Can anyone provide me with a basic reference on $A_\infty$ categories?
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3$\begingroup$ The book by Seidel on mirror symmetry has a very readable introduction. But beware that he uses sign conventions different from algebraists like Keller. $\endgroup$– Tom BachmannCommented Sep 27, 2013 at 7:32
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5 Answers
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I am currently trying to learn this, and this paper of B. Keller proved very useful. This one of J. Huebschmann seems a bit less basic, but might be usefull too. I must say I'm very interested in any other answers.
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$\begingroup$ How are $A_\infty$-algebras related to $A_\infty$-cateogries? $\endgroup$ Commented Sep 27, 2013 at 9:24
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$\begingroup$ The $R$-algebras that you learn about in a first course on algebra are one-point categories enriched over $R-\mathbf{Mod}$, the category of $R$-modules. If we drop the one-point restriction, we get an $R$-linear category. In the $A_{\infty}$ setting, an $A_{\infty}$-algebra is just an $A_{\infty}$ category with one object. (Note that the higher-$\mu$ data is still nontrivial in this setting!) $\endgroup$– dvitekCommented Jul 23, 2017 at 20:59
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Here's a list of references:
- BLM, 483 Pages, Pretriangulated 𝔸_∞-Categories;
- COS, 25 Pages, Localizations of the Category of 𝔸_∞-Categories and Internal Homs;
- Faonte, 156 Pages, Nerve construction, 𝔸_∞-functors and homotopy theory of differential graded categories;
- Faonte, 47 Pages, Simplicial nerve of an 𝔸_∞-category;
- Faonte, 53 Pages, 𝔸_∞-functors and homotopy theory of dg-categories;
- Fukaya, 97 Pages, Floer Homology and Mirror Symmetry;
- Horel, 10 Pages, The homotopy theory of 𝔸_∞-categories;
- Keller, 31 Pages, Introduction to 𝔸_∞-algebras and modules;
- KS, 70 Pages, Notes on 𝔸_∞-algebras, 𝔸_∞-categories and non-commutative geometry I;
- Lefèvre-Hasegawa, 230 Pages, Sur les 𝔸_∞-Catégories;
- Ornaghi, 113 Pages, Comparison Results About DG-Categories, 𝔸_∞-Categories, Stable ∞-Categories and Noncommutative Motives;
- Seidel, 334 Pages, Fukaya Categories and Picard–Lefschetz Theory.
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I believe the thesis of Kenji Lefèvre-Hasegawa is a remarkable piece of work, and is very readable (references spotted by Samuel Tinguely are very good, but they are about $A_\infty$-algebras):
https://arxiv.org/abs/math/0310337
Unfortunately it is in only available in French.
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1$\begingroup$ +1, I just don't agree with the first word of the last sentence! $\endgroup$ Commented Mar 27, 2012 at 14:41
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$\begingroup$ :-) I kind of always feel guilty to give a reference in French (it could be considered as self-promotion :-)) $\endgroup$– DamienCCommented Mar 27, 2012 at 15:51
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To me the best path to $A_\infty$-categories is via topology. One first gets familiar with the notion of $A_\infty$-space as a natural generalization of that of topological monoid (the classical reference by Jim Stasheff is probably still where one should have a look for a complete account on this). Next one considers topological categories as a natural generalization of topological monoids (the only difference being that the product is not defined on $M\times M$ but on a fibered product $M_1\times_{M_0}M_1$, the latter reducing to the first for the space of objects $M_0$ consisting of a single point). Then one mixes these two generalizations of topological monoid and gets the definition of "$A_\infty$ topological category" in the most natural possible way (in my opinion). Once one is familiar with this, one sees that nothing changes if instead of being in the topological setting one works in any setting where "homotopies" are meaningful. For instance one can work with dg-categories, and this gives the version of $A_\infty$-category one usually meets.
answered Mar 27, 2012 at 17:44
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$\begingroup$ @domenico this approach is of course deep and interesting, but it's also too abstract for somebody which is starting and whose motivations are more algebraic (fact that everyone so far has given for granted although the question doesn't make it precise). $\endgroup$ Commented Mar 27, 2012 at 19:07
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5$\begingroup$ That's right, but it is precisely since the question does not specify whether the motivations are algebraic or a reference is sought for the general idea of $A_\infty$ category that I suggested to start from where everything begun: my fear is that with no background on $A_\infty$-spaces at all, the whole of $A_\infty$-categories may remain too abstract and out of focus, a bit like the definition of homotopy between morphisms of chain complexes without having seen a proof of Poincare' lemma before. $\endgroup$ Commented Mar 27, 2012 at 20:02
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You will like this one, I believe. Have fun!
https://arxiv.org/abs/math/0306332
by the way, a short notes: https://arxiv.org/abs/1310.3718
