Double categories and base change in homotopy theory

We begin a combination of three existing theories: abstract homotopy theory, monoidal category theory, and indexed category theory. The first studies categories with weak equivalences, such as topological spaces and chain complexes, and their homotopy categories and derived functors. The second studies categories with a product, such as the smash product of based spaces or the tensor product of modules. The third studies families of categories indexed by objects of a base category, such as modules indexed by rings or bundles indexed by their base spaces.;Many situations in mathematics include two, or all three, of these structures. Based spaces or spectra with their smash products, and chain complexes with their tensor product, are examples of monoidal homotopy theories. Modules over rings, sheaves over spaces, and diagrams over enriched categories are examples of indexed monoidal categories. Sectioned spaces, parametrized spectra, chain complexes of sheaves, and homotopical diagrams are examples of indexed homotopy theories---and most of them also have a monoidal structure. Despite this ubiquity, however, 'indexed monoidal abstract homotopy theory' does not seem to have been widely studied.;Interesting phenomena already appear when we combine our three theories in pairs. Monoidal abstract homotopy theory is fairly well studied. But indexed abstract homotopy theory has to deal with derived base change functors, frequently including composites of left and right derived functors, which the standard technology is often insufficient to deal with. And in addition to a naive notion of indexed monoidal category, there is another sort of 'indexed monoidal structure' analogous to the tensor product of bimodules.;We deal with both of these subtleties by using double categories. Firstly, we show that passage to derived functors is a 'double pseudofunctor,' providing a general framework to compare composites of left and right derived functors. This is also useful in monoidal abstract homotopy theory, once we move beyond monoidal homotopical categories to consider monoidal derived functors and derived enriched categories. Secondly, we show that bimodule-like tensor products can be modeled by a special sort of double category which we call a framed bicategory. In good cases, a naive indexed monoidal category gives rise to a framed bicategory, but not all framed bicategories arise in this way; another important construction consists of monoids and bimodules in some other framed bicategory.;Finally, we combine all three theories to obtain indexed monoidal abstract homotopy theory, and in particular a homotopical theory of framed bicategories. This is not hard for framed bicategories arising from naive indexed monoidal categories, but for those composed of monoids and bimodules, we also need a bar construction to compose bimodules homotopically. We give a detailed analysis for distributors (bimodules between enriched categories), with applications to homotopy limits; the general case is left for future work.