Emily Riehl sees similarities between the viola, which she grew up playing, and the mathematical field of higher category theory, in which she is currently a leading participant. She thinks of the two as the “glue” of their respective domains; just as the viola creates a richer orchestral sound, “there’s a sense in which category theory makes mathematics deeper,” she said.

The categorical perspective emerged in mathematics in 1945 when Samuel Eilenberg and Saunders Mac Lane published their radical paper, “General Theory of Natural Equivalences.” It proposed a deeply unconventional idea, arguing that mathematics needed to do away with the equal sign, and the whole simplistic notion of equality, and replace it with the deeper, more sophisticated idea of “equivalence.”

Instead of calling two things exactly equal, Eilenberg and Mac Lane urged mathematicians to embrace sophisticated new mathematical structures that captured the many ways in which two things might be the same, or equivalent.

The proposal was received with skepticism. Riehl, an associate professor of mathematics at Johns Hopkins University, says that many early readers of Eilenberg and Mac Lane’s work wondered, “Is this even mathematics?”

But the doubts didn’t persist for long. Today, category theory and its next-generation version, higher category theory, are central to many fields of math, from algebraic geometry to mathematical physics. In those areas, Riehl said, “I think it would be impossible to describe the kind of basic objects of study without categorical language.”

In higher category theory, mathematicians like Riehl don’t just think about ways in which two objects are equivalent. They also think about equivalences between equivalences, and equivalences between equivalences between equivalences, and so on upward in a never-ending tower of relationships. These equivalence relationships are captured in an abstract mathematical object called an infinity category.

Riehl is currently working to expand the usefulness of infinity categories in mathematics. She and her longtime collaborator, Dominic Verity of Macquarie University in Australia, are nearly finished with a book that rewrites the massive, highly technical foundations of the field. Riehl hopes that their reframing will make higher category theory accessible to more mathematicians while offering new insights into why the mathematics of equivalence is so powerful. In part because of this work, Riehl was recently announced as the winner of the 2021 AWM-Joan and Joseph Birman Research Prize in Topology and Geometry.

Quanta Magazine recently spoke with Riehl about her forthcoming book as well as her years playing high-level Australian rules football, how her identity as a queer woman has been “protective” in mathematics, and the obligation mathematicians have to address the social justice issues of the moment. This interview is based on phone and email interviews and has been condensed and edited for clarity.