The Mathematical and Philosophical Legacy of Alexander Grothendieck

Preface
	Bibliography
Contents
Grothendieck\'s 40 Main Years (1949–1991): A Unitary Vision Through the TSK Models (Topos of Sheaves over Kripke Models)
	Contents
	1 Introduction
	2 A Brief Look at the TSK Models
		2.1 Phenomenological Layer (S)
		2.2 Historical Layer (K)
		2.3 Metaphysical Layer (T)
	3 First Epoch: SK, 1949–1953 and 1955–1957
		3.1 SK(1): Functional Analysis, 1949–1953
		3.2 SK(2): Homological Algebra, 1955–1957
	4 Second Epoch: TSK, 1958-1964 and 1964–1968
		4.1 (T)SK(3): Algebraic Geometry, 1958–1964
		4.2 TSK(4): Arithmetic Geometry, 1964–1968
	5 Third Epoch: TSK, 1981-1986 and 1983–1991
		5.1 TSK(5): Topological Algebra and Mathematical Philosophy, 1981–1986
		5.2 TSK(6): Universal Algebra and World View, 1983–1991
	6 Conclusions
	References
Grothendieck and Differential Equations
	Contents
	1 Foreword
	2 Differential Operators
	3 Singularities
	4 Gauss-Manin Connection
	5 Galois Aspects
	6 Conclusion
	References
The Birkhoff-Grothendieck Theorem
	Contents
	1 Introduction
	2 Vector Bundles on Riemann Surfaces
	3 The Grothendieck Theorem
		3.1 Some Further Works in the Spirit of Birkhoff-Grothendieck
	4 Riemann-Hilbert Problem and Birkhoff\'s Theorem
		4.1 Fuchsian Differential Equations on Riemann Surfaces
		4.2 The Bolibrukh Counterexamples: An Instance
		4.3 Birkhoff\'s Theorem
	References
Infinite Products and Congruent Numbers
	Contents
	1 Introduction
	2 A Piece of Correspondence
	3 Euler: The Emergence of the Functions  and  ζ
		3.1 Formal Calculations
		3.2 Reciprocity I
	4 Euler\'s Constants for Number Fields
	5 Reciprocity II: Euler\'s Criterion
	6 Arithmetic Hecke Products
		6.1 Explicit Formulas
		6.2 Artin (and Other) L-Functions
	7 An Example from the Last Entry in Gauss\'s Diary
		7.1 BSD Conjecture
		7.2 Higher Reciprocity Laws
	8 Emergence of the Zeta Function II
	9 Adèles and Idèles
		9.1 From Automorphic Forms to Automorphic Representations
		9.2 ζ- Function of a Scheme over Z
	10 The Congruent Number Problem and Higher Reciprocity
		10.1 History of the Problem
		10.2 Present and Future
		10.3 Density of Congruent Numbers
		10.4 Tate–Shafarevich Groups of the Congruent Number Elliptic Curves
	11 Aspect of Grothendieck\'s Work on L-Functions
		11.1 The Gross–Zagier Formula and Applications
	Appendix by Bill Allombert: Computational Aspect Using PARI/GP
		Solving the Fermat Equation Using the Dirichlet Class Number Formula
		Computing Hilbert Class Fields Using Dirichlet Class Number Formula
		Solving the Congruent Number Problem Using Analytical Methods
			The BSD Formula
			The Gross–Zagier Formula
	References
The Enduring Legacy of Grothendieck\'s Duality Theorem
	Contents
	1 Introduction
	2 Fantappié\'s Vision
	3 The Arrival of Topological Vector Spaces
	4 Continuing Implications
		4.1 Hyperfunctions and Analytic Functionals
		4.2 Duality in Hypercomplex Analysis
	References
Conjectures and Counterexamples in Grothendieck\'s Work in Functional Analysis
	Contents
	1 Introduction
	2 Background
		2.1 Dieudonné, Schwartz, and Functional Analysis in Nancy in 1949
	3 Grothendieck and Functional Analysis
		3.1 Topological Vector Spaces, Counterexamples, and Solutions to Open Problems
	4 Tensor Products
	5 Grothendieck\'s Produits Tensoriels Topologiques etEspaces Nucléaires
	6 1983: The Year of the Counterexamples
	7 A Few Final Remarks
	References
Tôhoku 65 Years After
	Contents
	1 Grothendieck\'s New Algebraic Geometry: Tôhoku 65 Years After, Introductory Elements
	2 Philosophical Intermezzo
	3 Spectral Sequence, First Introduction as Example of a Synthesis Mechanism
	4 Grothendieck in His First Chapter Gives Generalities on Abelian Categories
	5 Grothendieck\'s Vision of Sheaves, Brief Historical Reconstruction
	6 Homological Algebra in Abelian Categories
	7 First Introduction to Spectral Sequences
	8 Some Interesting Examples of Spectral Sequences
	9 Universal Functors
	10 Effaceable Functors
	11 Derived Functors
	12 In Two Variables
	13 Spectral Sequences, Spectral Functors
	14 Remark on Exactness
	15 Spectral Sequence
	16 Spectral Sequence of a Filtered Complex
	17 Two Spectral Sequences Convergent to the Same Graded Object
	18 Resolvent Functors
	19 Cohomology with Coefficients in a Sheaf
	20 Philosophical General Reflection
	21 Annex
		21.1 Examples with Details
	References
About Grothendieck Fibrations
	Contents
	1 Introduction
	2 Categories Varying over a Category
	3 Indexed Categories and Grothendieck Fibrations
	4 Logic in the Fibrational Framework
	Appendix
	References
Grothendieck Did Not Believe in Universes, He Believed in Topos and Schemes
	Contents
	1 I Think There Is Mathematics Behind All of This
		1.1 Generality and Unity
		1.2 The Mathematics of Injective Resolutions
		1.3 Naive Simplicity for Spectral Sequences
	2 Equivalence of Categories
	3 The Worst Joke I Ever Heard
		3.1 Strong Limit Cardinals
		3.2 What Is Required for Grothendieck\'s Large Structures
	References
The ``Unifying Notion\'\' of Topos
	Contents
	1 Introduction
	2 The Multiform Nature of Toposes
		2.1 Toposes as Generalised Topological Spaces
		2.2 Toposes as Universes
		2.3 Toposes as Classifying Spaces
	3 The Reception of Toposes
		3.1 The Vision and the Tool
		3.2 `Sites Without Toposes\', `Toposes Without Sites\'
	4 Toposes as `Bridges\': the Underlying Vision and Some Examples
		4.1 The Key Principles
		4.2 Some Examples of `Bridges\'
			Theories of Presheaf Type
			Topos-Theoretic Fraïssé Theorem
			Topological Galois Theory
			Stone-Type Dualities
	5 Future Perspectives
	References
Motivating Motives
	Contents
	1 Introduction
	2 The Riemann Hypothesis
	3 The Weil Conjectures
	4 Motives
	5 Further Reading
		5.1 Introduction
		5.2 The Riemann Hypothesis
		5.3 The Weil Conjectures
		5.4 Motives
My View on and Experience with Grothendieck\'sAnabelian Geometry
	Contents
	1 Introduction
	2 Following Galois
	3 Anabelian Geometry
	4 m-Step Solvable Anabelian Geometry
	5 The Grothendieck Philosophy
	References
Grothendieck\'s Use of Equality
	Contents
	1 Overview
	2 Introduction
	3 Universal Properties
	4 Products in Practice
	5 Universal Properties in Algebraic Geometry
	6 The Problem with Grothendieck\'s Use of Equality
	7 More on ``Canonical\'\' Maps
	8 Canonical Isomorphisms in More Advanced Mathematics
	9 Summary
	References
Boolean Valued Models, Sheafifications, and Boolean Ultrapowers of Tychonoff Spaces
	Contents
	1 Introduction
	2 Preliminaries and Notations
	3 From Boolean Valued Models to Presheaves, and Conversely
		3.1 Boolean Valued Models as Separated Presheaves
		3.2 Fullness, Mixing Property, and Sheaves
		3.3 The Duality Between Boolean Valued Models and Presheaves
	4 A Topological Description of Sheafifications
		4.1 The Mixified Model
	5 Boolean Ultrapowers as Sheafifications
		5.1 The Semantics of C(`3́9`42`\"̇613A``45`47`\"603ASt(B), Y) When Y Is Tychonoff
		5.2 The Degree of Elementarity of Y Inside C(`3́9`42`\"̇613A``45`47`\"603ASt(B), Y)+/G
	References
Toward a Geometry for Syntax
	Contents
	1 Introduction
		1.1 Type Theory and the Relative Point of View
		1.2 Universes in Type Theory and Category Theory
			Strict Base Change via Universal Objects
			Grothendieck\'s Universes
			Universes in a Category
			Grothendieck–Bénabou Universes Inside a Topos
		1.3 Abstract and Concrete Syntax of Type Theory
			Computerized Proof Assistants
			External vs. Internal Equality
			Decidability of External Equality
			Running Example: Injectivity of Type Constructors
		1.4 Normalization and Injectivity for Free Monoids
			The Theory of Monoids
			Constructing the Free Monoid on a Set
			Injectivity of Scalar Multiplication in the Free Monoid
	2 Free Models of Type Theory and Normalization
		2.1 Natural Models of Type Theory
			Representable Maps and Natural Models
			Function Spaces on a Natural Model
			The (2,1)-Category of Natural Models
			Free Natural Models: The Abstract Syntax of Type Theory
			From Universes to Natural Models
		2.2 Injectivity of Type Constructors in Free Natural Models
		2.3 Normal Forms Are not Functorial in Substitutions
		2.4 Models of Variables and the Method of Computability
			Models of Variables Over a Natural Model
			Why Is It Hard to Build a Model Based on Normal Forms?
			Tait\'s Method of Computability
			Freyd\'s Categorical Reconstruction of Tait Computability
	3 Normalization by Gluing for Free Natural Models
		3.1 Synthetic Tait Computability for Models of Type Theory
			The Topos of Computability Spaces Over a Model of Variables
			Recollement of Computability Spaces
			The Internal Language of Computability Spaces
			Internalizing the Model of Variables
			The Computability Space of Normal Forms
			Injectivity of Normal Type Constructors
			The Universe of Normalization Spaces
			Closure of Normalization Spaces Under Connectives
		3.2 From Normalization Spaces to a Natural Model ofType Theory
		3.3 The Normalization Result
			The Functors of Atomic and Canonical Points
			Hydration of Variables via Bocquet, Kaposi, and Sattler\'s inserter
			The Normalization Map and Its Injectivity
		3.4 Injectivity of Type Constructors
	4 Concluding Remarks
	References
Investigating Definability in Propositional Logic via Sheaves on Grothendieck Topologies
	Contents
	1 Introduction
	2 Intuitionistic Logic and Heyting Algebras
	3 A Route to Dualities
	4 Sheaf Representation
	5 Exactness Properties
	6 Applications to Proof Theory and Model Theory
	7 Fixpoints and Periodicity
	8 Solving Equations via Projectivity
	9 Conclusions
	References
Grothendieck and Model Theory: Five Charactersin Search of a Theme
	Contents
	1 Prelude: Grothendieck on Mysteries and Galois
	2 Introduction
	3 Stability: An Early Grothendieckian Theme?
		3.1 Classification Theory
		3.2 Model Theory: Perspective and Fine Grain
			Taxonomies
		3.3 The Main Dividing Line: Stability
			Grothendieck, 1952: Early Version of Stability
	4 Galois Theory of Model Theory
		4.1 Model Theory as a Natural Galois-Theoretic Framework
			Poizat Makes the Connection Explicit
			In First Order, the Key Role of Imaginaries
		4.2 Some Translations (Following Medvedev/Takloo-Bighash)
		4.3 Summary of the First Rapprochement: The Two Sources
	5 Galois à la Grothendieck, in Model Theory
		5.1 Interpretations and Stability
		5.2 Interpretation Functor Between Classes of Models
	6 Categories and Abstract Elementary Classes: The Great Reversal
		6.1 Abstract Elementary Classes: Model Theory\'sSemantic Essence
		6.2 Accessible Categories and Grothendieck
		6.3 Opening Toward the Future
	7 Two Ascents: Hrushovski\'s Core and Higher Stability
		7.1 Ascent 1: Definability Patterns (Some Features)
		7.2 Ascent 2: Higher Stability?
	8 Conclusive Remarks
	Appendices
	A Model Theoretic Galois Theory
		A.1 Normal and Splitting Extensions
		A.2 Some Differences (Lost in Translation)
		A.3 A Couple of Notions for the Translation
			Normal Extensions
			Splitting Extensions
		A.4 A Key Step: Coding Finite Sets
			The Crucial Notion
	B Makkai-Reyes, Stable Interpretations
		B.1 The Makkai-Reyes Approach: Models as Functors
		B.2 Interpretation Functor Between Classes of Models
		B.3 Examples: ACF, RCF
		B.4 Stable Interpretations: A Bit on Galois Theory
		B.5 The Galois Group of a First-Order Theory
	C On Hrushovski\'s Definability Patterns
		C.1 The Pattern Language: First Obstruction
		C.2 Possible Workarounds
		C.3 Galois Morleyizations
		C.4 Abstract Cores (Hrushovski)
		C.5 The Core of j?
	References
Context-Dependence and Descent Theory
	Contents
	1 Context-Dependence
	2 Grothendieck\'s Framework of Fibered Categories
	3 Pursuing Descent for Context-Dependence
	4 Conclusion
	Appendix
	References
Grothendieck and Teichmüller Modular Spaces
	Contents
	1 Origin of the Problem
	2 Algebraic Curve Relative to an Analytic Space
	3 Teichmüller Curves
	4 Opération of the Group γ
	5 Curves of Genus g
	6 Moduli Space of Curves of Genus g
	7 Ordinary Curves
	8 Foundations of Analytic Geometry
	9 Some Problems of Moduli
	10 Linear Rigidification
	11 Existence of the Teichmüller Space