In today’s post I will introduce the notion of pivotal categories, for which the double dual of an object is equivalent to the object itself, and the concept of a balanced structure or “twist”, which then leads to the notion of ribbon categories, where an equality of morphisms is true if it holds as an …
In this post I will focus on the concept of dual objects and dualizability, which in the context of categorical quantum mechanics captures the notion of maximally entangle states (Bell states, etc). In the graphical calculus, this enables wires to bend backwards in time.Here are the snake equations, the fundamental axioms of dual objects:
Over the last months, I took a break from this blog project in order to write my Master’s dissertation on Disorder and topology in spin systems. I since started my PhD at Aalto University in Finland and am working hard on other projects. I finally found some time to able to return with this third …
In this second blog post, I introduce monoidal categories which are categories with extra data which allows for their description in terms of diagrams. Equations for a monoidal category are correct if their diagrammatic representation holds up to planar (2D) isotopy. But by introducing braidings, and forming braided monoidal categories, one also has a diagrammatic …
Monoidal category theory: An introduction to diagrams Read More »
This first post is a basic introduction to the notion of a category, and accompanying definitions: objects, morphisms, functors, equivalences, natural transformations, and universal properties! Hope you enjoy! Here are the commutative diagrams which encode the axioms of a category: