Thank you! I hoped for something like this. Now, the second part of my question: is there an example of something natural, an object of CAT/T, defined up to this equivalence ? That is, where it is important that you consider this equivalence rather then something stronger...But perhaps it is better stated as a separate question.

Todd Timple: I am interested in the relation between functors : two functors H:S-->T and H':S-->T are "weakly equivalent" iff there is a self-equivalence s:S-->S such that H and $H'\circ s$ are equivalent. I wanted to ask whether there is a higher category theoretic view on this relation between functors? And I wanted to see an example of a class of functors where any two functors are weakly equivalent for non-trivial reasons; in other words, a theorem claiming that functors with certain properties are necessarily weakly equivalent. I shall update the question accordingly..Thank you!

Todd Trimble: yes, that is what I mean....But I removed the sentence your comment refers to.

Thanks for your comments. Then I shall just remove the opening sentence: it seems rather to confuse than clarify....Does the second paragraph seems unclear as well ?

Thank you! Are there any references where people discuss this sort of questions in detail?

Will Sawin: what does it mean 'the structure factors through the automorphism' $\sigma$ ; in what sense ? Then I am still confused whether Hodge numbers are preserved...

maybe this is just a confusion/silliness on my part, I admit...

Jason Starr: what is the answer to that part ? Do you mean to say that $dim_C H^a(X,Ω^b_X)$ is a definition of these numbers in algebraic de Rham cohomology ?

@Mahdi: be aware that the paper gets updated from time to time so you need to check ihes.fr/~gromov/topics/recent.html to find the most recent version (10 july as of now)...also, at the very end itmentions a result of Esnault-Viehweg ("Such ”rank inequalities” are reminiscent of inequalities for spaces of sections and (cohomologies in general) of positive vector bundles such e.g. as in the Khovanski-Teissier theorem and in the Esnault-Viehweg proof of the sharpened Dyson-Roth lemma, but a direct link is yet to be found") which appears close to your interests.

Are there (online) samples of music generated with the category theoretic techniques from the book?