Tininess and right adjoints to exponentials
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912900944232448 |
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| author | Hernández, Enrique Ruiz Solórzano, Pedro |
| author_facet | Hernández, Enrique Ruiz Solórzano, Pedro |
| contents | Objects $T$ whose exponential functor $(-)^T$ admits a right adjoint $(-)_T$ are known under different names. The fact that they exist, yet that the only set that satisfies this in the category of sets is the singleton made Lawvere suggest they ought to be ``amazingly tiny'' -- hence Lawvere's acronym ``A.T.O.M.''
This report explores how intuitively tiny any such object is. Evidences both in favor and to the contrary are produced by looking at their categorical behavior (subobjects, quotients, retracts, etc) when the ambient category is a topos.
The topological behavior (connectedness, contractibility, connected components, etc) of both $T$ and $(-)_T$ is further analyzed in toposes that satisfy certain precohesive conditions over their decidable objects, where this tininess is tested against parts of Lawvere's foundational proposal for Synthetic Differential Geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tininess and right adjoints to exponentials Hernández, Enrique Ruiz Solórzano, Pedro Category Theory Primary 18B25, Secondary 03G30, 03B38 Objects $T$ whose exponential functor $(-)^T$ admits a right adjoint $(-)_T$ are known under different names. The fact that they exist, yet that the only set that satisfies this in the category of sets is the singleton made Lawvere suggest they ought to be ``amazingly tiny'' -- hence Lawvere's acronym ``A.T.O.M.'' This report explores how intuitively tiny any such object is. Evidences both in favor and to the contrary are produced by looking at their categorical behavior (subobjects, quotients, retracts, etc) when the ambient category is a topos. The topological behavior (connectedness, contractibility, connected components, etc) of both $T$ and $(-)_T$ is further analyzed in toposes that satisfy certain precohesive conditions over their decidable objects, where this tininess is tested against parts of Lawvere's foundational proposal for Synthetic Differential Geometry. |
| title | Tininess and right adjoints to exponentials |
| topic | Category Theory Primary 18B25, Secondary 03G30, 03B38 |
| url | https://arxiv.org/abs/2602.12239 |