LAX COMMA CATEGORIES

• DOI: 10.48550/arxiv.2405.03773
• Publication Date: 2024-01-01
• Authors: Clementino, Maria...; Nunes, Fernando L...; Prezado, Rui
• Keywords: 18N10, 18N15, 18A05, 18A22, 18A40; 2-dimensional category theory; Grothendieck descent theory; cartesian closed category; Galois theory; topological functor; FOS: Mathematics; effective descent morphism; Mathematics - Category Theory; Category Theory (math.CT); exponentiability; lax comma categories

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.<br/>13 pages<br/>