Very-Well-Behaved Epireflections for Categories of Models of Sketches
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908526835662848 |
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| author | Xarez, João J. |
| author_facet | Xarez, João J. |
| contents | Firstly, precise conditions on how to obtain very-well-behaved epireflections are explored and improved from the author's previous papers; meaning that, beginning with a monad and a prefactorization system on a category, is produced a reflection with stable units (stronger than semi-left-exactness, also called admissibility in categorical Galois Theory) and an associated monotone-light factorization. Then, we were able to show that, for a pseudo-filtered category J in which every arrow is a monomorphism, the colimit functor on Set^J produces a very-well-behaved epireflection; if J = 2 the monotone-light factorization is non-trivial, as showed as an example. Then, new results are presented that grant very-well-behaved subreflections from the very-well-behaved reflections induced by an adjunction given by right Kan extensions for presheaves. These subreflections are obtained by restricting to the models of a sketch; it is showed finally that the known very-well-behaved reflection of n-categories into n-preorders is an example of this process (being n any positive integer). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Very-Well-Behaved Epireflections for Categories of Models of Sketches Xarez, João J. Category Theory 18A40, 18A32, 18E50, 18N50, 18F20, 18C30, 18N10 Firstly, precise conditions on how to obtain very-well-behaved epireflections are explored and improved from the author's previous papers; meaning that, beginning with a monad and a prefactorization system on a category, is produced a reflection with stable units (stronger than semi-left-exactness, also called admissibility in categorical Galois Theory) and an associated monotone-light factorization. Then, we were able to show that, for a pseudo-filtered category J in which every arrow is a monomorphism, the colimit functor on Set^J produces a very-well-behaved epireflection; if J = 2 the monotone-light factorization is non-trivial, as showed as an example. Then, new results are presented that grant very-well-behaved subreflections from the very-well-behaved reflections induced by an adjunction given by right Kan extensions for presheaves. These subreflections are obtained by restricting to the models of a sketch; it is showed finally that the known very-well-behaved reflection of n-categories into n-preorders is an example of this process (being n any positive integer). |
| title | Very-Well-Behaved Epireflections for Categories of Models of Sketches |
| topic | Category Theory 18A40, 18A32, 18E50, 18N50, 18F20, 18C30, 18N10 |
| url | https://arxiv.org/abs/2509.07241 |