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A-module scheme algebra axiom bicategory Boolean canonical map canonically isomorphic cartesian closed categories category of sets characteristic map cohomology combinatory logic commutative group scheme complete lattice composition condition construction continuous functions continuous lattice cotangent complex defined definition deforming G denote diagram element endomorphism equation example fiber finite limits flat formula function spaces functor given Grothendieck group scheme hence induced topology injective spaces intuitionistic inverse image inverse limit isomorphism Lawvere left adjoint left exact Lemma Let F map of ringed mathematics monomorphism monotone natural numbers notations object obstruction obtained ontology open sets open subsets partially ordered set product topology projection proof Proposition prove pullback quasi-coherent recursive result retract right adjoint ringed topoi satisfies scheme G Se|s sequence set theory sheaf sheaves simplicial split stack subobjects subspace Suppose Theorem toposes torsor U-topos Yoneda functor