algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, a commutative ring spectrum, roughly equivalent to a

E_\infty

-ring spectrum, is a

commutative 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 ...

in a goodsymmetric monoidal with respect to

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 ...

and perhaps some other conditions; one choice is the category of

symmetric spectra In algebraic topology, a symmetric spectrum ''X'' is a spectrum of pointed simplicial sets that comes with an action of the symmetric group \Sigma_n on X_n such that the composition of structure maps :S^1 \wedge \dots \wedge S^1 \wedge X_n \to S^1 ...

category of spectra. The category of commutative ring spectra over the field

\mathbb

of rational numbers is

Quillen equivalent In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the t ...

to the category of

differential graded algebra In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geo ...

s over

\mathbb

. Example: The

Witten genus In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the r ...

may be realized as a

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

of commutative ring spectra MString → tmf. See also:

simplicial commutative ring In algebra, a simplicial commutative ring is a monoid object, commutative monoid in the category (mathematics), category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplic ...

highly structured ring spectrum In mathematics, a highly structured ring spectrum or 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_\infty-ring is called an E_\infty-ring. W ...

and

derived scheme In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutativ ...

Terminology

Almost all reasonable categories of commutative ring spectra can be shown to be

to each other. Thus, from the point view of the

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 ...

, the term "commutative ring spectrum" may be used as a synonymous to an

E_\infty

-ring spectrum.

Notes

References

* * Algebraic topology {{topology-stub