Tietze transformations for sites of toposes

Question

I have read that people think of a site as a presentation of the corresponding sheaf topos. For instance, on page 7 of this text by Caramello: as Grothendieck observed himself, a site of definition for a given topos can be seen as a sort of presentation of it by generators and relations (one can think of the objects of the category underlying the site as defining the generators, and of the arrows and covering sieves as defining the relations).

Now, for presentations of monoids (or groups), Tietze transformations are a very powerful tool. Namely, any two presentations of a given monoid can be related by a sequence of the following transformations:

• add or remove a definable generator,

• add or remove a derivable relation.

(You can think of finite presentations, although larger ones can be handled by considering transfinite sequences of transformations).

Thus my question: are there similar "Tietze transformations" on sites, such that any two sites having equivalent categories of sheaves can be related by Tietze transformations?

Answer 1

There are two observation to be made that limit a little this kind of analogy:

1) Site are indeed in some sense presentations, but an infinity theory (I mean with operation of infinite arity) something like the theory of ininitary pretopos, i.e. categories satisfying all of Giraud axioms except being presentable, but including being co-complete. Because of that you can expect to describe the transformation that you need to go from a site to an other by a finite sequence move involving a finite number of objects.

2) Site are presentations, but more in the sense of "generators + ideal", not "generators" + "generator of an ideal". So "adding definable relation" does not really makes sense, essentially any valid covering relation is already part of your topology. But you could consider the notion of topology generated by some class of covering of course.

Once this said, Grothendieck Comparison lemma (see https://ncatlab.org/nlab/show/comparison+lemma) can be used to do something relatively close to what you have in mind:

It says something which can roughly be translated by "if I add a bunch of new object to my site that can be covered by the previous one then I don't change the topos of sheaves " (This is very rough, I refer you to the precise statement of theorem 3.2 on the nLab page).

Any equivalence between toposes of sheaves on two different sites can always be obtained by two application of this lemma (one going up and the other going down). Essentially, by taking a subcategory of the topos of sheaves large enough to contains the two sites that you have.