A Pedestrian Guide to Topos Theory

For a while now I’ve been slowly working on trying to figure out how to develop a more accessible introduction to an area of mathematics called topos theory.

Topos theory has lots of interesting connections to other areas of mathematics and related topics, including logic, geometry, topology (the study of spaces), and computer science. Amongst its many applications, topos theory offers: a logic for reasoning about situations that don’t involve sharp clear distinctions and yes/no questions; a formulation of calculus that justifies the naive approach we first learned in school; a language in which quantum mechanics looks less weird; and even a different way of understanding the foundations of mathematics itself.

Since it sits at the intersection of so many different mathematical topics, topos theory allows us to transfer ideas between different domains, and also to find surprising generalisations of familiar ideas.  For example, in topos theory we can talk about spaces like the real numbers or the Euclidian plane, but we can also find interesting things that behave like spaces, have non-trivial structure, but that don’t contain any points!  Topos theory also provides a way of thinking about logical reasoning that dispenses with a lot of basic assumptions that we might have thought were essential to logic — like the idea that every statement we deal with is either true or false.

These ideas should have lots of applications outside of mathematics — in particular in philosophy.  For example, problems related to vagueness (involving properties like ‘tall’ or ‘red’ that don’t have clearly defined sharp boundaries) seem naturally suited to a logic in which the answer to a question can be something in between Yes and No.

Many of these weird and exciting features of topos theory can be understood without too much mathematical sophistication.  But if you’ve ever tried to learn a bit about topos theory you may have found it impossible to get started, because most of the standard textbooks on the subject (such as those in this bibliography) tend not to be written for the beginner.  Historically, topos theory arose from a convergence of several sophisticated areas of mathematics, and so a lot of presentations of the subject lean heavily on these connections and assume that the reader is familiar with ideas from algebraic geometry or category theory.

I think it’s possible to make the ideas of topos theory accessible without assuming a lot of background knowledge, by starting with the connections between topos theory and logic, building up from simple examples and working slowly toward more sophisticated ideas.  My plan is to write at least some of the pieces of this here on the blog.

My preferred style of teaching and writing, when reasonably possible, is to introduce new ideas by showing how they follow from basic principles rather than having them appear from nowhere.  Sometimes this will involve exploring the most obvious first guess at an answer and then backtracking to fix a problem.  The goal is always to understand not just the definitions but the reason why things are defined as they are — to let you know that you could have found the answer yourself with a bit of guidance, and to get you to a position where you can re-construct the definition for yourself if you forget it.

In the bibliography there’s a list of standard reference materials on topos theory and related topics. The top of this list — the comprehensive encyclopaedia of topos theory — is Peter Johnstone’s Sketches of an Elephant, a multi-volume compendium that approaches the subject from every possible angle to see topos theory in all its diverse aspects.  The title comes from the parable of the blind men inspecting an elephant, each of them able to examine just one part of the animal: one finds it to be like a wall, another like a tree trunk, another like a fan, and so on. Likewise, topos theory can be understood in many different ways: Johnstone’s book is divided into parts with titles such as “Toposes as Categories”, “Toposes as Spaces”, “Toposes as Theories”, and “Toposes as Mathematical Universes”.

As this blog unfolds we’ll gradually explore some of these facets of topos theory.  But where Johnstone’s work is comprehensive, this blog is much more modest in scope; where his book is vast and authoritative, this is incremental and cautious; where Johnstone’s book is the Elephant, this blog is the elephant shrew.


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