  Tannaka duality and its extensions by Lurie, Sch\"appi et al. reveal that
many schemes as well as algebraic stacks may be identified with their tensor
categories of quasi-coherent sheaves. In this thesis we study constructions of
cocomplete tensor categories (resp. cocontinuous tensor functors) which usually
correspond to constructions of schemes (resp. their morphisms) in the case of
quasi-coherent sheaves. This means to globalize the usual local-global
algebraic geometry. For this we first have to develop basic commutative algebra
in an arbitrary cocomplete tensor category. We then discuss tensor categorical
globalizations of affine morphisms, projective morphisms, immersions, classical
projective embeddings (Segre, Pl\"ucker, Veronese), blow-ups, fiber products,
classifying stacks and finally tangent bundles. It turns out that the universal
properties of several moduli spaces or stacks translate to the corresponding
tensor categories.
