Suppose we are given a small (enriched) category $C$, and for $a,b \in C$ an isomorphism $m:a \to b$. It is always possible to find a functor $F: C \to C$, with $F(a)=b$ and a natural transformation $N$ of the identity functor to $F$ so that $N_a=m$. This is a nice simple exercise.

Can this always be done when $C$ is an $A_{\infty}$-category?

The intuition behind this is the following. Given a path $m$ from $a$ to $b$ in say a CW complex $X$, we can always find a homotopy $H: X \times [0,1] \to X$ of the identity map so that $H|_{\{a\} \times [0,1]} = m$.

The question can be generalized to ask when there is a "homotopy extension property" for $A_{\infty}$ subcategories.

Edit: For the first question the answer is affirmative by an abstract non-sense argument. Take the $A_{\infty}$ Yoneda embedding $Y$ of C into Ch-dg the category of chain complexes. Then find the functor $F$ and the natural transformation $N$ for $Y(m)$ inside Y(C) which is a dg-category. Then just pull-back both by Y. However it maybe nice to find a more direct argument.



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