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Mar 27, 2012 at 20:02 comment added domenico fiorenza 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.
Mar 27, 2012 at 19:07 comment added Fernando Muro @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).
Mar 27, 2012 at 17:44 history answered domenico fiorenza CC BY-SA 3.0