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fbd0a43ffe47cead6ab30ea66aad88c2d82a18e9 | subsection | 98 | 1,121 | Global families | A morphism of orthogonal spaces is:
an acyclic fibration in the {\mathcal {F}}-global model structure
if and only if it has the right lifting property
with respect to the set I_{{\mathcal {F}}};
a fibration in the {\mathcal {F}}-global model structure
if and only if it has the right lifting property
with respect to t... | {
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"raw": "J. A. Lind, Diagram spaces, diagram spectra and spectra of units. Algeb. Geom. Topol. 13 (2013), 1857–1935.",
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17265ea4657720e63d2a1ce39f0f666c8d5d2599 | subsection | 99 | 1,121 | Global families | Since f and j are {\mathcal {F}}-equivalences, so is q by
Proposition REF (iii);
so q is an {\mathcal {F}}-equivalence and a global fibration,
hence an {\mathcal {F}}-level equivalence by
Proposition REF (xiii).(i)\Longleftrightarrow (iii) The morphism f is an {\mathcal {F}}-equivalence
if and only if the {\mathcal {... | {
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cbf70a453e3d12c5c56892800baaf4ed6893232f | subsection | 100 | 1,121 | Global families | Given two morphisms f:A\longrightarrow B and g:X\longrightarrow Y of orthogonal spaces
we denote byf\Box g = (f\boxtimes Y)\cup (B\boxtimes g) \ : \ A\boxtimes Y\cup _{A\boxtimes X}B\boxtimes X \ \longrightarrow \ B\boxtimes Ythe pushout product morphism.pushout product
We recall that a model structure on a symmetric m... | {
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409b899b64f90524086a9c4a98f55b3ebd282d33 | subsection | 101 | 1,121 | Global families | The pushout product of two such generators is isomorphic to the morphism{\mathbf {L}}_{G\times K,V\oplus W}\times i_{k+m} \ : \ {\mathbf {L}}_{G\times K,V\oplus W}\times \partial D^{k+m}
\ \longrightarrow \ {\mathbf {L}}_{G\times K,V\oplus W}\times D^{k+m} \ ,compare Example REF .
Since G\times K belongs to the family ... | {
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9e2ea4b5368a2c126e201a9eed0481de93a5ae50 | subsection | 102 | 1,121 | Global families | For every flat cofibration i:A\longrightarrow B that is also an {\mathcal {F}}-equivalence
and every orthogonal space Y the morphismi\boxtimes Y \ : \ A\boxtimes Y \ \longrightarrow \ B\boxtimes Yis an h-cofibration and an {\mathcal {F}}-equivalence.h-cofibration
Moreover, the class of h-cofibrations that are also {\ma... | {
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b67c2ffc9cfa189d78179a8ca8feb2994ee0d795 | subsection | 103 | 1,121 | Global families | An orthogonal monoid space R is
commutativeorthogonal monoid space!commutative if moreover
\mu \circ \tau _{R,R}=\mu , where \tau _{R,R}:R\boxtimes R\longrightarrow R\boxtimes R
is the symmetry isomorphism of the box product.
A morphism of orthogonal monoid spaces is a morphism of orthogonal spaces
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2e668c1c88c7b442410d8f3edd3d4268417a4a6d | subsection | 104 | 1,121 | Global families | If the underlying orthogonal space of R is {\mathcal {F}}-cofibrant,
then every cofibration of R-modules is an {\mathcal {F}}-cofibration of underlying
orthogonal spaces.
If R is commutative, then with respect to \boxtimes _R
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31c1749fe48ec414d61bc3d23dd0f87236a19a2d | subsection | 105 | 1,121 | Global families | This concludes the proof that every cofibration of R-modules
is an {\mathcal {F}}-cofibration of underlying orthogonal spaces.Strictly speaking, Theorem 4.1 of
does not apply verbatim to the {\mathcal {F}}-global model structure
because the hypothesis that every object is small (with respect to some regular cardinal)
... | {
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b539ff1b296bb84e7aba62a8136133cc6ff0f582 | subsection | 106 | 1,121 | Global families | Then M_k\boxtimes _R \varphi
is obtained by passing to horizontal pushouts in the following commutative diagram
of orthogonal spaces:@C=15mm{
M_{k-1}\boxtimes _R X [d]_{M_{k-1}\boxtimes _R \varphi } &
A_k \boxtimes X[d]^{A_k\boxtimes \varphi }[l][r]^-{f_k\boxtimes X}
& B_k \boxtimes X[d]^{B_k\boxtimes \varphi }\\
M_{k... | {
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bda6bbffae974c0565435a8128de0c6059c6586d | subsection | 107 | 1,121 | Equivariant homotopy sets | In this section we define the equivariant homotopy sets \pi _0^G(Y)
of orthogonal spaces and relate them by restriction maps defined from
continuous homomorphisms between compact Lie groups.
As the Lie groups vary, the resulting structure is a `Rep-functor' {\underline{\pi }}_0(Y),
i.e., a contravariant functor
from th... | {
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22d1ed2fc48302a8b17246e68621d438975a618c | subsection | 108 | 1,121 | Equivariant homotopy sets | For every pair of orthogonal spaces X and Y and
every G-space A, the canonical map
([A,p_X]^G,[A,p_Y]^G)\ : \ [A,X\times Y]^G \ \longrightarrow \ [A,X]^G \times [A, Y]^G
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c2ccd1ea2b6df6668d69e9f1a19b4f2358371376 | subsection | 109 | 1,121 | Equivariant homotopy sets | A choice of such a homotopy specifies an equivariant lifting problem on the left:@C=12mm{
A\times \lbrace 0,1\rbrace [r]^-{g,g^{\prime }} [d] & X(V) [d]^{f(V)} &
A\times \lbrace 0,1\rbrace [r]^-{g,g^{\prime }}[d] & X(V) [r]^-{X(\varphi )} &
X(W) [d]^{f(W)}
\\
A\times [0,1][r]_-\beta & Y(V) &
A\times [0,1][r]_-\beta @{-... | {
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3ac5af85a49b2c38f2827c449c8d95a8d8c0c0e0 | subsection | 110 | 1,121 | Equivariant homotopy sets | We let A be a compact K-space and consider the composite[A, B_{\operatorname{gl}}G]^K \ \xrightarrow{} \ [A, (B_{\operatorname{gl}}G)({\mathcal {U}}_K) ]^K \ \xrightarrow{} \ \operatorname{Prin}_{(K,G)}(A)\ ,where the first map is the bijection of Proposition REF (i),
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"Stefan Schwede"
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] | 2,018 | en | Mathematics | [
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696ea47562d890ef9941e0c4e8d7986c7a2e53c1 | subsection | 111 | 1,121 | Equivariant homotopy sets | We denote by \alpha ^* the
restriction functor from G-spaces to K-spaces
(or from G-representations to K-representations)
along \alpha , i.e., \alpha ^* Z (respectively \alpha ^* V)
is the same topological space as Z
(respectively the same inner product space as V) endowed with
K-action viak\cdot z \ = \ \alpha (k)\cdo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e6d0f2b0d8795de934774205642ef5586039c356 | subsection | 112 | 1,121 | Equivariant homotopy sets | We let W be the span of V, V^{\prime } and V^{\prime \prime } inside {\mathcal {U}}_G.
We can then view j, j^{\prime } and j^{\prime \prime } as equivariant linear isometric embeddings
from U to W.Since the images of j and j^{\prime \prime } are orthogonal,
they are homotopic through G-equivariant linear isometric embe... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f4da82f203f0491dd2562ad5d430863d918fc686 | subsection | 113 | 1,121 | Equivariant homotopy sets | Here we consider a closed subgroup H of G, an element g\in G and denote byc_g \ : \ H \ \longrightarrow \ H^g\ , \quad c_g(h)\ =\ g^{-1}h g\the conjugation homomorphism, where
H^g=\lbrace g^{-1} h g\ | \ h\in H\rbrace is the conjugate subgroup.
As any group homomorphism,
c_g induces a restriction mapc_g - conjugation b... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e4133773481cb753889518141d76e944cef36deb | subsection | 114 | 1,121 | Equivariant homotopy sets | Then \alpha and \alpha ^{\prime } belong to the same path component of the space \hom (K,G)
of continuous homomorphisms, and so they are conjugate by an element of G,
compare Proposition REF .We denote by Rep \operatorname{Rep} - category of compact Lie groups and conjugacy classes of homomorphisms
the category whose o... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01446292",
"end": 2228,
"openalex_id": "https://openalex.org/W2030880744",
"raw": "R. K. Lashof, Equivariant bundles. Illinois J. Math. 26 (1982), no. 2, 257–271.",
"source_ref_id": "7498aab94f240104ecf0dc39d9c62983b2ac52e2"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9a74e212efe1f187e1cc02b145dd6bd2300e073c | subsection | 115 | 1,121 | Equivariant homotopy sets | For a continuous homomorphism \alpha :K\longrightarrow G, we let C(\alpha )
denote the centralizer, in G, of the image of \alpha , and we setE^\alpha \ = \ \lbrace x\in E\ |\ (k,\alpha (k))\cdot x = x\text{ for all $k\in K$}\rbrace \ ,the space of fixed points of the graph of \alpha .
Since the G-action on the universa... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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83d479e120ea8694683c25fb040644849c6ef648 | subsection | 116 | 1,121 | Equivariant homotopy sets | This morphism is a global equivalence of orthogonal spaces by
Proposition REF (ii),
as long as G acts faithfully on W.Proposition 5.13
Let {\mathcal {C}} be a category and{ \Lambda \ : \ spc\ @<.4ex>[r] & \ {\mathcal {C}}\ : \ U @<.4ex>[l] }an adjoint functor pair such that the composite functor
U\Lambda :spc\longrig... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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79dcdbfe3289409f78e64a246e3639dda37656f3 | subsection | 117 | 1,121 | Equivariant homotopy sets | This G-fixed point is adjoint to a morphism of orthogonal spaces\hat{x}\ : \ {\mathbf {L}}_{G,V\oplus W} \ \longrightarrow \ U Xand hence adjoint to a {\mathcal {C}}-morphismx^\flat \ : \ \Lambda ({\mathbf {L}}_{G,V\oplus W}) \ \longrightarrow \ Xthat satisfies\pi _0^G(U x^\flat )(u^{\mathcal {C}}_{G,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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82902897ab210fdac74d1fd4ed925fcbc771ea85 | subsection | 118 | 1,121 | Equivariant homotopy sets | Since[x] \ = \ \pi _0^G(U x^\flat )(\pi _0^G(U\Lambda (\rho _{G,V,W}))^{-1}(u^{\mathcal {C}}_{G,W})) \ ,naturality yields that\tau [x] \ &= \ \tau (\pi _0^G(U x^\flat )(\pi _0^G(U\Lambda (\rho _{G,V,W}))^{-1}(u^{\mathcal {C}}_{G,W})))\\
&= \ \pi _0^K(U x^\flat )(\pi _0^K(U\Lambda (\rho _{G,V,W}))^{-1}(\tau (u^{\mathcal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b38f7465a58eac35b8d2ceb4c15d2bd132b355af | subsection | 119 | 1,121 | Equivariant homotopy sets | Theny \ = \ (U X)(\varphi \oplus W)(x)\text{\qquad in\quad } (U X)(V^{\prime }\oplus W)^Gis another representative of the class [x].
The restriction morphism\varphi ^\sharp \ = \ {\mathbf {L}}(\varphi \oplus W,-)/G\ : \ {\mathbf {L}}_{G,V^{\prime }\oplus W} \ \longrightarrow \ {\mathbf {L}}_{G,V\oplus W}makes the follo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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31658aaf575ea6059ee80ae635724de06e64e524 | subsection | 120 | 1,121 | Equivariant homotopy sets | Moreover, the adjoint of (U\psi )(V\oplus W)(x) coincides with the composite\Lambda ({\mathbf {L}}_{G,V\oplus W}) \ \xrightarrow{}\ X \ \xrightarrow{}\ Y\ .So naturality follows:\tau (\pi _0^G(U\psi )[x])\ &= \ \pi _0^K(U\psi \circ U x^\flat )(\pi _0^K(U\Lambda (\rho _{G,V,W}))^{-1}(z)) \\
&= \ \pi _0^K(U\psi )\left( \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a068603b89b9f1f7bfd9bc9d4f4d424cf6803070 | subsection | 121 | 1,121 | Equivariant homotopy sets | We denote by x\times y the image of the G-fixed point (x,y)
under the G-mapi_{V,W}\ : \ X(V)\times Y(W)\ \longrightarrow \ (X\boxtimes Y)(V\oplus W)that is part of the universal bimorphism.
If \varphi :V\longrightarrow V^{\prime } and \psi :W\longrightarrow W^{\prime } are equivariant linear isometric embeddings,
thenX... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e0062fb3c0e7f0bbb95a36c6bc9e8142a4fc09c8 | subsection | 122 | 1,121 | Equivariant homotopy sets | Part (iii) exploits that the square@C=18mm{
X(V)\times Y(W) [r]^-{i_{V,W}}[d]_{\text{twist}}&
(X\boxtimes Y)(V\oplus W)[d]^{\tau (\chi _{V,W})} \\
Y(W)\times X(V) [r]_{i_{W,V}} &
(Y\boxtimes X)(W\oplus V)}commutes. The image of (x,y) under the upper right composite represents
\tau _*(x\times y), whereas the image of (y... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4541a301db4dd67e189215a698370dc6edf0f28b | subsection | 123 | 1,121 | Equivariant homotopy sets | We let I be an indexing set and \lbrace X_i\rbrace _{i\in I}
a family of based orthogonal spaces, i.e., each equipped with a
distinguished point x_i\in X_i(0).
If K\subset J are two nested, finite subsets of I, then
the basepoints of X_k for k\in J-K provide a morphism\boxtimes _{k\in K} X_k \ \longrightarrow \ \boxtim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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85f0391813e11eb91dab561642bbe1bcf2f7d59d | subsection | 124 | 1,121 | Equivariant homotopy sets | Then for every compact Lie group G the map (REF )
is bijective.For every k\in I we define a `projection'\Pi _k \ : \ \boxtimes ^{\prime }_{i\in I} X_i \ \longrightarrow \ X_kas follows. Since the infinite box product is defined as a colimit,
we must specify the `restriction' of \Pi _k to \boxtimes _{j\in J}X_j
for ever... | {
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"Stefan Schwede"
] | [
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83c38f84e82cc05cf33387b2e75dc6f9aa17a546 | subsection | 125 | 1,121 | Equivariant homotopy sets | In other words, the canonical map\operatorname{colim}_{J\subset I,|J|<\infty } \pi _0^G(\boxtimes _{j\in J} X_j) \ \longrightarrow \ \pi _0^G(\boxtimes ^{\prime }_{i\in I} X_i)is surjective.
For finite sets J the map \prod _{j\in J}\pi _0^G(X_j)\longrightarrow \pi _0^G(\boxtimes _{j\in J} X_j)
is bijective by Corollary... | {
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"Stefan Schwede"
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bcb89475485a241c99aebfb1fdfb9110ed6ef53c | subsection | 126 | 1,121 | Ultra-commutative monoids | Orthogonal monoid spaces are the lax monoidal continuous functors
from the linear isometries category {\mathbf {L}} to the category of spaces.
Orthogonal monoid spaces with strictly commutative multiplication
(i.e., the lax symmetric monoidal functors)
play a special role, and we honor this
by special terminology, refe... | {
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de0d0028343a0df93d26a089ae733cf8590794af | subsection | 127 | 1,121 | Ultra-commutative monoids | We refer to this structure as a `global power monoid';
it consist of an abelian monoid structure
on the set \pi _0^G(R) for every compact Lie group G,
natural for restriction along continuous homomorphisms,
and additional structure that can equivalently be encoded
as power operations (see Definition REF )
or as transfe... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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e1e37ced924b64b4eb042e2c67ffbce7265875b2 | subsection | 128 | 1,121 | Global model structure | In this section we formally define ultra-commutative monoids
and establish various formal properties of the category umon of
ultra-commutative monoids.
We introduce free ultra-commutative monoids in Example REF .
The main result is the model structure with global equivalences as the weak equivalences,
see Theorem REF .... | {
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00370921e7570cfe03056848ca9d869c1290189e | subsection | 129 | 1,121 | Global model structure | A linear isometric embedding \psi :{\mathcal {U}}\longrightarrow {\mathcal {U}}^{\prime } between
countably infinite dimensional inner product spaces induces a map
R(\psi ) : R({\mathcal {U}}) \longrightarrow R({\mathcal {U}}^{\prime }); the resulting `action map'{\mathbf {L}}({\mathcal {U}},{\mathcal {U}}^{\prime })\... | {
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b3e86de4cfaf838044005e1e045ee7d7cbbb5b3b | subsection | 130 | 1,121 | Global model structure | The forgetful functor from the category of ultra-commutative monoids
to the category of orthogonal spaces creates all limits,
all filtered colimits and those coequalizers that are reflexive in
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72dc539cbb4bc2ff60ef8e5034da649713c33360 | subsection | 131 | 1,121 | Global model structure | Since every inner product space is isometrically isomorphic
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"Stefan Schwede"
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e8c3bf6ff3fbbd863904be71915234693060998a | subsection | 132 | 1,121 | Global model structure | One way to construct a tensor R\otimes A is as a coequalizer,
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{\mathbb {P}}( ({\mathbb {P}}R)\times A)\quad @<.4ex>[r]^-{{\mathbb {P}}(\alpha \times A)}
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0adb6780a332985e7f278634aa166f52c650030b | subsection | 133 | 1,121 | Global model structure | We can also consider the realization |B|_\text{in}
internal to ultra-commutative monoids,
i.e., a coend, in the category of ultra-commutative monoids, of the functor{\mathbf {\Delta }}^{\operatorname{op}}\times {\mathbf {\Delta }}\ \longrightarrow \ umon \ , \quad ([m],[n])\ \longmapsto \ B_m\otimes \Delta ^n \ .We cal... | {
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8f6ba5fd206189e292dd6bfa2234b17c91704ccb | subsection | 134 | 1,121 | Global model structure | Indeed, since \boxtimes preserves colimits
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c7ba77ddeec094a5c40df12e368f7778659a498f | subsection | 135 | 1,121 | Global model structure | Moreover, the coequalizer is reflexive in the underlying category of orthogonal spaces,
by the morphisms{
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cc8e97129bf40b18d010052938e44dd6801f86b0 | subsection | 136 | 1,121 | Global model structure | We shall write R\rhd A for the tensor of an ultra-commutative monoid R
with a based space (A,a_0), in order to distinguish it
from the (objectwise) smash product of
the underlying based orthogonal space of R with A.
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efa336a43182ed4745ab938a3d822c5cff03f7d3 | subsection | 137 | 1,121 | Global model structure | Then R\rhd |A| is an internal realization of the simplicial ultra-commutative monoid
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a7ae5919c2619e26b1071ad10f2d1386d4447e01 | subsection | 138 | 1,121 | Global model structure | More explicitly, if S\subseteq \lbrace 1,2,\dots ,n\rbrace is a subset,
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5dd1bc32d016e1b27a9f8d09f40ac6761281783d | subsection | 139 | 1,121 | Global model structure | Indeed, for n=2 the cube K^2(i) is a square and looks like{
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58cbd0db693698cd84332a31ef8c7767452aaca9 | subsection | 140 | 1,121 | Global model structure | We will now proceed to prove that in the category of orthogonal spaces,
all cofibrations and acyclic cofibrations in the positive global model structure
are symmetrizable with respect to the box product.
The next proposition will be used to verify this
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e8230244c4dbdd6f0b5ef4c6da22aad54e45b188 | subsection | 141 | 1,121 | Global model structure | So {\mathbb {P}}^n preserves the homotopy relation, and hence also homotopy equivalences.(ii) We argue by induction on k.
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b4dcd56608592e4ae5792f7c8b39dffcd63b21df | subsection | 142 | 1,121 | Global model structure | The induced morphism{\mathbb {P}}^n(A\times D^{k+1})\ \longrightarrow \ {\mathbb {P}}^n( A\times D^{k+1}\cup _{A\times \partial D^{k+1}} B\times \partial D^{k+1})is then a weak equivalence by .
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fcf1af51ea5bb2dd4cf8801bf24ac0e2c8df6b95 | subsection | 143 | 1,121 | Global model structure | In other words, all acyclic cofibrations in the positive global model structure
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56f06dc5e08e9c49e12188389bd9d97c903b16c8 | subsection | 144 | 1,121 | Global model structure | Here the wreath product \Sigma _n\wr G acts on V^n by(\sigma ;\, g_1,\dots ,g_n)\cdot (v_1,\dots , v_n) \ = \ (g_{\sigma ^{-1}(1)}v_{\sigma ^{-1}(1)},\dots , g_{\sigma ^{-1}(n)}v_{\sigma ^{-1}(n)}) \ .The map i_k^{\Box n} is \Sigma _n-equivariant,
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7604421358324d0c69e5939e8c1bc58df835de2f | subsection | 145 | 1,121 | Global model structure | By or
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85f22d35d52a6ae2146988739f74fb0b458288d4 | subsection | 146 | 1,121 | Global model structure | And indeed, for no n\ge 2 is the morphism
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c43ac86136005eb053469cf03f7eda4e57dec95e | subsection | 147 | 1,121 | Global model structure | Theorem 3.2 of thus shows that the positive global
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a381a18c10b36fd7d51d106d8cd3c76b67e5322a | subsection | 148 | 1,121 | Global model structure | In other words,
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b3f761059b84b45de98633351ba2d081346eae5e | subsection | 149 | 1,121 | Global model structure | For ultra-commutative monoids this
means that a pushout square has the form@C=15mm{
R [d]_j [r]^-f_-\simeq & T[d]^{j\boxtimes _R T}\\
S[r]_-{S\boxtimes _R f} & S\boxtimes _R T}where S and T are
considered as R-modules by restriction along j respectively f.
For left properness we now suppose that j is a cofibration and ... | {
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ee848d041373bab98e45adf3bf806621bdbd2239 | subsection | 150 | 1,121 | Global model structure | \end{array} \right.All morphisms in the cube K^n(i) are smash products of identities and
copies of the morphism i:A\longrightarrow B.
We let Q^n(i) denote the colimit of the punctured n-cube,
i.e., the cube K^n(i) with the terminal vertex removed, and
i^{\Box n}:Q^n(i)\longrightarrow K^n(i)(\lbrace 1,\dots ,n\rbrace )=... | {
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88252fb456fc72b7d8dc7e8e16a7154122ac468e | subsection | 151 | 1,121 | Global model structure | In other words, all acyclic cofibrations in the positive global model structure
of orthogonal spectra are symmetrizable.(i) We recall from Theorem REF (iii)
the setI_{{\mathcal {A}}ll} \ = \ \lbrace \ G_m( ( O(m)/H\times i_k )_+) \ | \ m,k \ge 0, H\le O(m)\rbraceof generating flat cofibrations of orthogonal spectra, w... | {
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a68f17e457c647364d4132e2bcd8e1303e76c7fa | subsection | 152 | 1,121 | Global model structure | So the morphism (REF ) is a flat cofibration.(ii) Theorem REF (iii) describes a
set J_{{\mathcal {A}}ll}\cup K_{{\mathcal {A}}ll} of generating acyclic cofibrations for the global model structure
on the category of orthogonal spectra.
From this we obtain a set J^+\cup K^+
of generating acyclic cofibration for the posi... | {
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d752ac1d581d43a8159ff36fac63cae467dda434 | subsection | 153 | 1,121 | Global model structure | So the morphism\lambda _{\Sigma _n\wr G,V^n, W^n} \ : \ F_{\Sigma _n\wr G, V^n\oplus W^n}\, S^{V^n} \ \longrightarrow \ F_{\Sigma _n\wr G, W^n}is a global equivalence by Theorem REF .
The vertical morphisms in the commutative square@C=17mm{
F_{\Sigma _n\wr G, V^n\oplus W^n} \, S^{V^n}
[r]^-{ \lambda _{\Sigma _n\wr G,V^... | {
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a5ee2a19dad14e4291e70516ad3f93933d0175a2 | subsection | 154 | 1,121 | Global model structure | Cofibrations and acyclic cofibrations are symmetrizable by
Theorem REF ,
so the model structure satisfies the `commutative monoid axiom'
of .
The symmetric algebra functor {\mathbb {P}} commutes with filtered colimits
by the analog of Corollary REF
for ultra-commutative ring spectra.
Theorem 3.2 of thus shows that the... | {
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ed4107feb03260e21b74afd30b8926d230ab7db5 | subsection | 155 | 1,121 | Global model structure | The strategy is the one that we have employed several times before:
the functor \pi _0^G from ultra-commutative ring spectra to sets
is representable, namely by \Sigma ^\infty _+ {\mathbb {P}}(B_{\operatorname{gl}}G),
the unreduced suspension spectrum of the free ultra-commutative monoid
generated by B_{\operatorname{g... | {
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"Stefan Schwede"
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6839481aad48efe9698d2cd201dddd9fa7b539f1 | subsection | 156 | 1,121 | Global model structure | So the induced morphism of unreduced suspension spectra
\Sigma ^\infty _+{\mathbb {P}}(\rho _{G,V,W}) is a global equivalence by
Corollary REF .
In particular, the morphism of \operatorname{Rep}-functors
{\underline{\pi }}_0(\Sigma ^\infty _+{\mathbb {P}}(\rho _{G,V,W})) is an isomorphism. So
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f9ec28d790dac7a5f77db993b57578e51a5fdb0d | subsection | 157 | 1,121 | Global model structure | So together this shows that
\pi _0^K(\Sigma ^\infty _+ {\mathbb {P}}(B_{\operatorname{gl}} G)) is a free abelian group with
basis the classes\operatorname{tr}_L^K(\sigma ^L(\alpha ^*([m](u_G)))) \ &= \ \operatorname{tr}_L^K(\alpha ^*(\sigma ^{\Sigma _m\wr G}([m](u_G)))) \\
&= \ \operatorname{tr}_L^K(\alpha ^*(P^m(\sigm... | {
"cite_spans": []
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5e75c1ee775db817859a3642c61da9fb10152bc6 | subsection | 158 | 1,121 | Global model structure | We define T as the following pushout,
in the category of ultra-commutative ring spectra:{ {\mathbb {P}}A[r]^-{{\mathbb {P}}i}[d]_{\rho } & {\mathbb {P}}(C A) [d]\\
R [r]_\psi & T
}Then T is globally connective,
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bd94aee536496e60baa73e6d24fab72e3a09e6ad | subsection | 159 | 1,121 | Global model structure | So {\mathbb {P}}(B_\bullet (A,A,\ast )) is isomorphic,
as a simplicial ultra-commutative ring spectrum,
to B_\bullet ^{\wedge }({\mathbb {P}}A,{\mathbb {P}}A,{\mathbb {S}}),
the bar construction of {\mathbb {P}}A with respect to smash product.
realization!of simplicial ultra-commutative ring spectraSince colimits commu... | {
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d8cbe896a1544494675beea4338f65d5412060f5 | subsection | 160 | 1,121 | Global model structure | The realization |B_\bullet ^{\wedge }(R, {\mathbb {P}}A, {\mathbb {S}})|
is the colimit of the sequence of orthogonal spectraR\ =\ B^{[0]} \ \longrightarrow \ B^{[1]}\ \longrightarrow \ \dots \ \longrightarrow \ B^{[n]}\ \longrightarrow \ \dots \ .Moreover, the n-skeleton B^{[n]} is obtained from the previous stage
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32f0f48801817e86ae75b7e3b734674cfb8d95d2 | subsection | 161 | 1,121 | Global model structure | The pushout product property of the flat cofibrations
(Proposition REF )
thus shows that i^{\Box n}, and hence the latching morphism, is a flat cofibration.So both horizontal morphisms
in the pushout square (REF )
are h-cofibrations, and the cokernel of the inclusion B^{[n-1]}\longrightarrow B^{[n]}
is isomorphic to(B_... | {
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8cce6abfd1b98729f74439de075d3757913f7e0c | subsection | 162 | 1,121 | Global model structure | The latching object L_1^{\mathbf {\Delta }} is simply the spectrum B_0^{\wedge }(R,{\mathbb {P}}A,{\mathbb {S}})=R,
and for n=1 the pushout (REF )
specializes to a pushout of orthogonal spectra\begin{aligned}
@C=12mm{
R\wedge {\mathbb {P}}A\wedge \lbrace 0,1\rbrace _+[r]^-{\text{incl}}[d]_{d_0+ d_1} &
R\wedge {\mathbb ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0e9b92bd9a232f3e58e27e56171bb31de490b7ae | subsection | 163 | 1,121 | Global model structure | Since the two horizontal morphisms in the pushout square (REF )
are h-cofibrations, the equivariant homotopy groups of the pushout
B^{[1]} participate in an exact Mayer-Vietoris sequence{\underline{\pi }}_0(R\wedge {\mathbb {P}}A)\oplus {\underline{\pi }}_0(R\wedge {\mathbb {P}}A)
\ \longrightarrow \ {\underline{\pi }}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6e398a18ad9f8557c8c88ea03be1536b2bd96b5e | subsection | 164 | 1,121 | Global model structure | By exactness of the sequence (REF )
there is a unique morphism of global functors \bar{\varphi }:{\underline{\pi }}_0(T)\longrightarrow F
such that \bar{\varphi }\circ {\underline{\pi }}_0(\psi )=\varphi .
Since {\underline{\pi }}_0(\psi ) is a surjective homomorphism of global power functors
and the composite \bar{\va... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f31e8192ba610ab7bfe96d9828258a4cbbea8fd6 | subsection | 165 | 1,121 | Global model structure | We let G be a compact Lie group and V a non-zero faithful G-representation.
Then B_{\operatorname{gl}}G={\mathbf {L}}_{G,V} is a global classifying space for G.
The free ultra-commutative ring spectrum
{\mathbb {P}}(\Sigma ^\infty _+ B_{\operatorname{gl}}G) is isomorphic to the unreduced suspension spectrum
of the free... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e6b4be65921563ae473a7eeca8280447dfdac022 | subsection | 166 | 1,121 | Global model structure | So for every finitely supported function
f:I\longrightarrow {\mathbb {N}}, the morphism{\square }_{f(i)\ne 0}\, {\underline{\pi }}_0\left({\mathbb {P}}^{f(i)}(X_i)\right) \ \longrightarrow \ {\underline{\pi }}_0\left({\bigwedge }_{f(i)\ne 0}\, {\mathbb {P}}^{f(i)}(X_i) \right)from the iterated box product is an isomorp... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f5eecb75576b86c9685a57b0822b237eea8b2f3a | subsection | 167 | 1,121 | Global model structure | Equivariant homotopy groups take wedges to direct sums, so the canonical morphism{\bigoplus }_{i\in I} \, {\underline{\pi }}_0(X_i)\ \longrightarrow \ {\underline{\pi }}_0\left({\bigvee }_{i\in I} X_i \right)is an isomorphism of global functors.
So altogether we obtain that {\underline{\pi }}_0({\mathbb {P}}( \bigvee _... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3c5704476f3389ec3eab168fe66693decee1f071 | subsection | 168 | 1,121 | Global model structure | So Proposition REF applies
and shows that the morphism{\mathbb {P}}j\ : \ {\mathbb {P}}B \ \longrightarrow \ {\mathbb {P}}( C f)is a coequalizer, in the category of global power functors, of the two morphisms{\underline{\pi }}_0({\mathbb {P}}B)\diamond {\underline{\pi }}_0({\mathbb {P}}f)\ , \ {\underline{\pi }}_0({\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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90280f865db18ed112c2d0da3968fd9e5e5b96fb | subsection | 169 | 1,121 | Global model structure | Then Y is globally connective (i.e., globally (-1)-connected).
The adjoint q^\flat of q factors as the compositeY\wedge S^n \ \xrightarrow{} \ (\Omega ^n X)\wedge S^n \ \xrightarrow{}\ X \ .The first morphism is a global equivalence since suspension is fully homotopical,
and the second morphism is a global equivalence
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e328d9f9abe1b7d9cb05001624af878eaa73de47 | subsection | 170 | 1,121 | Global model structure | Since {\mathbb {P}}^{i_k}(Y) is a retract of {\mathbb {P}}Y,
it is also globally connective.
Since flat cofibrations are symmetrizable
(Theorem REF (i)),
each of the orthogonal spectra {\mathbb {P}}^{i_j}(Y) is again flat.
So the smash product
{\mathbb {P}}^{i_1}(Y)\wedge \dots \wedge {\mathbb {P}}^{i_k}(Y) is globall... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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922da3029c0fc88f4f41e791dbc4a9eafa849dba | subsection | 171 | 1,121 | Global model structure | We form the wedge of all these morphismsF \ : \ X \ = \ {\bigvee }_{j\in J}\, \Sigma ^\infty _+ B_{\operatorname{gl}} G_j \wedge S^n \ \longrightarrow \ R\ .All we need to remember about F is that its source X is globally (n-1)-connected
and positively flat, and that the morphism of global functors{\underline{\pi }}_n(... | {
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"raw": "D. White, Model structures on commutative monoids in general model categories. J. Pure Appl. Algebra 221 (2017), no. 12, 3124–3168.",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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d88daad7a63487dcddbcaa5033ba85afa256dcc0 | subsection | 172 | 1,121 | Global model structure | Moreover, the pushout square witnesses that the cokernel of \psi _m is isomorphic toR\wedge \text{coker}(i^{\Box m})/\Sigma _m\ \cong \ R\wedge (C X/X )^{\wedge m}/\Sigma _m \ \cong \ R\wedge {\mathbb {P}}^m(X\wedge S^1) \ .Since X is globally (n-1)-connected, its suspension X\wedge S^1
is globally n-connected, and so ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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d768f6acb292e78dfe5f891c2ec22bc577f8383a | subsection | 173 | 1,121 | Global model structure | The analogous result in equivariant stable homotopy theory for a fixed
finite group has been obtained by Ullman .
More is true: the next theorem effectively constructs a right adjoint functorH \ : \ \mathcal {G}l\mathcal {P}ow\ \longrightarrow \ \operatorname{Ho}^{\text{gl.\,connective}}(ucom)to the functor {\underline... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8560512ad10eb8aee5e810edcf808c1d31c23239 | subsection | 174 | 1,121 | Global model structure | For example, we can choose compact Lie groups G_j and
elements y_j \in \pi _0^{G_j}({\mathbb {P}}X)
that altogether generate this kernel as a global functor.
Then we represent each class y_j as a morphism of
orthogonal spectraf_j \ : \ \Sigma ^\infty _+ B_{\operatorname{gl}} G_j \ \longrightarrow \ {\mathbb {P}}Xthat s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cd609682f70f3b9d40b949d77e235014685463a5 | subsection | 175 | 1,121 | Global model structure | More precisely,
we construct a sequence of cofibrations of ultra-commutative ring spectraT \ = \ T_0 \ \longrightarrow \ T_1 \ \longrightarrow \ \dots \ \longrightarrow \ T_n \ \longrightarrow \ \dotsby induction on n. We obtain T_n by applying Theorem REF
in dimension n to R=T_{n-1}.
Then T_n is globally connective, ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
134995,
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4c7852125d227f58952cba386a5b11b6f3efb65b | subsection | 176 | 1,121 | Global power monoids | In this section we investigate the algebraic structure that an
ultra-commutative multiplication produces on the \operatorname{Rep}-functor {\underline{\pi }}_0(R).
Besides an abelian monoid structure on \pi _0^G(R) for every
compact Lie group G, this structure includes power operations and
transfer maps. We formalize t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2f1510788aa314c691b87f68baa7a0c3922f7994 | subsection | 177 | 1,121 | Global power monoids | An important special case will later be the
multiplicative ultra-commutative monoid \Omega ^\bullet R
arising from an ultra-commutative ring spectrum R.
In this situation the power operations satisfy further compatibility conditions
with respect to the addition and the transfer maps on {\underline{\pi }}_0(\Omega ^\bul... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6220885129c019e8d1b612be9afcecb9438b51c3 | subsection | 178 | 1,121 | Global power monoids | So its image under the map \mu _{V,\dots ,V}
is a (\Sigma _m\wr G)-fixed point of R(V^m),
representing an element[m]( [x] ) \ = \ \langle \mu _{V,\dots ,V}(x,\dots ,x) \rangle
\ \in \ \pi _0^{\Sigma _m\wr G} ( R )\ .If we stabilize x along a G-equivariant
linear isometric embedding \varphi :V\longrightarrow W
to R(\va... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
1061,
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b878a173adaaf9adeead69fbe26673e41e4b5109 | subsection | 179 | 1,121 | Global power monoids | An embedding of a product of wreath products is now defined by\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 +\sigma ^{\prime };\, g_1,\d... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7c4dae0dfbeafb8b27a4b9b5b3709758bc276617 | subsection | 180 | 1,121 | Global power monoids | This yields an embedding of an iterated wreath product\Psi _{k,m}\ : \ \Sigma _k\wr (\Sigma _m\wr G) \qquad &\longrightarrow \qquad \Sigma _{k m}\wr G \\
(\sigma ;\, (\tau _1;\, g^1),\dots ,(\tau _k;\, g^k)) \ &\longmapsto \ (\sigma \natural (\tau _1,\dots ,\tau _k);\, g^1+\dots +g^k)\ .Here each g^i=(g^i_1,\dots ,g^i_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
3293,
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971753eceaa66b2dc4dcdce2f3ead7e2f6b43a30 | subsection | 181 | 1,121 | Global power monoids | Since we will always hit \Psi _{k,m} with functors that are invariant
under conjugation, this should motivate that the construction is reasonably natural.Definition 2.6 global power monoid
A global power monoid is a functorabelian Rep-monoidM \ : \ \operatorname{Rep}^{\operatorname{op}} \ \longrightarrow \ {\mathcal {A... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c8b461fe596c4d8816560cd2e614c81dd88a7a64 | subsection | 182 | 1,121 | Global power monoids | (Additivity)
For all compact Lie groups G, all m>i>0 and all x\in M(G)
the relation
\Phi ^*_{i,m-i}( [m](x) )\ = \ p_1^*([i](x)) \ + \ p_2^*([m-i](x))
holds in M((\Sigma _i\wr G)\times (\Sigma _{m-i}\wr G)),
where \Phi _{i,m-i} is the monomorphism (REF ) and
p_1:(\Sigma _i\wr G)\times (\Sigma _{m-i}\wr G)\longrightar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a07adda0831f63089162302a8cbfc4e05fb49862 | subsection | 183 | 1,121 | Global power monoids | In this notation, the additivity requirement in Definition REF
becomes the relation\Phi ^*_{i,m-i}( [m](x) )\ = \ [i](x) \ \oplus [m-i](x)\ .In a global power monoid,
the power operations are also additive with respect to the external addition:
for all compact Lie groups G and K and all m\ge 1, and
all classes x\in M(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ea2ddb3ec671197c24e7f567095358c67d17dcba | subsection | 184 | 1,121 | Global power monoids | Restricting further to the
diagonal takes the m-fold external sum to m\cdot x in M(G).We will soon discuss that the power operations of an ultra-commutative monoid
define a global power monoid. One aspect of this is the additivity
of the power operations, which could be shown directly from the definition.
However, we w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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82325f9e2d8f357e645b8b7b1b7ddef39d4b3ab9 | subsection | 185 | 1,121 | Global power monoids | Similarly,G(0+\operatorname{Id}_X) ( \tau _{X\vee X}(F(i)(x)+ F(j)(y)))\ = \ G(0+\operatorname{Id}_X)( G(i)(\tau _X(x)) + G(j)(\tau _X(y)))\ .Since G is additive, this shows the relation (REF ).
We let \nabla =(\operatorname{Id}+\operatorname{Id}):X\vee X\longrightarrow X denote the fold morphism,
so thatF(\nabla )(F(i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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33b3fca33fa47e05d60e7225f7fd064a4e9a94f5 | subsection | 186 | 1,121 | Global power monoids | The power operation[m]\ :\ \pi _0(M) = \pi _0^G(\underline{M}) \longrightarrow \pi _0^{\Sigma _m\wr G}(\underline{M})
= \pi _0(M)then sends an element x to m\cdot x.Example 2.13 (Naive units of an orthogonal monoid space) units!of an orthogonal monoid space!naive
Every orthogonal monoid space R contains an interesting ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d942d0ba1659db9a6c5037888ce91e525b89b87a | subsection | 187 | 1,121 | Global power monoids | So for varying G, the subgroups M^\times (G)
indeed form a global power submonoid of M.We call a global power monoid N group-likegroup-like!global power monoid
if the abelian monoid N(G) is a group for every compact Lie group G.
If f:N\longrightarrow M is a homomorphism of global power monoids and N is group-like,
then... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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60d6f923ce15544d71e0cf5ba0d1c59e8c20e7a4 | subsection | 188 | 1,121 | Global power monoids | Since the restriction maps \alpha ^*:M(G)\longrightarrow M(K) and the power operations
[m]:M(G)\longrightarrow M(\Sigma _m\wr G) are monoid homomorphisms, the universal property
provides unique homomorphisms \alpha ^*:M^\star (G)\longrightarrow M^\star (K) and
[m]:M^\star (G)\longrightarrow M^\star (\Sigma _m\wr G) suc... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8b8101f87ef96cc27c012f6329b10cb8c44de5ca | subsection | 189 | 1,121 | Global power monoids | So in terms of global classifying spaces
the free ultra-commutative monoid generated by B_{\operatorname{gl}}G is given by{\mathbb {P}}( B_{\operatorname{gl}} G) \ = \ {\coprod }_{m\ge 0}\, B_{\operatorname{gl}}(\Sigma _m\wr G) \ .The tautological class u_G\in \pi _0^G(B_{\operatorname{gl}} G) is represented by the orb... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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95273c61bb32c00cf929b158c56c3199e0966392 | subsection | 190 | 1,121 | Global power monoids | So if the set I is infinite, then the underlying orthogonal space of
the coproduct is the `infinite box product'
in the sense of Construction REF ,box product!of orthogonal spaces!infinite i.e., the filtered colimit,
formed over the poset of finite subsets of I, of the finite coproducts,\boxtimes _{i\in I}^{\prime }\, ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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01c32d1c8dbb086da1f8423d070e1b213136c121 | subsection | 191 | 1,121 | Global power monoids | We establish the monoid structure and power operations first,
which also shows the uniqueness.
Since \operatorname{Rep}(e,A) has only one element, it is the additive unit.
Since restriction maps are monoid homomorphisms,
the trivial homomorphism is the neutral element of \operatorname{Rep}(G,A).
The sumq_1 + q_2 \ \in ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4e22255edb07d70e381faf4e789af2073397bfc6 | subsection | 192 | 1,121 | Global power monoids | Clearly, pointwise multiplication of homomorphisms makes \operatorname{Rep}(G,A) into
an abelian monoid (even an abelian group), and the monoid structure is
contravariantly functorial in G.
When we define [m](\alpha ) by the formula of the proposition,
then the remaining axioms of a global power monoid
(compare Definit... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e2018fede63b41437a46cf5b8daabbad633db174 | subsection | 193 | 1,121 | Global power monoids | The Yoneda lemma then provides a unique morphism of \operatorname{Rep}-functorsf \ : \ \operatorname{Rep}(-,A)\ \longrightarrow \ Msuch that f(A)(\operatorname{Id}_A)=x,
and this morphism is given by f(G)(\alpha )=\alpha ^*(x), for \alpha :G\longrightarrow A.
We need to show that f is a morphism of global power monoids... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ed4ee687cbef72fee06b6ecb11156e152c3712bd | subsection | 194 | 1,121 | Global power monoids | So ultimately we need to calculate \pi _0^K({\mathbb {P}}(B_{\operatorname{gl}}G)).The tautological class u_G in \pi _0^G(B_{\operatorname{gl}}G)
was defined in (REF ).
We setu_G^{umon}\ = \ \eta _*(u_G )\ \in \ \pi _0^G({\mathbb {P}}(B_{\operatorname{gl}} G))\ ,where \eta :B_{\operatorname{gl}} G\longrightarrow {\math... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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06642a8f742ee4cc4e1380a8a1a57caef5ec2d6e | subsection | 195 | 1,121 | Global power monoids | So part (i) follows from Proposition REF (ii)
and the fact that \pi _0^K commutes with disjoint unions.(ii) By (i) every element of \pi _0^K({\mathbb {P}}(B_{\operatorname{gl}} G)) is of the form \alpha ^*([m](u));
every morphism of global power monoids
f:{\underline{\pi }}_0({\mathbb {P}}(B_{\operatorname{gl}} G))\lo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6f7c0185f6c8bcc4afe184b8076f3433dd50c8b2 | subsection | 196 | 1,121 | Global power monoids | Given \alpha :K\longrightarrow \Sigma _m\wr G and \bar{\alpha }:K\longrightarrow \Sigma _n\wr G,
we have\alpha ^*([m](u))\ + \ \bar{\alpha }^*([n](u)) \ &= \ (\alpha ,\bar{\alpha })^*(p_1^*([m](u)) + p_2^*([n](u))) \\
&= \ (\alpha ,\bar{\alpha })^*(\Phi _{m,n}^*([m+n](u))) \\
&= \ (\Phi _{m,n}\circ (\alpha ,\bar{\alpha... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a08c6ca3da60b8ac0f99c149f71f6bead3e6b5c7 | subsection | 197 | 1,121 | Global power monoids | For k\ge 1 we have[k](\alpha ^*([m](u)))\ &= \ (\Sigma _k\wr \alpha ^*)([k]([m](u)))\\
&= \ (\Sigma _k\wr \alpha ^*)(\Psi _{k, m}^*([k m](u)))\ = \ (\Psi _{k,m}\circ (\Sigma _k\wr \alpha ))^*([k m](u))\ ;hencef(\Sigma _k\wr K)([k](\alpha ^*&([m](u))))\ = \ (\Psi _{k,m}\circ (\Sigma _k\wr \alpha ))^*([k m](x)) \\
&= \ (... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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