The following is an English translation of André Joyal's letter concluding the private debate that he had organized (see this section for more details). The French original of the letter can be found here.

*30 January 2015 *

*On some works of Olivia Caramello** *

*The purpose of this letter is to recognize the originality of the
duality theorem of Olivia Caramello as published in her paper “Lattices
of theories”. *

*Caramello’s duality theorem is a result in the theory of classifying
toposes. It states that the quotients of a geometric theory are in
bijection with the subtoposes of the classifying topos of this theory.
Although considered “folkloric” by some experts, the result does not
appear in the literature. I had believed that one could directly deduce
it from the theory of classifying toposes of Makkai and Reyes. It is
only recently, in the context of a discussion with Caramello, Johnstone
and Lafforgue, that the latter attracted my attention to an aspect of
Caramello’s proof which I had missed, namely that the category
underlying the syntactic site of a geometric theory varies if one
changes the axioms of the theory. Surprised by this observation, I tried
to exhibit the “folkloric” proof that I thought I had of this theorem.
With my great astonishment, it took me a night of work to construct a
proof based on my knowledge of the subject, and the proof depended only
partially on Makkai-Reyes’ theory! It is true that Johnstone
subsequently sketched a simpler proof based on a paper by Tierney.
Nonetheless, I draw from this experience the following conclusions: (1)
that I had misread Caramello’s paper; (2) that the duality theorem may
falsely appear straightforward; (3) that the result is non-trivial; (4)
that the result is original, since it does not appear in the literature.*

*In her article “The unification of Mathematics via Topos Theory”,
Caramello proposes to interpret several invariants of a classifying
topos in terms of the theories that it classifies. When two geometric
theories have the same classifying topos (Morita-equivalence between
theories), a kind of bridge is created between these theories. For
example, the quotients of one are in bijection with the quotients of the
other by virtue of the duality theorem.*

*Olivia Caramello shows a remarkable talent in discovering and
bringing to light certain general aspects of mathematics. Her
methodology “toposes as bridges” is a vast extension of Felix Klein’s
Erlangen Program.**
*

*André Joyal
Emeritus Professor
UQAM
*