Abstract. The Grothendieck construction gives an equivalence between fibrations and indexed categories. We will begin with a review of the classical story. We will then lift this correspondence to two monoidal variants, a global version and a fibre-wise version. Under certain conditions these are equivalent, so one can transfer fibre-wise monoidal structures to the total category. We will give some examples demonstrating the utility of this construction in applied category theory and categorical algebra.

This, in turn, had its roots in our work on network models, a setup for the compositional design of networked systems:

• John Baez, John Foley, Joe Moeller and Blake Pollard, Network models.

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