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

TL;DR

This paper studies lax comma categories over a base category, examining their topologicity, extensivity, cartesian closedness, and descent properties, and establishes conditions for these properties to hold.

Contribution

It provides new conditions under which lax comma categories are complete, cocomplete, extensive, and cartesian closed, advancing understanding of their structural properties.

Findings

The forgetful functor from $ ext{Cat}//X$ to $ ext{Cat}$ is topological iff $X$ is large-complete.

Conditions for $ ext{Cat}//X$ to be complete, cocomplete, extensive, and cartesian closed are established.

Analysis of descent in $ ext{Cat}//X$ identifies necessary conditions for effective descent morphisms.

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.