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# Homology, Homotopy and Applications

## Volume 9 (2007)

### Number 2

### Erratum to “Category of $A_{\infty}$-categories”

Pages: 163 – 164

DOI: https://dx.doi.org/10.4310/HHA.2007.v9.n2.a6

#### Author

#### Abstract

The erroneous statement (HHA 5 (2003), no. 1, 1-48) that the collection of unital $A_\infty$-categories, all $A_\infty$-functors, and all $A_\infty$-transformations (resp. equivalence classes of natural $A_\infty$-transformations) form a $\mathcal K$-2-category ${\mathcal K}{}^uA_\infty$ (resp. ordinary 2-category ${}^uA_\infty$) is corrected as follows. All 2-category axioms are satisfied, except that $1_e\cdot f$ does not necessarily equal $1_{ef}$ for all composable 1-morphisms $e , f$. The axiom $e\cdot1_f=1_{ef}$ does hold. The mistake does not affect results on invertible 2-morphisms and quasi-invertible 1-morphisms in ${}^uA_\infty$.

#### Keywords

$A_{\infty}$-categories, $A_{\infty}$-functors, $A_{\infty}$-transformations, unit $A_{\infty}$-transformation, 2-category

#### 2010 Mathematics Subject Classification

18D05, 18D20, 18G55, 57T30

Published 1 January 2007