This paper is devoted to the study of natural algebraic operations on derived categories arising in algebraic geometry. Our main goal is to identify the category of sheaves on a fiber product with two algebraic constructions: on the one hand, the tensor product of the categories of sheaves on the factors, and on the other hand, the category of linear functors between the categories of sheaves on the factors (thereby realizing functors as integral transforms). Among the varied applications of our main results are the calculation of Drinfeld centers (and higher centers) of monoidal categories of sheaves and the construction of topological field theories.
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