Information for prospective students

On this page, you can find brief and informal descriptions of my research subject, so that you can get an idea of what I do. I also describe the courses which are offered by the logic group in the math department, so that you can find out what types of subjects we teach here. Finally, I have some suggested research projects for graduate students or summer students.

Category theory and Logic

What is category theory?

Category theory is a branch of pure mathematics which is concerned with abstract properties of structures and mappings between them. Many people regard category theory in the first place as a language for mathematics, or as a powerful organizational tool. It places strong emphasis on studying the mappings, or morphisms, between mathematical objects rather than studying these objects in isolation. Thus instead of looking at, say, groups we consider the category of groups, i.e. the groups and the homomorphisms of groups. Another typical feature of category theory is that it often uses commutative diagrams to explain, emphasize or proof matters.

Category theory is in part useful because it allows for clean and conceptual statements and descriptions which bring out the underlying structure. A traditional example: when studying groups, you learn that a group is a set with some additional structure on it. Similarly, a ring is a set with additional structure on it, as are topological spaces, partial orderings, vector spaces, and so on. Category theory describes these situations by saying that there is a "forgetful functor" from the category of groups (rings, spaces,...) to the category of sets.
What is more, given a set S we may construct the free group on that set; it is characterized up to isomorphism by a universal property (given a function from S to another group it uniquely factors through the free group via a group homomorphism). Similarly, one may construct the free ring on a set S, or the vector space with basis S, or the discrete topological space on the set S, and so on. Each of these free constructions is expressed in categorical terms by saying that the forgetful functor has a left adjoint.

Apart from being a concise formulation for stating the existence of a free construction, general results about adjoint functors now allow us to conclude things about these free constructions without having to prove them for groups, rings, spaces,... separately. For example, a well known general theorem from category theory says that left adjoints preserve sums and right adjoints preserve products. In the instance of groups, this says that the product of a family of groups has as its underlying set the product of the underlying sets. It also says that the free group on the set S+T is the direct sum of the free groups on S and on T. Sometimes it is said that such categorical results are not really mathematical results. Well, that is part of the point: category theory tells us which parts of mathematics are trivial for trivial reasons.

Categories and logic

However, category theory has more to it than this organizational function. It is also studied as a topic in its own right, much in the same way as one might study, say, ring theory. There are deep and interesting connections between category theory, topology and logic. Just to give a few pointers: categories serve as models for various kinds of logic, ranging from equational logic (used to describe algebraic structures), first order logic (the kind studied in traditional mathematical logic), higher order logic, intuitionistic logic (where we avoid the law of the excluded middle) and even other kinds such as linear logic.

Conversely, one may think of certain categories (called toposes) as universes of mathematics: their objects are set-like, and their morphisms are function-like, but the logic which governs them is not classical. Typically the axiom of choice is not true, the logic is not classical (excluded middle fails) and there may be more than two, even infinitely many truth-values. In spite of this different logic, we may still develop mathematics inside such a category. For example, this can be used to provide settings for non-standard analysis and differential geometry.

Categories and topology

Categories are also closely related to topology. One facet of this relation is the fact that to every space X we may associate a category Sh(X) of sheaves on X. The category Sh(X) is where topology and logic meet: we can use it to study X using logical techniques, or we can use spaces to find interesting models of certain logical theories. For example, sheaf models have been used to give elegant proofs of milestone results in mathematical logic, such as the independence of the continuum hypothesis.

Homotopy theory is also heavily involved in category theory, in at least two ways. First, just as in topology one studies not just spaces, continuous maps and homotopies, one also considers higher homotopies: these are homotopies between homotopies between homotopies... and so on. A similar thing happens in category theory: instead of just looking at objects and morphisms, one can consider morphisms between morphisms between morphisms... . Such higher-dimensional structures are called higher categories, and it is not hard to imagine that understanding those would be very useful for homotopy, and vice versa.
In addition, there is a class of categories called Quillen model categories, which aim at abstracting away from the stretchable rubber you learned spaces are made of. The result is a setting which allows for many of the constructions available in homotopy theory, but in a "neutral" category. One of the big advantages is that this gives us ways of comparing different models of homotopy, by looking at how the resulting model categories relate.

In addition, category theory and logic are heavily used in theoretical computer science and mathematical physics.

Why study category theory?

Even if you're not planning on doing research directly in these areas, you will benefit greatly from understanding the basics: it will help you see the relevant structural features of whatever area you're studying more clearly, help you ask the right kind of questions and help you understand which results are a consequence of general structural considerations and which are more specific in nature. In other words, learning some category theory is an excellent way of mastering some of valuable skills needed to become a successful researcher in pure mathematics!

Courses

Every year the logic group in Ottawa offers courses in these areas both on the undergraduate and on the graduate level. In addition to the courses listed below, we sometimes offer directed study courses or seminars. Here's a list of all the courses we offer, with an explanation of what you can expect to learn in them.

Thesis projects

If you are interested in doing a summer project, an undergraduate thesis, an MSc or a PhD in a topic related to logic, category theory or mathematical physics, there are plenty of opportunities here in Ottawa. Our group is one of the main centers of category theory in North America, with three researchers (Prof. Blute, Prof. Scott and myself) working in the area and several others working in neighbouring areas.

We are always looking for motivated new students and we'd be happy to answer any further questions you may have!

Examples of possible projects

Below I list a few possibilities, with brief descriptions. Also check out the pages of my colleagues in the logic group to see what projects they work on!