In some notes on derived stacks, in describing categories of fibrant objects, the author drops this parenthetical:
(Grothendieck said in his famous letter to Quillen that the choice of $\mathscr F$ is like the choice of a basis of a module.)
Question 1: Where can I find the quote that this refers to? What is the wording of this quote?
Question 2: Can someone explain this analogy? My understanding of model category related stuff isn't good enough for me to understand this.
I guess that 'the letter' is meant to be Grothendieck's Pursuing Stacks, which started as a letter to Quillen (as one can read in the document) and then evolved in a kind of book/diary addressed to the reader. I haven't found that precise quote (the djvu file is not searchable), but there is a section (125, starting at page 530) which deals with this topic. The big paragraph in section 532 answers your second question (and maybe the second part of the first one). It should be what you're looking for. It essentially says that what really matters are weak equivalences. There are tons of examples of different model structures on the same category (e.g. chain complexes) with the same weak equivalences, and it is well known (after the work of Quillen, Grothendieck, Dwyer, Kan...) that all these have exactly the same (higher) homotopy theory. I'd say that a choice of (co)fibrations amounts to a choice of resolutions to carry out explicit derived functor computations, but in homological algebra we learned even earlier that different kinds of resolutions lead to the same results. Grothendieck was actually interested in categories with weak equivalences not admitting suitable fibrations or cofibrations satisfying some of the usual axioms. His proposal was dealing with derivators.
You can also find this quote in the first paragrah of section 1 of the letter of Grothendieck to Thomason (a pdf can be found at http://webusers.imj-prg.fr/~georges.maltsiniotis/groth/ps/lettreder.pdf).
"Les constructions homotopiques essentielles sont indépendantes de toutes structures supplémentaires, tel un ensemble C de cofibrations ou un ensemble F de fibrations ou les deux à la fois. De telles structures supplémentaires sont utiles, dans la mesure où elles permettent d'expliciter les constructions essentielles, et d'en établir l'existence. Mais elles ne sont pas plus essentielles pour le sens intrinsèque des opérations (qu'elles auraient tendance plutôt à obscurcir, jusqu'à présent) que le choix d'une base plus ou moins arbitraire d'un module, en algèbre linéaire."
