A dagger category is a category equipped with a contravariant involution: it is a foundational construct in categorical quantum mechanics. The involution (dagger) is strict in the sense of being the identity-on-objects. However, there exist models with non-strict involution.

In this talk, I will present an alternative formulation of dagger categories as ‘inner product’ categories with a non-strict involution. This requires an intermediate step of considering unitary categories, a weaker version of a dagger category. Dagger, unitary, and inner product categories have a common structure which is an ‘achiral involution’ which is used to prove a three-way equivalence between the 2-categories of dagger categories, unitary categories, and inner product categories.

This work [1] is motivated by Paul-André Melliès’ work on chirality [2].

Pre-requisites: Basic category theory (definitions of functors, natural transformations, and 2-categories).

References:

[1]. Robin Cockett, Durgesh Kumar, and Priyaa Varshinee Srinivasan. "Unitary, Inner product, and Dagger categories." arXiv preprint arXiv:2603.27614 (2026).

[2]. Paul-André Melliès. Dialogue categories and chiralities. Publications of the Research Institute for Mathematical Sciences, 52:359–412, 2016.

The "graphical configuration space" X on a space consists of collections of points indexed by the vertices of a simple graph G, with the condition that points corresponding to adjacent vertices must be distinct. When X = Rd, the union of these spaces over all possible graphs forms an algebraic structure that we call the "little discs contractad", which extends the classical operad of little discs.
In this talk, I will provide a short overview of the theory of "contractads", focusing on the little discs contractad. I will also present various other basic examples of contractads and discuss some applications in enumerative combinatorics.
The talk is based on joint work with Denis Lyskov: https://arxiv.org/abs/2406.05909, https://arxiv.org/abs/2509.21255.