Category Theory

Applied Category Theory - The Emerging Science of Compositionality





(via Brian McKenna)

Seven Sketches in Compositionality: An Invitation to Applied Category Theory

Seven Sketches in Compositionality: An Invitation to Applied Category Theory

2018 by Brendan Fong and David I. Spivak


Comprehending Ringads

Comprehending Ringads

2016 by Jeremy Gibbons

Ringad comprehensions represent a convenient notation for expressing database queries. The ringad structure alone does not provide a good explanation or an efficient implementation of relational joins; but by allowing heterogeneous comprehensions, involving both bag and indexed table ringads, we show how to accommodate these too.
Indexed/parametric/graded monads are the key (read the paper to understand the pun).

Databases from finite categories

Spivak and Kent (2011). Ologs: A categorical framework for knowledge representation:

In this paper we introduce the olog, or ontology log, a category-theoretic model for knowledge representation (KR). Grounded in formal mathematics, ologs can be rigorously formulated and cross-compared in ways that other KR models (such as semantic networks) cannot. An olog is similar to a relational database schema; in fact an olog can serve as a data repository if desired. Unlike database schemas, which are generally difficult to create or modify, ologs are designed to be user-friendly enough that authoring or reconfiguring an olog is a matter of course rather than a difficult chore. It is hoped that learning to author ologs is much simpler than learning a database definition language, despite their similarity. We describe ologs carefully and illustrate with many examples. As an application we show that any primitive recursive function can be described by an olog. We also show that ologs can be aligned or connected together into a larger network using functors. The various methods of information flow and institutions can then be used to integrate local and global world-views. We finish by providing several different avenues for future research.

Ologs are essentially RDFs extended to encompass commuting diagrams, so a visual little language. The paper talks about how database schema can automatically be extracted from ologs.

Philip Wadler: Category Theory for the Working Hacker



Cakes, Custard, and Category Theory

Eugenia Cheng's new popular coscience book is out, in the U.K. under the title Cakes, Custard and Category Theory: Easy recipes for understanding complex maths, and in the U.S. under the title How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics:

Most people imagine maths is something like a slow cooker: very useful, but pretty limited in what it can do. Maths, though, isn't just a tool for solving a specific problem - and it's definitely not something to be afraid of. Whether you're a maths glutton or have forgotten how long division works (or never really knew in the first place), the chances are you've missed what really makes maths exciting. Calling on a baker's dozen of entertaining, puzzling examples and mathematically illuminating culinary analogies - including chocolate brownies, iterated Battenberg cakes, sandwich sandwiches, Yorkshire puddings and Möbius bagels - brilliant young academic and mathematical crusader Eugenia Cheng is here to tell us why we should all love maths.

From simple numeracy to category theory ('the mathematics of mathematics'), Cheng takes us through the joys of the mathematical world. Packed with recipes, puzzles to surprise and delight even the innumerate, Cake, Custard & Category Theory will whet the appetite of maths whizzes and arithmophobes alike. (Not to mention aspiring cooks: did you know you can use that slow cooker to make clotted cream?) This is maths at its absolute tastiest.

Cheng, one of the Catsters, gives a guided tour of mathematical thinking and research activities, and through the core philosophy underlying category theory. This is the kind of book you can give to your grandma and grandpa so they can boast to their friends what her grandchildren are doing (and bake you a nice dessert when you come and visit :) ). A pleasant weekend reading.

Conservation laws for free!




Seemingly impossible programs


Programming language semantics is typically applied to
prove compiler correctness and allow (manual or automatic) program
verification. Certain kinds of semantics can also be applied to
discover programs that one wouldn't have otherwise thought of. This is
the case, in particular, for semantics that incorporate topological
compact) do have decidable equality, contradicting perhaps popular

and introduce the theory behind their derivation and working. In
optimally plays sequential games of unbounded length, (ii) implements
the Tychonoff Theorem from topology (and builds finite-time search
from proof theory (and allows us to extract programs from classical
examples in the languages Haskell and Agda.


Luca Cardelli Festschrift


Luca Cardelli has made exceptional contributions to the world of programming
many areas: software; language design, including experimental languages;
programming language foundations; and the interaction of programming languages
and his wide impact.
they will all interest Luca.

Hopefully the videos will be posted soon.

Dependently-Typed Metaprogramming (in Agda)





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