On the Art of Giving the Same Name to Different Things

Mathematics has developed an increasingly “higher dimensional” point of view of when different things deserve the same name, categorifying the traditional logical notion of equality to isomorphism (from Greek isos “equal” and morphe “form” or “shape”) and equivalence (from Latin aequus “equal” and valere “be well, be worth”). In practice, mathematicians tend to become more flexible in determining when different things deserve the same name as those things become more complicated, as measured by the dimensions of the categories to which they belong. Unfortunately, these pervasive notions of sameness no longer satisfy Leibniz’s identity of indiscernibles — the assertion that two objects are identical just when they share the same properties — essentially because the traditional set theoretical foundations of mathematics make it too easy to formulate “evil” statements. However, in a new proposed foundation system there are common rules that govern the meaning of identity for mathematical objects of any type that allow one to “transport” information along any identification. Moreover, as a consequence of Voevodsky’s univalence axiom, these identity types are faithful to the meanings of sameness that have emerged from centuries of mathematical practice.

Emily Riehl is Professor of Mathematics at Johns Hopkins University, working on higher category theory, abstract homotopy theory, and homotopy type theory. She studied at Harvard and Cambridge Universities, earned her Ph.D. at the University of Chicago, and was a Benjamin Pierce and NSF postdoctoral fellow at Harvard University. She has published over thirty papers and written three books: Categorical Homotopy Theory (Cambridge 2014), Category Theory in Context (Dover 2016), and Elements of ∞-Category Theory (Cambridge 2022, joint with Dominic Verity). She was recently elected as a member at large of the Council of the American Mathematical Society. In addition to her research, Dr. Riehl is active in promoting access to the world of mathematics through popular writing and in interviews and podcasts. She was also a co-founder of Spectra: the Association for LGBT Mathematicians.

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