LAUSR

We develop the basic theory of curved $$A_{\infty }$$A∞-categories ($$cA_{\infty }$$cA∞-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 $$A_{\infty }$$A∞-categories in the sense of Positselski (Weakly curved $$A_\infty $$A∞ algebras over a topological local ring, 2012. arxiv:1202.2697v3). Between two $$cA_{\infty }$$cA∞-categories $$\mathfrak {a}$$a and $$\mathfrak {b}$$b, we introduce a $$cA_{\infty }$$cA∞-category $$\mathsf {qFun}(\mathfrak {a}, \mathfrak {b})$$qFun(a,b) of so-called $$qA_{\infty }$$qA∞-functors in which the uncurved objects are precisely the $$cA_{\infty }$$cA∞-functors from $$\mathfrak {a}$$a to $$\mathfrak {b}$$b. The more general $$qA_{\infty }$$qA∞-functors allow us to consider representable modules, a feature which is lost if one restricts attention to $$cA_{\infty }$$cA∞-functors. We formulate a version of the Yoneda Lemma which shows every $$cA_{\infty }$$cA∞-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.