We develop the basic theory of curved $$A_{infty }$$ A ∞ -categories ( $$cA_{infty }$$ c A ∞ -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 $$cA_{infty }$$ c A ∞ -categories $$mathfrak {a}$$ a and $$mathfrak {b}$$ b , we introduce a $$cA_{infty }$$ c A ∞ -category $$mathsf {qFun}(mathfrak {a}, mathfrak {b})$$ qFun ( a , b ) of so-called $$qA_{infty }$$ q A ∞ -functors in which the uncurved objects are precisely the $$cA_{infty }$$ c A ∞ -functors from $$mathfrak {a}$$ a to $$mathfrak {b}$$ b . The more general $$qA_{infty }$$ q A ∞ -functors allow us to consider representable modules, a feature which is lost if one restricts attention to $$cA_{infty }$$ c A ∞ -functors. We formulate a version of the Yoneda Lemma which shows every $$cA_{infty }$$ c A ∞ -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.
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机译：我们开发了曲线A _ { infty}的基本理论$$ a∞-categories（$$ CA _ { inftations $$ c a∞-categorous），包含Fukaya类别的框架（Fukaya et AL。在第一部分，AMS / IP研究中的高级数学，Vol？46，美国数学社会，普罗维登斯，RI，2009）和弱弯曲的$$ A _ { infty} $$ a∞-categories在positselski的意义上（弱弯曲$$ A_ infty $$ A∞代数在拓扑本地戒指，2012年。ARXIV：1202.2697v3）。在两个$$ ca _ { idty} $$ c a∞-categories $$ mathfrak {a} $$ a和$$ mathfrak {b} $$ b，我们介绍一个$$ ca _ { infty} $$ c a∞-category $$ mathsf {qfun}（ mathfrak {a}， mathfrak {b}）$$ qfun（a，b）所谓的$$ qa _ { infty} $$ q a∞ - 函数，其中uncurved对象正准确地是$$ ca _ { infty} $$ c a -functors从$$ mathfrak {a} $$ a to $$ mathfrak {b} $$ b。 QA _ { infty} $$ Q a-ockers允许我们允许我们考虑可代表模块，这是一个丢失的功能，如果一个人限制为$$ ca _ { infty} $$ c a∞functors。我们制定了yoneda引理的一个版本，它显示了每种$$ ca _ { infty} $$ c a∞-category，以同情相当于弯曲的dg类别，与未经修正的情况。我们还提供了杠铃的弯曲版本。
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