So u is a local isomorphism. We obtain another two functors to the category of topological spaces, which are equivalent to the analytic topology on A . U i , where I is a cofiltered category. Let W i be the pullback of V along t i . Y be the localization of Y at V .
U t is the inclusion .
Clearly U is disjoint with V as A is spatial. We prove that the fraction U is a complement of V . So W U is not initial. A mono is analytic iff it is a finitely copresentable fraction. X is a finitely copresentable fraction.