Data types with symmetries and polynomial functors over groupoids

Polynomial functors (over Set or other locally cartesian closed categories) are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect is often prominent and justifies the terminology. For example, Tambara’s theorem states that the category of finite polynomial functors is the Lawvere theory for commutative semirings [45], [18]. In this talk I will explain how an upgrade of the theory from sets to groupoids (or other locally cartesian closed 2-categories) is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (in the sense of Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez and Dolan. The multi-variate setting also includes relations and spans, multispans, and stuff operators. An attractive feature of this theory is that with the correct homotopical approach — homotopy slices, homotopy pullbacks, homotopy colimits, etc. — the groupoid case looks exactly like the set case. After some standard examples, I will illustrate the notion of data-types-with-symmetries with examples from perturbative quantum field theory, where the symmetries of complicated tree structures of graphs play a crucial role, and can be handled elegantly using polynomial functors over groupoids. (These examples, although beyond species, are purely combinatorial and can be appreciated without background in quantum field theory.) Locally cartesian closed 2-categories provide semantics for a 2-truncated version of Martin-Löf intensional type theory. For a fullfledged type theory, locally cartesian closed ∞-categories seem to be needed. The theory of these is being developed by David Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of ∞-results obtained in joint work with Gepner. Details will appear elsewhere.