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arXiv · 2504.09520

Hofmann-Streicher lifting of fibred categories

Andrew Slattery·Jonathan Sterling
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Abstract

In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe from the category of sets into the category of set-valued presheaves on a small category. 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 small discrete fibration gives the generic family of the universe'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 given by postcomposition with a fibration. Finally, we construct a new 2-bifibration of fibrations in which the opcartesian and cartesian lifts arise from these pseudo-adjunctions.

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Andrew Slattery, Jonathan Sterling. 2025-04-13. Hofmann-Streicher lifting of fibred categories. https://doi.org/10.46298/lmcs-22(2%3A30)2026

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