Partial monoid actions on objects in categories with pullbacks and their globalizations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915773274914816 |
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| author | Khrypchenko, Mykola Klock, Francisco |
| author_facet | Khrypchenko, Mykola Klock, Francisco |
| contents | Let $M$ be a monoid, $\mathscr{C}$ a category with pullbacks and $X$ an object of $\mathscr{C}$. We introduce the notion of a partial action $α$ of $M$ on $X$ and study the globalization question for $α$. If $α$ admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in $\mathscr{C}$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $\mathscr{C}$. We specify this construction to the case of categories admitting certain coproducts and coequalizers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11300 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partial monoid actions on objects in categories with pullbacks and their globalizations Khrypchenko, Mykola Klock, Francisco Category Theory Group Theory Primary: 16W22, 18B10, secondary: 20M30, 18A40 Let $M$ be a monoid, $\mathscr{C}$ a category with pullbacks and $X$ an object of $\mathscr{C}$. We introduce the notion of a partial action $α$ of $M$ on $X$ and study the globalization question for $α$. If $α$ admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in $\mathscr{C}$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $\mathscr{C}$. We specify this construction to the case of categories admitting certain coproducts and coequalizers. |
| title | Partial monoid actions on objects in categories with pullbacks and their globalizations |
| topic | Category Theory Group Theory Primary: 16W22, 18B10, secondary: 20M30, 18A40 |
| url | https://arxiv.org/abs/2309.11300 |