Presheaves and cocompletions in formal category theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| author | Arkor, Nathanael McDermott, Dylan |
| author_facet | Arkor, Nathanael McDermott, Dylan |
| contents | We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions. We furthermore extend the theory of weighted colimits from enriched category theory to this setting, developing the concepts of atomicity and rank, and providing recognition theorems for presheaf objects, free cocompletions, and cocomplete objects. As an application of our methods, we construct free cocompletions, under arbitrary classes of colimit-small weights, of (possibly large) categories enriched in (not necessarily symmetric) monoidal categories and bicategories; this resolves a longstanding omission in the literature on enriched category theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22370 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Presheaves and cocompletions in formal category theory Arkor, Nathanael McDermott, Dylan Category Theory 18D70, 18D65, 18D60, 18A30, 18A35, 18F20, 18D20, 18N10 We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions. We furthermore extend the theory of weighted colimits from enriched category theory to this setting, developing the concepts of atomicity and rank, and providing recognition theorems for presheaf objects, free cocompletions, and cocomplete objects. As an application of our methods, we construct free cocompletions, under arbitrary classes of colimit-small weights, of (possibly large) categories enriched in (not necessarily symmetric) monoidal categories and bicategories; this resolves a longstanding omission in the literature on enriched category theory. |
| title | Presheaves and cocompletions in formal category theory |
| topic | Category Theory 18D70, 18D65, 18D60, 18A30, 18A35, 18F20, 18D20, 18N10 |
| url | https://arxiv.org/abs/2604.22370 |