I think that we have been quite successful in doing stuff with category theory, having made it widely used in quantum industry, by companies like IBM, Google, ourselves, and may others.
Photonic quantum computing now uses string diagrams as their main language.
Soon we'll report on an educational experiment we did over the summer, showing that category theory can actually make inaccessible things accessible, and likely will affect how physics is thought in secondary school.
We even got Jim Lambek into Forbes! 🙂 (the last place they expected to see him one of his sons told me)
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Senior Analyst, AI & Quantum Computing, Paul Smith-Goodson, dives in as Cambridge Quantum ("CQ") today announced a quantum computing first. It released an open-source toolkit and library for a Quantum Natural Language Processing (QNLP) toolkit called lambeq.
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Senior Analyst, AI & Quantum Computing, Paul Smith-Goodson, dives in as Quantinuum is an integrated software-hardware quantum computing company that uses trapped-ion for its compute technology. It recently released a significant update to its Lambeq open-source
Python library and toolkit.
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Honestly...naturality, adjunctions, Yoneda...have never played any direct role in anything we did.
Our starting point is that monoidal categories are the direct description of processes, something that first started in computer science, and we then carried over to physics, and then cognition and AI. I'd say that it enables an Heraclites+Schrodinger picture
of reality.
But there are of course a ton of other uses of CT, which are completely unrelated, if not orthogonal, and rely on
naturality, adjunctions, Yoneda...
Dogmatic slogans about what category theory is supposed to be have done nothing but damage to the field. It seems to me that the culture has become more healthy and tolerant now, with the growing use of CT in many areas outside of mathematics, where not ideology,
but use matters.
Cheers, Bob.
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"Yoneda showed that maps in any category can be
represented as natural transformations" (Lawvere & Schanuel,
Conceptual Mathematics, p. 378). Isn't this reason enough to think of
category theory as the theory of naturality?
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That would be like saying group theory is the theory of permutations
(because of the Cayley theorem).
Perhaps my little colloquium talk entitled
``The natural transformation in mathematics''
at
would be of some interest in this connexion. I am sure lots of us have
given similar talks. The goal of the paper considered the first in category
theory was to define natural transformation. That required functor, and
that required category.
Ross
