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...a branch of [[mathematics]], a '''connective spectrum''' is a [[spectrum (topology)|spectrum]] whose [[homotopy]] sets <math>\pi_k</math> of negative degrees *[https://mathoverflow.net/q/62086 Why are connective spectra called “connective”?] ...

..., the '''mapping spectrum''' <math>F(X, Y)</math> of [[spectrum (topology)|spectra]] ''X'', ''Y'' is characterized by {{topology-stub}} ...

...heaves]] <math>\pi_* F \to \pi_* G</math> is an isomorphism. A '''sheaf of spectra''' is then a fibrant/cofibrant object in that category. [[Category:Algebraic topology]] ...

...etric spectrum|symmetric spectra]]</ref> category of [[spectrum (topology)|spectra]]. The category of commutative ring spectra over the field <math>\mathbb{Q}</math> of rational numbers is [[Quillen equ ...

In algebraic topology, a '''symmetric spectrum''' ''X'' is a [[spectrum (topology)|spectrum]] of pointed [[simplicial set]]s that comes with an action of the ...mes \Sigma_n</math>. A morphism between symmetric spectra is a morphism of spectra that is equivariant with respect to the actions of symmetric groups. ...

...of, say, a [[spectrum (topology)|spectrum]] ''X'' is the set of all (say) spectra ''Y'' whose [[smash product]] with ''X'' is zero: <math>X \otimes Y = 0</ma The notion applies to [[module spectrum|module spectra]] and in that case one usually qualifies a ring spectrum over which the sma ...

...scipline of [[topology]], the '''Brown–Gitler spectrum''' is a [[Spectrum (topology)|spectrum]] whose [[cohomology]] is a certain [[cyclic module]] over the [[ ...=https://ncatlab.org/nlab/files/GoerssOnBrownGitler.pdf|title=Brown–Gitler Spectra}}</ref> ...

...''S'' is the monoidal unit in the category of [[spectrum (homotopy theory)|spectra]]. It is the [[suspension spectrum]] of ''S''⁰, i.e., a set of t [[Category:Algebraic topology]] ...

In [[algebraic topology]], a '''G-spectrum''' is a [[spectrum (topology)|spectrum]] with an [[group action|action]] of a (finite) group. ...ear=2015 |doi=10.1112/jtopol/jtv005 |volume=8 |issue=2 |journal=Journal of Topology |pages=476–528}} ...

...Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra <math>BP\langle{n}\rangle</math>.<ref>Burklund, Schlank, Yuan (2022). ''The ...d |title=A higher chromatic analogue of the image of J |journal=Geometry & Topology |volume=21 |issue=2 |pages=1033–93 |date=2017 |doi=10.2140/gt.2017.21.1033 ...

...993}}. Similar constructions are defined for maps of [[Spectrum (topology)|spectra]].<ref>{{Cite web |last=Lurie |first=Jacob |date=2010-04-27 |title=Phantom ...grightarrow y</math> in the [[homotopy category]] of [[Spectrum (topology)|spectra]] is called an <math>\alpha</math>-phantom map if, for any spectrum s with ...

...;\mathbb{Z}/p)</math> using [[Eilenberg–Maclane spectrum|Eilenberg–MacLane spectra]]. ...lenberg–Maclane spectra]] giving generators for the cohomology of resolved spectra<ref name=":0">{{Cite book|last=Ravenel, Douglas C.|url=https://www.worldcat ...

...dridge Bousfield, ''The localization of spaces with respect to homology'', Topology vol. 14 (1975)</ref> ===Stable model structure on spectra=== ...

...s |date= |title=Lectures on Topological Hochschild Homology and Cyclotomic Spectra |url=https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/papers.html |a ...lex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, <math>\wedge_\mathbb{S}</math> acts formally like the derived tens ...

...uspension''' is an operation [[Inverse function|inverse]] to [[suspension (topology)|suspension]].<ref>{{cite conference| ...ed desuspension.<ref>{{cite book|author=Margolis|first=Harvey Robert|title=Spectra and the Steenrod Algebra|publisher=[[Elsevier|North-Holland]]|year=1983|isb ...

...y]] of an [[abelian category]] and the ∞-category of [[spectrum (topology)|spectra]] are both stable. ...rresponding notion ([[stabilization (topology)]]) in classical [[algebraic topology]]. ...

{{short description|Theorem relating to algebraic topology}} ...r theorem''', named after [[Peter Landweber]], is a theorem in [[algebraic topology]]. It is known that a [[complex orientation]] of a [[homology theory]] lead ...

|field = [[Mathematics]], [[Algebraic Topology]] ...ref> known as "Pete", was an American mathematician working in [[algebraic topology]], known for the concept of [[Bousfield localization]]. ...

{{Short description|Concept in geometric topology}} ...mathematics]], '''assembly maps''' are an important concept in [[geometric topology]]. From the [[homotopy]]-theoretical viewpoint, an assembly map is a [[Univ ...

In [[algebraic topology]], an '''<math>\mathbb{S}</math>-object''' (also called a '''symmetric sequ ...r-known model for [[Highly structured ring spectrum|highly structured ring spectra]] due to Elmendorf, Kriz, Mandell and May.{{clarify|what’s “considerably be ...