Categorical spacetime

John Baez is preparing a colloquium at the Perimeter Institute where the application of Category Theory (CT) to quantum spacetime will be discussed. This page contains links to (a few) introductory and (many) advanced papers on how the ideas of CT (and its generalization to n-categories) can be used to analyze the category of Hilbert spaces.

I’ve developed a lively interest in CT during my computer science wanderings , and even wrote an elementary introduction (geared to programmers) that you may find useful (if you’re new to the field and don’t mind digressions into computer programming).

Categories are deceptively simple: they abstract the notion of sets and their morphisms, or, if you prefer, of objects and their transformations. Thus, a category is defined as a set of objects related by arrows that are composable (in the functional sense). Composition is associative and there’s an identity arrow for each object in the category. As you can easily see, virtually any mathematical (and, by extension, physical) structure can be modeled as a category. For instance, your objects can be groups, with arrows representing group morphisms. Or you could model any discrete physical system’s evolution, by taking as objects its states and as arrows its transitions. With the latter example, we begin to see how CT relates to physics: think of arrows as processes or object transformations. CT somehow captures their essence: one moves from the basic category definition to an exploration of morphisms (called functors) between categories, and from here to constructing categories whose objects are categories, with functors as arrows. Iterate and you’re soon talking about processes consisting of processes consisting of processes consisting of… But of course that’s not all: one also studies morphisms between functors (called natural transformations), and obtains a very precise statement of how any object is equivalent to the set its transformations: objects just disappear on behalf of their relationships! This result, called Yoneda’s Lemma, is beautifully presented in Barry Mazur’s When is a thing equal to some other thing? and further explored in Brown and Porter’s Category Theory: an abstract setting for analogy and comparison (both of them being also an excellent introduction to CT). Although i’m not privy with the applications of CT to LQG and other ‘relationist’ approaches, i think that this blurring of objects in favor of their transformations are at the heart of CT’s appeal to some physicists.

Even discounting its (highly tentative) applications to physics, CT raises deep issues in pure mathematics and philosophy: for instance, this very interesting entry of the Stanford Encyclopedia of Philosophy ends its introduction with these words:

Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called “categorical doctrines” at the syntactic, proof-theoretic, and semantic levels. Category theory is an alternative to set theory as a foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon.

All that said, i’m a bit skeptical on the usefulness of CT as a means of synthesizing new laws. For what i’ve seen, it is very powerful for analyzing and discovering properties of already known structures, but i have yet to find a convincing case where it is used to reveal brand new laws. For instance, CT is used in Axiomatic QFT to formalize the theory axioms (see, e.g., Buchholz’s Algebraic quantum field theory: a status report or any other of the review papers on Local Quantum Physics), but it’s not clear to me whether CT is, in this context, more than a convenient mathematical language. Let me, however, rush to say that i’m by no means an expert, and that reading Baez’s writings above may well change my mind!

At any rate, i do like Category Theory, if only for its mathematical beauty, and i think it’s very worthwhile spending a little time learning a bit about it if you are interested in mathematical physics (see here for more of my recommended readings).

Update: Over at A Neighborhood of Infinity, there’s a discussion about “two kinds of maths” (namely, structural and content-providing) which somehow coincides with my half-baked feelings about CT (and, as a bonus, provides a link to yet another proof of Yoneda’s lemma):

Most branches of mathematics have a mixture of the two types of theorem. Typically the structure theorems are used as a tool to discover content. In some sense the content is the end and the structure is the means. But category theory seems different in this regard – it seems to be mainly about structure. Every time I read category theory I see ever more ingenious and abstract tools but I don’t really see what I think of as content. What’s hard in category theory is getting your head around the definitions and often the theorems are trivial consequences of these. For example consider the Yoneda lemma as described here. This is a pretty challenging result for beginning category theorists. And yet these notes say: “once you have thoroughly understood the statement, you should find the proof straightforward”. This exactly fits with what I’m getting at: the theorem is an immediate consequence of the definitions and the proof essentially shows that the machinery does what you’d expect it to do.

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