Is the Equal Sign Overrated? Mathematicians Hash It Out

“You have this situation where [papers] either come back from journals with absurd referee reports that reflect deep misunderstandings, or they just take several years to publish,” Barwick said. “It can make people’s lives uncomfortable because an unpublished paper sitting on your website for years and years starts to look a little funny.”

Yet the biggest problem was not papers that went unpublished, but papers that used infinity categories and did get published — with errors.

Lurie’s books are the single, authoritative text on infinity categories. They are completely rigorous, but hard to completely grasp. They’re especially poorly suited to serving as reference manuals — it’s difficult to look up specific theorems, or to check that a specific application of infinity categories that one might encounter in someone else’s paper really works out.

“Most people working in this field have not read Lurie systematically,” said André Joyal, a mathematician at the University of Quebec in Montreal whose earlier work was a key ingredient in Lurie’s books. “It would take a lot of time and energy, so we sort of assume what’s in his book is correct because almost every time we check on something it is correct. Actually, all the time.”

The inaccessibility of Lurie’s books has led to an imprecision in some of the subsequent research based on them. Lurie’s books are hard to read, they’re hard to cite, and they’re hard to use to check other people’s work.

“There is a feeling of sloppiness around the general infinity categorical literature,” Zakharevich said.

Despite all its formalism, math is not meant to have sacred texts that only the priests can read. The field needs pamphlets as well as tomes, it needs interpretive writing in addition to original revelation. And right now, infinity category theory still exists largely as a few large books on the shelf.

“You can take the attitude that ‘Jacob tells you what to do, it’s fine,’” Rezk said. “Or you can take the attitude that ‘We don’t know how to present our subject well enough that people can pick it up and run with it.’”

Yet a few mathematicians have taken up the challenge of making infinity categories a technique that more people in their field can run with.

A User-Friendly Theory

In order to translate infinity categories into objects that could do real mathematical work, Lurie had to prove theorems about them. And to do that, he had to choose a landscape in which to create those proofs, just as someone doing geometry has to choose a coordinate system in which to work. Mathematicians refer to this as choosing a model.

Lurie developed infinity categories in the model of quasi-categories. Other mathematicians had previously developed infinity categories in different models. While those efforts were far less comprehensive than Lurie’s, they’re easier to work with in some situations. “Jacob picked a model and checked that everything worked in that model, but often that’s not the easiest model to work in,” Zakharevich said.

In geometry, mathematicians understand exactly how to move between coordinate systems. They’ve also proved that theorems proved in one setting work in the others.

With infinity categories, there are no such guarantees. Yet when mathematicians write papers using infinity categories, they often move breezily between models, assuming (but not proving) that their results carry over. “People don’t specify what they’re doing, and they switch between all these different models and say, ‘Oh, it’s all the same,’” Haine said. “But that’s not a proof.”

For the past six years, a pair of mathematicians have been trying to make those guarantees. Riehl and Dominic Verity, of Macquarie University in Australia, have been developing a way of describing infinity categories that moves beyond the difficulties created in previous model-specific frameworks. Their work, which builds on previous work by Barwick and others, has proved that many of the theorems in Higher Topos Theory hold regardless of which model you apply them in. They prove this compatibility in a fitting way: “We’re studying infinity categories whose objects are themselves these infinity categories,” Riehl said. “Category theory is kind of eating itself here.”

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