We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category C, we apply this construction to a fibration whose fibers are the homotopy categories of the slice categories C/A, and we show that in the case C=Top, our construction applied to this fibration recovers the usual 2-category of spaces.
Keywords: Grothendieck fibrations, hyperdoctrine, 2-category
2020 MSC: Primary: 18D30. Secondary: 55U35
Theory and Applications of Categories, Vol. 35, 2020, No. 35, pp 1312-1378.