Suppose I have an $A\_{\infty}$-space $X$, such that its unit is only a unit up to homotopy. When the space is well-behaved (well-pointed? What is the weakest condition possible?), I can replace it with a homotopy equivalent version of $X$ that has an honest unit. I read the definition for the classifying space of an $A\_{\infty}$-space in Stasheff's papers. He uses honest units. Is it possible to circumvent this somehow? 



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> Is there a functorial definition of the classifying space $BX$ that does not strictify the homotopy unit?
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Homotopy Units in $A_\infty$-spaces 
For your first question: 


If $X$ has the homotopy type of a CW space, then you can replace $X$ by any CW space $Y$ that
is homotopy equivalent to it (in the unbased sense).


Then $Y$ is also $A\_\infty$ with a good basepoint. Then you can use the homotopy extension property to make the basepoint a strict unit for the multiplication.


As to your last question:


I think the answer is yes. For example, one can use the Boardman and Vogt description of the classifying space; this is a model which doesn't require that the unit be strict.


