William Lawvere

About

From Wikipedia

Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.Biography Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley's textbook General Topology. Lawvere found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. Truesdell supported Lawvere's application to study further with Samuel Eilenberg, a founder of category theory, at Columbia University in 1960.

Published Works

• FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES
• Metric spaces, generalized logic, and closed categories
• Adjointness in Foundations
• AN ELEMENTARY THEORY OF THE CATEGORY OF SETS
• Equality in hyperdoctrines and comprehension schema as an adjoint functor
• Sets for Mathematics
• Diagonal arguments and cartesian closed categories
• The Category of Categories as a Foundation for Mathematics
• Conceptual Mathematics
• Ordinal sums and equational doctrines
Continuously Variable Sets; Algebraic Geometry = Geometric Logic
Cohesive Toposes and Cantor's 'lauter Einsen'
Some thoughts on the future of category theory
Some algebraic problems in the context of functorial semantics of algebraic theories
Introduction
Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes
Foundations and Applications: Axiomatization and Education
Variable Quantities and Variable Structures in Topoi
Qualitative distinctions between some toposes of generalized graphs
On the duality between varieties and algebraic theories
Unity and identity of opposites in calculus and physics
Introduction
Introduction to Part I
Algebraic Theories
Categories in Continuum Physics
Seminar on Triples and Categorical Homology Theory
Linearization of graphic toposes via Coxeter groups
Co-existence of Krull filtrations
Categories of Space and of Quantity
Grassmann’s Dialectics and Category Theory
Left and right adjoint operations on spaces and data types
Comments on the Development of Topos Theory
Euler’s Continuum Functorially Vindicated
ALGEBRAIC THEORIES, ALGEBRAIC CATEGORIES, AND ALGEBRAIC FUNCTORS
Review
Categorical algebra for continuum micro physics
Sifted and filtered colimits
Reports of the Midwest Category Seminar II
Category Theory, Homology Theory and their Applications II
Introduction
Foreword
Preface
ABSTRACT ALGEBRAIC CATEGORIES
Preliminaries
Algebraic theories and algebraic categories
Reflexive coequalizers
Algebraic categories as free completions
Properties of algebras
A characterization of algebraic categories
From filtered to sifted
Canonical theories
Algebraic functors
Birkhoff's variety theorem
CONCRETE ALGEBRAIC CATEGORIES
One-sorted algebraic categories
Algebras for an endofunctor
Equational categories of Σ-algebras
S-sorted algebraic categories
SPECIAL TOPICS
Morita equivalence
Free exact categories
Exact completion and reflexive-coequalizer completion
Finitary localizations of algebraic categories
Postscript
Monads
Abelian categories
More about dualities for one-sorted algebraic categories
References
Preface
Organisation of the book
Galileo and multiplication of objects
The category of sets
Sets, maps, composition
Sets, maps, and composition
Composing maps and counting maps
The algebra of composition
Isomorphisms
Division of maps: Isomorphisms
Division of maps: Sections and retractions
Two general aspects or uses of maps
Isomorphisms and coordinates
Pictures of a map making its features evident
Retracts and idempotents
Quiz
How to solve the quiz problems
Composition of opposed maps
Summary/quiz on pairs of ‘opposed’ maps
Summary: On the equation p ∘ j = 1A
Review of ‘I-words’
Test 1
Brouwer's theorems
Categories of structured sets
Examples of categories
Ascending to categories of richer structures
Categories of diagrams
Monoids
Maps preserve positive properties
Objectification of properties in dynamical systems
Idempotents, involutions, and graphs
Some uses of graphs
Test 2
Review of Test 2
Elementary universal mapping properties
Universal mapping properties
Terminal objects
Points of an object
Products in categories
Universal mapping properties and incidence relations
More on universal mapping properties
Uniqueness of products and definition of sum
Labelings and products of graphs
Distributive categories and linear categories
Examples of universal constructions
The category of pointed sets
Test 3
Test 4
Test 5
Binary operations and diagonal arguments
Higher universal mapping properties
Map objects
Exponentiation
Map object versus product
The contravariant parts functor
Subobject, logic, and truth
Parts of an object: Toposes
The Connected Components Functor
Group theory and the number of types of connected objects
Constants, codiscrete objects, and many connected objects
APPENDICES
Geometery of figures and algebra of functions
Adjoint functors with examples from graphs and dynamical systems
The emergence of category theory within mathematics
Annotated Bibliography
Adjoints in and among Bicategories
Conceptual Mathematics: A First Introduction to Categories.
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Other Resources

• en.wikipedia.org

Schools

University of California, Berkeley

Public research university in Berkeley, California, United States

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University of Chicago

Private research university in Chicago, Illinois, United States

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Columbia University

Private Ivy League research university in New York City

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University at Buffalo

University with campuses in and around Buffalo, NY and Amherst, NY

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