Cartesian closed varieties (part 1)

Richard Garner – 30 September 2020

In a 1990 paper, Peter Johnstone gave various characterisations of the class of algebraic theories whose category of models in Set is cartesian closed. One of these was syntactic, and a little involved; another was semantic, characterising them as those obtained as the so-called "two-valued collapse" of a Grothendieck topos with a set of projective generators.

The objective of these talks (for there will be more than one) is to improve on Johnstone's characterisation results: in the end, we will prove to the semantic side that the category of cartesian closed algebraic theories is equivalent to the category of strongly zero-dimensional source-etale localic categories, and cofunctors between them.

However, the focus this time will be on the syntactic side: we show that Johnstone's slightly involved conditions can be simplified, proving in the end that cartesian closed theories are precisely which decompose as a kind of semidirect product of a unary theory and a hyperaffine one.