Erratum to `Category of $A_\infty$-categories'

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 $\mathcal{K}^u A_\infty$(resp. ordinary 2-category $^u A_\infty$) 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 $e \cdot 1_f = 1_{ef}$ does hold. The mistake does not affect results on invertible 2-morphisms and quasi-invertible 1-morphisms in $^u A_\infty$.