Chapter 60: Limits, Sites, and the Presheaf Topos

the relevant chapter earned the arrows of Cat_τ from the monoid of τ-holomorphic programs, and the relevant chapter verified the category axioms (the relevant theorem, I.T22) and constructed the Yoneda embedding (the relevant definition, I.D54). This chapter completes the categorical infrastructure by constructing finite limits (the relevant definition, I.D55), equipping Cat_τ with a Grothendieck topology via the primorial coverage (the relevant definition, I.D56), and forming the presheaf topos PSh(Cat_τ) (the relevant definition, I.D57). The primorial coverage categorifies the CRT decomposition (Section [subsec:ch30-crt], I.D29): at each primorial stage M_k, the Chinese Remainder Theorem gives a covering family that resolves an object into its prime-factor components. The key results are that PSh(Cat_τ) is a Grothendieck topos (the relevant theorem, I.T24) and that it is countable (Proposition [prop:countable-topos], I.P26) — both consequences of the smallness and countability of Cat_τ (Proposition [prop:set-countable], I.P12). The earned topos of the relevant chapter will be carved from this presheaf topos by imposing the sheaf condition with respect to the primorial coverage.