Large-structure tools like toposes and derived categories in cohomology never go far from arithmetic in practice, yet existing formalizations are stronger than ZFC. We formalize the practical insight by grounding the entire toolkit of EGA and SGA at the level of finite order arithmetic. Grothendieck pre-empted many set theoretic issues in cohomology by positing a universe: “a set ‘large enough’ that the habitual operations of set theory do not go outside it” (SGA 1 VI.1 p. 146). His universes… CONTINUE READING