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I am quite curious:

What is the precise definition of "homotopy equivalence" or "isomorphism" of two curved $A_\infty$ algebra $A$ and $B$?

What is the condition to set for the morphism $f:A \rightarrow B$ and $g: B \rightarrow A$ ?

Will such relation keep the Hochschild (co)-homology invariant?

Thanks!

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As far as I am aware a good notion only exists when you make some restriction... e.g. consider curved $A_{\infty}$ algebras over $C[[t]]$ and consider curvatures which vanish modulo $t$ or consider a curvature $W$ which is strictly central ... the general theory of curved $A_{\infty}$ algebra is very badly behaved... –  Daniel Pomerleano Sep 26 '13 at 11:56
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I should add that, if one wants to consider the first case, one also wants to also take into account the topological structure when defining $HH^*$ of such curved $A_\infty$ algebras and morphisms of these algebras etc. The theory becomes essentially that of deformation theory of uncurved $A_\infty$ algebras. –  Daniel Pomerleano Sep 26 '13 at 12:18
    
P.S. I just found this earlier answer, mathoverflow.net/questions/86821/… which says some of the stuff above in more detail. –  Daniel Pomerleano Sep 26 '13 at 13:23
    
Greal, thank a lot! By the way, do you have any reference about the curvature strictly central case? –  Jay Sep 27 '13 at 6:12
    
The notion is considered here for instance math.northwestern.edu/~getzler/Papers/cyclic2.pdf, though HH^* invariants are not defined for these gadgets. –  Daniel Pomerleano Sep 27 '13 at 12:43
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