Filtered colimit elimination from Birkhoff's variety theorem
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916693971828736 |
|---|---|
| author | Kawase, Yuto |
| author_facet | Kawase, Yuto |
| contents | Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon "filtered colimit elimination," and study a sufficient condition for it. We show that if a locally finitely presentable category $\mathscr{A}$ satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to $\mathscr{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05304 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Filtered colimit elimination from Birkhoff's variety theorem Kawase, Yuto Category Theory 18C10, 18C35, 18E45 Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon "filtered colimit elimination," and study a sufficient condition for it. We show that if a locally finitely presentable category $\mathscr{A}$ satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to $\mathscr{A}$. |
| title | Filtered colimit elimination from Birkhoff's variety theorem |
| topic | Category Theory 18C10, 18C35, 18E45 |
| url | https://arxiv.org/abs/2309.05304 |