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On the tensor product of well generated dg categories

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arxiv 2002.09991 v2 pith:V63KNO7F submitted 2020-02-23 math.CT

classification math.CT
keywords categoriesproducttensorderivedgeneratedwellalpha-cocompletealpha-continuous
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We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of To\"en [26]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [21]. Given a regular cardinal alpha, we define and construct a tensor product of homotopically alpha-cocomplete dg categories and prove that the well generated tensor product of alpha-continuous derived dg categories (in the sense of [21]) is the alpha-continuous dg derived category of the homotopically alpha-cocomplete tensor product. In particular, this shows that the tensor product of well generated dg categories preserves alpha-compactness.

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  1. The category of necklaces is a test category

    math.CT 2026-07 accept novelty 6.0 of 10

    The category of necklaces is a test category, so its presheaf category is a model for homotopy types.

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