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724748c621ed78327ec63544cf17e8ec64299c18 | subsection | 598 | 1,121 | Global model structures for orthogonal spectra | We let G be a group from {\mathcal {F}} and V a finite-dimensional
faithful G-subrepresentation of the complete G-universe {\mathcal {U}}_G.
By Proposition REF (i)
the map f(V):X(V)\longrightarrow Y(V) is a G-weak equivalence.
Since the representation sphere S^V
can be given a G-CW-structure, the induced map on G-homo... | {
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"Stefan Schwede"
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3c9acffc86f1a7386fea2e0fe095470e381d21b0 | subsection | 599 | 1,121 | Global model structures for orthogonal spectra | The cofiber sequence of G-CW-complexesS(W)_+ \ \longrightarrow \ D(W)_+ \ \longrightarrow \ S^Winduces a fiber sequence of equivariant mapping spaces\operatorname{map}_*^G(S^W,X(V)) \ \longrightarrow \ \operatorname{map}^G(D(W),X(V)) \ \longrightarrow \ \operatorname{map}^G(S(W),X(V)) \ .Since W^G=0, the G-fixed points... | {
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"Stefan Schwede"
] | [
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92a804f9643e6e1710b9c213436c9f8cfe397db7 | subsection | 600 | 1,121 | Global model structures for orthogonal spectra | So the map f(V) is G-weakly equivalent to\Omega f(V\oplus {\mathbb {R}}) \ : \ \Omega X(V\oplus {\mathbb {R}})
\ \longrightarrow \ \Omega X(V\oplus {\mathbb {R}})\ .Hence we have a homotopy fiber sequence of G-spacesX(V)\ \xrightarrow{}\ Y(V) \ \longrightarrow \ F(V\oplus {\mathbb {R}}) \ .Since F is an {\mathcal {F}}-... | {
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"Stefan Schwede"
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0b5138bba91f6325ce299dc90d0ad87e4ec5af4d | subsection | 601 | 1,121 | Global model structures for orthogonal spectra | We setK_{\mathcal {F}}\ = \ \bigcup _{G,V,W} {\mathcal {Z}}(\lambda _{G,V,W}) \ ,the set of all pushout products of sphere inclusions i_m:\partial D^m\longrightarrow D^m
with the mapping cylinder inclusions of the morphisms \lambda _{G,V,W}
(compare Construction REF );
here the union is over a set of representatives
of... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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f7d8541b2ef9ef79f78b0fd0954476eae7a3c2b7 | subsection | 602 | 1,121 | Global model structures for orthogonal spectra | The morphism f is thus an {\mathcal {F}}-global fibration and an {\mathcal {F}}-equivalence,
so this provides one of the factorizations as required by MC5.The morphism \lambda _{G,V,W} represents the map(\tilde{\sigma }_{V,W})^G\ :\ X(W)^G\ \longrightarrow \ \operatorname{map}_*^G(S^V,X(V\oplus W))^G\ ;by Proposition R... | {
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fbc247437ee2b6aeca5d4786b97bdf25c6768b87 | subsection | 603 | 1,121 | Global model structures for orthogonal spectra | Now we let j:A\longrightarrow B be an {\mathcal {F}}-cofibration that is also an {\mathcal {F}}-equivalence and
we show that it has the left lifting property with respect
to {\mathcal {F}}-global fibrations.
We factor j=q\circ i,
via the small object argument for J_{\mathcal {F}}\cup K_{\mathcal {F}},
where i:A\longrig... | {
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750e1c3d3ec09cb99eb7bdc3fbefc93cd8f962cc | subsection | 604 | 1,121 | Global model structures for orthogonal spectra | For easier reference we spell out the special case {\mathcal {F}}={\mathcal {A}}ll
for the maximal family of all compact Lie groups,
resulting in the global model structure
on the category of orthogonal spectra.Theorem 3.18 (Global model structure)
The global equivalences, global fibrations and flat cofibrations
form ... | {
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847682f7dd0c78a52cfd72e6eaec43318f0de902 | subsection | 605 | 1,121 | Global model structures for orthogonal spectra | The {\mathcal {E}}-mixed {\mathcal {F}}-global model structure is again proper
(Propositions 4.1 and 4.2 of ).When {\mathcal {F}}=\langle e\rangle is the minimal family of trivial groups,
this provides infinitely many {\mathcal {E}}-mixed model structures on
the category of orthogonal spectra that are all Quillen equiv... | {
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d5cfc512f734801ef1a80c75657ad1c6e92ce9a8 | subsection | 606 | 1,121 | Global model structures for orthogonal spectra | Since equivariant homotopy groups take wedges of orthogonal spectra
to direct sums (Corollary REF (i)),
any wedge of {\mathcal {F}}-equivalences is again an {\mathcal {F}}-equivalence.
So the pointset level wedge maps by
an {\mathcal {F}}-equivalence to the wedge of the {\mathcal {F}}-cofibrant approximations,
and the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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2c51dfc58736ecf2fec17710ef3c200dca97264f | subsection | 607 | 1,121 | Global model structures for orthogonal spectra | Since the right hand side is a product in {\mathcal {GH}}_{\mathcal {F}} of the family \lbrace X_i\rbrace _{i\in I},
so is the left hand side.We turn to the interaction of the smash product of orthogonal spectra
with the level and global model structures.
Given two morphisms f:A\longrightarrow B and g:X\longrightarrow ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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ba84736918e738f6017806e28f6afb244c5fd273 | subsection | 608 | 1,121 | Global model structures for orthogonal spectra | Then the following relations hold for the sets of generating
cofibrations and acyclic cofibrations:I_{\mathcal {E}}\, \Box \, I_{\mathcal {F}}\ \sqsubset \ I_{{\mathcal {E}}\times {\mathcal {F}}} \ ,\quad I_{\mathcal {E}}\, \Box \, J_{\mathcal {F}}\ \sqsubset \ J_{{\mathcal {E}}\times {\mathcal {F}}} \text{\quad and\qu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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ae8f3d6c35e56e82ebbc240c8fd38b754fadbe1a | subsection | 609 | 1,121 | Global model structures for orthogonal spectra | In the special case where {\mathcal {E}}={\mathcal {F}}=\langle e\rangle ={\mathcal {E}}\times {\mathcal {F}}
are the trivial global families, part (iii) of the previous proposition
specializes to Proposition 12.6 of .Proposition 3.24
Let {\mathcal {E}} and {\mathcal {F}} be two global families.The pushout product of ... | {
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c7e08f878690a0a649041a8d07efc4386f463945 | subsection | 610 | 1,121 | Global model structures for orthogonal spectra | So if {\mathcal {F}} is multiplicative, then with respect to the smash product,
the {\mathcal {F}}-global model structure is a
symmetric monoidal model category
in the sense of .
A corollary is that the
homotopy category {\mathcal {GH}}_{\mathcal {F}}, i.e., the localization of the category
of orthogonal spectra at the... | {
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57c60ee0709c5f76926111b64654c5e15c5ec374 | subsection | 611 | 1,121 | Global model structures for orthogonal spectra | So X\wedge f:X\wedge Y\longrightarrow X\wedge Z
is a {\underline{\pi }}_*-isomorphism of orthogonal G-spectra by
Theorem REF , because X is G-flat.
This proves that X\wedge f is again an {\mathcal {F}}-equivalence.(ii)
We let f:X\longrightarrow Y be an {\mathcal {F}}-equivalence between flat orthogonal spectra.
We choo... | {
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413b85957b9d8014f65313d790c4db7dd647b2b4 | subsection | 612 | 1,121 | Global model structures for orthogonal spectra | Since i\wedge Y is an h-cofibration with cokernel isomorphic to (B/A)\wedge Y,
the long exact homotopy group sequence then shows that i\wedge Y is an {\mathcal {F}}-equivalence.The proof that the class of h-cofibrations that are also {\mathcal {F}}-equivalences
is closed under cobase change, coproducts and sequential c... | {
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"Stefan Schwede"
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c6abeb5147c838876a4db9cc8efaa8a6b8f32b2f | subsection | 613 | 1,121 | Global model structures for orthogonal spectra | Then for every global family {\mathcal {F}},
the functor N\wedge _R - takes {\mathcal {F}}-equivalences of left R-modules
to {\mathcal {F}}-equivalences of orthogonal spectra.The argument is completely parallel to the unstable
precursor in Proposition REF .
We call a right R-module N homotopical
if the functor N\wedge ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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aaaf82e480f3f99c677abc4002afb1a97fabadc2 | subsection | 614 | 1,121 | Global model structures for orthogonal spectra | Moreover, since the morphism f_k is a flat cofibration, it is an h-cofibration
(by Corollary REF (iii)),
and so the morphisms f_k\wedge X and f_k\wedge Y are h-cofibrations.
Corollary REF (i)
then implies that the induced morphism on horizontal
pushouts M_k\wedge _R \varphi is again an {\mathcal {F}}-equivalence.Now ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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cdc4f7969b128c9660084f472d1f63d773e13c66 | subsection | 615 | 1,121 | Global model structures for orthogonal spectra | A morphism f:X\longrightarrow Y of orthogonal spectra is a fibration
in the positive global model structure if and only if
for every compact Lie group G, every G-representation V
and every faithful G-representation W with W\ne 0
the map f(W)^G:X(W)^G\longrightarrow Y(W)^G is a Serre fibration and the square@C=13mm{ X(W... | {
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bc95df42fa29c0b3641189a4724bfd6810675189 | subsection | 616 | 1,121 | Global model structures for orthogonal spectra | By (or rather its dual formulation),
an orthogonal spectrum is fibrant in the positive global model structure
if it is equivalent in the positive strong level model structure to
a global \Omega -spectrum; this is equivalent to
being a positive global \Omega -spectrum.
The proof that the positive global model structure ... | {
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"doi": "10.1016/j.topol.2005.02.004",
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dce1d073efddd36dcdb1ee70ff0124f9d2080332 | subsection | 617 | 1,121 | Triangulated global stable homotopy categories | global stable homotopy category|(As the homotopy category of a stable model structure,
the global stable homotopy category {\mathcal {GH}}
comes with the structure of a triangulated category.
The shift functor is the suspension of orthogonal spectra,
and the distinguished triangles arise from mapping cone sequences.
In... | {
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"doi": "10.1090/surv/099",
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"Stefan Schwede"
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7bdba2546d36404765aa86731c4779f75df7155e | subsection | 618 | 1,121 | Triangulated global stable homotopy categories | The distinguished triangles are defined from mapping cone sequences,
i.e., a triangle is distinguished if and only if it is isomorphic,
in {\mathcal {GH}}_{\mathcal {F}}, to a sequence of the formX \ \xrightarrow{}\ Y \ \xrightarrow{} \ C f \ \xrightarrow{}\ X\wedge S^1for some morphism of orthogonal spectra f:X\longri... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2f235ded348b746b788357c88706031d8ac8b5e3 | subsection | 619 | 1,121 | Triangulated global stable homotopy categories | We will now argue that shifting also preserves distinguished
triangles on the nose; equivalently, the derived shift is an exact
functor of triangulated categories if we equip it with the identity isomorphism
\operatorname{sh}\circ (-\wedge S^1)=(-\wedge S^1)\circ \operatorname{sh}.To prove our claim we consider a disti... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4b5a830c990e2db4370030a44a4de37c0375e7e8 | subsection | 620 | 1,121 | Triangulated global stable homotopy categories | An object C of {\mathcal {T}} is compactcompact!object in a triangulated category
if for every family \lbrace X_i\rbrace _{i\in I} of objects the canonical map{\bigoplus }_{i\in I}\, {\mathcal {T}}(C,\, X_i) \ \longrightarrow \ {\mathcal {T}}(C,\, {\oplus }_{i\in I}\, X_i)is an isomorphism.
A set {\mathcal {G}} of obje... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ec0e2b39fc244b1e209a8acc82c20ac3fc8e5521 | subsection | 621 | 1,121 | Triangulated global stable homotopy categories | The suspension spectrum of B_{\operatorname{gl}} G comes with a stable tautological classe_G \ = \ e_{G,V} \in \ \pi _0^G(\Sigma ^\infty _+ B_{\operatorname{gl}} G)defined in (REF ).
stable tautological classIn the proof of the next theorem
we will start using the shorthand notation\llbracket X,Y\rrbracket _{\mathcal {... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6da265a3698c75e22aa576aed0291b9ec9730754 | subsection | 622 | 1,121 | Triangulated global stable homotopy categories | So the localization functor induces a bijection{\mathcal {S}}p(\Sigma ^\infty _+ B_{\operatorname{gl}}G, X) / \text{homotopy} \ \longrightarrow \ \llbracket \Sigma ^\infty _+ B_{\operatorname{gl}}G, X \rrbracket _{\mathcal {F}}from the set of homotopy classes of morphisms of orthogonal spectra
to the set of morphisms i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7ec73e0149d8571fb13dbcdac69b168205832edf | subsection | 623 | 1,121 | Triangulated global stable homotopy categories | This proves that the
spectra \Sigma ^\infty _+ B_{\operatorname{gl}}G form a set of weak generators
for {\mathcal {GH}}_{\mathcal {F}} as G varies over {\mathcal {F}}.A covariant functor E from a triangulated category {\mathcal {T}}
to the category of abelian groups is called
homologicalhomological functor if
for every... | {
"cite_spans": [
{
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"doi": "",
"end": 1881,
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"raw": "A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9 (1996), 205–236.",
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"Stefan Schwede"
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8a7d30e0e1366857e32603af176ff5b2d8680fec | subsection | 624 | 1,121 | Triangulated global stable homotopy categories | Once this representing data is chosen, the assignment X\mapsto R X extends
canonically to a functor R:{\mathcal {S}}\longrightarrow {\mathcal {T}} that is right adjoint to F.
In much the same way, part (iv) is a formal consequence of part (ii).We let {\mathcal {T}} be a triangulated category with sums.
A localizing sub... | {
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"Stefan Schwede"
] | [
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4f3fa34ba098173c49ab57e92339f5e854313f15 | subsection | 625 | 1,121 | Triangulated global stable homotopy categories | For an infinite indexing set, the morphism{\prod }_{i\in I}\, f_i\ : \ {\prod }_{i\in I}\, X_i\ \longrightarrow \ {\prod }_{i\in I}\, X_i^\text{f}may fail to be a global equivalence, and then the target,
but not the source, of this map is a product in {\mathcal {GH}}
of the family \lbrace X_i\rbrace _{i\in I}.
So when ... | {
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"Stefan Schwede"
] | [
"math.AT"
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67639d3b5e759ac1ca997fb6e418be7d3f6ac4b6 | subsection | 626 | 1,121 | Triangulated global stable homotopy categories | For every object X of {\mathcal {T}} there is a distinguished triangle
A \ \longrightarrow \ X \ \longrightarrow \ B \ \longrightarrow \ A[1]
such that A\in {\mathcal {T}}_{\ge 0} and B\in {\mathcal {T}}_{\le -1}.The t-structure is non-degenerate
if every object in the intersection \bigcap _{n\in {\mathbb {Z}}} {\mat... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1878296a60fca610c120fae42977c9df94ab537c | subsection | 627 | 1,121 | Triangulated global stable homotopy categories | \mathcal {GF}_{\mathcal {F}} - category of {\mathcal {F}}-global functorsTheorem 4.9
For every global family {\mathcal {F}},
the classes of {\mathcal {F}}-connective spectra and {\mathcal {F}}-coconnective spectra
form a non-degenerate t-structure on {\mathcal {GH}}_{\mathcal {F}}.
The heart of this t-structure
consis... | {
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09dee6f50e82d143f5716dbd28622fba5935fa67 | subsection | 628 | 1,121 | Triangulated global stable homotopy categories | Representability (Theorem REF (i))
and the suspension isomorphism provide an isomorphism\llbracket \Sigma ^\infty _+ B_{\operatorname{gl}} G, \Sigma ^\infty _+ B_{\operatorname{gl}} K[n]\rrbracket _{\mathcal {F}}\ \cong \ \pi _0^G(\Sigma ^\infty _+ B_{\operatorname{gl}}K[n]) \ \cong \ \pi _{-n}^G(\Sigma ^\infty _+ B_{... | {
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"Stefan Schwede"
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7268291cff3c38f49db916e063e50eb5a083250e | subsection | 629 | 1,121 | Triangulated global stable homotopy categories | Because {\mathcal {P}} is a set of positive, compact generators
for the triangulated category {\mathcal {GH}}_{\mathcal {F}},
shows that the restriction of the tautological functor (REF )
to the heart is an equivalence of categories\ H \ \xrightarrow{} \ \operatorname{mod-}\operatorname{End}({\mathcal {P}}) \ .So to e... | {
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"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6cad9fe5c71a91172844857224efff95bb3b719d | subsection | 630 | 1,121 | Triangulated global stable homotopy categories | Indeed, a choice of inverse to the equivalence {\underline{\pi }}_0 of
Theorem REF , composed with the inclusion of
the heart, provides an Eilenberg-Mac Lane functorH \ : \ \mathcal {GF}\ \longrightarrow \ {\mathcal {GH}}to the global stable homotopy category.We let {\mathcal {T}} be a triangulated category with infin... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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05dffe23c0e65a15dbe98389d78b2c6bf1ed257b | subsection | 631 | 1,121 | Triangulated global stable homotopy categories | We start withA_0 \ = \ \bigoplus _{P\in {\mathcal {P}}, x\in \llbracket P,X\rrbracket } \, P\ .Then A_0 belongs to \langle {\mathcal {P}}\rangle _+ and the canonical morphism
u_0:A_0\longrightarrow X (i.e., the morphism x on the summand indexed by x)
induces a surjection \llbracket P,u_0\rrbracket :\llbracket P,A_0\rrb... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d3c22cbf9b371cc23190981c48b0b6d06e7f6944 | subsection | 632 | 1,121 | Triangulated global stable homotopy categories | The map\llbracket P,u_0\rrbracket \ = \ \llbracket P,u\rrbracket \circ \llbracket P,\varphi _0\rrbracket \ : \ \llbracket P,A_0\rrbracket \ \longrightarrow \ \llbracket P,X\rrbracketis surjective for P\in {\mathcal {P}},
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"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f2e679829909be77ef1598b2e44e0a3a31c30433 | subsection | 633 | 1,121 | Triangulated global stable homotopy categories | The universal property of the box product produces a morphism of
global functors{\underline{\pi }}_0(X) \, \Box \, {\underline{\pi }}_0 (Y) \ \longrightarrow \ {\underline{\pi }}_0(X\wedge Y) \ .We recall from (REF ) that
the symmetric monoidal derived smash product \wedge ^{\mathbb {L}}
on the global stable homotopy c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4ea1df2fc3f8982bae30ce1e80d09cb3d8b083f8 | subsection | 634 | 1,121 | Triangulated global stable homotopy categories | Since -\Box {\underline{\pi }}_0(\Sigma ^\infty _+ B_{\operatorname{gl}} K) is right exact
(by Remark REF ),
the upper row in the commutative diagram@C=5mm{
({\underline{\pi }}_0 A) \Box {\underline{\pi }}_0(\Sigma ^\infty _+ B_{\operatorname{gl}} K) [r][d] &
({\underline{\pi }}_0 B) \Box {\underline{\pi }}_0(\Sigma ^\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.047406043857336044,
... |
3e2997a65ac0e4ab900d2d9d541a4625120fc1c8 | subsection | 635 | 1,121 | Triangulated global stable homotopy categories | Proposition REF
then shows that {\mathcal {X}} is the class of all globally connective orthogonal spectra.
This proves the proposition in the special case Y=\Sigma ^\infty _+ B_{\operatorname{gl}} KNow we perform the same argument in the other variable.
We fix a globally connective spectrum X and let {\mathcal {Y}} de... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ab49179d0afc33ecae1b895fbd3df934ec9d3452 | subsection | 636 | 1,121 | Triangulated global stable homotopy categories | The following lemma is a direct consequence of
Proposition REF (i).Lemma 4.17
The morphism \lambda ^V_{F_{G,V}}:F_{G,V}\wedge S^V\longrightarrow \operatorname{sh}^V F_{G,V}
takes the element a_{G,V} to the class in \pi _0^G(\operatorname{sh}^V F_{G,V})
that is represented by the G-fixed
point (0,\operatorname{Id}_V)\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fd1c502c365d074543a80c02467d1ba8ad59e599 | subsection | 637 | 1,121 | Triangulated global stable homotopy categories | The orthogonal spectrum F_{G,V} is flat, and hence cofibrant
in the global model structure.
So the localization functor induces a bijection{\mathcal {S}}p(F_{G,V}, E) / \text{homotopy} \ \longrightarrow \ \llbracket F_{G,V}, E \rrbracketfrom the set of homotopy classes of morphisms of orthogonal spectra
to the set of m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b21ad579d742600cfd8ab5b706aaecb7b3d7699b | subsection | 638 | 1,121 | Triangulated global stable homotopy categories | Other examples are U(m)unitary group or S U(m)special unitary group
(the latter for m\ge 2)
acting on the underlying {\mathbb {R}}-vector
space of {\mathbb {C}}^{m}, with stabilizer groups U(m-1) respectively S U(m-1).
Similarly, we can consider the tautological representation of S p(m)symplectic group
on the underlyin... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-74311-8",
"end": 932,
"openalex_id": "https://openalex.org/W4242522601",
"raw": "A. L. Besse, Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Springer-Verlag, Berlin, 1987. xii+510 pp.",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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873fa9bdc4beddf83fb3febcfcd527590ef675b3 | subsection | 639 | 1,121 | Triangulated global stable homotopy categories | The value of the morphism i at an inner product space W is then the mapi(W) \ : \ F_{H,L}(W) = {\mathbf {O}}(L,W)/H
\ &\longrightarrow \ {\mathbf {O}}(V,W\oplus {\mathbb {R}})/G \ = \ (\operatorname{sh}F_{G,V})(W) \\
(w,\varphi )\cdot H \qquad &\longmapsto \qquad ((w,0),(\varphi \oplus {\mathbb {R}})\circ \psi ^{-1})\c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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-0... |
c9e83a2a04f7b5ba2bbe3b906194cef9c98e9be2 | subsection | 640 | 1,121 | Triangulated global stable homotopy categories | We identify the mapping cone of q
with the sphere S^{V} via the G-equivariant homeomorphismh \ : \ C q\ \cong \ S^{V}that is induced by the mapG/H\times [0,1] \ \longrightarrow \ S^V \ , \quad (g H,x) \ \longmapsto \ g \cdot (1-x)/x \cdot v \ .Under this identification the mapping cone inclusion i:S^0\longrightarrow C ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
2717a22151647d82ac740c2c4122b51a0424e409 | subsection | 641 | 1,121 | Triangulated global stable homotopy categories | The composite\Sigma ^\infty _+ B_{\operatorname{gl}} G\ \xrightarrow{}\ F_{G,V} S^V\ \xrightarrow{} \ F_{G,V}(G/H_+\wedge S^1)
\ \xrightarrow{}\ F_{H,L}coincides with the morphism T, by direct inspection
of the effects at the inner product space V.Our next claim is that the following diagram
of orthogonal spectra commu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
92a31e09c4646127aba21d09eb4009ff18b0c561 | subsection | 642 | 1,121 | Triangulated global stable homotopy categories | Since G-maps out of
G/H correspond to H-fixed points,
this in turn reduces to the claim that the two mapsS^1 \ \longrightarrow \ \left( {\mathbf {O}}(V,V\oplus {\mathbb {R}})/G\right)^{H}
\ = \ ( (\operatorname{sh}F_{G,V})(V))^{H}that send t\in S^1 to( (-t\cdot v,0), i_0)\cdot G \text{\qquad respectively\qquad }
( (0,t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.002... |
174b24dcfd7ec0326f7605b122f29868c36992b2 | subsection | 643 | 1,121 | Triangulated global stable homotopy categories | We consider the wide G-equivariant embeddingG/H \ \longrightarrow \ V \ , \quad g H\ \longmapsto \ g v\ .This embedding was already used to identify the tangent space T_{e H}(G/H)
at the preferred coset with the subspace L inside V;
the inclusion L\longrightarrow V corresponds to the differential at e H.
The orthogonal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
75a6d801e3d2152e6ff6c321e08d48cfe1dfae32 | subsection | 644 | 1,121 | Triangulated global stable homotopy categories | We denote by f\diamond {\mathbb {R}} the composite H-mapS^U\wedge S^1 \ \xrightarrow{} \ E(U\oplus L)\wedge S^1\ \xrightarrow{} \ E(U\oplus L\oplus {\mathbb {R}})\ \xrightarrow[\cong ]{E(U\oplus \psi )} \ E(U\oplus V) \ .The class G\boxtimes _H\langle f\rangle in \pi _0^G(G\ltimes _H E)
is then represented by the compo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
9ef9352107c2f0954912497b099573e91261a7d4 | subsection | 645 | 1,121 | Triangulated global stable homotopy categories | Any relative homotopy between these two functions
induces a based G-equivariant homotopy between the representative
of the class \operatorname{Tr}_H^G(a_{H,L}) and the map r
defined in (REF ). So r itself is a representative
of the class \operatorname{Tr}_H^G(a_{H,L}).
Since T:\Sigma ^\infty _+ B_{\operatorname{gl}}G\l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
298994a6e27fc65818e60e4ba9ebd4fcd710e2c8 | subsection | 646 | 1,121 | Change of families | In this section we compare the global stable homotopy categories for two
different global families {\mathcal {F}} and {\mathcal {E}}, where we suppose that {\mathcal {F}}\subseteq {\mathcal {E}}.
Then every {\mathcal {E}}-equivalence is also an {\mathcal {F}}-equivalence,
so we get a `forgetful' functor on the homotopy... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0007... |
833cba71f484b404868dafddec1441a45a45796b | subsection | 647 | 1,121 | Change of families | Here, however, the adjoints are not fully faithful as soon
as the group G is non-trivial.The global family {{\mathcal {F}}in} of finite groups is an important example
to which the discussion of this section applies.
We will show that rationally, the associated {{\mathcal {F}}in}-global stable homotopy category
admits a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aba94b62b2be71bca21f95601e54c2a95ad8cf65 | subsection | 648 | 1,121 | Change of families | But instead of arguing by hand that U preserves products,
we give an alternative construction
of the left adjoint by model category theory.
Indeed, it is immediate from the definitions of {\mathcal {F}}-equivalences
and {\mathcal {F}}-global fibrations that the identity functor is a right Quillen functor
from the {\mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0097438",
"end": 536,
"openalex_id": "https://openalex.org/W4236256974",
"raw": "D. Quillen, Homotopical algebra. Lecture Notes in Mathematics, Vol. 43, Springer-Verlag, 1967. iv+156 pp.",
"source_ref_id": "4c0900e5d5eb23dd... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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87e7bd5af4deed2cf9f01700ae344962ce9b9350 | subsection | 649 | 1,121 | Change of families | Indeed, for every pair of orthogonal spectra A and B
this strong monoidal structure and the adjunction counits provide a morphismU \left( ( R A ) \wedge ^{\mathbb {L}}_{\mathcal {E}}( R B ) \right) \ \cong \ U (R A ) \wedge _{\mathcal {F}}^{\mathbb {L}}U (R B)
\ \xrightarrow{}\ A \wedge _{\mathcal {F}}^{\mathbb {L}}Bwh... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1215/s0012-7094-96-08317-9",
"end": 1873,
"openalex_id": "https://openalex.org/W2055982028",
"raw": "A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astéris... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d7be4fdfdea94bf7ccca3bb1c03ff20cc64266cc | subsection | 650 | 1,121 | Change of families | Here {\mathcal {GH}}({\mathcal {E}};{\mathcal {F}}) denotes the `{\mathcal {E}}-global homotopy category with
support outside {\mathcal {F}}', i.e., the full subcategory of {\mathcal {GH}}_{\mathcal {E}}
of spectra all of whose {\mathcal {F}}-equivariant homotopy groups vanish.
The functor i_*:{\mathcal {GH}}({\mathcal... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0002-9947-1973-0341469-9",
"end": 1757,
"openalex_id": "https://openalex.org/W2008020084",
"raw": "K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc. 186 (1974), 419–458.",
"source... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
af5ba98fafe84948dcf5afdf80515a2f2135e2c5 | subsection | 651 | 1,121 | Change of families | The result is that there
are no `exotic' invertible objects, i.e., the only smash invertible
objects of {\mathcal {GH}} are the suspensions and desuspensions of the global sphere spectrum.
The same is true more generally for the {\mathcal {F}}-global stable homotopy
category relative to any multiplicative global family... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
e7594a353e8d42b6c7dbb30dbf9fba428d7cff68 | subsection | 652 | 1,121 | Change of families | So \epsilon _X is an isomorphism, and hence \operatorname{Pic}(P)[X]=[P X]=[X].Now we have all necessary ingredients to determine the Picard
group of the {\mathcal {F}}-global stable homotopy category.Theorem 5.5
For every multiplicative global family {\mathcal {F}},
the Picard group of the {\mathcal {F}}-global stabl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.00... |
7c1768cad4402af2bd8292d00ead90eca7842ae0 | subsection | 653 | 1,121 | Change of families | Similarly, an orthogonal spectrum is
right inducedright induced
from {\mathcal {F}} if it is in the essential image of the right adjoint R_{\mathcal {F}}:{\mathcal {GH}}_{\mathcal {F}}\longrightarrow {\mathcal {GH}}.We start with a criterion, for certain `reflexive' global families,
that characterizes the left induced ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.006574448198080063,... |
3677c97f8e4fcb79bcae68753feac3a2eca9d8ad | subsection | 654 | 1,121 | Change of families | We need to show that {\mathcal {X}} coincides with the class of spectra left induced from {\mathcal {F}}.Geometric fixed point homotopy groups commute with sums and take exact triangles
to long exact sequences. So {\mathcal {X}} is closed under sums and triangles, i.e.,
it is a localizing subcategory of the global homo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.045647989958524704,
0.0... |
f408a3406a5b040d70797e3dfc2049f81699b6fa | subsection | 655 | 1,121 | Change of families | The adjunction counit \epsilon _X:L( U X )\longrightarrow X is an {\mathcal {F}}-equivalence,
so it induces isomorphisms of geometric fixed point groups for all groups in {\mathcal {F}}.
By the hypothesis on X and naturality of the inflation maps p^*,
the morphism \epsilon _X induces isomorphisms
of geometric fixed poi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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588c8a5fd1920e0bf24fb77f2a5a4a3edc06f110 | subsection | 656 | 1,121 | Change of families | Indeed, geometric fixed points commute with suspension spectra in the
following sense:
if A has trivial G-action, then\pi _* ( \Sigma ^\infty A ) \ \cong \ \Phi ^G_*(\Sigma ^\infty A)\ ,compare Example REF .
So the suspension spectrum \Sigma ^\infty A has `constant geometric fixed points',
and it is left induced from t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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47c2395cc2d8c4e89ee2def0c21e7293b9a43ad8 | subsection | 657 | 1,121 | Change of families | This abuse of notation is justified by the fact that the value of the prolongation
at n_+ is canonically homeomorphic to the original value,
see Remark REF .
The coend can be calculated by a familiar quotient
space construction in the ambient category of all topological spaces,
compare Proposition REF :
F(K) can be obt... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ae054357357f39118d8859cb2f30999084cd860d | subsection | 658 | 1,121 | Change of families | The O(V)-action on F({\mathbb {S}})(V) is via the action on S^V and the continuous
functoriality of F.Proposition 5.15
Let F be a {\mathbf {\Gamma }}-space and G a compact Lie group.The projection p:G\longrightarrow \pi _0 G=\bar{G} to the group of path components
induces an isomorphism of geometric fixed point homoto... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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af93758c0f3f32fc86ebeca527570394fc33eb3c | subsection | 659 | 1,121 | Change of families | The argument for k<0 is similar.(ii)
The global family {{\mathcal {F}}in} of finite groups is reflexive, and
for every compact Lie group K the projection K\longrightarrow \pi _0 K
to the finite group of path components is universal with respect to {{\mathcal {F}}in}.
Part (i) verifies the geometric fixed point criterio... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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71a2201de9b01c2076c88454171e8c460fdd1cb6 | subsection | 660 | 1,121 | Change of families | HenceX^k_G( A\times &E({\mathcal {F}}\cap G)) \ = \ \llbracket \Sigma ^\infty _+ {\mathbf {L}}_{G,V}( A\times E({\mathcal {F}}\cap G)),X[k]\rrbracket \\
&\cong \ \llbracket L_{\mathcal {F}}(U_{\mathcal {F}}(\Sigma ^\infty _+ {\mathbf {L}}_{G,V} A)) ,X[k]\rrbracket \ \cong \ \llbracket \Sigma ^\infty _+ {\mathbf {L}}_{G... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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56b5af715a5ce7dc4afe6062bb2270e6c9805b82 | subsection | 661 | 1,121 | Change of families | Then an orthogonal spectrum X is in the essential image
of the relative right adjoint R:{\mathcal {GH}}_{\mathcal {F}}\longrightarrow {\mathcal {GH}}_{\mathcal {E}} if and only if
for every group G in {\mathcal {E}} and every cofibrant G-space A the mapX^*_G(A)\ \longrightarrow \ X^*_G(A\times E({\mathcal {F}}\cap G))i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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74471fd4ecb2d892920bf666b9e67349b427daab | subsection | 662 | 1,121 | Change of families | So by the previous paragraph the map
X_G^0(\Pi ):X^0_G(A)\longrightarrow X^0_G(A\times E G)
is bijective.For k>0 we apply the same argument to the
global \Omega -spectrum \operatorname{sh}^k X (which also has cofree levels)
and exploit the natural isomorphism(\operatorname{sh}^k X)^0_G(A)\ = \ \llbracket \Sigma ^\infty... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0040-9383(88)90002-x",
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"raw": "J. F. Adams, J.-P. Haeberly, S. Jackowski, J. P. May, A generalization of the Atiyah-Segal completion theorem. Topology 27 (1988), no. 1,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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85e4e65ce97ae43a9128cbdfc9793ff96b4d3c95 | subsection | 663 | 1,121 | Change of families | For a compact Lie group G and a cofibrant G-space A,
its value isE^*(E G\times _G A) \ ,the E-cohomology of the Borel construction
(also known as homotopy orbit construction).
Here E G is a universal free G-space, which is unique up to equivariant
homotopy equivalence.
We claim that these Borel cohomology theories asso... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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33030abdae6ca5478c8f26e6f550fa4c764bc4fe | subsection | 664 | 1,121 | Change of families | This data gives rise to a composite morphism of global classifying spacesB_{\operatorname{gl}}\alpha \ : \ B_{\operatorname{gl}} K \ = \ {\mathbf {L}}_{K,\alpha ^*(V)\oplus W} \ \xrightarrow{}\ {\mathbf {L}}_{K,\alpha ^*(V)} \ \xrightarrow{}\ {\mathbf {L}}_{G,V} \ = \ B_{\operatorname{gl}}G\ .The first morphism restric... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0040-9383(75)90029-4",
"end": 1811,
"openalex_id": "https://openalex.org/W1982228919",
"raw": "J. C. Becker, D. H. Gottlieb, The transfer map and fiber bundles. Topology 14 (1975), 1–12.",
"source_ref_id": "9f9624c068e2f0a04ca... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0eeee8cfba5b86d3cb47fcdb0b54a022038b7ecc | subsection | 665 | 1,121 | Change of families | This proves the claim that the `global Borel theories' are
precisely the ones right induced from non-equivariant stable homotopy theory.Construction 5.21
We introduce a specific pointset level liftb \ : \ {\mathcal {S}}p\ \longrightarrow \ {\mathcal {S}}pof the right adjoint R:{\mathcal {SH}}\longrightarrow {\mathcal ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1ac22ce9ffc79185f1587612e729dbd0fcb18821 | subsection | 666 | 1,121 | Change of families | Since {\mathbf {L}}(V,{\mathbb {R}}^\infty ) is contractible, the morphism
i_E:E\longrightarrow b E is a non-equivariant level equivalence,
hence a non-equivariant stable equivalence.The next result shows that the global Borel construction b
takes \Omega -spectra to global \Omega -spectra,
and that the functor b realiz... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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76d264288800257f69185a00292dd40d1da4b82a | subsection | 667 | 1,121 | Change of families | Moreover, the restriction map
\operatorname{res}_W:{\mathbf {L}}(V\oplus W,{\mathbb {R}}^\infty )\longrightarrow {\mathbf {L}}(W,{\mathbb {R}}^\infty )
is a G-homotopy equivalence (by Proposition REF (ii)),
hence it induces another G-homotopy equivalence\operatorname{map}(\operatorname{res}_W,\Omega ^V E(V\oplus W))\ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5267e63a8224d6774771f2286e26dec2142f0256 | subsection | 668 | 1,121 | Change of families | Since the morphism i_E:E\longrightarrow b E
becomes an isomorphism in {\mathcal {SH}}, it induces another bijection on {\mathcal {SH}}(A,-).We endow the functor b with a lax symmetric monoidal transformation\mu _{E,F}\ : \ b E\wedge b F \ \longrightarrow \ b(E\wedge F)\ .To construct \mu _{E,F} we start from the (O(V)\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cd46c1c64e9ec89a31308ebc7a866d3c063ac5df | subsection | 669 | 1,121 | Change of families | For every compact Lie group G it induces a ring homomorphism
of G-equivariant homotopy groups\pi _0^G(E) \ \longrightarrow \ \pi _0^G( b E ) \ \cong \ E^0( B G ) \ .When E={\mathbb {S}} is the sphere spectrum and G is finite,
Carlsson's theorem
(proving the Segal conjecture) shows that the map{\mathbb {A}}(G)\cong \pi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 921,
"openalex_id": "",
"raw": "G. Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjecture. Ann. of Math. (2) 120 (1984), no. 2, 189–224.",
"source_ref_id": "16728a11aac6fa12ff0a85e5f3badf0f7f805142",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7c49af160e24823dd5a4cdf3afdb5578e653239e | subsection | 670 | 1,121 | Change of families | Since \hat{{\mathbb {S}}} is non-equivariantly stably equivalent to {\mathbb {S}},
this shows that for every group G the equivariant homotopy group\pi _k^G(b E) \ \cong \ E^{-k}(B G)is naturally a module over the commutative ring \pi _0^G( \hat{{\mathbb {S}}}).For the global K-theory spectrum
(compare Construction REF ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 725,
"openalex_id": "",
"raw": "M. F. Atiyah, G. B. Segal, Equivariant K-theory and completion. J. Differential Geom. 3 (1969), 1–18.",
"source_ref_id": "0800d99a1e2db92a7f0b6cfa213361818bf604b0",
"start": 261
},
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e277512a7f84984766d20320649de8975a9495a8 | subsection | 671 | 1,121 | Change of families | We will show now that the forgetful functor has both a left and a right adjoint.The `equivariant' smash product of orthogonal G-spectra
is simply the smash product of the underlying non-equivariant
orthogonal spectra with diagonal G-action.
So the trivial action functor (-)_G:{\mathcal {S}}p\longrightarrow G{\mathcal {... | {
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{
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"doi": "",
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"source_ref_id": "1fe5cf0f78b8a5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d73fd38d01013489e7af41a44193754808626e14 | subsection | 672 | 1,121 | Change of families | Since global \Omega -spectra are the fibrant objects
in a model structure underlying {\mathcal {GH}},
the pointset level product \prod _{i\in I} X_i then represents
the product in {\mathcal {GH}}.Even though X_i is a global \Omega -spectrum,
the underlying orthogonal G-spectrum (X_i)_G need not
be a G-\Omega -spectrum.... | {
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{
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"source_ref_id": "1fe5cf0f78b8a5a... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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696a20de96539795e971283e47b99d4bfe8a6968 | subsection | 673 | 1,121 | Change of families | We choose a faithful G-representation V and let\Omega ^V \operatorname{sh}^V \ : \ {\mathcal {S}}p\ \longrightarrow \ G{\mathcal {S}}pdenote the functor that takes an orthogonal spectrum X to the
orthogonal G-spectrum with U-th level( \Omega ^V \operatorname{sh}^V X )(U) \ = \ \operatorname{map}_*(S^V, X(U\oplus V))\ .... | {
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{
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"source_ref_id": "1fe5cf0f78b8a5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1b8a16a4c6cb1a59241446a5c1ee4fb30f004c66 | subsection | 674 | 1,121 | Change of families | The sequence of natural bijections{\mathcal {GH}}(L(\Sigma ^\infty _+ G/H), X) \ \cong \ G\text{-}{\mathcal {SH}}(\Sigma ^\infty _+ G/H,\, U X) \ \cong \ \pi _0^H (X) \ \cong \ {\mathcal {GH}}(\Sigma ^\infty _+ B_{\operatorname{gl}}H , X)shows that the left adjoint L takes the unreduced suspension spectrum
of the coset... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a9e1d2ed411e370e9750ed54b3a46020a15d6555 | subsection | 675 | 1,121 | Change of families | The same arguments as
in Theorem REF show the existence of both
adjoints to this forgetful functor, with the same kind of monoidal properties.Theorem REF discusses the maximal case
of the global family {\mathcal {F}}={\mathcal {A}}ll of all compact Lie groups.
The minimal case is the global family \langle G\rangle gene... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4e6932b8491a744befa9a546793e1f27c3ac0624 | subsection | 676 | 1,121 | Change of families | We recall from Proposition REF
that the evaluation map{\mathbf {A}}(G,K) \ \longrightarrow \ \pi _0^K(\Sigma ^\infty _+ B_{\operatorname{gl}} G) \ , \quad \tau \longmapsto \tau (e_G)is an isomorphism, where e_G\in \pi _0^G(\Sigma ^\infty _+ B_{\operatorname{gl}} G)
is the stable tautological class.
More precisely, the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4a24e2f53546e1995116e955e35747f40f7e4251 | subsection | 677 | 1,121 | Change of families | So when both G and K are finite, then
also the equivariant homotopy group \pi _k^K(\Sigma ^\infty _+ B_{\operatorname{gl}} G) is torsion
for all k>0.The conclusion of Proposition REF
is no longer true if we drop the finiteness hypothesis on one
of the two groups G or K.
For example, for G=e we have \Sigma ^\infty _+ B... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s0305004102006126",
"end": 1842,
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"raw": "J. D. Christensen, M. Hovey, Quillen model structures for relative homological algebra. Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c8f8eab2ac0ea6545f3a5be41e23d010f704d4d3 | subsection | 678 | 1,121 | Change of families | We call a morphism f:X\longrightarrow Y of orthogonal spectra a
rational {\mathcal {F}}-equivalenceF-equivalence@{\mathcal {F}}-equivalence!rational
if the map{\mathbb {Q}}\otimes \pi _k(f)\ : \ {\mathbb {Q}}\otimes \pi _k^G(X)\ \longrightarrow \ {\mathbb {Q}}\otimes \pi _k^G(Y)is an isomorphism for all integers k and ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s002090100347",
"end": 1276,
"openalex_id": "https://openalex.org/W3105110639",
"raw": "S. Schwede, B. Shipley, A uniqueness theorem for stable homotopy theory. Math. Z. 239 (2002), 803–828.",
"source_ref_id": "0ecc9f5d45cf5ff... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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19da38efaab9104b186d9b18411b737098529876 | subsection | 679 | 1,121 | Change of families | If k is any integer, then
the morphism vector spaces between two such objects are given by\llbracket (\Sigma ^\infty _+ B_{\operatorname{gl}} K)_{\mathbb {Q}}[k],\ (\Sigma ^\infty _+ B_{\operatorname{gl}} G)_{\mathbb {Q}}\rrbracket \ &\cong \ \pi _k^K((\Sigma ^\infty _+ B_{\operatorname{gl}} G)_{\mathbb {Q}}) \ \cong \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0040-9383(02)00006-x",
"end": 844,
"openalex_id": "https://openalex.org/W1976086282",
"raw": "S. Schwede, B. Shipley, Stable model categories are categories of modules. Topology 42 (2003), no. 1, 103–153.",
"source_ref_id": "... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4b084fdf4964e5baf348804c33f017307c3d43c6 | subsection | 680 | 1,121 | Change of families | In the constant global functor \underline{{\mathbb {Q}}} we have\operatorname{tr}_e^{C_2}(1)\ =\ 2\cdot p^*(1)
\text{\qquad in\qquad $\underline{{\mathbb {Q}}}(C_2)={\mathbb {Q}}$\ ,}where p:C_2\longrightarrow e is the unique group homomorphism.
So for every morphism of global functors \varphi :\underline{{\mathbb {Q}}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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818952cc4922543779394cf6e268a15c09be3eaa | subsection | 681 | 1,121 | Change of families | If \alpha :K\longrightarrow G is a surjective group homomorphism and
H\le G a proper subgroup, then L=\alpha ^{-1}(H) is a proper
subgroup of K and the relation\alpha ^*\circ \operatorname{tr}_H^G \ = \ \operatorname{tr}_L^K \circ (\alpha |_L)^*\ : \ M(H)\longrightarrow M(K)shows that the inflation
map \alpha ^*:M(G)\l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6a77a906a7841d0e6c33cda8aa98c2e8248665af | subsection | 682 | 1,121 | Change of families | For every closed subgroup H of G the restriction map \operatorname{res}_H^G
is a morphism in {\mathbf {A}}(G,H). If H is finite, the
Yoneda lemma provides a unique morphism\operatorname{Out}_H \ \longrightarrow \tau ({\mathbf {A}}_G)of \operatorname{Out}-functors from the representable functor
\operatorname{Out}_H={\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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dfb9c192cc1e66bf1aac159825a57324d98027a5 | subsection | 683 | 1,121 | Change of families | So (\tau {\mathbf {A}}_G)(K) is a free abelian group with basis the
classes of \alpha ^* for all conjugacy classes of
homomorphisms \alpha :K\longrightarrow G.On the other hand, the group
( \operatorname{Out}_H / W_G H )(K) is free abelian with basis
given by W_G H-orbits of conjugacy classes of epimorphisms \alpha :K\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fdaed0db5f6ae0e79af3ed9884bfbed356ccb419 | subsection | 684 | 1,121 | Change of families | The global functoriality in G is via \tau , i.e., as the composite{\mathbf {A}}(G,K)\otimes (\rho X)(G)\ &\longrightarrow \ \mathcal {GF}({\mathbf {A}}_K,{\mathbf {A}}_G)\otimes (\rho X)(G)\\
&\xrightarrow{} \ \operatorname{mod-}\operatorname{Out}(\tau ({\mathbf {A}}_K),\tau ({\mathbf {A}}_G))\otimes (\rho X)(G)\ \xrig... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
a446fdd1c25e14bcb86c1edcd142251b1ca7e952 | subsection | 685 | 1,121 | Change of families | So the group\operatorname{mod-}\operatorname{Out}(\operatorname{Out}_G,X) \ \cong \ X(G)is a direct summand of the group\operatorname{mod-}\operatorname{Out}(\tau ({\mathbf {A}}_G),X) \ \cong \ \mathcal {GF}({\mathbf {A}}_G,\rho X) \ \cong \ (\rho X)(G) \ ,and this splitting is natural for morphisms of \operatorname{Ou... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c8d358bf2508d57604021ab8dad2961a3c72abb3 | subsection | 686 | 1,121 | Change of families | When we apply Proposition REF
to the underlying orthogonal G-spectrum of an orthogonal spectrum,
it specializes to the following:Corollary 5.37
For every orthogonal spectrum X, every finite group G and every integer k
the mapgeometric fixed points!rationally\bar{\Phi }\ : \ \tau ( {\underline{\pi }}_k(X) ) (G) \ \lon... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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047b9917a164c39e45063f48831df885c5b55240 | subsection | 687 | 1,121 | Ultra-commutative ring spectra | This chapter is devoted to ultra-commutative ring spectra,
our model for extremely highly structured, multiplicative global stable
homotopy types.
On the point set level, these objects are simply
commutative orthogonal ring spectra;
we use the term `ultra-commutative' to emphasize
that we care about their homotopy theo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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085a50a17b49afdc2105b2af272c8e8275459673 | subsection | 688 | 1,121 | Power operations | global power functor|(In this section we introduce the formal setup for encoding
the power operations on ultra-commutative ring spectra.
In Definition REF we define global power functors,
which are global Green functors equipped with additional power operations,
satisfying a list of axioms reminiscent of the properties... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1105,
"openalex_id": "https://openalex.org/W1559076581",
"raw": "N. Ganter, Global Mackey functors with operations and n-special lambda rings. arXiv:1301.4616",
"source_ref_id": "972fe53c7ddbdf54e94048311040ecd489e03ddc",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9ba40a63de400e0b71ab6aeeaf3ab3f5a4c5112d | subsection | 689 | 1,121 | Power operations | Since we will work with power operations a lot,
we take the time to expand the definition: the operation P^m takes the class
represented by a based G-map f:S^V\longrightarrow R(V), for some G-representation V,
to the class of the (\Sigma _m\wr G)-mapS^{V^m} = (S^V)^{\wedge m} \ \xrightarrow{} \ R(V)^{\wedge m} \ \xrigh... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0a6feef1c18ca3eb6acfe26afa6e43dc7576df8e | subsection | 690 | 1,121 | Power operations | The definition makes use of certain embeddings between products
and wreath products:\Phi _{i,j}\ : \ (\Sigma _i\wr G)\ \times \ (\Sigma _j\wr G)\hspace*{36.98866pt} &\longrightarrow \quad \Sigma _{i+j}\wr G \\
((\sigma ;\, g_1,\dots ,g_i),\, (\sigma ^{\prime };\, g_{i+1},\dots ,g_{i+j})) \ &\longmapsto \ (\sigma +\sigm... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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01417bc220af55a2f8e1db2537202bdb097211db | subsection | 691 | 1,121 | Power operations | (Additivity)
For all compact Lie groups G, all m\ge 1, and all x,y\in R(G)
the relation
P^m(x+y) \ = \ \sum _{k=0}^m\ \operatorname{tr}_{k,m-k} (P^k(x)\times P^{m-k}(y))
holds in R(\Sigma _m\wr G), where \operatorname{tr}_{k,m-k} is the transfer associated
to the embedding \Phi _{k,m-k}:(\Sigma _k\wr G)\ \times \ (\S... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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953460e1f074723a55f32d36919bda234022585e | subsection | 692 | 1,121 | Power operations | We suppose that [G:H]=m,
and we choose an H-basis of G, i.e., an ordered m-tuple
\bar{g}=(g_1,\dots ,g_m) of elements in disjoint H-orbits such thatG \ = \ {\bigcup }_{i=1}^m \ g_i H \ .The wreath product \Sigma _m\wr H acts freely and transitively
from the right on the set of all such H-bases of G, by the formula(g_1,... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a5c1f3b858cf97d3e0b7ab13d7c1cbe659697831 | subsection | 693 | 1,121 | Power operations | Here [g] runs over a set of representatives of the finite
set of K-H-double cosets.
(Inflation)
For every continuous epimorphism \alpha :K\longrightarrow G of compact Lie groups
the relation
\alpha ^*\circ N_H^G \ =\ N_L^K\circ (\alpha |_L)^*
holds as maps from R(H)\longrightarrow R(K), where L=\alpha ^{-1}(H).
For... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1080/00927879308824627",
"end": 1109,
"openalex_id": "https://openalex.org/W2148871441",
"raw": "D. Tambara, On multiplicative transfer. Comm. Algebra 21 (1993), no. 4, 1393–1420.",
"source_ref_id": "9f46095e1c34837144cf06b100e0a1f... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
6c8573e4b62263bdb3ccae68faa248f2a28390e6 | subsection | 694 | 1,121 | Power operations | We define the G-equivariant R-homology group of A
as the groupequivariant homology group!of an orthogonal spectrumR_0^G(A) \ = \ \pi _0^G(R\wedge A)\ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)} \, [S^V, R(V)\wedge A ]^G\ .Every continuous group homomorphism \alpha :K\longrightarrow G induces a
restriction homom... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6d79880eacb41632303a72e1c0ebe8236fe4c0a0 | subsection | 695 | 1,121 | Power operations | Then we define the m-th power operationP^m \ : \ R_0^G(A)\ \longrightarrow \ R_0^{\Sigma _m\wr G}(A^{(m)})by the obvious generalization of (REF ):
the operation P^m takes the class
represented by a based G-map f:S^V\longrightarrow R(V)\wedge A, for some G-representation V,
to the class of the (\Sigma _m\wr G)-mapS^{V^m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
8a607aa30567b12e8de8ac921856373850d5da92 | subsection | 696 | 1,121 | Power operations | The differential of \gamma at (e H,\dots ,e H)
is a (\Sigma _m\wr H)-equivariant linear isometry(d \gamma )_{(e H,\dots ,e H)}\ : \ L^m \ \cong \ T_{e(\Sigma _m\wr H)} \big ( (\Sigma _m\wr G) / (\Sigma _m\wr H) \big )\ .In the next proposition and its corollaries, we will use this
equivariant isometry to identify L^m w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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65b331b1d06a67b32f8db48a118cec6d2d0de946 | subsection | 697 | 1,121 | Power operations | The collapse map\lambda _H^G\ = \ l_H^G/H \ : \ G/H_+ \ \longrightarrow \ S^Lis then given by the formula\lambda _H^G(g H) \ = \ {\left\lbrace \begin{array}{ll}
l / (1-|l|) & \text{ if $g=s(l)\cdot h$ with $(l,h)\in D(L)\times H$, and }\\
\quad \ast & \text{ if $g$ is not of this form.}
\end{array}\right.}We define a s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
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