Weak cartesian properties of simplicial sets
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category Δ to...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2023
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| _version_ | 1817931298324348928 |
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| author | Constantin, C Fritz, T Perrone, P Shapiro, BT |
| author_facet | Constantin, C Fritz, T Perrone, P Shapiro, BT |
| author_sort | Constantin, C |
| collection | OXFORD |
| description | Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category Δ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in Δ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in Δ , which we characterize completely, along with several other classes of squares in Δ . Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example. |
| first_indexed | 2024-09-25T04:02:57Z |
| format | Journal article |
| id | oxford-uuid:07805bf9-e286-4633-95c5-7ee5603ef3e7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:19:48Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:07805bf9-e286-4633-95c5-7ee5603ef3e72024-11-11T08:52:29ZWeak cartesian properties of simplicial setsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:07805bf9-e286-4633-95c5-7ee5603ef3e7EnglishSymplectic ElementsSpringer2023Constantin, CFritz, TPerrone, PShapiro, BTMany special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category Δ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in Δ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in Δ , which we characterize completely, along with several other classes of squares in Δ . Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example. |
| spellingShingle | Constantin, C Fritz, T Perrone, P Shapiro, BT Weak cartesian properties of simplicial sets |
| title | Weak cartesian properties of simplicial sets |
| title_full | Weak cartesian properties of simplicial sets |
| title_fullStr | Weak cartesian properties of simplicial sets |
| title_full_unstemmed | Weak cartesian properties of simplicial sets |
| title_short | Weak cartesian properties of simplicial sets |
| title_sort | weak cartesian properties of simplicial sets |
| work_keys_str_mv | AT constantinc weakcartesianpropertiesofsimplicialsets AT fritzt weakcartesianpropertiesofsimplicialsets AT perronep weakcartesianpropertiesofsimplicialsets AT shapirobt weakcartesianpropertiesofsimplicialsets |