Making Structural Mathematics Work:  

A Philosophical and Logical Introduction to Categories and Functors

A Short Course on Category Theory by Prof. Colin McLarty

15:00-18:00, June 27-30, 2016

Room B112, Building 2, Department of Philosophy, Peking University

Course Abstract

Mathematicians today often describe abstract spaces and algebraic structures in purely structural terms.  For example, instead of defining a space S by its points, they will define S just by its geometric relations to other spaces, and to algebraic structures like groups.  Categories and functors were created in the 1940s as the standard tools for doing this in topology and algebra.  They spread into functional analysis and algebraic geometry in the 1950s, and today are standard across most of pure mathematics.  Category theory can even produce independent foundations for mathematics which are naturally close to the way most mathematicians normally think of mathematics.  The lectures will introduce the basic ideas of category, functor, and natural transformation in terms of axioms, and examples, and graphic depiction.  We will begin on topos theory, and close with discussion of many examples in mathematics and in philosophy,

There will be no textbook but there will be pdf lecture slides.

References

Steve Awodey Category Theory.

Saunders Mac Lane Categories for the Working Mathematician.

Colin McLarty Elementary Categories, Elementary Toposes.

Tentative plan

Lecture 1 (June 27):  Classical mathematical motives for category theory.   Axioms for a category, plus mathematical and philosophical examples, and how to picture a category visually.  What isomorphism means in current mathematics.  Comparing different ideas of structuralism from Bourbaki and from recent philosophers.

Lecture 2 (June 28): Defining structure up to isomorphism in a category.  Explain what is natural about natural transformations.   Then define functors so that we can give the modern definition of natural transformations.  Then give examples and show how to picture them visually.  Introduce the category of categories.

Lecture 3 (June 29)  Foundations.  Lawvere’s Elementary Theory of the Category of Sets, and his Category of Categories as a Foundation.  Show how close these are to what most mathematics textbooks actually say about their basic assumptions, than Zermelo Frankel set theory is. Relations of categorical foundations to Zermelo Frankel set theory.

Lecture 4 (June 30): Topos theory in Grothendieck’s sense, and the Lawvere-Tierney theory of elementary toposes.  Applications in mathematical and philosophical logic.

Speaker Introduction

Colin McLarty is the Truman P Handy Professor of Philosophy and of Mathematics at Case Western Reserve University, and is the current President of the Philosophy of Mathematics Association.   Besides working in pure category theory, he has published on general foundations for mathematics, and specific foundations for modern number theory.  He has also published and spoken on history of modern mathematics, especially Henri Poincare, Emmy Noether, the original members of the group Bourbaki, and such founders of category theory as Saunders Mac Lane, Alexander Grothendieck, and William Lawvere.