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have started model structure for L-infinity algebras
“There exists a model category structure on a category of dg-coalgebras whose fibrant objects are precisely L-∞ algebras.”
Isn’t there that the operad for L-∞ algebras is the cofibrant replacement of the operad for dg-algebras in a model structure on the category of dg-operads ? Now the algebras over this copfibrant replacements are “the same as” fibrant objects in a model structure on the category of algebras over the original dg-operad. Is there a general pattern here ?
I am finally working on adding some substance to the entry model structure for L-infinity algebras (long overdue).
My main purpose currently is to highlight some useful statements that are contained but maybe a bit hard to spot in the article
(For instance there is a prop. 4.42 in that article whose proof contains a bunch of important facts which are not advertized in the statement of the proposition.)
For a survey, check out the Summary table.
But beware, I am still editing and working on the entry…
I have added a brief commented cross-link between model structure for L-infinity algebras (in the section On cosimplicial algebras) and monoidal Dold-Kan correspondence (in the section dual monoidal DK – Quillen equivalences) on how Jonathan Pridham’s article has a Quillen equivalence between dg-algebras and cosimplicial algebras by the inverse to the normalized-cochains-functor, relating two models for $L_\infty$-algebras.
(Just a pointer so far, more as a note to myself. Will try to add more details later on.)
(edit: have now added a little bit of content )
@Urs: Jon Pridham told me of the following notes that
Stefano Maggiolo took from a course Jonathan gave in Rome:
http://poisson.phc.unipi.it/~maggiolo/wp-content/uploads/2008/12/WDTII_Pridham.pdf
I would put a link on the model structure for L-infinity algebras page but am not sure if that is the best place for it.
Hi Tim,
ah, thanks for the link, that looks like a useful survey of his article. Sure, that would fit wll at model structure for L-infinity algebras, also at deformation theory etc, I presume.
I have finally added to model structure for L-infinity algebras right at the beginning the statement of the “default” model structure, that induced by $L_\infty$-algebras being homotopy algebras over an operad.
I have split the definition-section now into one piece with this “default” model structure and then all the other model structures together as ingredients of the Quillen equivalence of that to infinitesimal derived $\infty$-stacks.
(The entry still deserves more details to be addedd, of course.)
added
the remark that the class of weak equivalences (for $L_\infty$) on $dgCoAlg$ is not the evident one,
the proposition that instead it is a proper subclass;
the analogous remark for the weak equivalence on $dgAlg$.
Chevalley-Eilenberg assumes finite-dimensional graded components, isn’t it ? What are the finiteness conditions on the graded components of the objects when definining the model categories in question ? I do not see any finiteness conditions when skimming the entry…(as well as for the entry model structure on dg-coalgebras).
Pridham uses pro-dg-algebras to get around the finiteness condition, that’s why.
Interesting.
Very interesting even.
I think it’s known since
that every coalgebra is the filtered colimit of its finite-dimensional sub-coalgebras, and hence that its linear dual is a pro-fin-dim-algebra. See for instance p. 7 of
In
it was observed that this remains true in the presence of differentials.
I learned this first from Todd Trimble some years back. I seemed to recall that then Todd had written some notes related to this, but now I don’t find them.
@Urs
they might be in Todd’s private web but not published to be viewable by the general public.
I’ve been meaning to complete a long post to the Café on this topic, since it was discussed recently between John and Mike (how are spaces of measures coalgebras). But in the meantime, here are some past scribblings, roughly in backwards chronological order:
Mainly these were in view of giving an explicit description of the closed structure of the category of cocommutative coalgebras (and the enrichment of algebras therein). Urs has mentioned in the nLab that this fact can be deduced cleanly and abstractly from the Gabriel-Ulmer description of CocommCoalg as the category of lex functors from the opposite of the category of finite-dimensional cocommutative algebras to $Set$ (i.e. from the category of finite-dimensional algebras to $Set$), which follows straightaway from what Urs just wrote on colimits of finite-dimensional pieces. This fact is also mentioned at cofree coalgebra, and called the “Fundamental Theorem of Coalgebras”, with a reference to a Handbook of Algebras article by Walter Michaelis often mentioned by Jim Stasheff. And also at CocommCoalg under Local Presentability.
In my personal web I had begun looking more generally at the category of commutative comonoids for a locally presentable symmetric monoidal closed category, but that takes us a little further off-topic I guess. I can probably think of more instances; it’s something I come back to with some frequency.
13: Even with new moments coming from Sweedler the very duality you talk about is known far before Sweedler, in the works of Cartier and Dieudonné (instead of pro-finite dimensional vector spaces/modules they work with linearly compact topological vector spaces/modules/rings, some related notions include linearly compact module, Cartier duality, Gabriel-Oberst duality), but I did not know that it was effectively applied to the theory of $L_\infty$-algebras and model categories in question.
P.S. The Abrams-Weibel reference is new to me; I recorded it into dual gebra.
Cartier duality (in its original sense) is general if linearly compact modules are used, not only for finite group schemes as presently stated in the entry. This is equivalent to working with pro-finite dimensional vector spaces. Unfortunately, I will have no time today to write the amendment (which is nicely explained in the Dieudonné’s book cited in 16.
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