In mathematics, a highly structured ring spectrum or A ∞ {\displaystyle A_{\infty }} -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A ∞ {\displaystyle A_{\infty }} -ring is called an E ∞ {\displaystyle E_{\infty }} -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.
Background Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory. Beside their formal properties, E ∞ {\displaystyle E_{\infty }} -structures are also important in calculations, since they allow for operations in the underlying cohomology theory, analogous to (and generalizing) the well-known Steenrod operations in ordinary cohomology. As not every cohomology theory allows such operations, not every multiplicative structure may be refined to an E ∞ {\displaystyle E_{\infty }} -structure and even in cases where this is possible, it may be a formidable task to prove that. The rough idea of highly structured ring spectra is the following: If multiplication in a cohomology theory (analogous to the multiplication in singular cohomology, inducing the cup product) fulfills associativity (and commutativity) only up to homotopy, this is too lax for many constructions (e.g. for limits and colimits in the sense of category theory). On the other hand, requiring strict associativity (or commutativity) in a naive way is too restrictive for many of the wanted examples. A basic idea is that the relations need only hold up to homotopy, but these homotopies should fulfill again some homotopy relations, whose homotopies again fulfill some further homotopy conditions; and so on. The classical approach organizes this structure via operads, while the recent approach of Jacob Lurie deals with it using ∞ {\displaystyle \infty } -operads in ∞ {\displaystyle \infty } -categories. The most widely used approaches today employ the language of model categories. All these approaches depend on building carefully an underlying category of spectra.
Approaches for the definition
Operads The theory of operads is motivated by the study of loop spaces. A loop space ΩX has a multiplication
Ω X × Ω X → Ω X {\displaystyle \Omega X\times \Omega X\to \Omega X}
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