Geodesic
Monoidal Grothendieck Construction
Theory and applications of categories, Tome 35 (2020), pp. 1159-1207.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely lax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigate the relation between this `global' monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
Classification : 18D30, 18M05
Keywords: Fibrations, indexed categories, Grothendieck construction, monoidal 2-categories, monoidal pseudofunctors 
@article{TAC_2020_35_a30,
     author = {Joe Moeller and Christina Vasilakopoulou},
     title = {Monoidal {Grothendieck} {Construction}},
     journal = {Theory and applications of categories},
     pages = {1159--1207},
     publisher = {mathdoc},
     volume = {35},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2020_35_a30/}
}
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Joe Moeller; Christina Vasilakopoulou. Monoidal Grothendieck Construction. Theory and applications of categories, Tome 35 (2020), pp. 1159-1207. http://geodesic.mathdoc.fr/item/TAC_2020_35_a30/