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# Homology, Homotopy and Applications

## Volume 8 (2006)

### Number 2

### A construction of quotient $A_{\infty}$-categories

Pages: 157 – 203

DOI: http://dx.doi.org/10.4310/HHA.2006.v8.n2.a9

#### Authors

#### Abstract

We construct an $A_\infty$-category ${\mathsf D}({\mathcal C}|{\mathcal B})$ from a given $A_\infty$-category ${\mathcal C}$ and its full subcategory ${\mathcal B}$. The construction is similar to a particular case of Drinfeld's construction of the quotient of differential graded categories. We use ${\mathsf D}({\mathcal C}|{\mathcal B})$ to construct an $A_\infty$-functor of K-injective resolutions of a complex, when the ground ring is a field. The conventional derived category is obtained as the 0-th cohomology of the quotient of the differential graded category of complexes over acyclic complexes. This result follows also from Drinfeld's theory of quotients of differential graded categories.

#### Keywords

$A_{\infty}$-category; $A_{\infty}$-functor; $A_{\infty}$-transformation; $K$-injective resolution; quotient $A_{\infty}$-category

#### 2010 Mathematics Subject Classification

16E45, 18G10, 18G55, 57T30