Hofmann-Streicher lifting of fibred categories
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914511619883008 |
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| author | Slattery, Andrew Sterling, Jonathan |
| author_facet | Slattery, Andrew Sterling, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hofmann-Streicher lifting of fibred categories Slattery, Andrew Sterling, Jonathan Category Theory Logic in Computer Science 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. |
| title | Hofmann-Streicher lifting of fibred categories |
| topic | Category Theory Logic in Computer Science |
| url | https://arxiv.org/abs/2504.09520 |