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

Loading next page...
 
/lp/springer_journal/filtered-ca-infty-c-a-categories-and-functor-categories-xChuLX1imh
Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off