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"Four men, who had been blind from birth, wanted to know what an elephant was like; so they asked an elephant-driver for information. He led them to an elephant, and invited them to examine it; so one man felt the elephant's leg, another its trunk, another its tail and the fourth its ear. Then they attempted to describe the elephant to one another. The first man said "The elephant is like a tree". "No," said the second, "the elephant is like a snake". "Nonsense!" said the third, "the elephant is like a broom". "You are all wrong," said the fourth, "the elephant is like a fan". And so they went on arguing amongst themselves, while the elephant stood watching them quietly."
The Indian folklore story of the blind men and the elephant, as adapted from E. J. Robinson’s Tales and Poems of South India by P. T. Johnstone in the Preface of "Sketches of an Elephant"
A topos is a category with some basic structure admitting a wide range of interesting interpretations. It can be a set-theoretical universe where the alternative types of mathematics take place, a generalized notion of space that lifts geometrical intuition far beyond the usual topological spaces or a syntax-free presentation of a first-order theory of some sort.
In this course on topos theory, we mainly focus on its first interpretation and its role in model construction, unifying techniques from forcing and Heyting-valued models to different types of realizability. We start with a very short and gentle introduction to category theory. Then, we introduce elementary toposes and as their concrete examples, we present the categories of sheaves and Heyting-valued sets, on the one hand, and the effective topos, on the other. Then, we move to the connection between topoi and logic to present a topos-theoretical version of some independence results including the independence of the continuum hypothesis and the axiom of choice. We also use toposes to realize some exotic, yet coherent possibilities including the so-called computable (resp. Brouwerian) universe in which all the functions on natural numbers (resp. real line) are computable (resp. continuous). These models are used to prove the consistency of Church-Turing thesis (resp. Brouwer's theorem on continuity of all functions on the reals) from intuitionistic arithmetic (resp. analysis).
Despite what it may appear at the first glance, the course only assumes familiarity with some basic concepts in first-order logic, algebra and topology and hence it must be accessible to all mathematics and computer science students. The rest, including the preliminaries on category theory, will be built during the course, whenever it’s needed.
References:
The main references are:
Category Theory, Steve Awodey, see also the raw and incomplete draft of my book here.
Sheaves in Geometry and Logic - A first introduction to topos heory, Suanders Mac Lane and Ieke Moerdijk.
Handbook of Categorical Algebra, Volume 3, Francis Borcuex.
Sketches of an Elephant: A Topos Theory Compendium, Peter Johnstone.
For an extensive collection of related books, lecture notes, video lectures and expository notes, see here.
Recordings:
They can be found on Youtube and Aparat after each session! Note that the channels are not exclusive to this course and they contain other series of lectures, as well.
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