E-infinity Ring Spectrum
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In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
encoding a refinement of a multiplicative structure on a
cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. A commutative version of an A_\infty-ring is called an E_\infty-ring. While originally motivated by questions of
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
and
bundle theory Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction pa ...
, they are today most often used in
stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...
.


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 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 \operatorname^, and these spaces are equipped with certai ...
, and which has allowed also new constructions of more classical objects such as
Morava K-theory In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is s ...
. Beside their formal properties, 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_\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 In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p+q. This defines an associative (and distributive) graded commutative product opera ...
) fulfills associativity (and commutativity) only up to homotopy, this is too lax for many constructions (e.g. for
limits and colimits In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions suc ...
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
operad In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
s, while the recent approach of
Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. In 2014, Lurie received a MacArthur Fellowship. Lurie's research interests are algebraic geometry, topology, and ...
deals with it using \infty-operads in \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
operad In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
s is motivated by the study of
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ...
s. A loop space ΩX has a multiplication :\Omega X \times \Omega X \to \Omega X by composition of loops. Here the two loops are sped up by a factor of 2 and the first takes the interval ,1/2and the second /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 A_\infty-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 A_\infty-ring spectrum can now be imagined as an algebra over an A_\infty-operad in a suitable category of spectra and suitable compatibility conditions (see May, 1977). For the definition of E_\infty-ring spectra essentially the same approach works, where one replaces the A_\infty-operad by an E_\infty-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 \Omega^2X 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 \infty-cube operad of infinite-dimensional cubes in infinite-dimensional space, which is an example of an E_\infty-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-module In algebraic topology, an \mathbb-object (also called a symmetric sequence) is a sequence \ of objects such that each X(n) comes with an actionAn action of a group ''G'' on an object ''X'' in a category ''C'' is a functor from ''G'' viewed as a cate ...
s (see Elmendorf et al., 2007). S-modules possess a
model structure A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided into ...
, whose homotopy category is the
stable homotopy category A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use toda ...
. In S-modules the category of modules over an A_\infty-operad and the category of
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
s are Quillen equivalent and likewise the category of modules over an E_\infty-operad and the category of commutative monoids. Therefore, is it possible to define A_\infty-ring spectra and E_\infty-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.


Diagram spectra

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 (X_0, X_1, \dots) together with maps \Sigma X_i \to X_, where ΣX denotes the
suspension Suspension or suspended may refer to: Science and engineering * Car suspension * Cell suspension or suspension culture, in biology * Guarded suspension, a software design pattern in concurrent programming suspending a method call and the calling ...
. Another viewpoint is the following: one considers the category of sequences of spaces together with the monoidal structure given by a
smash product In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash prod ...
. Then the sphere sequence (S^0, S^1,\dots) 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 Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
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 (X_0, X_1, \dots) together with an action of the ''n''-th
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
on X_n. 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 (X_0, X_1, \dots) together with an action of the ''n''-th
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
on X_n and maps \Sigma X_i \to X_ satisfying suitable equivariance conditions. The category of symmetric spectra has a monoidal product denoted by \wedge. 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 :X_n\wedge X_m \to X_, 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 A_\infty-operad and the category of
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
s are Quillen equivalent and likewise the category of modules over an E_\infty-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 set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...
s. Symmetric and orthogonal spectra are arguably the simplest ways to construct a sensible
symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...
of spectra.


Infinity-categories

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 Lurie is often a Jewish surname, but also an Irish and English surname. The name is sometimes transliterated from/to other languages as Lurye, Luriye and Lure (from Russian), Lourié (in French). Other variants include: Lurey, Loria, Luria, ...
). One can also define infinity-versions of (commutative) monoids and then define A_\infty-ring spectra as monoids in spectra and E_\infty-ring spectra as commutative monoids in spectra. This is worked out in Lurie's book ''Higher Algebra''.


Comparison

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.


Examples

It is easiest to write down concrete examples of E_\infty-ring spectra in symmetric/orthogonal spectra. The most fundamental example is the sphere spectrum with the (canonical) multiplication map S^n\wedge S^m \to S^. It is also not hard to write down multiplication maps for Eilenberg-MacLane spectra (representing ordinary
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
) and certain Thom spectra (representing
bordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same d ...
theories). Topological (real or complex) K-theory is also an example, but harder to obtain: in symmetric spectra one uses a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
interpretation of K-theory, in the operad approach one uses a machine of multiplicative
infinite loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology ...
theory. A more recent approach for finding E_\infty-refinements of multiplicative cohomology theories is Goerss–Hopkins obstruction theory. It succeeded in finding E_\infty-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 In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is s ...
and also other variants of Brown-Peterson cohomology possess an A_\infty-ring structure (see e.g. Baker and Jeanneret, 2002). Basterra and Mandell have shown that Brown–Peterson cohomology has even an E_4-ring structure, where an E_4-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 E_\infty-ring spectra. On the other hand, Tyler Lawson has shown that
Brown–Peterson cohomology In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by , depending on a choice of prime ''p''. It is described in detail by . Its representing spectrum is denoted by BP. Complex cobordism and Quillen's idempot ...
does not have an E_\infty structure.


Constructions

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 E_2-structure, the category of modules has a monoidal
smash product In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash prod ...
; if it is at least E_4, then it has a symmetric monoidal (smash) product. *One can form group ring spectra. *One can define the
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
, topological
Hochschild homology In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a fiel ...
, 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.


See also

*
Commutative ring spectrum In algebraic topology, a commutative ring spectrum, roughly equivalent to a E_\infty-ring spectrum, is a commutative monoid in a goodsymmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of ...
*
En-ring In mathematics, an \mathcal_n-algebra in a symmetric monoidal infinity category ''C'' consists of the following data: *An object A(U) for any open subset ''U'' of Rn homeomorphic to an ''n''-disk. *A multiplication map: *:\mu: A(U_1) \otimes \cdo ...


References


References on E-ring spectra

* * *


References about structure of E-ring spectra

* Basterra, M.; Mandell, M.A. (2005). " Homology and Cohomology of E-infinity Ring Spectra" (PDF) *


References about specific examples

* *


General references on related spectra

* * * * *{{cite web , first=S. , last=Schwede S. Schwede , url=http://www.math.uni-bonn.de/~schwede/SymSpec.pdf , title=An untitled book project about symmetric spectra , year=2007 Algebraic topology Spectra (topology)