hence the only input of it is from . Therefore represents a differential. d=2 Unpacking the coherence condition for gives the degree map . In the sum there are the inequalities of indices giving equal to . Unpacking the coherence sum gives the relation , which when rewritten with and as the differential and multiplication, it is , which is the Leibniz rule for differential graded algebras. d=3 In this degree the associativity structure comes to light. Note if then there is a differential graded algebra structure, which becomes transparent after expanding out the coherence condition and multiplying by an appropriate factor of , the coherence condition reads something like Notice that the left hand side of the equation is the failure for to be an associative algebra on the nose. One of the inputs for the first three maps are coboundaries since is the differential, so on the cohomology algebra these elements would all vanish since . This includes the final term since it is also a coboundary, giving a zero element in the cohomology algebra. From these relations we can interpret the map as a failure for the associativity of , meaning it is associative only up to homotopy. d=4 and higher order terms Moreover, the higher order terms, for , the coherent conditions give many different terms combining a string of consecutive into some and inserting that term into an along with the rest of the 's in the elements . When combining the terms, there is a part of the coherence condition which reads similarly to the right hand side of , namely, there are terms In degree the other terms can be written out as showing how elements in the image of and interact. This means the homotopy of elements, including one that's in the image of minus the multiplication of elements where one is a homotopy input, differ by a boundary. For higher order , these middle terms can be seen how the middle maps behave with respect to terms coming from the image of another higher homotopy map. Diagrammatic interpretation of axioms There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad explaining how to visually think about this higher homotopies. This intuition is encapsulated with the discussion above algebraically, but it is useful to visualize it as well. Examples Associative algebras Every associative algebra has an -infinity structure by