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tensor category

Many local notions and constructions for ${}_{R}{\mathsf{Mod}} $ over some $R\in\mathsf{CRing}$ can be carried out in an arbitrary (linear) cocomplete tensor category.

Classification of thick tensor ideals and localizing tensor ideals as the key to capturing the global structure of tensor categories; construction of novel support theories.

Important: the tensor-triangulated category.

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Tensor ideals

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Stable module categories

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Compact and compactly generated

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Links to this page

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tensor-triangulated category in Unsorted/tensor category
Day convolution

Equivalently, take the 2-category of cocomplete tensor categories $\mathsf{Cat}_{c\otimes}$, \begin{align*} \mathsf{Cat}_{c\otimes}( {\mathsf{Fun}}_{\widehat{\otimes}}(\mathsf{C}^{\operatorname{op}}, \mathsf{D}), ?) \cong \mathsf{Cat}_{c\otimes}(\mathsf{C}, ?) \times \mathsf{Cat}_{\otimes}(\mathsf{D}, ?) .\end{align*}