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July 20, 2024
What Is Entropy?
Posted by John Baez
I wrote a little book about entropy; here’s the current draft:
If you see typos and other mistakes, or have trouble understanding things, please let me know!
An alternative title would be 92 Tweets on Entropy, but people convinced me that title wouldn’t age well: in decade or two few people may remember what ‘tweets’ were.
Here is the foreword, which explains the basic idea.
July 15, 2024
Skew-Monoidal Categories: Logical and Graphical Calculi
Posted by Emily Riehl
guest post by Wilf Offord
One of the earliest and most well-studied definitions in “higher” category theory is that of a monoidal category. These have found ubiquitous applications in pure mathematics, physics, and computer science; from type theory to topological quantum field theory. The machine making them tick is MacLane’s coherence theorem: if anything deserves to be called “the fundamental theorem” of monoidal categories, it is this. As such, numerous other proofs have sprung up in recent years, complementing MacLane’s original one. One strategy with a particularly operational flavour uses rewriting systems: the morphisms of a free monoidal category are identified with normal forms for some rewriting system, which can take the form of a logical system as in (UVZ20,Oli23), or a diagrammatic calculus as in (WGZ22). In this post, we turn to skew-monoidal categories, which no longer satisfy a coherence theorem, but nonetheless can be better understood using rewriting methods.
July 12, 2024
Double Limits: A User’s Guide
Posted by Emily Riehl
Guest post by Matt Kukla and Tanjona Ralaivaosaona
Double limits capture the notion of limits in double categories. In ordinary category theory, a limit is the best way to construct new objects from a given collection of objects related in a certain way. Double limits, extend this idea to the richer structure of double categories. For each of the limits we can think of in an ordinary category, we can ask ourselves: how do these limits look in double categories?
July 10, 2024
An Operational Semantics of Simply-Typed Lambda Calculus With String Diagrams
Posted by Emily Riehl
guest post by Leonardo Luis Torres Villegas and Guillaume Sabbagh
Introduction
String diagrams are ubiquitous in applied category theory. They originate as a graphical notation for representing terms in monoidal categories and since their origins, they have been used not just as a tool for researchers to make reasoning easier but also to formalize and give algebraic semantics to previous graphical formalisms.
On the other hand, it is well known the relationship between simply typed lambda calculus and Cartesian Closed Categories(CCC) throughout Curry-Howard-Lambeck isomorphism. By adding the necessary notation for the extra structure of CCC, we could also represent terms of Cartesian Closed Categories using string diagrams. By mixing these two ideas, it is not crazy to think that if we represent terms of CCC with string diagrams, we should be able to represent computation using string diagrams. This is the goal of this blog, we will use string diagrams to represent simply-typed lambda calculus terms, and computation will be modeled by the idea of a sequence of rewriting steps of string diagrams (i.e. an operational semantics!).
July 9, 2024
Imprecise Probabilities: Towards a Categorical Perspective
Posted by Emily Riehl
guest post by Laura González-Bravo and Luis López
In this blog post for the Applied Category Theory Adjoint School 2024, we discuss some of the limitations that the measure-theoretic probability framework has in handling uncertainty and present some other formal approaches to modelling it. With this blog post, we would like to initiate ourselves into the study of imprecise probabilities from a mathematical perspective.