DOCSLIB.ORG

18.786: Number Theory II (Lecture Notes)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
18.786: Number Theory II (Lecture Notes) 18.786: Number Theory II (lecture notes) Taught by Bjorn Poonen Spring 2015, MIT Last updated: May 14, 2015 1 Disclaimer These are my notes from Prof. Poonen's course on number theory, given at MIT in spring 2015. I have made them public in the hope that they might be useful to others, but these are not official notes in any way. In particular, mistakes are my fault; if you find any, please report them to: Eva Belmont ekbelmont at gmail.com 2 Contents 1 February 36 Analysis review: Γ function, Fourier transform, θ function, measure theory; proof of the analytic continuation of ξ (and ζ) using the functional equation for θ 2 February 5 10 More analysis review: Lp spaces, Riesz representation theorem, Haar measures, LCA groups, characters, Pontryagin duality 3 February 12 14 Fourier inversion formula for LCA groups; equivalent definition of local fields; all characters are gotten by shifting a \standard" one, and Fb =∼ F 4 February 19 17 Conductor of a character; self-dual Haar measure of F (under Fourier transform) with standard ; unramified characters are j js; all characters are η · j js; twisted dual; L(χ) and local zeta integral Z(f; χ); statement of meromorphic continuation and functional equation for Z(f; χ) 5 February 24 21 Proof of meromorphic continuation and functional equation of the local zeta integral Z(f; χ) 6 February 26 25 Standard additive character ; Tamagawa measure on A; vol(A=K) = 1; Schwarz- Bruhat functions; Fourier transforms for A $ A and A=K $ K; Poisson summation formula 7 March 3 28 Proof of the Poisson summation formula; proof of Riemann-Roch using Poisson summation formula; norm of an id`ele;id`eleclass characters 8 March 5 32 Meromorphic continuation and functional equation for global zeta integrals Z(f; χ) 9 March 10 36 Local to global information; G-modules; definition(s) of group cohomology 3 10 March 12 41 More equivalent definitions of group cohomology; restriction, inflation, corestriction maps; Shapiro's lemma; homology 11 March 17 44 Tate cohomology Hb ∗ (for a finite group); Tate cohomology is simple for Z=NZ; cup products on H∗ and Hb ∗; definition/examples of profinite groups 12 March 19 48 Supernatural numbers; pro-p-groups and Sylow p-subgroups; discrete G-modules; cohomology of profinite groups; cohomological dimension 13 March 31 52 Nonabelian cohomology (H0 and H1); L=K-twists; Hilbert Theorem 90; Galois ´etale k-algebras 14 April 2 55 Equivalence of categories between k-vector spaces and L-vector spaces with semilinear 1 G-action; H (G; GLr(L)) = f0g (general version of Hilbert theorem 90); examples: K¨ummer theory, Artin-Schreier theory, elliptic curves 17 April 14 59 Galois representations ρ : GK ! GLn(Qp); semisimplifications of representations and Brauer-Nesbitt theorem; Tate twists; geometric Frobenius; image of Frobenius in Gk; Weil group 18 April 16 62 Grothendieck's `-adic monodromy theorem (formula for ρ : W ! GL(V ) on some open subgroup in terms of some nilpotent twisted matrix); Weil-Deligne representations 19 April 23 66 Deligne's classification of `-adic representations of W (they are $ Weil-Deligne representations); attempting to extend representations from W to G 20 April 28 69 More on conditions for representations of W to be representations of G; unipotent and semisimple matrices; an analogue of Jordan decomposition for non-algebraically closed fields; Frobenius-semisimple WD representations 4 21 April 30 73 Characterization of indecomposable Frobenius-semisimple WD representations; admissible representations of GLn(K); statement of local Langlands correspondence 22 May 5 76 Big diagram depicting relationships between various representation groups studied so far; local L-factors for representations of W , and for Weil-Deligne representations 23 May 7 80 24 May 12 84 25 May 14 87 5 Number theory Lecture 1 Lecture 1: February 3 Topics: • Tate's thesis (what led to the Langlands program) • Galois cohomology • Introduction to Galois representation theory (also tied in to the Langlands program) Reference (for Tate): Deitmar Introduction to harmonic analysis. Also, Bjorn will be posting official notes! Before Tate, there was Riemann { the prototype for Tate's thesis is the analytic continuation and functional equation for the Riemann zeta function. For Re s > 1, recall Y X ζ(s) = (1 − p−s)−1 = n−s: n≥1 Theorem 1.1 (Riemann, 1860). (Analytic continuation) The function ζ(s) extends to a meromorphic function on C, which is holomorphic except for a simple pole at 1. (Functional equation) The completed zeta function1 ξ(s) = π−s=2Γ(s=2)ζ(s), where Γ is the gamma function, satisfies ξ(s) = ξ(1 − s). Hecke (1918, 1920) generalized this to ζK (s) and other L-functions. Margaret Matchett (1946) started reinterpreting this in adelic terms, and John Tate (1950) finished this in his Ph.D. thesis. (It's not just a reinterpretation: it gives more information than Hecke's proofs.) Analytic prerequisites. Recall that the Gamma function is Z 1 dt Γ(s) = e−tt−s : 0 t −t × −s × × dt Note that e is a function R ! C and t (as a function of t) takes R>0 ! C and t is a × Haar measure on R>0 (it's translation-invariant on the multiplicative group). (If you do this sort of thing over a finite field you get Gauss sums!) This is convergent for Re s > 0. Proposition 1.2. (1) Γ(s + 1) = sΓ(s) (use integration by parts) (2) Γ(s) extends to a meromorphic function on C with simple poles at 0; −1; −2;::: , and no zeros (3) Γ(n) = (n − 1)! for n 2 Z≥1 1 p R 1 −x2 p (4) Γ 2 = π (by a change of variables this is equivalent to −∞ e dx = π) Now I'll review the Fourier transform. 1think of ξ as ζ with extra factors corresponding to the infinite places 6 Number theory Lecture 1 n Definition 1.3. f : R ! C tends to zero rapidly if for every n ≥ 1, x f(x) ! 0 as jxj ! 1 1 (i.e. jf(x)j = O jxjn ). (r) Definition 1.4. Call f : R ! C a Schwartz function if for every r ≥ 0, f tends to zero rapidly. Write S = S (R) for the set of Schwartz functions. Examples: e−x2 , zero, bump functions (any C1 function with compact support). Definition 1.5. Given f 2 S , define the Fourier transform Z fb = f(x)e−2πixydx: R Because f is a Schwartz function, this converges, and it turns out that fb is also a Schwartz function. 2 Example 1.6. If f(x) = e−πx , then f 2 S and fb = f. You can also define Fourier transforms on L2. Eventually we'll have to generalize all of this from R to compact abelian groups. Theorem 1.7 (Fourier inversion formula). If f 2 S then Z f(x) = feb 2πixydy: R In particular, fbb(x) = f(−x). Theorem 1.8 (Poisson summation formula). If f 2 S , then X X f(n) = fb(n): n2Z n2Z Definition 1.9. For real t > 0, define X 2 θ(t) = e−πn t: n2Z This actually makes sense for any t in the right half plane. You can also define θ(it) = θ(t), defined in the upper half plane; it is a modular form. In general, theta functions are associated to lattices. In this case, the lattice is just Z ⊂ R. Theorem 1.10 (Functional equation of θ). For every real t > 0, 1 − 2 1 θ(t) = t θ t : 7 Number theory Lecture 1 x Proof. If f 2 S and c 6= 0, then the Fourier transform of f c is cfb(cy). Apply −πx2 − 1 −πtx2 this to f(x) = e and c = t 2 to get that the Fourier transform of ft(x) = e is −π 1 y2 − 1 e ( t ) fbt(y) = t 2 . Now apply the Poisson summation formula to ft(x) to get −πn2t X − 1 −πn2· 1 e = t 2 e t : n2Z P −πn2t Note that θ(t) = 1 + 2 n≥1 e , so X 2 θ(t) − 1 e−πn t = : 2 n≥1 P −s=2 s −s Proof of Theorem 1.1. Recall ξ(s) = n π Γ( 2 )n ; start by looking at an indi- vidual summand: 1 s s Z dx − 2 s −s − 2 −s −x s=2 π Γ 2 n = π n e x 0 x 2 dx Make a change of variables x = πn t and recall that x is translation-invariant: 1 Z 2 dt = e−πn tts=2 0 t Now sum over n. As long as you can justify interchanging the sum and integral, Z 1 θ(t) − 1 dt ξ(s) = ts=2 0 2 t Why can you interchange the sum and integral? For s 2 R>1, this is OK because everything is nonnegative and the sum on the left converges. In fact, P R ::: converges absolutely for any complex s: changing the imaginary part does not affect the absolute value jts=2j. If s < 0, you can't expect this to make any sense: if t is close to zero, then the ts=2 part won't converge, and the θ part doesn't help enough. Now I want to replace this expression with something that does make sense for all s. Plan: we have Z 1 θ(t) − 1 dt Z 1 θ(t) − 1 dt ξ(s) = ts=2 + ts=2 1 2 t 0 2 t I(s) θ(t)−1 P −πn2t where I(s) converges for all s 2 C: this is because 2 = n≥1 e , and as t ! 1, the first term (n = 1) dominates.
Load more

Recommended publications
  • A Twisted Nonabelian Hodge Correspondence
    A TWISTED NONABELIAN HODGE CORRESPONDENCE Alberto Garc´ıa-Raboso A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2013 Tony Pantev David Harbater Professor of Mathematics Professor of Mathematics Supervisor of Dissertation Graduate Group Chairperson Dissertation Committee: Tony Pantev, Professor of Mathematics Ron Y. Donagi, Professor of Mathematics Jonathan L. Block, Professor of Mathematics UMI Number: 3594796 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3594796 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 A TWISTED NONABELIAN HODGE CORRESPONDENCE COPYRIGHT 2013 Alberto Garc´ıa-Raboso Acknowledgements I want to thank my advisor, Tony Pantev, for suggesting the problem to me and for his constant support and encouragement; the University of Pennsylvania, for its generous financial support in the form of a Benjamin Franklin fellowship; Carlos Simpson, for his continued interest in my work; Urs Schreiber, for patiently listening to me, and for creating and maintaining a resource as useful as the nLab; Marc Hoyois, for ever mentioning the words 1-localic 1-topoi and for answering some of my questions; Angelo Vistoli, for his help with Lemma 5.2.2; Tyler Kelly, Dragos, Deliu, Umut Isik, Pranav Pandit and Ana Pe´on-Nieto, for many helpful conversations.
    [Show full text]
  • Rational Points and Obstructions to Their Existence 2015 Arizona Winter School Problem Set Extended Version
    RATIONAL POINTS AND OBSTRUCTIONS TO THEIR EXISTENCE 2015 ARIZONA WINTER SCHOOL PROBLEM SET EXTENDED VERSION KĘSTUTIS ČESNAVIČIUS The primary goal of the problems below is to build up familiarity with some useful lemmas and examples that are related to the theme of the Winter School. Notation. For a field k, we denote by k a choice of its algebraic closure, and by ks Ă k the resulting separable closure. If k is a number field and v is its place, we write kv for the corresponding completion. If k “ Q, we write p ¤ 8 to emphasize that p is allowed be the infinite place; for this particular p, we write Qp to mean R. For a base scheme S and S-schemes X and Y , we write XpY q for the set of S-morphisms Y Ñ X. When dealing with affine schemes we sometimes omit Spec for brevity: for instance, we write Br R in place of BrpSpec Rq. A ‘torsor’ always means a ‘right torsor.’ Acknowledgements. I thank the organizers of the Arizona Winter School 2015 for the opportunity to design this problem set. I thank Alena Pirutka for helpful comments. 1. Rational points In this section, k is a field and X is a k-scheme. A rational point of X is an element x P Xpkq, i.e., a section x: Spec k Ñ X of the structure map X Ñ Spec k. 1.1. Suppose that X “ Spec krT1;:::;Tns . Find a natural bijection pf1;:::;fmq n Xpkq ÐÑ tpx1; : : : ; xnq P k such that fipx1; : : : ; xnq “ 0 for every i “ 1; : : : ; mu: krT1;:::;Tns Hint.
    [Show full text]
  • Introduction
    Introduction The theory we describe in this book was developed over a long period, starting about 1965, and always with the aim of developing groupoid methods in homotopy theory of dimension greater than 1. Algebraic work made substantial progress in the early 1970s, in work with Chris Spencer. A substantial step forward in 1974 by Brown and Higgins led us over the years into many fruitful areas of homotopy theory and what is now called ‘higher dimensional algebra’2. We published detailed reports on all we found as the journey proceeded, but the overall picture of the theory is still not well known. So the aim of this book is to give a full, connected account of this work in one place, so that it can be more readily evaluated, used appropriately, and, we hope, developed. Structure of the subject There are several features of the theory and so of our exposition which divert from standard practice in algebraic topology, but are essential for the full success of our methods. Sets of base points: Enter groupoids The notion of a ‘space with base point’ is standard in algebraic topology and homotopy theory, but in many situations we are unsure which base point to choose. One example is if p W Y ! X is a covering map of spaces. Then X may have a chosen base point x, but it is not clear which base point to choose in the discrete inverse image space p1.x/. It makes sense then to take p1.x/ as a set of base points. Choosing a set of base points according to the geometry of the situation has the implication that we deal with fundamental groupoids 1.X; X0/ on a set X0 of base points rather than with the family of fundamental groups 1.X; x/, x 2 X0.
    [Show full text]
  • Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9)
    Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9) February 22, 2014 In the previous lecture, we stated the following result: Theorem 1 (Nonabelian Poincare Duality in Topology). Let M be a manifold of dimension n and let (Y; y) be a pointed space which is (n − 1)-connected. Then there exists a (homotopy) cosheaf F of spaces on the Ran space Ran(M) with the following property: for every collection of disjoint connected open sets U1;:::;Uk ⊆ M, there is a canonical homotopy equivalence Y F(Ran(U1;:::;Uk)) ' Mapc(Ui;Y ): 1≤i≤k To get a feeling for what Theorem 1 is saying, it is useful to consider another example of space-valued cosheaves: Example 2. Let f : E ! B be a continuous function between topological spaces. Then the construction U 7! f −1(U) determines a functor from the partially ordered set of open subsets of B to the category of topological spaces. This functor determines a cosheaf with values in the 1-category of spaces: for example, for every pair of open sets U; V ⊆ B, the diagram of topological spaces f −1(U \ V ) / f −1(U) f −1(V ) / f −1(U [ V ) is a homotopy pushout square (this is a mild strengthening of the existence of a Mayer-Vietoris sequence for computing the homology of U [ V ). Let us imagine that every homotopy cosheaf on a topological space B arises via the construction of Example 2 (I do not know if this is actually true). Then Theorem 1 suggests that there should exist a continuous map of topological spaces f : E ! Ran(M) with the property that for every collection of disjoint connected open sets U1;:::;Uk ⊆ M, the inverse image −1 Q f Ran(U1;:::;Uk) is homotopy equivalent to Mapc(Ui;M).
    [Show full text]
  • Brauer-Manin for Homogeneous Spaces
    Université Paris Sud Faculté des Sciences d’Orsay Département de Mathématiques M2 Arithmétique, Analyse, Géométrie Mémoire Master 2 presented by Gregorio Baldi Brauer-Manin for Homogeneous Spaces advisor Professor David Harari Academic year 2015/2016 M2 Arithmétique, Analyse, Géométrie Département d’Enseignement de Mathématiques, Bât. 425 Université Paris-Sud 11 91405 Orsay CEDEX Contents 1 Arithmetic of Linear Algebraic Groups6 1.1 Homogeneous spaces over a field................................6 1.2 Kernels and quotients......................................8 1.3 Connectedness..........................................8 1.4 Unipotent groups.........................................9 1.5 Reductive groups......................................... 11 1.6 Algebraic tori........................................... 12 1.6.1 The variety of maximal tori, weak approximation................... 12 1.7 Groups of multiplicative type.................................. 13 1.8 More about semisimple groups................................. 14 1.9 Borel and parabolic subgroups................................. 15 1.10 Split reductive groups...................................... 15 1.10.1 Kneser-Tits conjecure.................................. 15 1.11 Root Data from algebraic group................................. 16 1.11.1 Lie algebra associated and the adjoint representation................. 16 1.11.2 Root Data......................................... 16 1.11.3 Isogeny, More Root Data................................ 17 1.12 Important properties of semisimple
    [Show full text]
  • The Life of Alexandre Grothendieck, Volume 51, Number 9
    Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre Grothendieck Allyn Jackson This is the first part of a two-part article about the life of Alexandre Grothendieck. The second part of the article will appear in the next issue of the Notices. Et toute science, quand nous l’enten- the Institut des Hautes Études Scientifiques (IHÉS) dons non comme un instrument de pou- and received the Fields Medal in 1966—suffice to voir et de domination, mais comme secure his place in the pantheon of twentieth cen- aventure de connaissance de notre es- tury mathematics. But such details cannot capture pèce à travers les âges, n’est autre chose the essence of his work, which is rooted in some- que cette harmonie, plus ou moins vaste thing far more organic and humble. As he wrote in et plus ou moins riche d’une époque à his long memoir, Récoltes et Semailles (Reapings and l’autre, qui se déploie au cours des Sowings, R&S), “What makes the quality of a re- générations et des siècles, par le délicat searcher’s inventiveness and imagination is the contrepoint de tous les thèmes apparus quality of his attention to hearing the voices of tour à tour, comme appelés du néant. things” (emphasis in the original, page P27). Today Grothendieck’s own voice, embodied in his written And every science, when we understand works, reaches us as if through a void: now seventy- it not as an instrument of power and six years old, he has for more than a decade lived domination but as an adventure in in seclusion in a remote hamlet in the south of knowledge pursued by our species France.
    [Show full text]
  • Principal Infinity-Bundles-General Theory
    Principal ∞-bundles – General theory Thomas Nikolaus, Urs Schreiber, Danny Stevenson November 8, 2016 Abstract The theory of principal bundles makes sense in any ∞-topos, such as the ∞-topos of topological, of smooth, or of otherwise geometric ∞-groupoids/∞-stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bun- dles. For suitable choices of structure ∞-group G these G-principal ∞-bundles repro- duce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomol- ogy. We discuss here this general abstract theory of principal ∞-bundles, observing that it is intimately related to the axioms that characterize ∞-toposes. A central result is a natural equivalence between principal ∞-bundles and intrinsic nonabelian cocycles, implying the classification of principal ∞-bundles by nonabelian sheaf hyper- cohomology. We observe that the theory of geometric fiber ∞-bundles associated to principal ∞-bundles subsumes a theory of ∞-gerbes and of twisted ∞-bundles, with twists deriving from local coefficient ∞-bundles, which we define, relate to extensions of principal ∞-bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice ∞-topos. arXiv:1207.0248v2 [math.AT] 4 Nov 2016 1 Contents 1 Overview 3 2 Preliminaries 8 2.1 Epimorphismsandmonomorphisms . 11 2.2 GroupoidsandGroups .............................. 12 2.3 Cohomology.................................... 15 3 Principal bundles 18 3.1 Introductionandsurvey ............................. 18 3.2 Definition and classification . 20 4 Twisted bundles and twisted cohomology 26 4.1 Actions and associated ∞-bundles.......................
    [Show full text]
  • Nonabelian Poincare Duality (Lecture 8)
    Nonabelian Poincare Duality (Lecture 8) February 19, 2014 Let M be a compact oriented manifold of dimension n. Then Poincare duality asserts the existence of an isomorphism ∗ H (M; A) ' Hn−∗(M; A) for any abelian group A. Here the essential hypothesis is that M is a manifold: that is, that every point of n M admits a neighborhood which is homeomorphic to Euclidean space R . It is therefore natural to begin by n asking what Poincare duality tells us in the special case M = R . Since Euclidean space is not compact, we should consider instead a more general formulation of Poincare duality which applies to noncompact oriented manifolds: ∗ Hc (M; A) ' Hn−∗(M; A): n In the case M = R , the right hand side of this isomorphism is easy to understand (since M is contractible), and Poincare duality comes down to a calculation ( ∗ n A if ∗ = n Hc (R ; A) ' 0 otherwise. From this isomorphism, one can deduce Poincare duality for more general manifolds by means of a local-to- global principle. To formulate it, let U(M) denote the collection of all open subsets of M, partially ordered by inclusion. For each object U 2 U(M), let C∗(U; A) denote the singular chain complex of U, and let ∗ Cc (U; A) denote the compactly supported cochain complex of U. Then the constructions ∗ U 7! C∗(U; A) U 7! Cc (U; A) can be regarded as functors from U(M) to the category Chain of chain complexes of abelian groups, or to the associated 1-category ModZ described in the previous lecture.
    [Show full text]
  • Topics in Galois Cohomology
    Topics in Galois Cohomology Lambert A'Campo 01570095 Supervised by Alexei Skorobogatov 2018/2019 Abstract Group cohomology appears in many different number theoretic questions and has various interpretations which make the theory very versatile and give rise to sur- prising connections. We will see some general methods for working with group cohomology as shown in [9] and [21]. Central results are Hilbert's Theorem 90, restriction-corestriction and the inflation-restriction sequence. These methods then accompany us throughout the rest of the text where we see how to use them in number theory. In most cases concrete consequences about diophantine equations, for example the weak Mordell-Weil theorem, can be derived from the theory. Further, we outline some of the connections between group coho- mology, Galois Descent and central simple algebras, our main source being [9]. Moving on we prove local class field theory like in [16] and [5] and we state what carries over to the global case where we encounter certain local-global principles and Artin reciprocity which is a vast generalisation of the famous law of quadratic reciprocity. In particular, we show how these generalised reciprocity laws can be used to give restrictions on hypothetical solutions to the Fermat equation as in [15]. Finally we give an original proof of the Artin-Schreier theorem based on the much deeper Merkurjev-Suslin theorem. Acknowledgements I thank Alexei Skorobogatov for giving me so much freedom while working on this project, for suggesting great books and for spending many hours on finding a mistake I couldn't manage to find myself.
    [Show full text]
  • Nonabelian Cohomology
    Nonabelian Cohomology Arber Selimi Institut Superior Tecnico 17.06.2020 ShortTitle Arber Selimi Outline 1 Introduction 2 Crossed Modules over a Group (Groupoid) 3 Crossed Complexes 4 Nonabelian Cohomology 5 Crossed n-fold extensions and Cohomology 6 Further Applications ShortTitle Arber Selimi Introduction For a topological space X, and an abelian group G, there is a bijection Hn(X; G) = [X; K(G; n)], using this result we will get the intuition to define the nonabelian cohomology. We start the talk by introducing the notion of Crossed modules, and give the first examples. We continue by introducing Crossed complexes and giving the constructions that we need. We finish the talk by defining the nonabelian cohomology using the intuition coming from the bijection given above, and we interpret cohomology classes via extensions. ShortTitle Arber Selimi Definition of Crossed Modules 1 A crossed module over a group M = (µ : M ! P ) is a morphism of groups µ : M ! P , called the boundary of M together with an action (m; p) ! mp of P on m satisfying the following two axioms p −1 1 µ(m ) = p µ(m)p −1 µ(n) 2 n mn = m for all m; n 2 M and p 2 P . We call M a crossed P -module. 2 A morphism between two crossed modules is defined to be a pair of homomorphism (f; g), such that they commute with the action. 3 We denote by XMod=Grp the category of crossed modules 4 kerµ is abelian, and imµ is normal subgroup of P . ShortTitle Arber Selimi Examples of Crossed Modules 1 If M E P , the boundary map given by the inclusion i : M ! P , and the action via conjugation, then M is a P crossed module.
    [Show full text]
  • LECTURES on N-CATEGORIES and COHOMOLOGY
    LECTURES ON n-CATEGORIES AND COHOMOLOGY TALKS BY JOHN BAEZ, NOTES BY MICHAEL SHULMAN Contents Preface 2 1. The Basic Principle of Galois Theory 3 1.1. Galois theory 3 1.2. The fundamental group 4 1.3. The fundamental groupoid 5 1.4. Eilenberg{Mac Lane Spaces 5 1.5. Grothendieck's dream 6 2. The Power of Negative Thinking 10 2.1. Extending the periodic table 10 2.2. The categorical approach 12 2.3. Homotopy n-types 13 2.4. Stuff, structure, and properties 15 2.5. Questions and comments 16 3. Cohomology: The Layer-Cake Philosophy 17 3.1. Factorizations 17 3.2. Cohomology and Postnikov towers 20 3.3. Questions and comments 23 4. A Low-Dimensional Example 25 4.1. Review of Postnikov towers 25 4.2. Example: the classification of 2-groupoids 27 4.3. Relation to the general case 29 4.4. Questions and comments 31 5. Appendix: Posets, Fibers, and n-Topoi 33 5.1. Enrichment and posets 33 5.2. Fibers and fibrations 36 5.3. n-Topoi 40 5.4. Geometric morphisms, classifying topoi, and n-stuff 43 5.5. Monomorphisms and epimorphisms 46 5.6. Pointedness versus connectedness 50 6. Annotated Bibliography 55 1 2 TALKS BY JOHN BAEZ, NOTES BY MICHAEL SHULMAN Preface The goal of these talks was to explain how cohomology and other tools of al- gebraic topology are seen through the lens of n-category theory. The talks were extremely informal, glossing over the difficulties involved in making certain things precise, just trying to sketch the big picture in an elementary way.
    [Show full text]