Conversations over the summer led to sharp improvements on my February ArXiv article on weak foundations for Grothendieck's number theory. I now formalize the entire Grothendieck tool kit at the strength of finite order arithmetic. In other words at the strength of ETCS (with NNO) or simple type theory with infinity. This includes even the largest scale structures: derived categories for duality, and the 2-category of geometric morphisms between toposes. This does not get as low as n-th order arithmetic for some fixed finite n, but is reasonably weak and yet quite general. This foundation is natural for the Grothendieck toolkit as a whole, while of course no single application uses the whole. Besides that the argument is now simpler, the key advances since my talks in July are: 1) a proof using only bounded separation that every sheaf of modules over any Grothedieck site has an infinite-length injective resolution. Of course the resolution can be 0 from some point on in some cases, but in general it cannot. 2) a conservative extension of Mac~Lane set theory with proper classes and collections of them, modelled on Takeuti's conservative higher order extension of PA. My ArXiv article has been revised. The new version re-titled "A finite order arithmetic foundation for cohomology," is on my web page at http://www.cwru.edu/artsci/phil/Derived_functor.pdf and is on the math arXiv as http://arxiv.org/abs/1102.1773 It is substantially shorter than the February version, while proving a stronger result. I take this as a case where it is easier to prove the right theorem than the wrong one. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]