# Homotopy Units in $A_\infty$-spaces

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?

Is there a functorial definition of the classifying space $BX$ that does not strictify the homotopy unit?

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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.

Boardman and Vogt use what they call WA-spaces and they write in Homotopy-Everything H-spaces: "A WA-space (...) is approximately an $A_{\infty}$-space." I don't understand the "approximately"-part. How are these two notions related? –  Ulrich Pennig Nov 8 '11 at 23:01