2025 10 24 https://julesh.com/forest/000D/ 000D /forest/000D/ Costrong profunctors Let be a monoidal category and let and be -actegories. A costrong profunctor or Tambara comodule is a functor equipped with a natural transformation called the costrength, The dual concept of costrong profunctors is strong profunctors. References Context Backlinks 2025 10 24 https://julesh.com/forest/000C/ 000C /forest/000C/ Strong profunctors Let be a monoidal category and let and be -actegories. A strong profunctor or Tambara module is a functor equipped with a natural transformation called the strength, The dual of strong profunctors is costrong profunctors. Related 2025 10 24 https://julesh.com/forest/000C/ 000C /forest/000C/ Strong profunctors Let be a monoidal category and let and be -actegories. A strong profunctor or Tambara module is a functor equipped with a natural transformation called the strength, The dual of strong profunctors is costrong profunctors. 2025 10 21 https://julesh.com/forest/0004/ 0004 /forest/0004/ Actegories Let be a monoidal category and a category. An action of on , also called an -actegory structure on , is given by a functor , together with the following coherence data: A natural transformation called the unitor A natural transformation called the actor Coherence equations 2025 10 21 https://julesh.com/forest/0006/ 0006 /forest/0006/ From strong monoidal functors to actegories Let and be monoidal categories, and let be a strong monoidal functor. Then we can define an -actegory structure on by . 2025 10 21 https://julesh.com/forest/0007/ 0007 /forest/0007/ The golden theorem of actegories Let be a symmetric monoidal category. Every -actegory that is also a symmetric monoidal category in a compatible way is necessarily defined by a strong symmetric monoidal functor , up to natural isomorphism. Specifically, if is a symmetric monoidal actegory then where is defined by . This means that in the vast majority of cases, actegories are redundant and strong symmetric monoidal functors suffice. Reference: Actegories for the working amthematician, section 5 2025 10 21 https://julesh.com/forest/0008/ 0008 /forest/0008/ From actegories to locally graded categories Let be a monoidal category and an -actegory. We construct a locally graded category called , which we regard as a Para construction, as follows: The objects of are objects of For objects of , the hom-object is given by The identity morphism in is given by the unitor The composition of and is given by Contributions