David Corfield
Narrative notes

Contents

Contents

The typical descriptive unit of great scientific achievements is not an isolated hypothesis but rather a research programme.

Translated into mathematics.

Narratives category theorist tell themselves

Category theory is an attempt to provide general tools for all of mathematics. Its history, dating back to the 1940s, is characterised by ambitious attempts to reformulate branches of mathematics and even mathematics as a whole. It has since moved on to influence theoretical computer science and mathematical physics. Resistance to this movement over the years has taken the form of accusations of engaging in abstraction for abstraction’s sake. Here we explore the role of narrative in forming the self-identity of category theorists.

A self-narrative. Of course, any practitioner of any discipline identifies with a narrative to frame what they take themselves to me doing, to set goals, standards, etc.

“my life is always embedded in the story of those communities from which I derive my identity […][t]he possession of an historical identity and the possession of a social identity coincide” (After Virtue, 1981, p. 221).

Years ago on this blog, I was exploring the way narrative may be used to give direction to a tradition of intellectual enquiry. This led eventually to a book chapter, Narrative and the Rationality of Mathematical Practice in B. Mazur and A. Doxiades (eds), Circles Disturbed, Princeton, 2012.

Someone reading this piece recently has invited to me to speak at a workshop, Narrative and mathematical argument, listed here. Reflecting on what I might discuss, I settled on the following:

Years of hanging around this place have given me plenty to talk about, but perhaps people have some insights they’d care to share.

I know some would rather avoid direct identification as a category theorist, instead describing themselves indirectly as a mathematician/mathematical physicist/computer scientist, etc. who does research in/looks to use the tools of category theory. But as the kind of person who shows up to CT2019, Applied Category Theory 2019 or SYCO 4, do you have narratival ways of thinking about your longer term research path? This might be in relation to achievements historical figures of the tradition (Mac Lane, Kan, Grothendieck, Lawvere, etc.), things your supervisor told you, or perhaps in relation to alternative ways of doing research in your area.

Isolating the essence of an idea manifested across different situations; providing a common language; guidance for theory construction.

Tom’s Basic Category Theory:

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level. How is the lowest common multiple of two numbers like the direct sum of two vector spaces? What do discrete topological spaces, free groups, and fields of fractions have in common? We will discover answers to these and many similar questions, seeing patterns in mathematics that you may never have seen before.

Or Emily’s Category theory in context:

Atiyah described mathematics as the “science of analogy.” In this vein, the purview of category theory is mathematical analogy. Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another. The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline.

Category theory is rather interesting for being divisive, making what some take as overblown claims.

Used in many branches without it being much of a big deal. You can’t work in a large number of areas without using it, but you will probably not call yourself a category theorist.

What is divisive is the zeal. Abstract nonsense. Abstraction for abstraction sake.

Diaspora into computer science departments.

Founding act

1945 First publication by Eilenberg Mac Lane

1952 Put to use in algebraic topology text, Eilenberg and Steenrod.

1958 Daniel Kan adjoint functors

1960s Great revision of algebraic geometry by Grothendieck

1971 Categories for the working mathematician.

1970s Categorical logic: Lambek, Lawvere, etc.

Higher categories. Jacob Lurie.

Rise of ‘applied category theory’

Yoneda

The identity of an entity of a kind is determined by the pattern of the arrows probing it.

Better a nice category with ‘odd’ entities, than a category of only nice objects.

Worries about pointlessness

Norman Steenrod to coin the term abstract nonsense or general abstract nonsense for it. It is being used as in “This property is not specific to this context, it already follows from general abstract nonsense”. Peter Freyd expressed a similar feeling by his witticism:

“Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial.”

But abstract nonsense still tends to meet with some resistance. In the preface of his 1965 book Theory of Categories Barry Mitchell writes:

A number of sophisticated people tend to disparage category theory as consistently as others disparage certain kinds of classical music. When obliged to speak of a category they do so in an apologetic tone, similar to the way some say, “It was a gift – I’ve never even played it” when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisite that the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind.

Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.

An Introduction to Algebraic Topology

Joseph J. Rotman - 2013

Having illustrated the technique, let us now give the appropriate setting for algebraic topology.

Definition. A category C…

Mathematical abstraction (esp. categorical air guitar playing) is not a goal in itself: Bart jacobs,

Miles Reid described the study of category theory for its own sake as

surely one of the most sterile of all intellectual pursuits

(Undergraduate Algebraic Geometry, p.116).

‘Working’ in the title

Terminology

Category (Kant), functor (Carnap), natural transformation.

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.

Application

  1. Physics
  2. Computer science
  3. ‘Applied’ category theory

Needed to develop the twisted equivariant differential cohomology theories of modern gauge field theory.

Computational trinitarianism.

In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?

As a (slowly) recovering category-phobe, allow me to suggest that you change the way you think of category theory. Specifically, don’t think of category theory as a “theory”. A theory in mathematics generally consists of three components: a collection of related definitions, a collection of nontrivial theorems about the objects defined, and a collection of interesting examples to which the theorems apply. To learn a theory is to understand the proofs of the main theorems and how to apply them to the examples.

Category theory is different: there is an incredibly rich supply of definitions of examples, but very few theorems compared to other established “theories” like group theory or algebraic topology. Moreover, the proofs of the theorems are almost trivial (the Yoneda lemma is one of the most important theorems in category theory and it is not even called a “theorem”). A consequence of this is that you don’t have to sit around reading a category theory book before you make contact with the language of categories: the very act of understanding how people express results from “ordinary” mathematics in the language of categories and functors is learning category theory.

So now I’ll try to answer your question. It is possible to work in nearly any area of pure mathematics without much category theory, and most areas have people everywhere on the category theory spectrum (with the possible exception of algebraic geometry, wherein the language of derived functors is basically built into the foundations). Analysis in particular seems to be somewhat resistant (but not immune) to categorification, and if you are really committed to avoiding categories then you might consider exploring the more analytic aspects of what interests you (e.g. geometric PDE’s or analytic number theory). https://mathoverflow.net/q/165405/447

I was drawn to category theory (around when I was 18) because it seemed to have answers to a lot of questions I had, like “what is the commonality between a quotient group in algebra and a quotient space in topology?” or “what is duality?”, etc. And I think also in the beginning that I was enthralled by the generality and abstraction. But what I mainly want to emphasize now is that if category theory still has a fearsome reputation, then it probably shouldn’t: people should think of the inner techniques as no more of a big deal than a lot of things in algebra.

Last revised on September 26, 2019 at 01:38:17. See the history of this page for a list of all contributions to it.