ABSTRACT
Write Ho ( S ∗ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math16_1.jpg"/> for the homotopy category of pointed spaces. The tools of algebraic topology often amount to studying spaces by applying a variety of functors F : Ho ( S ∗ ) → Ho ( A ) , https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/umath16_1.jpg"/> for A https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math16_2.jpg"/> a homotopy theory that is ‘algebraic’ in nature. A typical example of such an F is the functor C ∗ ( − : R ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math16_3.jpg"/> taking cochains with values in a commutative ring R. In this case A https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math16_4.jpg"/> can be taken to be (the opposite of) the category of differential graded algebras over R, using the cup product of cochains as multiplication. If one wishes to take the rather subtle commutativity properties of the cup product into account, a more refined choice for A https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math16_5.jpg"/> would be the category of E ∞-algebras over R.
