## Monoidal extensivity and the comodule fibration

### Matthew Di Meglio – 18 May 2022

My talk will discuss ongoing work to develop the theory of comonoids and comodules in a symmetric monoidal category. In my talk, I will introduce the new notion of a monoidal-extensive symmetric monoidal fibration. This notion specialises to the familiar notion of extensive fibration when the monoidal product on the total category of the fibration is cartesian monoidal. I will also explain why the functor that sends a symmetric monoidal category with nice equalisers to its associated comodule fibration is right adjoint to the functor that sends a monoidal-extensive symmetric monoidal fibration with nice fibred equalisers to its underlying symmetric monoidal category.

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