Filtered cA∞-Categories and Functor Categories

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