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John Klein
John Klein
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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 to givespace; this is a description of BXmodel which doesn't require that the unit be strict.

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 Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.

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.

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John Klein
John Klein
  • 18.9k
  • 53
  • 109

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 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 Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.

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 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 Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.

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 Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.

Source Link
John Klein
John Klein
  • 18.9k
  • 53
  • 109

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 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 Boardman and Vogt description of the classifying space to give a description of BX which doesn't require that the unit be strict.