Timeline for Abelianization of monoids in arbitrary (symmetric) monoidal categories
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2021 at 17:08 | vote | accept | Stefano D'Alesio | ||
| Mar 21, 2021 at 15:37 | answer | added | Simon Henry | timeline score: 3 | |
| Mar 21, 2021 at 14:43 | comment | added | Chris Schommer-Pries | An example where the abelianization functor does not exist: take $\mathcal{C}$ to be the category of sets whose cardinality is either 1 or infinite with the product monoidal structure. Infinite groups are examples of monoids in this category. There are infinite groups whose (standard) abelianization is a non-trivial finite group, and these won't exist as commutative monoid objects in $\mathcal{C}$. | |
| Mar 21, 2021 at 12:02 | comment | added | fosco | In full generality, the existence of left adjoints is linked to the existence of colimits; all in all, abelianisation will exist in every category of internal monoids in $C$ such that $U : Ab(C) \to Mon(C)$ satisfies the assumptions of the adjoint functor theorem. | |
| Mar 21, 2021 at 10:17 | review | First posts | |||
| Mar 21, 2021 at 12:37 | |||||
| Mar 21, 2021 at 10:14 | history | asked | Stefano D'Alesio | CC BY-SA 4.0 |