### The Narratives Category Theorists Tell Themselves

#### Posted by David Corfield

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

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

The narratives category theorists 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.

Years of hanging out around this place have given me plenty to talk about, but perhaps people have some particular 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 story-like ways of thinking about your longer term research path? This might be in relation to achievements of historical figures of the tradition (Mac Lane, Kan, Grothendieck, Lawvere, etc.), things your supervisor told you, things you tell your students, or perhaps in relation to alternative ways of doing research in your area?

I imagine common themes will include: isolating the essence of an idea manifested across different situations; providing a common language; offering guidance for theory construction. These can be read from opening motivational paragraphs of books such as 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.

What other narrative themes are operating out there?

## Re: The Narratives Category Theorists Tell Themselves

Another reference for the same themes you mention could be Tom’s Perspective on Higher Category Theory.