Hofmann-Streicher lifting of fibred categories

2025-04-13

AI Summary

This paper addresses the relativization problem of Hofmann–Streicher universe lifting in fibred categories. It tackles the lack of functoriality and stability of universe lifting in higher-order type theory and homotopical semantics. Methodologically, the paper introduces a novel relative lifting mechanism constructed via the right pseudo-adjoint of a 2-functor—generalizing Awodey’s nerve functor approach to arbitrary fibrations (p colon mathcal{A} o mathcal{B}). By integrating Grothendieck universes, elementary topoi, and fibred structure, it establishes a universal, stable, and functorial lifting framework that yields (U)-small discrete fibrations. The resulting construction systematically extends universes from the base category to the total category, providing a parameterized foundational tool with well-behaved categorical properties—thereby supporting homotopy type theory and sheaf semantics.

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Abstract

In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe $mathcal{U}$ from $mathbf{Set}$ into the category of $mathbf{Set}$-valued presheaves on a $mathcal{U}$-small category $B$. More recently, Awodey presented an elegant functorial analysis of this construction in terms of the categorical nerve, the right adjoint to the functor that takes a presheaf to its category of elements; in particular, the categorical nerve's functorial action on the universal $mathcal{U}$-small discrete fibration gives the generic family of $mathcal{U}$'s Hofmann-Streicher lifting. Inspired by Awodey's analysis, we define a relative version of Hofmann-Streicher lifting in terms of the right pseudo-adjoint to the 2-functor $mathbf{Fib}_{A} omathbf{Fib}_{B}$ given by postcomposition with a fibration $pcolon A o B$.

Problem

• Extends Hofmann-Streicher lifting to fibred categories
• Analyzes right pseudo-adjoint of 2-functor Fib_A → Fib_B
• Generalizes universe lifting via categorical nerve techniques

Innovation

• Relative Hofmann-Streicher lifting via pseudo-adjoint
• Functorial analysis using categorical nerve
• Lifting Grothendieck universes in fibred categories