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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 28th 2015
• (edited May 28th 2015)

There was an ancient query box discussion sitting in the entry dg-Lie algebra which hereby I am moving from there to here.

begin of ancient discussion

+–{: .query}

Tim: I have changed the wording that Zoran suggested slightly. Of course, a dgla is an internal Lie algebra, a term that needs making precise in an entry, but then we must make precise the tensor product, and the symmetry. All that abstract baggage is, of course, in other entries, but I think it best to avoid the term ’simply’. I have heard it expressed that category theorists tend to use the term ’simply’ aand other similar terms too much from the point of view others working in neighbouring disciplines.

For instance, if someone knows de Rham theory from a geometric viewpoint, we know that in the long run it will be useful for them to understand the differential graded algebra from a categorical viewpoint as that is one of the most fruitful approaches for geometrically significant generalisations and applications BUT the debutant can get very put off by thinking that they have to understand lots of category theory before they can start understanding the de Rham complex. In fact coming from that direction they can understand the category theory via the de Rham theory. So I suggest that we simply avoid ’simply’!!

I know some researchers in other subject areas are looking with interest to the nLab as a quick means of entry into some interesting mathematics and a handy reference for definitions and background. That is great but it perhaps means that we have to be a bit careful about our natural feeling that the categorical approach is nearly always the ’best’. ’Simply’ is one problem, another is, I think, use of diagrams rather than formulae. My feeling is that both should be given (though the diagrams are more difficult to get looking nice).

Urs: these are all good points. In general I believe it will be good to offer different perspectives in an $n$Lab entry, and explain what they are useful for, each. I take the point that the word “simply” for the categorical perspective may raise unintended feelings, so maybe it should be avoided or at least not left uncommented.

But we should also not hide the important point here, which is hinted at by the word simply: I think that the important point is that the abstract category-theoretic formulation which packages a long list of detailed definitions in a single statement such as “internal Lie algebra” allows us to recognize that that list of definitions is right.

There are many definitions that one can dream up. But some are better than others and category theory can explain why.

For instance I have seen experts who calucalted with differential graded algebra all day long be mystified by why exactly all the sign rules are as they are. The best explanation they had was: it works and yields interesting results. They were positively interested to learn that all these signs follow automatically and consistently by realizing that differential graded algebra is algebra internal to the category of chain complexes.

This doesn’t mean that it is best to introduce DGCA in this internal language. But it does mean that it is worthwhile pointting out that lots of nitty-gritty details of definitions can “simply” be derived by starting with an abstract internal definition and then turning the crank.

Tim: I could not have put it better myself. I was wondering if there might not be some way in which this viewpoint might not be expressed explicitly. Perhaps David C has some thoughts.. sort of ’the unreasonable effectiveness of categorical language’?

My intention for my own contribution (with help hopefully) is to gradually add glosses in the lexicon entries so as to help interpret in both directions, categorically,and geometrically.

For instance, in the construction of the cobar one take the tensor algebra of the suspension of the cokernel (is it?) of the augmentation. WHY?!!!!!!! How can one understand this? Magic? It works? In fact it is still a bit of a mystery to me and saying that it comes from such and such a categorical property still needs spelling out for me. I have asked rational homotopy theorists and have partially understood things from their point of view but there are still gaps in my understanding of it and some of them worry me!

Toby: One should be able to say something like, ’From a category-theoretic perpsective, a differential graded Lie algebra is simply an internal Lie algebra in an appropriate category of chain complexes.’. This advertises what Urs says, that definitions come automatically from the category-theoretic perspective, without pretending that this will be simple to anyone coming from outside that perpsective.

Zoran Škoda: Tim, your question about the intricacies of cobar construction in the category of chain complexes is an interesting one, which I can not fully answer, specially in a short answer. However, still the categorical picture simplifies the viewpoint and the definition at least,and gives a direction how to proceed there as well. Given a dgca C one looks at the functor Tw(C,A) assigning to an algebra A the set of solution of the Maurer-Cartan equation $d t + t*t = 0$ where $*$ is the convolution product. Cobar construction is the (co)representative of this covariant functor. If you take Tw(C,A) as a contravariant functor on the coalgebras, for fixed A, then its representative is the bar construction (this is said in different words in entry twisting cochain). So bar and cobar construction are simply representatives of very natural functors; accidents of the realization of these functors by formulas in Ch are a bit unfortunate as you pointed out.

=–

end of ancient discussion

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 28th 2015
• (edited May 28th 2015)

Then I have slightly expanded the Idea-section such as to read now as follows:

A differential graded Lie algebra, or dg-Lie algebra for short, is equivalently

• a graded Lie algebra equipped with a differential that acts as a graded derivation with respect to the Lie bracket;

• a strict L-∞-algebra, i.e. an $L_\infty$-algebra in which only the unary and the binary brackets may be nontrivial.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 28th 2015
• (edited May 28th 2015)

I have restructured the Definition-section.