Timeline for Functors that preserve monoids
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 1, 2021 at 13:40 | vote | accept | Javi | ||
| Jul 1, 2021 at 11:26 | answer | added | Martin Brandenburg | timeline score: 6 | |
| Jul 1, 2021 at 10:52 | comment | added | Javi | I guess that is a more correct formulation of the question. I am not assuming anything else about $F'$. | |
| Jul 1, 2021 at 10:09 | comment | added | Martin Brandenburg | Question. Is there a lax monoidal functor $(F,\eta,\mu) : (\mathcal{C},\otimes) \to (\mathcal{D},\otimes)$ (with underlying functor $F$) such that $\mathrm{Mon}(F,\eta,\mu) : \mathrm{Mon}(\mathcal{C},\otimes) \to \mathrm{Mon}(\mathcal{D},\otimes)$ is equal to $F'$? | |
| Jul 1, 2021 at 10:08 | comment | added | Martin Brandenburg | "monoid" is an extra structure, not property, so that "maps monoids to monoids" is not defined; same for "lax monoidal". Here is one version of your question how I understand it: Let $(\mathcal{C},\otimes),(\mathcal{D},\otimes)$ be two monoidal categories, let $F : \mathcal{C} \to \mathcal{D}$ and $F ' : \mathrm{Mon}(\mathcal{C},\otimes) \to \mathrm{Mon}(\mathcal{D},\otimes)$ be two functors such that $U_{\mathcal{D}} F'=FU_{\mathcal{C}}$, where $U_{\mathcal{C}}$ denotes the forgetful functor $\mathrm{Mon}(\mathcal{C},\otimes) \to \mathcal{C}$. (Do you want to assume more about $F'$?). [...] | |
| Jul 1, 2021 at 9:14 | comment | added | Javi | @MartinBrandenburg maybe it would be more accurate saying that the functor sends monoids to monoids. But I don't see why it doesn't make sense to say that the functor is lax monoidal. I have a functor between (symmetric) monoidal categories so it makes sense to say whether it is (symmetric) lax monoidal or not. I know this functor sends monoids to monoids (or operads to operads). | |
| Jun 30, 2021 at 17:55 | comment | added | Martin Brandenburg | Also there are some technical inaccuracies here. It does not make sense to say that a functor preserves monoids, since being a monoid is not a property. Similarly, it does not make much sense to say that the functor is lax monoidal. You probably want to have a functor between monoid objects which lifts the given functor, maybe with some properties which need to be specified. | |
| Jun 30, 2021 at 17:49 | comment | added | Martin Brandenburg | The problem is that monoids only talk about tensor powers of an object, but a lax monoidal structure incorporates all tensor products. Surely there will be counterexamples. Maybe you can do something when coproducts exist which are compatible with tensors. | |
| Jun 30, 2021 at 17:29 | comment | added | Noah Snyder | It seems unlikely to me, surely you could cook up a monoidal category that had no non-trivial monoids, and then the preserves monoids condition tells you almost nothing and just about any functor will do. | |
| Jun 30, 2021 at 15:30 | history | asked | Javi | CC BY-SA 4.0 |