Relative monadicity
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929547131224064 |
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| author | Arkor, Nathanael McDermott, Dylan |
| author_facet | Arkor, Nathanael McDermott, Dylan |
| contents | We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense $\mathbb V$-functor $j \colon A \to E$, a $\mathbb V$-functor $r \colon D \to E$ is $j$-monadic if and only if $r$ admits a left $j$-relative adjoint and creates $j$-absolute colimits. This provides a refinement of the classical monadicity theorem -- characterising those categories whose objects are given by those of $E$ equipped with algebraic structure -- in which the arities of the algebraic operations are valued in $A$. In particular, when $j = 1$, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the ($j$-relative) monadicity of the composite of a $\mathbb V$-functor with a ($j$-relatively) monadic $\mathbb V$-functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10405 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative monadicity Arkor, Nathanael McDermott, Dylan Category Theory 18D70, 18D65, 18C15, 18C20, 18A40, 18C10, 18D20, 18N10 We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense $\mathbb V$-functor $j \colon A \to E$, a $\mathbb V$-functor $r \colon D \to E$ is $j$-monadic if and only if $r$ admits a left $j$-relative adjoint and creates $j$-absolute colimits. This provides a refinement of the classical monadicity theorem -- characterising those categories whose objects are given by those of $E$ equipped with algebraic structure -- in which the arities of the algebraic operations are valued in $A$. In particular, when $j = 1$, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the ($j$-relative) monadicity of the composite of a $\mathbb V$-functor with a ($j$-relatively) monadic $\mathbb V$-functor. |
| title | Relative monadicity |
| topic | Category Theory 18D70, 18D65, 18C15, 18C20, 18A40, 18C10, 18D20, 18N10 |
| url | https://arxiv.org/abs/2305.10405 |