Classifying strict discrete opfibrations with lax morphisms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917462700720128 |
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| author | Capucci, Matteo Myers, David Jaz |
| author_facet | Capucci, Matteo Myers, David Jaz |
| contents | We study discrete opfibration classifiers in enhanced 2-categories and show how, under suitable hypotheses, such classifiers can be endowed with the structure of a (lax or pseudo-)T-algebra and classify strict discrete opfibrations in 2-categories of (lax or pseudo-)T-algebras and lax morphisms.
This leads to a notion of discrete opfibration classifier in the enhanced setting, in which `small' (e.g. strict) discrete opfibrations are classified by `loose' (e.g. lax) maps.
We identify conditions on an enhanced 2-monad T and on a discrete opfibration classifier ensuring that this lifting to algebras is possible.
These conditions hold in a broad range of examples, including double categories, monoidal and symmetric monoidal categories, orthogonal factorization systems, and, more generally, structures encoded by opfamilial 2-monads.
In particular, this recovers and explains the role of Span(Set) as a classifier for strict double discrete opfibrations via lax double functors.
We also characterize when representable copresheaves are pseudo rather than merely lax in terms of `cartesianness at the representing object', for an abstract notion of cartesianness we introduce. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classifying strict discrete opfibrations with lax morphisms Capucci, Matteo Myers, David Jaz Category Theory 18N15 We study discrete opfibration classifiers in enhanced 2-categories and show how, under suitable hypotheses, such classifiers can be endowed with the structure of a (lax or pseudo-)T-algebra and classify strict discrete opfibrations in 2-categories of (lax or pseudo-)T-algebras and lax morphisms. This leads to a notion of discrete opfibration classifier in the enhanced setting, in which `small' (e.g. strict) discrete opfibrations are classified by `loose' (e.g. lax) maps. We identify conditions on an enhanced 2-monad T and on a discrete opfibration classifier ensuring that this lifting to algebras is possible. These conditions hold in a broad range of examples, including double categories, monoidal and symmetric monoidal categories, orthogonal factorization systems, and, more generally, structures encoded by opfamilial 2-monads. In particular, this recovers and explains the role of Span(Set) as a classifier for strict double discrete opfibrations via lax double functors. We also characterize when representable copresheaves are pseudo rather than merely lax in terms of `cartesianness at the representing object', for an abstract notion of cartesianness we introduce. |
| title | Classifying strict discrete opfibrations with lax morphisms |
| topic | Category Theory 18N15 |
| url | https://arxiv.org/abs/2510.23531 |