Orthogonal Spectra

Lasttime : sequential spectraEfo-

Spaces Spectra Quillen pair

to play nicely with homotopy theory

Spectra# Ab

HA T Awith nth

space Klan)

H faithfully embeds Ab into spectra

H

Spectra c- AbgC. Rings

A- ring is a monoid in ( Ab,④,1)

Need to define a symmetric monoidal product on Sp

Prolate there is a finite list of desirable

(properties for a category C withhold

soon asthe stable homotopy category

a monoidalproduct

Lewisproves you can only ever have 4 off

properties . Impossible to have all of them .

Remote Actually , you can with co- cats

Models of spectra←

a category C that has-

: 4 of 5 properties- symmetric spectra- orthogonal spectra- EKMM spectra- T - spaces

-

a spectrumhis connective

if inX -Ofor no

T - spacesonly give

connective spectra

G

Def: An orthogonal spectrum X consists of the followingGxoln)-orthogonal gp ofdata :

- pointed spacesXn with an) - action

"n matrices

- maps on : Xn AS'

→ Anti

such that the composite onem.io - -- o on

Gx #Olm) Crim)

Xn n SM → Xntm

is Oln) x Olm) E Olntm) - equivariantGx Oln) xocm)

0cm) Asm if we think of SM as the l - pt compactificationof th?

Ocn) × 0cm) ↳ Olntm)

block diagonal [oh AIDEI . A morphism of orthogonal spectra f :X-Y

is a

sequenceof Cn) - equivariant maps Xn try

sodh that the following diagram connotes :f. aid

Ann SM-yn nsm

to fwm doXntm→ Ynem

Examples: $ nthspace

Sn = I -pt compactification

ring spectrumof IRN

g. gm→5" let ON AIR

" ES 's Bioscopee.my Yu) init;¥g spectrum basepoint

Eilenberg - Maclane Spectra HA

d?! HA Han = AEST = { Eaisilaietfn }is

; fun think

AfiRn3oAo@3SuspensimSpedra.E:X

nthspace

Snax

00cm trivial action

Def : An orthogonal ring spectrum R is an orthogonalspectrum R together with

ugraded Mmm : Rn n Rm→ Them OHXQemqluiua.intmultiplication

"

in .

.

gn→ Rn for nzo 0cm - equivarianthas associativity ,

unit properties

Associativeinµ

Rn n Rm n Rp → Rn n Rm tp

Lum daM

Rum " Rp- Rntmtp

unitid n io Mn . o

Rn I Rp AS°-7 Rn n Ro- Rn

( and the symmetric thing)

multiplicativeTy Mn ,mo inn in = in em

g n Sms Rn n Rm A- Rn -in

centrality µ

Rmnsn id Rmn Rn → Rnnenype.IE!!I

aterm

Sna Rm¥4 Rn n Rm → Rnem

2-nm

= [Inku

&U re

-

Properties of stable Homotopy Category :-Hols

p) is the stable homotopy category-

it is abelian

finite products = finite coproductshas all kernels f- fibers)

all cokernels f- co fibers)fiber seqs A →B → C one cofiber seqs

←associated w/

-it is triangulated chain complexeshas a shift functor E

shift in the other direction Dhas a notion of "distinguished triangle "which yields long exact sequences

n = ④§- it is closed symmetric monoidal f

has a monoidal product A ← spreadsheetw/ unit $

has an internal horn Fix ,y)← fusiform

map(Xi'D

FIX Miz) t FCK, FLYZH

,

Spaces Spectrato

poop = comin R"

An

EIN - Ey)

Homen,

( A.,Bo ) = ¥

,

degree n horns

Hom ( KY ) includes maps of degree ¥00hdsp)

X- EY

oE -IR on

X. as'

→ * , TED.imXo→ Map!?:X )

DX,

look = colin (X. Era ,Essex

,Es - - - - - )

In Book = af CX) ⇒ * ( E X)

Cory Malkiewicz has a good overview on his webpageStefan Schwed - Lectures on Equivariant stable

homotopy theory

Birgit Richter - Commutative Ring spectra

Handbook of Htry Theory (on nlab)

( real , orthogonal)

G - equivariant spectra should be indexed ont G - reps

let V be a G -rep ,

X on orthogonal spectrum

Xlv ) = linllthn, VI. A Xn

OLD

htpy "groups" of G - spectra are Mackey functors