On Lax Monoidal Functors
Published:
What is the difference between
- a lax monoidal functor
- a monoid in a Day-monoidal category
- a morphism of lax-algebras for the free monoid 2-monad, and
- a codistributive law with the tensor product?
Blog posts
2025
2023
Bisimulations, Equality and Traces
Published:
Strong bisimulation for CCS is the preferred equivalence method in concurrency because it relates less programs than trace equality. However, the reality is that is strong bisimulation and trace equality ought to be regarded as equivalent. This is the essence behind proof assistant’s like (e.g.) Isabelle. So what is going here?
The mini Yoneda lemma for Type Theorists
Published:
I have managed to teach the Yoneda lemma to students who knew very little about category theory, here’s how you do it.
CCCs and the complete models of STLC
Published:
Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why.
2022
The Axiom of Choice in Type Theory
Published:
The Axiom of Choice (AC) is an axiom that states that the product of a family of non-empty sets is itself non-empty. This is a rather controversial axiom amongst mathematicians but in type theory this axiom is provable within the logic.
Inconsistencies in Cartesian Closed Categories with fixed-points
Published:
There is a very nice paper out there stating that “Any Cartesian Closed Category (CCC) with an initial object and a fixed-point operator is trivial”. Essentially this means that in languages like (e.g.) Haskell the empty type is not actually empty as it contains the non-terminating computation. Perhaps this is obvious, but here’s the categorical explanation.