In mathematics, a **highly structured ring spectrum** or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring is called an -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.

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, -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 -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 -operads in -categories. The most widely used approaches today employ the language of model categories.^{[ citation needed ]}

All these approaches depend on building carefully an underlying category of spectra.

The theory of operads is motivated by the study of loop spaces. A loop space ΩX has a multiplication

by composition of loops. Here the two loops are sped up by a factor of 2 and the first takes the interval [0,1/2] and the second [1/2,1]. This product is not associative since the scalings are not compatible, but it is associative up to homotopy and the homotopies are coherent up to higher homotopies and so on. This situation can be made precise by saying that ΩX is an algebra over the little interval operad. This is an example of an -operad, i.e. an operad of topological spaces which is homotopy equivalent to the associative operad but which has appropriate "freeness" to allow things only to hold up to homotopy (succinctly: any cofibrant replacement of the associative operad). An **-ring spectrum** can now be imagined as an algebra over an -operad in a suitable category of spectra and suitable compatibility conditions (see May, 1977).

For the definition of **-ring spectra** essentially the same approach works, where one replaces the -operad by an -operad, i.e. an operad of contractible topological spaces with analogous "freeness" conditions. An example of such an operad can be again motivated by the study of loop spaces. The product of the double loop space is already commutative up to homotopy, but this homotopy fulfills no higher conditions. To get full coherence of higher homotopies one must assume that the space is (equivalent to) an *n*-fold loopspace for all *n*. This leads to the in -cube operad of infinite-dimensional cubes in infinite-dimensional space, which is an example of an -operad.

The above approach was pioneered by J. Peter May. Together with Elmendorf, Kriz and Mandell he developed in the 90s a variant of his older definition of spectra, so called **S-modules** (see Elmendorf et al., 2007). S-modules possess a model structure, whose homotopy category is the stable homotopy category. In S-modules the category of modules over an -operad and the category of monoids are Quillen equivalent and likewise the category of modules over an -operad and the category of commutative monoids. Therefore, is it possible to define -ring spectra and -ring spectra as (commutative) monoids in the category of S-modules, so called *(commutative) S-algebras*. Since (commutative) monoids are easier to deal with than algebras over complicated operads, this new approach is for many purposes more convenient. It should, however, be noted that the actual construction of the category of S-modules is technically quite complicated.

Another approach to the goal of seeing highly structured ring spectra as monoids in a suitable category of spectra are categories of diagram spectra. Probably the most famous one of these is the category of symmetric spectra, pioneered by Jeff Smith. Its basic idea is the following:

In the most naive sense, a *spectrum* is a sequence of (pointed) spaces together with maps , where ΣX denotes the suspension. Another viewpoint is the following: one considers the category of sequences of spaces together with the monoidal structure given by a smash product. Then the sphere sequence has the structure of a monoid and spectra are just modules over this monoid. If this monoid was commutative, then a monoidal structure on the category of modules over it would arise (as in algebra the modules over a commutative ring have a tensor product). But the monoid structure of the sphere sequence is not commutative due to different orderings of the coordinates.

The idea is now that one can build the coordinate changes into the definition of a sequence: a *symmetric sequence* is a sequence of spaces together with an action of the *n*-th symmetric group on . If one equips this with a suitable monoidal product, one gets that the sphere sequence is a *commutative* monoid. Now ** symmetric spectra ** are modules over the sphere sequence, i.e. a sequence of spaces together with an action of the *n*-th symmetric group on and maps satisfying suitable equivariance conditions. The category of symmetric spectra has a monoidal product denoted by . A **highly structured (commutative) ring spectrum** is now defined to be a (commutative) monoid in symmetric spectra, called a *(commutative) symmetric ring spectrum*. This boils down to giving maps

which satisfy suitable equivariance, unitality and associativity (and commutativity) conditions (see Schwede 2007).

There are several model structures on symmetric spectra, which have as homotopy the stable homotopy category. Also here it is true that the category of modules over an -operad and the category of monoids are Quillen equivalent and likewise the category of modules over an -operad and the category of commutative monoids.

A variant of symmetric spectra are **orthogonal spectra**, where one substitutes the symmetric group by the orthogonal group (see Mandell et al., 2001). They have the advantage that the naively defined homotopy groups coincide with those in the stable homotopy category, which is not the case for symmetric spectra. (I.e., the sphere spectrum is now cofibrant.) On the other hand, symmetric spectra have the advantage that they can also be defined for simplicial sets. Symmetric and orthogonal spectra are arguably the simplest ways to construct a sensible symmetric monoidal category of spectra.

Infinity-categories are a variant of classical categories where composition of morphisms is not uniquely defined, but only up to contractible choice. In general, it does not make sense to say that a diagram commutes strictly in an infinity-category, but only that it commutes up to coherent homotopy. One can define an infinity-category of spectra (as done by Lurie). One can also define infinity-versions of (commutative) monoids and then define **-ring spectra** as monoids in spectra and **-ring spectra** as commutative monoids in spectra. This is worked out in Lurie's book *Higher Algebra*.

The categories of S-modules, symmetric and orthogonal spectra and their categories of (commutative) monoids admit comparisons via Quillen equivalences due to work of several mathematicians (including Schwede). In spite of this the model category of S-modules and the model category of symmetric spectra have quite different behaviour: in S-modules every object is fibrant (which is not true in symmetric spectra), while in symmetric spectra the sphere spectrum is cofibrant (which is not true in S-modules). By a theorem of Lewis, it is not possible to construct one category of spectra, which has all desired properties. A comparison of the infinity category approach to spectra with the more classical model category approach of symmetric spectra can be found in Lurie's *Higher Algebra* 4.4.4.9.

It is easiest to write down concrete examples of -ring spectra in symmetric/orthogonal spectra. The most fundamental example is the sphere spectrum with the (canonical) multiplication map . It is also not hard to write down multiplication maps for Eilenberg-MacLane spectra (representing ordinary cohomology) and certain Thom spectra (representing bordism theories). Topological (real or complex) K-theory is also an example, but harder to obtain: in symmetric spectra one uses a C*-algebra interpretation of K-theory, in the operad approach one uses a machine of multiplicative infinite loop space theory.

A more recent approach for finding -refinements of multiplicative cohomology theories is Goerss–Hopkins obstruction theory. It succeeded in finding -ring structures on Lubin–Tate spectra and on elliptic spectra. By a similar (but older) method, it could also be shown that Morava K-theory and also other variants of Brown-Peterson cohomology possess an -ring structure (see e.g. Baker and Jeanneret, 2002). Basterra and Mandell have shown that Brown–Peterson cohomology has even an -ring structure, where an -structure is defined by replacing the operad of infinite-dimensional cubes in infinite-dimensional space by 4-dimensional cubes in 4-dimensional space in the definition of -ring spectra. On the other hand, Tyler Lawson has shown that Brown–Peterson cohomology does not have an structure.

Highly structured ring spectra allow many constructions.

- They form a model category, and therefore (homotopy) limits and colimits exist.
- Modules over a highly structured ring spectrum form a stable model category. In particular, their homotopy category is triangulated. If the ring spectrum has an -structure, the category of modules has a monoidal smash product; if it is at least , then it has a symmetric monoidal (smash) product.
- One can form group ring spectra.
- One can define the algebraic K-theory, topological Hochschild homology, and so on, of a highly structured ring spectrum.
- One can define the space of units, which is crucial for some questions of orientability of bundles.

In mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a *ring* is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, in particular abstract algebra, a **graded ring** is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as **gradation** or **grading**.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, a **Hopf algebra**, named after Heinz Hopf, is a structure that is simultaneously a algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

In mathematics, a **monoidal category** is a category equipped with a bifunctor

In algebraic topology, a branch of mathematics, a **spectrum** is an object representing a generalized cohomology theory. This means given a cohomology theory

In mathematics, an **operad** is concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others. They form a category-theoretic analog of universal algebra.

In category theory, a branch of mathematics, a **monoid** in a monoidal category is an object *M* together with two morphisms

In category theory, a branch of mathematics, a **PROP** is a symmetric strict monoidal category whose objects are the natural numbers *n* identified with the finite sets and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each *n*, the symmetric group on *n* letters is given as a subgroup of the automorphism group of *n*. The name PROP is an abbreviation of "PROduct and Permutation category".

In mathematics, **topological modular forms (tmf)** is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer *n* there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space *X*, one obtains an abelian group structure on the set of homotopy classes of continuous maps from *X* to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.

In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg-Maclane spectra for any Abelian group ^{pg 134}. Note, this construction can be generalized to commutative rings as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence.

In mathematics, the **field with one element** is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted **F**_{1}, or, in a French–English pun, **F**_{un}. The name "field with one element" and the notation **F**_{1} are only suggestive, as there is no field with one element in classical abstract algebra. Instead, **F**_{1} refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of **F**_{1} have been proposed, but it is not clear which, if any, of them give **F**_{1} all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.

In the theory of operads in algebra and algebraic topology, an **A _{∞}-operad** is a parameter space for a multiplication map that is homotopy coherently associative.

In the mathematical field of algebraic topology, a **commutative ring spectrum**, roughly equivalent to a -ring spectrum, is a commutative monoid in a good category of spectra.

**Derived algebraic geometry** is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

In algebraic topology, a **symmetric spectrum***X* is a spectrum of pointed simplicial sets that comes with an action of the symmetric group on such that the composition of structure maps

In algebra, an **operad algebra** is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring *R*, with an operad replacing *R*.

In mathematics, **Koszul duality**, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras as well as topology. The prototype example, due to Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, is the rough duality between the derived category of a symmetric algebra and that of an exterior algebra. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.

In mathematics, an **-algebra** in a symmetric monoidal infinity category *C* consists of the following data:

This is a glossary of properties and concepts in algebraic topology in mathematics.

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*Rings, modules, and algebras in stable homotopy theory*. AMS. ISBN 978-0-8218-4303-1. - May, J. Peter (1977).
*-ring spaces and -ring spectra*. Springer. - May, J. Peter (2009). "What precisely are ring spaces and ring spectra?".
*Geometry & Topology Monographs*.**16**: 215–282. arXiv: 0903.2813 .

- Basterra, M.; Mandell, M.A. (2005). "Homology and Cohomology of E-infinity Ring Spectra" (PDF)
- Lawson, T. (2017). "Calculating obstruction groups for -ring spectra". arXiv: 1709.09629 [math.AT].

- Baker, A.; Jeanneret, A. (2002). "Brave new Hopf algebroids and extensions of
*MU*-algebras".*Homology, Homotopy and Applications*.**4**(1): 163–173. doi: 10.4310/HHA.2002.v4.n1.a9 . - Basterra, M.; Mandell, M.A. (June 2013). "The multiplication on BP" (PDF).
*Journal of Topology*.**6**(2): 285–310. arXiv: 1101.0023 . doi:10.1112/jtopol/jts032. S2CID 119652118. Archived from the original (PDF) on 2015-02-06.

- Lurie, J. "Higher Algebra" (PDF). Archived from the original (PDF) on 2015-02-06.
- Mandell, M.A.; May, J.P.; Schwede, S.; Shipley, B. (2001). "Model Categories of Diagram Spectra" (PDF).
*Proc. London Math. Soc*.**82**(2): 441–512. doi:10.1112/S0024611501012692. - Richter, B. (2017). "Commutative ring spectra". arXiv: 1710.02328 [math.AT].
- Schwede, S. (2001). "S-modules and symmetric spectra" (PDF).
*Math. Ann*.**319**(3): 517–532. doi:10.1007/PL00004446. S2CID 6866612. - Schwede S. Schwede, S. (2007). "An untitled book project about symmetric spectra" (PDF).

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