Timeline for What is known about the category of monads on Set?
Current License: CC BY-SA 3.0
12 events
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| Aug 21, 2015 at 14:18 | history | edited | David Spivak | CC BY-SA 3.0 |
I changed the notation $Mon$ to $Mnd$
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| Feb 14, 2011 at 5:10 | comment | added | Mike Shulman | @Harry: The nLab page is really only about monoids in monoidal categories whose tensor product preserves colimits on both sides, although I observe that it is not always maximally explicit on that point. The monoidal category of endofunctors does not have that property: it preserves colimits on one side, but not the other. | |
| Feb 13, 2011 at 23:25 | answer | added | Steve Lack | timeline score: 25 | |
| Feb 12, 2011 at 14:02 | vote | accept | David Spivak | ||
| Feb 12, 2011 at 14:02 | vote | accept | David Spivak | ||
| Feb 12, 2011 at 14:02 | |||||
| Feb 12, 2011 at 12:43 | comment | added | Buschi Sergio | I know that V. Trnková studied and try do a classification of $Set$ endofunctors, in the book "Automata and ALgebra in Categories" (J Adamek, V. Trnkova) there is a deep study of these arguments in connection by (abstract formal) automata theory. at pag.317-19 of this book there are some observations about $Set$ monads too... MAy be you can write to J. Adamek. | |
| Feb 12, 2011 at 12:18 | answer | added | Tom Leinster | timeline score: 22 | |
| Feb 12, 2011 at 9:19 | comment | added | Harry Gindi | The category of affine schemes is categorically well-behaved.. on its own. The trouble with it comes from embedding it into the category of Zariski/étale/fppf sheaves living over it, then restricting to geometric objects (schemes or algebraic spaces). In particular, it's complete-cocomplete. But anyway, what does that have to do with monoidal categories? | |
| Feb 12, 2011 at 5:57 | comment | added | Theo Johnson-Freyd | @Harry Gindi: "It's my understanding that such categories [of monoids] are, as it were, as nice as you could possibly want". Because as everyone knows, the category of affine schemes is particularly well-behaved :). Anyway, I'd say "yes and no" to this statement. In many common situations (e.g. presentable category with cocontinuous tensor product), it's the category of comonoids that's particularly nice. The category of monoids is generally not nice (it includes algebraic geometry, for example), but rather "rich". | |
| Feb 12, 2011 at 4:56 | comment | added | Harry Gindi | You should check out this page on the nLab: ncatlab.org/nlab/show/category+of+monoids . However, to talk about $Mon$ at all, you have to use universes or doubly-large categories (might as well use universes at that point). I hope the nLab reference gives you some information you didn't know (for instance that $Mon$ has all pushouts). | |
| Feb 12, 2011 at 4:43 | comment | added | Harry Gindi | @David: $Mon$ is the category of monoids in the strict monoidal category of endofunctors on $Set$ under composition. It's my understanding that such categories are, as it were, as nice as you could possibly want, provided the underlying monoidal category has good properties. It's a well-known defect, however, that ordinary morphisms of monads are somehow not good enough to support a very rich theory. Fioré-Kock have some papers that you might find interesting, where they give a good characterization of a double category of monads, where the new vertical morphisms repair the defect. | |
| Feb 12, 2011 at 4:31 | history | asked | David Spivak | CC BY-SA 2.5 |