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started entry on category of monoids. Spelled out the free monoid construction. Stated the construction of pushouts of monoids along free maps with reference to Schwede-Shipley. Will fill in the proof in a moment.
added the algorithm for computing pushouts along free monoid morphisms in $Mon(C)$ to category of monoids
added a bunch of references on constructions of free monoids, also a general reference by Porst. Added the statement that under mild conditions $Mon(C)$ is monadic and locally presentable if $C$ is.
That’s got to be the longest “Proposition” environment I’ve ever seen… (-:
Also, for monadicity of Mon(C) you surely at least need C to have countable coproducts. If I read correctly, Horst assumes all his categories are locally presentable.
you surely at least need C to have countable coproducts.
I’ll try to clean it up later. Am on a shaky connection right now…
Okay, I have added in the local presentabability-clause.
I have tried to clarify at category of monoids that all the constructions currently mentioned there assume the tensor product preserves colimits in both variables, which fails in some examples (notably the example of monads as monoids in an endofunctor category).
Thanks.
added section on filtered colimits with the statement that $U : CMon(C) \to C$ creates filtered colimits, for $C$ closed symmetric monoidal
I created the pages monoid in a monoidal category, commutative monoid in a symmetric monoidal category, and tried to improve their (infinity,1)-categorical versions monoid in a monoidal (infinity,1)-category and commutative monoid in a symmetric monoidal (infinity,1)-category. (I merged commutative algebra in an (infinity,1)-category into the latter page.
The entry monoid is predominantly about the same concept, it is definitely neither thought of nor written about monoid in Sets. Just it says early on that the motivating classical example is in Set, but the whole entry is about several definitions, all but that one example in generality of a monoidal category (or higher category). Other entries like monad refer to monoid, and I think this is how it should stay (the unadorned link should be about the most standard generality of the term). This is not against existence of another entry, though I still think a merge like between monad and comonad seem to me maybe more practical. But never mind.
created model structure on commutative monoids. also added to model structure on algebras over an operad the statement (from Spitzweck’s thesis) that if the operad is cofibrant and the monoid axiom holds, then the model structure on algebras over the operad exists. i don’t know why this wasn’t there already, and i’m also not quite sure what the stuff about G-objects and coloured operads is doing there…
Thanks! (But check your link, the above is broken.)
I see that indeed there is some stuff on that page which is not properly connected. The discussion of coloured operads belongs to the existence statement further below, by Berger-Moerdijk, on model structures on algebras over colored operads. Similarly the discussion of the $G$-objects is part of a discussion of model structures on algebras over symmetric operads, also by Berger-Moerdijk, which however seems missing from the page. Not sure what happened there, maybe I got interrupted.
Right now I am not in position to edit, will try to add more glue later. But in this spirit I would like to ask you to add qualifiers to your statement, since there are different conventions for what “operad” is to mean by default: add whether the given statement is for the symmetric/non-symmetric version and for colored or single object.
Then the stuff about commutative monoids we should make sure to cross-link with model structure on monoids in a monoidal model category.
Never mind the link, maybe that was my fault.
Adeel, could I ask you to make all environments for theorems/definitions/etc. that you add be numbered. By changing
+-- {: .un_theorem}
to
+-- {: .num_theorem}
etc. ? Thanks!
Sorry, I did have the wrong link and fixed it later. I confess to knowing almost nothing about operads, so I just copied in the statement from Spitzweck. He says he uses the definition of Kriz-May, would that just mean symmetric and non-coloured?
ANd sure, I will switch to numbered environments.
Also I just added E-infinity monoid in a symmetric monoidal model category. I hope I’m not adding stuff which is already in some form on the nLab. I should also add some examples and crosslinks.
Thanks.
Regarding model structures on $E_\infty$-algebras we have a general comment on that hidden at E-infinity algebra. Should be cross-linked with what you just added.
We also have something rudimentary at commutative monoid in a symmetric monoidal (infinity,1)-category.
I hope I’m not adding stuff which is already in some form on the nLab.
Yes, that’s an issue that we have to deal with. That’s why I am always after adding cross-links. We need to try to make sure that it is possible to find all possibly relevant articles on the $n$Lab without, of course, knowing in advance what they are titled.
In any case, thanks for all your additions!
Concerning rectification of E_∞-monoids to commutative monoids, one can actually give an if-and-only-if criterion for general monoidal model categories, see Proposition 10.1.2 and the much more general operadic statement in Theorem 9.3.6 in http://arxiv.org/abs/1410.5675.
@Urs, oh I see, this is going under the name E-infinity algebra on the nLab. I will try to merge these two entries then.
@Dmitri, oh cool, would you mind adding this to the page?
@Adeel: I added a statement here: http://ncatlab.org/nlab/show/commutative+monoid+in+a+symmetric+monoidal+model+category#rectification.
Great, thanks!
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