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

Abstract

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 \(\mathsf{Cat}//X\) to \(\mathsf{Cat}\) is topological if and only if \(X\) is large-complete. Moreover, we provide conditions for \(\mathsf{Cat}//X\) to be complete, cocomplete, extensive and cartesian closed. We analyze descent in \(\mathsf{Cat}//X\) 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.