Data types with symmetries and polynomial functors over groupoids

Polynomial functors 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. In this talk I will explain how an upgrade of the theory from sets to groupoids is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (cf. Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez-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 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 2-truncated intensional type theory. For a fullfledged type theory, locally cartesian closed \infty-categories seem to be needed. The theory of these is being developed by D.Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of \infty-results obtained in joint work with Gepner. Details will appear elsewhere.

💡 Research Summary

The paper develops a systematic extension of polynomial functors from the familiar setting of sets to the richer world of groupoids, thereby providing a natural framework for handling data types that possess intrinsic symmetries. In the classical “container” viewpoint, a polynomial functor is described by a pair of sets: a shape set (S) and, for each shape (s\in S), a position set (P_s). This captures many familiar data structures—lists, trees, options—by interpreting an element of the functor as a choice of shape together with a filling of its positions. However, this set‑based formulation cannot express the presence of non‑trivial automorphisms of shapes or positions, which are essential when the data type carries symmetry (for example, unordered collections, graphs up to isomorphism, or quantum field‑theoretic Feynman diagrams).

Groupoids, being categories in which every morphism is invertible, encode both objects and their symmetry groups in a single structure. By redefining polynomial functors over groupoids, the author introduces “symmetry containers”: the shape and position data are now groupoids, and the functor’s action respects the homotopical structure (i.e., it is defined via homotopy pullbacks, slices, and colimits). The crucial technical observation is that, with the appropriate homotopical machinery, the calculus of polynomial functors in the groupoid case mirrors the set case verbatim: homotopy slices replace ordinary slices, homotopy pullbacks replace ordinary pullbacks, and homotopy colimits replace ordinary colimits. Consequently, all familiar constructions—composition, substitution, differentiation—carry over unchanged, while the additional symmetry information is automatically retained.

The paper then shows how this unified perspective subsumes several previously distinct notions:

• Quotient containers (Abbott et al.) – containers modulo an equivalence relation – appear as a special case where the shape groupoid is a set equipped with a group action.
• Species and analytic functors (Joyal) – combinatorial structures counted up to isomorphism – are recovered by taking finite groupoids as shapes and positions.
• Stuff types (Baez‑Dolan) – “stuff” attached to objects of a groupoid – correspond to polynomial functors whose position groupoid is a discrete fibration over the shape groupoid.

Moreover, the multivariate setting naturally accommodates relations, spans, multispans, and “stuff operators”, thereby linking the theory to higher‑categorical constructions such as bicategories of spans and double categories.

A particularly compelling application is presented in the context of quantum field theory. Feynman diagrams form highly symmetric tree‑like graphs; their automorphism groups encode physical redundancies that must be quotiented out in perturbative expansions. By modeling such diagrams as objects of a suitable groupoid and applying a polynomial functor over that groupoid, one obtains a combinatorial description that automatically factors out symmetries. This yields cleaner enumeration formulas and clarifies the role of symmetry factors in the perturbative series, all without leaving the purely combinatorial realm.

The author also discusses the categorical semantics of the theory. Locally cartesian closed (LCC) 2‑categories provide a model for 2‑truncated intensional type theory, while fully homotopical models require LCC (\infty)-categories. Ongoing joint work with David Gepner is developing the (\infty)-categorical analogue, leading to “homotopical species” – a higher‑dimensional generalisation of Joyal’s species where the underlying indexing objects are (\infty)-groupoids. The results presented in the talk are essentially 2‑truncations of those (\infty)-level theorems, indicating that the full power of the theory lies in the homotopy‑coherent setting.

In summary, by upgrading polynomial functors to the groupoid level, the paper furnishes a unifying, homotopy‑theoretic language for data types with symmetry, bridges several strands of combinatorial and categorical research, and opens a pathway toward homotopical semantics for advanced type theories and quantum‑field‑theoretic combinatorics.