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

Filtered $$cA_\infty$$ c A ∞ -Categories and Functor Categories 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Categorical Structures Springer Journals

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

, Volume 26 (5) – May 28, 2018
54 pages

/lp/springer_journal/filtered-ca-infty-c-a-categories-and-functor-categories-xChuLX1imh
Publisher
Springer Journals
Subject
Mathematics; Mathematical Logic and Foundations; Theory of Computation; Convex and Discrete Geometry; Geometry
ISSN
0927-2852
eISSN
1572-9095
D.O.I.
10.1007/s10485-018-9526-2
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Applied Categorical StructuresSpringer Journals

Published: May 28, 2018

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