The Road from Circuits to Category Theory (qtcat2026)

String diagrams are often traced back to Richard Penrose's work in mathematical physics, but they also have an early and largely overlooked history in computer science through Günter Hotz. Following in the footsteps of Hotz, this talk will provide an introduction to string diagrams through the lens of circuit design. Rather than beginning with category theory, we start from circuits and their computational properties, and then work backwards to the categorical structures they describe. For example, we will see how modelling feedback motivates compact closed categories and reversibility suggests the structure of a monoidal dagger category. We will then see how the ability to copy data gives rise to Cartesian categories, with the no-cloning theorem identifying precisely why quantum circuits lack this structure. When circuits are no longer Cartesian, basic operations such as conditional statements become harder to reason about graphically. We conclude with a look at ongoing work on controlled monoidal categories and controlled string diagrams, which aim to restore graphical reasoning for conditional statements in non-Cartesian circuits. Licensed to the public under https://creativecommons.org/licenses/by/4.0/ about this event: https://pretalx.c3voc.de/qtcat-2026/talk/X8NKSX/