Lax comma categories: cartesian closedness, extensivity, topologicity, and descent

Maria Manuel Clementino, Fernando Lucatelli Nunes and Rui Prezado

We investigate the properties of lax comma categories over a base category X, focusing on topologicity, extensivity, cartesian closedness, and descent. We establish that the forgetful functor from Cat//X to Cat is topological if and only if X is large-complete. Moreover, we provide conditions for Cat//X to be complete, cocomplete, extensive and cartesian closed. We analyze descent in Cat//X and identify necessary conditions for effective descent morphisms. Our findings contribute to the literature on lax comma categories and provide a foundation for further research in 2-dimensional Janelidze's Galois theory.

Keywords: lax comma categories, Grothendieck descent theory, Galois theory, 2-dimensional category theory, topological functor, effective descent morphism, cartesian closed category, exponentiability

2020 MSC: 18N10, 18N15, 18A05, 18A22, 18A40

Theory and Applications of Categories, Vol. 41, 2024, No. 16, pp 516-530.

Published 2024-05-17.

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