## Perspectives on V

### Richard Garner – 1 February 2023

In the 1970's, Graham Higman gave a description of Thompson's group V as
the automorphism group of the free Jonsson--Tarski algebra with one
generator, where a Jonsson--Tarski algebra is a set X endowed with an
isomorphism to X * X.

Higman's original approach was to calculate this free algebra
explicitly, and via some combinatorics present its automorphism group V
in a more tractable way. Subsequently, it has become popular to describe
V in terms of certain maps between ideals of the free monoid on two
generators {l,r}, and this requires further combinatorics to equate with
Higman's description. In this talk, we explain why the combinatorics are
unneccessary: we can get the ideal-theoretic description almost
trivially from Freyd's description of the category of Jonsson--Tarski
algebras as a category of sheaves.

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