Filtered $$cA_\infty $$ c A ∞ -Categories and Functor Categories

Abstract

We develop the basic theory of curved $$A_{\infty }$$ -categories ( $$cA_{\infty }$$ -categories) in a filtered setting, encompassing the frameworks of Fukaya categories (Fukaya et al. in Part I, AMS/IP studies in advanced mathematics, vol 46, American Mathematical Society, Providence, RI, 2009) and weakly curved $$A_{\infty }$$ -categories in the sense of Positselski (Weakly curved $$A_\infty $$ algebras over a topological local ring, 2012. arxiv:1202.2697v3 ). Between two $$cA_{\infty }$$ -categories $$\mathfrak {a}$$ and $$\mathfrak {b}$$ , we introduce a $$cA_{\infty }$$ -category $$\mathsf {qFun}(\mathfrak {a}, \mathfrak {b})$$ of so-called $$qA_{\infty }$$ -functors in which the uncurved objects are precisely the $$cA_{\infty }$$ -functors from $$\mathfrak {a}$$ to $$\mathfrak {b}$$ . The more general $$qA_{\infty }$$ -functors allow us to consider representable modules, a feature which is lost if one restricts attention to $$cA_{\infty }$$ -functors. We formulate a version of the Yoneda Lemma which shows every $$cA_{\infty }$$ -category to be homotopy equivalent to a curved dg category, in analogy with the uncurved situation. We also present a curved version of the bar-cobar adjunction.