Birkhoff's variety theorem for relative algebraic theories
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916693324857344 |
|---|---|
| author | Kawase, Yuto |
| author_facet | Kawase, Yuto |
| contents | An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathscr{A}$, we define an "algebraic concept" relative to $\mathscr{A}$, which will be called an $\mathscr{A}$-relative algebraic theory, and show that $\mathscr{A}$-relative algebraic theories are equivalent to finitary monads on $\mathscr{A}$. In establishing such equivalence, a generalized Birkhoff's variety theorem plays an important role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04382 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Birkhoff's variety theorem for relative algebraic theories Kawase, Yuto Category Theory 18C10, 18C15, 18C35, 18E45 An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathscr{A}$, we define an "algebraic concept" relative to $\mathscr{A}$, which will be called an $\mathscr{A}$-relative algebraic theory, and show that $\mathscr{A}$-relative algebraic theories are equivalent to finitary monads on $\mathscr{A}$. In establishing such equivalence, a generalized Birkhoff's variety theorem plays an important role. |
| title | Birkhoff's variety theorem for relative algebraic theories |
| topic | Category Theory 18C10, 18C15, 18C35, 18E45 |
| url | https://arxiv.org/abs/2304.04382 |