California Riverside, California 92521 USA email: baez@math.ucr.edu April 7, 2004 To appear in Structural Foundations of Quantum Gravity, eds. ...
noncompact_miguel.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Physics and analysis on noncompact manifolds1 Miguel Carrin Alvarez2 o Department of Mathematics University of California Riverside, CA 92521, USA miguel@math.ucr.edu November 5, 2004 1 2 UCR Math department geometry seminar, November 5, 2004 Joint...
perturbation_miguel.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Math 260: Perturbation theory Miguel Carrin Alvarez o 1. The perturbation series. Starting with t tn t2 (t) = n0 0 0 0 ei(ttn )H0 (iV )ei(tn tn1 )H0 ei(t2 t1 )H0 (iV )eit1 H0 dt1 dtn1 dtn we get t tn1 t2 t (t) = n1 0 0 t tn 0 (iV )ei(t...
highwordpb.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Higher dimensional word problems with applications to equational logic Albert Burroni Universit PARIS 7, UFR de maths e 2 place Jussieu, 75005 Paris The word problem on a monoid admits two natural generalizations: The rst one is the extension from ...
5.14.02.pdf
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w07week06b.pdf
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vita.pdf
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Description: Garett M. Leskowitz 2429 Pierce Hall Deparment of Chemistry University of California Riverside, CA 92521 (909) 787-7365 Current Position Education 1790 West Arrow Route #202 Upland , CA 91786 (909) 981-6709 garett@citrus.ucr.edu Postdoctoral Resear...
classical5.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Classical Mechanics, Lecture 5 January 24, 2008 lecture by John Baez notes by Alex Honung 1 Symmetries and Conserved Quantities the n-Body Problem Today we will begin our quest to see where symmetries come from. For this, let us talk about symmet...
lambda_calculi.pdf
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gravitational_old.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: The Kepler Problem Math 241 Homework John Baez The goal of this problem is to see why two particles interacting via gravity move along nice curves like ellipses, parabolas and hyperbolas. This is called the Kepler problem since it was Kepler who dis...
classifying.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: The Classifying Space of a Topological 2-Group John C. Baez Danny Stevenson June 6, 2008 Abstract Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of principal 2-bundles ge...
connection.pdf
Path: UC Riverside >> PHYS >> 291 Fall, 2008
Description: Connections as Functors John C. Baez, October 21, 2004 In this homework well see how a vector bundle E equipped with a connection over a manifold X gives a functor F : P(X) Vect where P(X) is the category of paths in X, dened in the last homework. ...
f04week03.pdf
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cyclic2_miguel.pdf
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Description: Math 260: Categorifying the Riemann Zeta Function Miguel Carrin Alvarez o 1. Subcategories have faithful inclusions. Assume that C is a subcategory of D, that is, there is a functor i: C D which is an inclusion of objects and morphisms, preserving u...
cm05week06.pdf
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s04week02.pdf
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galilei_old.pdf
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Description: The Euclidean Group, the Galilei Group and the Free Particle Math 241 Homework John Baez This homework gives you some choice in which problems you do, depending on how ambitious you feel. In the section on the Euclidean group you can either do prob...
hw3.pdf
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