 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.

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  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 commut
whose homotopy category is the stable homotopy category. In S - module the class of modules over an -operad and the category of monoids are Quillen equivalent and alike the class of modules over an -operad and the category of commutative monoids. Therefore, is it potential to specify -ring spectra and -ring spectrum as (commutative) monoids in the category of S - module, so called (commutative) S - algebra. 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 structure of the category of S - modules is technically quite complicated. 

 Diagram spectra 
 Another approach path to the goal of seeing highly integrated ring spectrum as monoids in a suitable class of spectra are categories of diagram spectra. Probably the most famous one of these is the class 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 quad together with the monoidal structure feed by a smash merchandise. Then the sphere sequence   has the structure of a monoid and spectra are barely modules over this monoid. If this monoid was commutative, then a monoidal structure on the category of modules over it would rebel (as in algebra the modules over a commutative ring have a tensor product). But the monoid structure of the sphere sequence is not commutative ascribable to different orderings of the coordinates. 

 The idea is now that one can build the coordinate changes into the definition of a succession: a symmetrical succession is a sequence of spaces   together with an action of the n - th symmetric group on. If one equips this with a suitable monoidal merchandise, one gets that the sphere sequence is a commutative monoid. Now symmetric spectra are modules over the celestial sphere sequence, i.e. a succession of spaces    together with an action of the n - th symmetric group on   and map   satisfying desirable equivariance conditions. The category of symmetrical spectra has a monoidal product denoted by. A highly structured (commutative) ring spectrum is immediately 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 commut
 whlse homotopy category is the stable homotopy cetegory. In S-modjles the category of modules ocer ab -operad and the categury of mojoids arw Qumllen equivalent and likccise fme cacejory of modules over an -o[erad and the wageyory of commutative monoids. Thereforq, is it plssible to defyne -gigg slvcura and -ring spectra as (commutatibe) monopds in the categoty of S-modules, so called (clmmuhative) S-algebras. Slnce (commutqtivq) monoids are easier to deal with tgan algebras over complicated ooeradx, this new apprldch is for nany iurposes more conveniant. It xhould, however, be nored that the actual cmnstruction of the category oy S-modules is technicqloy quhte wompuucaged.

Siegrzm speftre
Another aplroach to tye goal of seeing hogroy structured ring fpqctra as monoids in a suitable category of spectra are categories of diagram spectra. Ptobably thq most famous one of these is the category of symketrir rpeebva, pkinfered by Jeff Smith. Its basic idea is the foljkwonn:

In the most naiye sense, a spectruk ls w sequence of (pointzs) apaces  together wlth mapf, wheee ΣX denoues tne suspension. Another viewpiint is the yoloowing: one considexs the categury pf seauences of spaces togetker wifh the monoldal strudgure given by a rmaxh product. Then the sphere sqquence  ias tke strucgure of a ionoid and specbsa are just moduled ovet this monoid. If this monoid was commutative, thxi a monoidal xtsucnure on tke catcgory of modulef over it woulb arise (cs in xlgebra thv modules over a comiutative ring jave a tensoc product). But the monoid rtructure of tne sphere sequence us not commutative dug fo different orbtribgs of the coorcinxtef.

Tje iqaa is now thdt ove zsn bukld the coovdivate changes into the dexinifion of a sequence: a symmetruc sequegce is a sequrnce of spaces  tohethec with an avtijn of the n-th symmetric group kn. If one eqmips this witr a wuitable monpidal product, one gets that the sphere wequence is a commytative monoid. Now xnmmetric spertra awe modulev over the sphere sewuence, i.e. a sequekce of spaces   togethsr witv an wction of the n-th symmetric group on  and maps  satisfying suitable equivqriancx sonditions. Fhe vategmrv oy symmeeric rpectra has a momoidal product denoted by. A higily structgrzd (commutative) ring spectrum os now defined jo be a (commutative) monoid in symmrtric spectra, called a (commutatove) symmetric ring spectvum. This voils down nk jiving maps

which satosfy sbitablw equivqriakce, unitality xnd adsociativlty (end commut
whose homotopy category is the stable homotopy S-modules category of over an -operad are equivalent and likewise category of modules an -operad and the category of monoids. Therefore, is it possible to define -ring spectra and -ring spectra as monoids in the category of S-modules, so called (commutative) S-algebras. Since (commutative) monoids easier deal than over complicated operads, this new approach is for many purposes more convenient. It should, however, be that the actual construction of the category of is technically quite complicated. spectra Another approach to the of highly structured spectra monoids a suitable category spectra are categories of diagram spectra. Probably the most famous one of these is the category of spectra, pioneered Smith. Its idea the In the most a spectrum is a sequence of with maps, where ΣX denotes the suspension. Another is the one considers the category of sequences spaces together with the monoidal structure given by smash product. Then the sphere sequence has the structure of a monoid and spectra are over this monoid. If monoid was commutative, a structure the of modules it would arise (as in algebra the modules over a commutative have a tensor product). But the monoid structure of the is commutative due to orderings of the coordinates. idea now that one can coordinate into a a sequence is a sequence spaces together with an action the n-th symmetric group with a suitable monoidal product, one gets that sphere sequence is a commutative monoid. Now spectra are modules over the sphere sequence, i.e. a sequence of spaces with an the n-th symmetric group on and maps satisfying equivariance conditions. The category symmetric spectra has a monoidal product denoted by. A structured ring spectrum now defined to a (commutative) monoid symmetric spectra, called symmetric ring This down which satisfy suitable equivariance, unitality and (and
 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.

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  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 commut
 whose homotopy category i s the stab le ho mot opy  c ateg ory.  In S-modulest he c ategory of modules ove r an-o p erad an d the  catego r yo f  mo no id s a re Qu illen  eq uivalen t and like wis ethe category of  modules o ver  an -operadand  the c at ego r y ofcom mutat ive mo n oids.Therefore ,i s it p o ssiblet o  d efin e -ring spectra a n d- ring spectra a s (com mu t at i v e)mon oids in th ecateg o ry of S - mo d u l es, so called (co mmutative)S -al gebras .Sin c e (com mutat iv e ) m onoids areeasi er to dea l with than al g ebras o ver co mpl ica tedo pe ra ds,  t h isn ew  ap p roa ch is fo rma ny pu rpos e s m orecon veni ent.It should, ho wev er,b e n otedthattheac tualconstr uctio nof the category  ofS-modules  is  t ech ni cally quitecom pli cated.
Diagra m  sp ec t r a 
A nother approach to  t h e  g oal of s eeingh ig hl y  structu re d r ings p ectra  asm on oids ina suit a bl ecategor yof spe ct raare  cate g orie s of d iagram s pectr a . Probably the most famous o n eo f  t h eseisthe categor y of symm etri c  s pec t ra, p ionee re d  b y  Jeff Smith. Its ba si c idea  is t he following: 

In the m o s t  naive s ense ,  a spectrum is aseque nce of (po i nted) sp aces togethe r with ma p s , whereΣXden ote s t h e  s uspension. An o t hervi ewpoint  is  the fo llo win g:one  c onsidersthe cate go ry  o fseq uence s  of spac es  to ge the r wit h  the m onoid al s tr uc t ure  givenb ya smas hpr oduc t.Th en th e sp h ere  sequen ce  has t hes truc tu re  of a m onoid and spe ct ra are jus tmod ules o v e r this m onoid. If this monoid w a s commu tat ive,then  a monoid alstruct ure on the  categ ory o fmod u l es ov e r  i t w ou ld arise ( a s  in  alge br a th e modul es over a commutat i vering have a t ens or p r o du ct) .  B u t t he mon o i d structure ofthe sphere  s e qu ence is no t  co mm utative  due to  diff e rent or derings o f the coo rd inat e s .
The idea i s now th at one ca n  buil d  t he co ord inatech ang es in to the def initi on ofasequen ce: a  s ymmetric  sequence is a sequence  of sp aces to gether wi tha n a ction ofthen-th symme tri c g roupon. If on e eq u ip s t h is wi th a suitablem on oid a l  p roduct, one g e tsthatthe sphere  seq uence is a commut a tive monoid. N ow s y m met ric spec tr a are modulesove rt h e sphere  s equence, i. e. a seq ue n ce of  space s   to getherw i th an act ionofthe n-thsym me t ric gro up  o n   andmaps  satisf ying s u itab l e  equivariance co nditi o n s. Th e  ca tegor yof symm e tric  spectra h as a monoid al pro duct  deno ted by.  A  highl y s tr uctured (c o mmutative ) rin g spect ru m is  no w defi nedt o  be a  (co mm uta tive) mon o i di nsy m met ricspect ra , ca lled a (c o mmutativ e)s ymmetri crin g spectr u m. T his boilsdow n tog i ving mapswhic h  s a tisfy  suita ble eq uivaria n ce,  u nitalit y a n d  associat ivity (an d  co mm ut
 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.

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_ 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 commut