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3b2b6f3ca477f403dc0947fee08906f9633835f5 | subsection | 32 | 34 | Example derivations | Here is the derivation of this equation:((x \mathop {{}_{1}\!{?}\!_{1}} y) \mathop {?_{p}} x) \mathop {?_{p}} (y \mathop {?_{p}} (x \mathop {{}_{1}\!{?}\!_{1}} y))
& = ((x \mathop {{}_{1}\!{?}\!_{1}} y) \mathop {?_{p}} (x \mathop {{}_{1}\!{?}\!_{1}} x)) \mathop {?_{p}} ((y \mathop {{}_{1}\!{?}\!_{1}} y) \mathop {?_{p}}... | {
"cite_spans": []
} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
"Sam Staton",
"Dario Stein",
"Hongseok Yang",
"Nathanael L. Ackerman",
"Cameron E. Freer",
"Daniel M. Roy"
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d4f03e3155ed8a615b13b696b7ffdd318a3f80e0 | subsection | 33 | 34 | Example derivations | Making (t \mathop {?_{p}} x) \mathop {?_{p}} (y \mathop {?_{p}} t) permutation-invariant means computing the average(t \mathop {?_{p}} x) \mathop {?_{p}} (y \mathop {?_{p}} t)&=
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} | 10.4230/LIPIcs.ICALP.2018.141 | 1802.09598 | The Beta-Bernoulli process and algebraic effects | [
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"Hongseok Yang",
"Nathanael L. Ackerman",
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730321149c8cd56a1f5ae352161b30f0c600235f | abstract | 0 | 1,121 | Abstract | This book introduces a new context for global homotopy theory, i.e.,
equivariant homotopy theory with universal symmetries. Many important
equivariant theories naturally exist not just for a particular group, but in a
uniform way for all groups in a specific class. Prominent examples are
equivariant stable homotopy, eq... | {
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dcc5ae2873cc384ba460e7b5f9f7df986b440493 | subsection | 1 | 1,121 | Preface | Equivariant stable homotopy theory has a long tradition,
starting from geometrically motivated questions about symmetries of manifolds.
The homotopy theoretic foundations of the
subject were laid by tom Dieck, Segal and May and their students and
collaborators in the 70's, and during the last decades
equivariant stable... | {
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1fc9d75d5c16e9d273efff287115c7d4af88f855 | subsection | 2 | 1,121 | Preface | By Theorem REF this forgetful functor
has a left adjoint L and a right adjoint R, both fully faithful,
that participate in a recollement of triangulated categories:@C=15mm{
{\mathcal {GH}}^+ @<-.3ex>[r]^-{i_*} &
{\mathcal {GH}}\ @<-.3ex>[r]^-U \ @<.4ex>@/^1pc/[l]^-{i^!} @<-.4ex>@/_1pc/[l]_-{i^*} &
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7c1202906ee00a67539876390e34ba8d0528f92e | subsection | 3 | 1,121 | Preface | We discuss these different global forms of B O
is some detail in Section ,
and the associated Thom spectra in Section .In the stable global world, every non-equivariant homotopy type
has two extreme global refinements, the `left induced'
(the global analog of a constant orthogonal space,
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3eac0e58dc498e6de631cb93b5b3197aa7960042 | subsection | 4 | 1,121 | Preface | The class of trivial groups
is also admissible here, but then we just recover the `traditional' stable category.
If the family {\mathcal {F}} is multiplicative, then the
{\mathcal {F}}-global model structure is monoidal with respect to
the smash product of orthogonal spectra and satisfies the monoid axiom
(Proposition ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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f5706f4523561633433e2a1312f71110388b406e | subsection | 5 | 1,121 | Preface | However, since the group G is not intrinsic and can vary, one needs
equivariant cohomology theories for all groups G, with some compatibility.Part of the compatibility can be deduced from the fact that
the same orbifold can be presented in different ways; for example, if G is a closed
subgroup of K, then the global quo... | {
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86cdbf3601e3960b4e17a0e99ecc328c1fe83f81 | subsection | 6 | 1,121 | Preface | By the adjunction relating the global and G-equivariant stable homotopy
categories (see Theorem REF ),
the morphisms \llbracket \Sigma ^\infty _+ {\mathbf {L}}_{G,V}M, E\rrbracket in the global stable
homotopy category biject with the G-equivariant E-cohomology groups of M.
In other words, when evaluated on a global qu... | {
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7ad17905db9844045d36520b1b47e401aad9c2d6 | subsection | 7 | 1,121 | Preface | The study of natural operations on \pi _0^G(Y)
is a recurring theme throughout this book;
in the later chapters we return to it in the contexts of
ultra-commutative monoids, orthogonal spectra and ultra-commutative ring spectra.Chapter is devoted to ultra-commutative monoids
(a.k.a. commutative monoids with respect to ... | {
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909e0695e299837664322d07448d422c31586109 | subsection | 8 | 1,121 | Preface | Here we work more generally relative to a global family {\mathcal {F}} and consider the
{\mathcal {F}}-equivalences (i.e., equivariant stable equivalences for all compact
Lie groups in the family {\mathcal {F}}).
We follow the familiar outline: a certain {\mathcal {F}}-level model structure
is Bousfield localized to an... | {
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e4d143ea1151d49b0f945ebc2d1518c8e375a5f2 | subsection | 9 | 1,121 | Preface | The equivariant homology theory represented by {\mathbf {MO}} can be obtained
from the one represented by {\mathbf {mO}} in an algebraic fashion, by inverting
the collection of `inverse Thom classes',
compare Corollary REF .
Section recalls the geometrically defined
equivariant bordism theories.
The Thom-Pontryagin con... | {
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5f84fba65dd36581252c2ce1cc1670da608856f9 | subsection | 10 | 1,121 | Preface | However, we do prove in Theorem REF
that evaluating a G-cofibrant special {\mathbf {\Gamma }}-G-space
on spheres yields a `G^\circ -trivial positive G-\Omega -spectrum',
where G^\circ is the identity component of G.
Our Appendix REF substantially overlaps
with the paper by May, Merling and Osorno that
provides compari... | {
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486a6e4572a2bb4d8a80a014516ad527e51827ce | subsection | 11 | 1,121 | Preface | In particular, by simply ignoring all group actions,
the examples presented in this book give models for many interesting
and prominent non-equivariant stable homotopy types.Since actions of compact Lie groups are central to this book,
some familiarity with the structure and representation theory of
compact Lie groups ... | {
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"doi": "10.1007/bfb0097438",
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"source_ref_id": "4c0900e5d5eb23dd... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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3e9290aa1f5f311836ba4121a50ce2a82af3595f | subsection | 12 | 1,121 | Unstable global homotopy theory | In this chapter we develop a framework for unstable global homotopy theory
via orthogonal spaces, i.e., continuous functors from the
linear isometries category {\mathbf {L}} to spaces.
In Section we define
global equivalences of orthogonal spaces and establish many basic properties
of this class of morphisms.
We also i... | {
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fc59378ba2ab62a8525afae3b66aa2a7d6ac06ef | subsection | 13 | 1,121 | Orthogonal spaces and global equivalences | In this section we introduce orthogonal spaces,
along with the notion of global equivalences,
our setup to rigorously formulate the idea of
`compatible equivariant homotopy types for all compact Lie groups'.
We introduce various basic techniques to manipulate global equivalences
of orthogonal spaces, such as recognitio... | {
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"source_ref_i... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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9040898e3384beaaf2fa185842ae45a40f0e7f69 | subsection | 14 | 1,121 | Orthogonal spaces and global equivalences | So the full topological subcategory with objects
the {\mathbb {R}}^n is a small skeleton of {\mathbf {L}}.Definition 1.1
An orthogonal spaceorthogonal space
is a continuous functor Y:{\mathbf {L}}\longrightarrow {\mathbf {T}} to the category of spaces.
A morphism of orthogonal spaces is a natural transformation.
We de... | {
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7204c9ba5a1711695c19f4198bb250ba7462a6f6 | subsection | 15 | 1,121 | Orthogonal spaces and global equivalences | In particular, D^0=\lbrace 0\rbrace is a one-point space and \partial D^0=\emptyset
is empty.Definition 1.2
A morphism f:X\longrightarrow Y
of orthogonal spaces is a global equivalenceglobal equivalence!of orthogonal spaces
if the following condition holds: for every compact Lie group G,
every G-representation V, eve... | {
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f6e7e8aac8cdc734bfb625fa797017238b739a76 | subsection | 16 | 1,121 | Orthogonal spaces and global equivalences | So \underline{g} is a global equivalence
if and only if g is a weak equivalence.Remark 1.4 The notion of global equivalence is meant to capture the idea that
for every compact Lie group G,
some induced morphism\operatorname{hocolim}_V f(V)\ :\ \operatorname{hocolim}_V X(V)\ \longrightarrow \ \operatorname{hocolim}_V Y(... | {
"cite_spans": []
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8085535cfd206b2c50a6eda485ddbb7ee98884c9 | subsection | 17 | 1,121 | Orthogonal spaces and global equivalences | We suppose that the lifting problem (H_0,K_0) has a solution
consisting of a continuous map \lambda :B\longrightarrow X
such that \lambda |_A=H_0 and a homotopy G:B\times [0,1]\longrightarrow Y
such thatG_0 \ = \ f\circ \lambda \ , \quad G_1\ = \ K_0 \text{\qquad and\qquad }
(G_t)|_A \ = \ f\circ H_0for all t\in [0,1].... | {
"cite_spans": []
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"Stefan Schwede"
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0111843a6c846809ce3cfe682c220db2757c9757 | subsection | 18 | 1,121 | Orthogonal spaces and global equivalences | We define a continuous map L:A\times [0,3]\times [0,1]\longrightarrow Y byL(-,t,s) \ = \ {\left\lbrace \begin{array}{ll}
f\circ H_{1-t} & \text{ for $0\le t\le s$, }\\
f\circ H_{1-s} & \text{ for $s\le t\le 3-s$, and}\\
f\circ H_{t-2} & \text{ for $3-s\le t\le 3$.}
\end{array}\right.}Then L(-,-,0) is the constant homot... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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731a35e559475715dd0870c17ec39b5f7ac72284 | subsection | 19 | 1,121 | Orthogonal spaces and global equivalences | The map \bar{J}=\bar{L}(-,-,0):B\times [0,3]\longrightarrow Y then satisfies\bar{J}|_{A\times [0,3]}\ = \ \bar{L}(-,-,0)|_{A\times [0,3]}\ = \ L(-,-,0)\ ,which is the constant homotopy at the map f\circ H_1;
so \bar{J} is a homotopy (parametrized by [0,3] instead of [0,1])
relative to A.
Because\bar{J}_0 \ = \ \bar{L}(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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4efbc266fadd7172e6f841680da633cefe016c8c | subsection | 20 | 1,121 | Orthogonal spaces and global equivalences | For every compact Lie group G and every exhaustive sequence \lbrace V_i\rbrace _{i\ge 1}
of G-representations the induced map
\operatorname{tel}_i f(V_i)\ : \ \operatorname{tel}_i X(V_i)\ \longrightarrow \ \operatorname{tel}_i Y(V_i)
is a G-weak equivalence.At various places in the proof we use without explicit menti... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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1ee228407990ed2cee450dfb222ed08d243e66a4 | subsection | 21 | 1,121 | Orthogonal spaces and global equivalences | Since f is a global equivalence, there is an H-equivariant
linear isometric embedding \psi :U\longrightarrow W and a continuous map
\lambda :D^k\longrightarrow X(W)^H such that
\lambda |_{\partial D^k}=X(\psi )^H\circ (\lambda ^{\prime })^H\circ \bar{\chi }
and f(W)^H\circ \lambda is homotopic, relative \partial D^k, t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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ed6b1e88bc48f9625ad2f9d4b40f7204d0d9fd7a | subsection | 22 | 1,121 | Orthogonal spaces and global equivalences | Lemma REF
(or rather its G-equivariant generalization)
applies to these homotopies, so instead of the original lifting problem
we may solve the homotopic lifting problem@C=15mm{
A[r]^-{\alpha ^{\prime }} [d]_{\text{incl}} & X(V_n)[r]^-{i_n}[d]_{f(V_n)} &
\operatorname{tel}_i X(V_i) [d]^{\operatorname{tel}_i f(V_i)} \\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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d5c21b5736fd92a3cbea021ee9365771bb1274bb | subsection | 23 | 1,121 | Orthogonal spaces and global equivalences | But there is a G-equivariant homotopy H:X(V_n)\times [0,1]\longrightarrow \operatorname{tel}_i X(V_i)
between i_m\circ X(i) and i_n, and a similar homotopy
K:Y(V_n)\times [0,1]\longrightarrow \operatorname{tel}_i Y(V_i) for Y instead of X.
These homotopies satisfyK\circ (f(V_n)\times [0,1]) \ = \ (\operatorname{tel}_i ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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168314cbea743c79f9aada86bc166cdf75893054 | subsection | 24 | 1,121 | Orthogonal spaces and global equivalences | The following diagram commutes@C=14mm{
X(V_n)^G [r]_-{\text{can}} [d]_{f(V_n)^G} @/^1pc/[rrr]^(.3){X(\text{incl})^G}&
\operatorname{tel}_{[0,n]} X(V_i)^G [r]_-{\text{incl}} [d]_{\operatorname{tel}f(V_i)^G} &
\operatorname{tel}_{[0,m]} X(V_i)^G [r]_-{\text{proj}} [d]^{\operatorname{tel}f(V_i)^G} &
X(V_m)^G [d]^{f(V_m)^G... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ec27211af8299f4f64adef45b958bd26209f3694 | subsection | 25 | 1,121 | Orthogonal spaces and global equivalences | So every homotopy equivalence is in particular a strong level equivalence.
By the following proposition, strong level equivalences are global equivalences.A continuous map \varphi :A\longrightarrow B is a
closed embeddingclosed embeddingembedding!closed|seeclosed embedding
if it is injective and a closed map.
Such a ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4f4e202cb03b77f07edee3178115c52736c4c812 | subsection | 26 | 1,121 | Orthogonal spaces and global equivalences | If the morphisms \alpha ,\beta and \gamma are global equivalences,
then so is the induced morphism of pushouts
\gamma \cup \beta \ : \ C\cup _A B \ \longrightarrow \ C^{\prime }\cup _{A^{\prime }} B^{\prime }\ .
Let
{ A [r]^-f [d]_g & B [d]^h\\
C [r]_-k & D }
be a pushout square of orthogonal spaces such that f is... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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99db3041082aab4352b34dbe7bafcf3cdff73340 | subsection | 27 | 1,121 | Orthogonal spaces and global equivalences | Since f is a global equivalence,
the equivariant lifting problem (X(\varphi )\circ \alpha ,\lambda )
has a solution (\psi :W\longrightarrow U,\, \lambda ^{\prime }:B\longrightarrow X(U)) such that\lambda ^{\prime }|_A\ =\ X(\psi )\circ X(\varphi )\circ \alphaand such that f(U)\circ \lambda ^{\prime }
is G-homotopic to ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0587b2d3cb1901f5bad383a5a63cfdb398f1e4a5 | subsection | 28 | 1,121 | Orthogonal spaces and global equivalences | So there is a
commutative diagram{
X[r]^-i [d]_f & \bar{X}[r]^-r[d]^g & X[d]^f \\
Y [r]_-j & \bar{Y} [r]_-s & Y}such that r i=\operatorname{Id}_X and s j=\operatorname{Id}_Y.
We let G be a compact Lie group, V a G-representation,
(B,A) a finite G-CW-pair and \alpha :A\longrightarrow X(V)
and \beta :B\longrightarrow Y(V... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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23b8f264d49472aefc4b20a48272c68003ae21cf | subsection | 29 | 1,121 | Orthogonal spaces and global equivalences | But this is straightforward:
we let G be a compact Lie group, V a G-representation,
(B,A) a finite G-CW-pair and \alpha :A\longrightarrow (X\times Z)(V)
and \beta :B\longrightarrow (Y\times Z)(V) continuous G-maps
such that (f\times Z)(V)\circ \alpha =\beta |_A.
Because(X\times Z)(V) \ = \ X(V) \times Z(V)and similarly... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c461d70eddb044bf7ddf0f5201ed66d03eeb941b | subsection | 30 | 1,121 | Orthogonal spaces and global equivalences | Moreover, the relation \psi _n(V)\circ \bar{\alpha }=\bar{\beta }|_A:A\longrightarrow Y_n(V)
holds because it holds after composition with the injective map Y_n(V)\longrightarrow Y_\infty (V).Since \psi _n is a global equivalence,
there is a G-equivariant linear isometric embedding \varphi :V\longrightarrow W
and a con... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b89ac35d2daaed035cbf05531fb59c1565d066bf | subsection | 31 | 1,121 | Orthogonal spaces and global equivalences | Since \alpha , \beta and \gamma are global equivalences, the three
vertical maps in the following commutative diagram of G-spaces
are G-weak equivalences, by Proposition REF :@C=15mm{
\operatorname{tel}_i C(V_i) [d]_{\operatorname{tel}\gamma (V_i)} &
\operatorname{tel}_i A(V_i) [l]_-{\operatorname{tel}g(V_i)} [d]^{\ope... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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821cf42e435743b5f04a436db90ffd9186725710 | subsection | 32 | 1,121 | Orthogonal spaces and global equivalences | Since f is a global equivalence, there is
a G-equivariant linear isometric embedding \varphi :V\longrightarrow W
and a continuous G-map \lambda :B\longrightarrow X(W)
such that
\lambda |_A=X(\varphi )\circ k(V)\circ \alpha and
such that f(W)\circ \lambda
is G-homotopic, relative to A, to Y(\varphi )\circ h(V)\circ \be... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4fb2b97dc34e6974a10ea1c6b5b5d68fff7a5bbd | subsection | 33 | 1,121 | Orthogonal spaces and global equivalences | Now we can choose a lift H^{\prime } in the square@C30mm{
B\times 1\cup _{A\times 1} A\times [0,1][d]_\sim [r]^-{\lambda \cup K^{\prime }} &
X(W) @{->>}[d]^{f(W)} \\
B\times [0,1] [r]_-H @{-->}[ur]_{H^{\prime }} & Y(W)
}where K^{\prime }:A\times [0,1]\longrightarrow X(W) is the constant homotopy
from X(\varphi )\circ k... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2fbe456a47bacb6dd5a12ceafd3a915102416092 | subsection | 34 | 1,121 | Orthogonal spaces and global equivalences | The unique morphism from the initial (i.e., empty)
to a terminal orthogonal space is not a global equivalence.The following proposition provides a lot of flexibility
for changing an orthogonal space into a globally equivalent one by modifying
the input variable. We will use it multiple times in this book.Theorem 1.10
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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43dd10a0293fa80dc92d700b3caecfbf4ca583b9 | subsection | 35 | 1,121 | Orthogonal spaces and global equivalences | Then \beta |_A=Y(i_V)\circ \alpha by hypothesis,
and we claim that Y(i_{F(V)})\circ \beta
is G-homotopic to Y(F(i_V))\circ \beta = (Y\circ F)(i_V)\circ \beta ,
relative A; granting this for the moment, we conclude that
the pair (i_V:V\longrightarrow F(V), \beta ) solves the lifting problem.It remains to construct the ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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363aa62241c5361aa4b9f331fd9d2834582e5e18 | subsection | 36 | 1,121 | Orthogonal spaces and global equivalences | Y)(V)\ = \ Y(V\otimes W)\ .Here, and in the rest of the book, we endow
the tensor product V\otimes W of two inner product spaces V and W
with the inner product characterized by\langle v\otimes w, \bar{v}\otimes \bar{w}\rangle \ = \ \langle v, \bar{v}\rangle \cdot \langle w, \bar{w}\ranglefor all v,\bar{v}\in V and w,\b... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-642-66243-0",
"end": 2172,
"openalex_id": "https://openalex.org/W1515431404",
"raw": "A. A. Kirillov, Elements of the theory of representations. Translated from the Russian by E. Hewitt. Grundlehren der Mathematischen Wissensc... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2a1d3c04f2eea40364fbdc55fe18d62e1e13f1c8 | subsection | 37 | 1,121 | Orthogonal spaces and global equivalences | By Schur's lemma, the endomorphism algebra \operatorname{Hom}_{\mathbb {R}}^G(\lambda ,\lambda )
is a finite-dimensional skew-field extension of {\mathbb {R}}, hence isomorphic
to {\mathbb {R}}, {\mathbb {C}} or {\mathbb {H}}.If \operatorname{Hom}_{\mathbb {R}}^G(\lambda ,\lambda ) is isomorphic to {\mathbb {R}},
then ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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9bb2bccfb40635802533898ecd3c39406e95e626 | subsection | 38 | 1,121 | Orthogonal spaces and global equivalences | Since there are only countably many isomorphism classes of irreducible orthogonal
G-representations, a complete G-universe exists.Remark 1.13
We let H be a closed subgroup of a compact Lie group G.
We will frequently use that the underlying H-representation of a complete
G-universe {\mathcal {U}} is a complete H-unive... | {
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"doi": "",
"end": 486,
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"st... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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f26b35bbbc0dd21f220adeca987a5af4a54d0f26 | subsection | 39 | 1,121 | Orthogonal spaces and global equivalences | Then {\mathcal {U}}_G is filtered by the finite sums n\cdot \rho _G, and we getY({\mathcal {U}}_G)\ = \ \operatorname{colim}_n \, Y(n\cdot \rho _G) \ ,where the colimit is taken along the inclusions
n\cdot \rho _G\longrightarrow (n+1)\cdot \rho _G that miss the last summand.Definition 1.16 An orthogonal space Y is clos... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
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6e0de220d3792ff5959a6e442e1e3c4fed76adfe | subsection | 40 | 1,121 | Orthogonal spaces and global equivalences | We turn these two actions into a single left action of the group K\times G
by defining((k,g)\cdot \varphi )(v) \ = \ k\cdot \varphi (g^{-1} \cdot v)for \varphi \in {\mathbf {L}}(V,{\mathcal {U}}) and (k,g)\in K\times G.
We recall that a continuous (K\times G)-equivariant map is a
(K\times G)-cofibration if it has the r... | {
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02ff11c217eee440b4c9cb3d16c29093e8b654da | subsection | 41 | 1,121 | Orthogonal spaces and global equivalences | The map{\mathbf {L}}(V,{\mathbb {R}}^m)\ \longrightarrow \ {\mathbf {L}}(V,{\mathbb {R}}^{m+n}) \ ,induced by the embedding {\mathbb {R}}^m\longrightarrow {\mathbb {R}}^{m+n}
as the first m coordinates,
is a homeomorphism from {\mathbf {L}}(V,{\mathbb {R}}^m) onto the N-fixed points
of {\mathbf {L}}(V,{\mathbb {R}}^{m+... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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1c8026eac1a2baea3c072acbb0d78dfda019a9f1 | subsection | 42 | 1,121 | Orthogonal spaces and global equivalences | Part (i) then implies that
{\mathbf {L}}(V,{\mathcal {U}}) is (K\times G)-cofibrant
and {\mathbf {L}}(V,{\mathcal {U}})/G is K-cofibrant.The following fundamental contractibility property goes back,
at least, to Boardman and Vogt .
The equivariant version that we need can be found in .Proposition 1.21
Let G be a compa... | {
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23ce341685b57af94b831492f73f2c53195ecf69 | subsection | 43 | 1,121 | Orthogonal spaces and global equivalences | The space {\mathbf {L}}(V,{\mathcal {U}}) comes with a (G\times G)-action
as in (REF ), and it is
(G\times G)-cofibrant by Proposition REF (ii).
Then {\mathbf {L}}(V,{\mathcal {U}}) is also cofibrant as a G-space for the
diagonal action, by Proposition REF (i).
Since {\mathbf {L}}(V,{\mathcal {U}}) is G-cofibrant and... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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2275f2d12eb8a60f6bb42ddfff7b7aa85fa9ecdd | subsection | 44 | 1,121 | Orthogonal spaces and global equivalences | Indeed, the map f^\flat (W) is the composite{\mathbf {L}}(V,W) \times _G A \ \xrightarrow{} \ {\mathbf {L}}(V,W) \times _G Y(V) \ \xrightarrow{} \ Y(W)\ .Example 1.24
For every compact Lie group G, every G-representation V
and every G-space A the semifree orthogonal space {\mathbf {L}}_{G,V} A
is closed.orthogonal spa... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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4bcfa61ad0174e7719291e4e709e868ee3d2ca24 | subsection | 45 | 1,121 | Orthogonal spaces and global equivalences | Since \Gamma \cap (1\times G)=\lbrace (1,1)\rbrace , every element l\in L
then has a unique preimage (l,\alpha (l)) under the projection,
and the assignment l\mapsto \alpha (l) is a continuous homomorphism
from L to G whose graph is \Gamma .We recall that a
universal G-spaceuniversal space!for a set of subgroups
for a ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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014801fdf8985f46e657a3763e7164bbc3277362 | subsection | 46 | 1,121 | Orthogonal spaces and global equivalences | On the other hand, if \Gamma \cap (1\times G)=\lbrace (1,1)\rbrace ,
then \Gamma is the graph of a unique continuous homomorphism \alpha :L\longrightarrow G,
where L is the projection of \Gamma to K. Then{\mathbf {L}}(V,{\mathcal {U}}_K)^\Gamma \ = \ {\mathbf {L}}^L(\alpha ^* V, {\mathcal {U}}_K)is the space of L-equiv... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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c4c45b00a4a59da29cbaeedb049a90896ef74125 | subsection | 47 | 1,121 | Orthogonal spaces and global equivalences | Indeed, if V and
\bar{V} are two faithful G-representations, then V\oplus \bar{V} is
yet another one, and the two restriction morphisms{\mathbf {L}}_{G,V}\ \xleftarrow{}\ {\mathbf {L}}_{G,V\oplus \bar{V}}\ \xrightarrow{}\ {\mathbf {L}}_{G,\bar{V}}are global equivalences by
Proposition REF (ii).Example 1.28 global clas... | {
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"Stefan Schwede"
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5ae345867c1f08d48652aaa4a522d3275d7b469d | subsection | 48 | 1,121 | Orthogonal spaces and global equivalences | Indeed, the total space {\mathbf {L}}(V,{\mathcal {U}}_K)
is homeomorphic to the Stiefel manifold of \dim (V)-frames in {\mathbb {R}}^\infty ,
and hence it admits a CW-structure. Every CW-complex is a normal Hausdorff space
(see for example or ),
hence completely regular. So {\mathbf {L}}(V,{\mathcal {U}}_K) is complet... | {
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9208ab6e46752e28a1cffacb239abaecbbbe32e9 | subsection | 49 | 1,121 | Orthogonal spaces and global equivalences | So we are done if we can show
that q:{\mathbf {L}}(V,{\mathcal {U}}_K)\longrightarrow {\mathbf {L}}(V,{\mathcal {U}}_K)/G is a universal (K,G)-principal bundle
in the above sense.Universal (K,G)-principal bundles can be built in different ways;
the most common construction is a version of
Milnor's infinite join ,
see f... | {
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724ebd112d92b816fd99932a383b8fe616d6e345 | subsection | 50 | 1,121 | Global model structure for orthogonal spaces | In this section we establish the global model structure on the category of orthogonal spaces, see Theorem REF .
Towards this aim we first discuss a `strong level model structure',
which we then localize.
In Proposition REF
we use the global model structure to relate unstable global homotopy
theory to the homotopy theo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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016125279333ae46b4ed0908af1bc6c31fcab33d | subsection | 51 | 1,121 | Global model structure for orthogonal spaces | The extension l_m(Z) of a continuous functor Z:{\mathbf {L}}^{\le m}\longrightarrow {\mathbf {T}}
is a coequalizer of the two morphisms of orthogonal spaces@C=7mm{
\coprod _{0\le j\le k\le m} {\mathbf {L}}({\mathbb {R}}^k,-)\times {\mathbf {L}}({\mathbb {R}}^j,{\mathbb {R}}^k)\times Z({\mathbb {R}}^j)
\ @<-.4ex>[r] @<.... | {
"cite_spans": [
{
"arxiv_id": "",
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"raw": "G. M. Kelly, Basic concepts of enriched category theory. Reprint of the 1982 original. Repr. Theory Appl. Categ. No. 10 (2005), vi+137 pp.",
"source_ref_id": "... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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de0bf4322698f6ac654785116bda0b52c24ef264 | subsection | 52 | 1,121 | Global model structure for orthogonal spaces | The m-th latching spacelatching space!of an orthogonal space
of Y is the O(m)-spaceL_m Y \ = \ (\operatorname{sk}^{m-1} Y)({\mathbb {R}}^m) \ ;it comes with a natural O(m)-equivariant map\nu _m=i_{m-1}({\mathbb {R}}^m)\ :\ L_m Y\ \longrightarrow \ Y({\mathbb {R}}^m) \ ,the m-th latching map.We agree to set \operatornam... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6e06da63888ffc8b9a1354ed25dd315314933527 | subsection | 53 | 1,121 | Global model structure for orthogonal spaces | We have\operatorname{sk}^0 Y \ = \ \text{const}(Y(0)) \ ,the constant orthogonal space with value Y(0); the latching map\nu _1 \ : \ L_1 Y \ = \ (\operatorname{sk}^0 Y)({\mathbb {R}}) \ = \ Y(0) \ \xrightarrow{} \ Y({\mathbb {R}})is the map induced by the unique linear isometric embedding u:0\longrightarrow {\mathbb {R... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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defd76b962dc0997cad63521aa3b421b89066c40 | subsection | 54 | 1,121 | Global model structure for orthogonal spaces | The evaluation functor\operatorname{ev}_{G,V}\ : \ spc\ \longrightarrow \ G{\mathbf {T}}factors through the category {\mathbf {L}}^{\le n} as the compositespc\ \longrightarrow \ spc^{\le n} \ \xrightarrow{} \ G{\mathbf {T}}\ .So the left adjoint semifree functor {\mathbf {L}}_{G,V} can be chosen as the composite
of the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3e851f3be8a79e15229e3485e3aaf914d6728127 | subsection | 55 | 1,121 | Global model structure for orthogonal spaces | The map f({\mathbb {R}}^m):X({\mathbb {R}}^m)\longrightarrow Y({\mathbb {R}}^m) is an O(m)-weak equivalence
for every m\ge 0.Clearly, condition (i) implies condition (ii), and that implies condition (iii)
(because the tautological action of O(m) on {\mathbb {R}}^m is faithful).
So we suppose that f({\mathbb {R}}^m) is ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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535439febb008eef8bdd191628a95749d47cebb9 | subsection | 56 | 1,121 | Global model structure for orthogonal spaces | Equivalently, for every m\ge 0 the latching map \nu _m:L_m B \longrightarrow B({\mathbb {R}}^m)
is an O(m)-cofibration.We are ready to establish the strong level model structure.Proposition 2.10
The strong level equivalences, strong level fibrations
and flat cofibrations form a topological cofibrantly generated model ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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797985d6d1ddb2f0221dbea0fe8188a091bc4905 | subsection | 57 | 1,121 | Global model structure for orthogonal spaces | Again by Proposition REF (iii),
J^{\operatorname{str}} detects the fibrations in the strong level model structure.The model structure is topological by Proposition REF ,
where we take {\mathcal {G}} as the set of orthogonal spaces L_{H,{\mathbb {R}}^m}
for all m\ge 0 and all closed subgroups H of O(m), and we take {\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0542daca1c8dc5093a04007b75920e5cb2ada29a | subsection | 58 | 1,121 | Global model structure for orthogonal spaces | Similarly, J^{\operatorname{str}} is a set of representatives of the
isomorphism classes of morphisms{\mathbf {L}}_{G,V}\times j_k\ : \ {\mathbf {L}}_{G,V}\times D^k\times \lbrace 0\rbrace
\ \longrightarrow \ {\mathbf {L}}_{G,V} \times D^k\times [0,1]for G a compact Lie group, V a faithful G-representation and k\ge 0.... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bfc742bfe670e1214bf3a416dedfb05f2c8aa720 | subsection | 59 | 1,121 | Global model structure for orthogonal spaces | Part (iii) is the special case of (ii) where K
is a trivial group, using that cofibrations of spaces are in particular
h-cofibrations (Corollary REF )
and hence closed embeddings (Proposition REF ).Now we proceed towards the global model structure
on the category of orthogonal spaces,
see Theorem REF .
The weak equival... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6204c0f7f482aab7d5726f48d7fd4ed6f120b1d3 | subsection | 60 | 1,121 | Global model structure for orthogonal spaces | We consider a compact Lie group G,
a faithful G-representation V, a finite G-CW-pair (B,A) and a commutative square:{
A[r]^-\alpha [d]_{\text{incl}} & X(V) [d]^{f(V)} \\
B[r]_-\beta & Y(V) }We will exhibit a continuous G-map \mu :B\longrightarrow X(V) such that
\mu |_A=\alpha and such that f(V)\circ \mu
is homotopic, ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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460c93a7f448655753b765b04d04a23b2a62f099 | subsection | 61 | 1,121 | Global model structure for orthogonal spaces | Since f is a global equivalence, there is
a G-equivariant linear isometric embedding \varphi :V\longrightarrow W and
a continuous G-map \lambda :B\longrightarrow X(W) such that
\lambda |_A=X(\varphi )\circ \alpha and f(W)\circ \lambda is G-homotopic to
Y(\varphi )\circ \beta relative A.
Since X is static, the map X(\va... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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74a346fbe1144ad95ef1edc0ca71d70cd371af91 | subsection | 62 | 1,121 | Global model structure for orthogonal spaces | Then the following two conditions are equivalent:The square of spaces
\begin{aligned}
@C=15mm{
\operatorname{map}(B,X)[r]^-{\operatorname{map}(j,X)} [d]_{\operatorname{map}(B,f)} &
\operatorname{map}(A,X) [d]^{\operatorname{map}(A,f)} \\
\operatorname{map}(B,Y)[r]_-{\operatorname{map}(j,Y)} & \operatorname{map}(A,Y) }
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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431a06bbff96fac070e87d3e9114d248adfe2925 | subsection | 63 | 1,121 | Global model structure for orthogonal spaces | Given any compact Lie group G and G-representations V and W,
the restriction morphism\rho _{G,V,W} \ = \ \rho _{V,W}/G \ : \ {\mathbf {L}}_{G,V\oplus W} \ \longrightarrow \ {\mathbf {L}}_{G,V}restricts (the G-orbit of) a linear isometric embedding
from V\oplus W to V.
If the representation V is faithful,
then this morp... | {
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"raw": "W. G. Dwyer, J. Spalinski, Homotopy theories and model categories. In: Handbook of algebraic topology, ed. I. M. James, Elsevier (1995)... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c4568ec745b50e142535b01ac13765d0008c41b0 | subsection | 64 | 1,121 | Global model structure for orthogonal spaces | For the other half of the factorization axiom MC5
we apply the small object argumentsmall object argument
(see for example or )
to the set J^{\operatorname{str}}\cup K.
All morphisms in J^{\operatorname{str}} are flat cofibrations and strong level equivalences.
Since {\mathbf {L}}_{G,V\oplus W} and {\mathbf {L}}_{G,V} ... | {
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b5b8fa6886ef0c02146a8cb545617726946ecaeb | subsection | 65 | 1,121 | Global model structure for orthogonal spaces | Right properness follows from
Proposition REF (xii) because
global fibrations are in particular strong level fibrations.The global model structure is topological by
Proposition REF ,
with {\mathcal {G}} the set of semifree orthogonal spaces {\mathbf {L}}_{G,V}
indexed by a set of representatives (G,V) of the isomorphi... | {
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8f02aac37854cb99ccb4d69bc622feeee99fa459 | subsection | 66 | 1,121 | Global model structure for orthogonal spaces | A morphism f:X\longrightarrow Y of orthogonal spaces is a fibration
in the positive global model structure if and only if
for every compact Lie group G,
every faithful G-representation V with V\ne 0
and every equivariant linear isometric embedding \varphi :V\longrightarrow W
the map f(V)^G:X(V)^G\longrightarrow Y(V)^G ... | {
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e3d23376811d8eb06f7a9d2fc1c43b8cf13b4e60 | subsection | 67 | 1,121 | Global model structure for orthogonal spaces | Then evaluation at a faithful G-representation V
and the semifree functor at (G,V) are a pair of adjoint functors{ {\mathbf {L}}_{G,V}\ : \ G{\mathbf {T}}\ @<.4ex>[r] &
\ spc\ : \ \operatorname{ev}_{G,V} @<.4ex>[l] }between the categories of G-spaces and orthogonal spaces.
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c95e73b6a1be0fc9d3ac977491f528ee5a5253b8 | subsection | 68 | 1,121 | Global model structure for orthogonal spaces | So the canonical mapZ(V) \ \longrightarrow \ \operatorname{colim}_{n\ge 1}\, Z(V_n) \ = \ Z({\mathcal {U}}_G)is also a G-weak equivalence. Since Z is a globally fibrant
replacement of Y, the G-space Z(V) calculates the right derived functor
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a75fce6b4d1a0256ea869d342ac01a3fc1fefda3 | subsection | 69 | 1,121 | Global model structure for orthogonal spaces | This data makes the functors{
(-)({\mathcal {U}}_K)\ : \ spc\quad @<.4ex>[r] &
\quad K{\mathbf {T}}\ : \ R_K @<.4ex>[l] }into an adjoint pair.Proposition 2.26
Let K be a compact Lie group.The adjoint functor pair ((-)({\mathcal {U}}_K),R_K) is a Quillen pair for
the global model structure of orthogonal spaces and the ... | {
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fecb51e7dbfff2a04e9f6c88f190f1301daf2913 | subsection | 70 | 1,121 | Global model structure for orthogonal spaces | So in the commutative square@C=20mm{
(R_K(X)(V))^G[r]^-{R_K(X)(\varphi )^G} [d]_{(R_K(f)(V))^G} &
(R_K(X)(V\oplus W))^G [d]^{(R_K(f)(V\oplus W))^G} \\
(R_K(Y)(V))^G[r]_-{R_K(Y)(\varphi )^G} & (R_K(Y)(V\oplus W))^G }both vertical maps are Serre fibrations and both horizontal maps
are weak equivalences.
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7940114778659f6c3c0bcd845102f04a676c9cd7 | subsection | 71 | 1,121 | Global model structure for orthogonal spaces | Since R A is static (Proposition REF (ii)) and closed,
the map (R A)(\varphi ):(R A)(V)\longrightarrow (R A)(W)
induced by any linear isometric embedding \varphi :V\longrightarrow W
is a weak equivalence and a closed embedding.
So the canonical mapA \ \cong \ \operatorname{map}({\mathbf {L}}(0,{\mathbb {R}}^\infty ),A... | {
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6098814c7c25d9ac178279affca9f0c9268545bf | subsection | 72 | 1,121 | Global model structure for orthogonal spaces | Then {\mathbf {L}}(W,{\mathbb {R}}^\infty ) is a universal
free K-space by Proposition REF (i).
So the projection from E K\times {\mathbf {L}}(W,{\mathbb {R}}^\infty ) to the second factor
is a K-weak equivalence between cofibrant K-spaces, hence
a K-homotopy equivalence. So the induced map\text{const}\ : \ (R A)(W) \... | {
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e3c8df6ecd82eab8324df38c2d70da447e01050f | subsection | 73 | 1,121 | Global model structure for orthogonal spaces | Indeed, for every compact Lie group K the two vertical maps in
the commutative square of K-spaces@C=25mm{
Y({\mathcal {U}}_K)[d]_{\text{const}}[r]^-{f({\mathcal {U}}_K)} &
Z({\mathcal {U}}_K)[d]^{\text{const}}\\
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1e704db5b509e7820886b3eb46cf9e528e37b8eb | subsection | 74 | 1,121 | Global model structure for orthogonal spaces | Then {\mathbf {L}}_{G,V}({\mathcal {U}}_K)={\mathbf {L}}(V,{\mathcal {U}}_K)/G is a classifying space
for principal (K,G)-bundles, by Proposition REF .
Since G has abelian identity component, it is 1-truncated
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fbbabddfef13da17142d358f94e9d0f5a5af449c | subsection | 75 | 1,121 | Global model structure for orthogonal spaces | As n varies, these topological simplices assemble into a covariant functor{\mathbf {\Delta }}\ \longrightarrow {\mathbf {T}}\ , \quad [n]\ \longmapsto \ \Delta ^n\ ;the coface maps are given by(d_i)_*(t_1,\dots ,t_n) \ = \left\lbrace \begin{array}{ll}
\quad (0,t_1,\dots ,t_n) & \mbox{for $i=0$,} \\
(t_1,\dots ,t_i,t_i,... | {
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45bcbb444abbd7f11a203e0ea760ef6f06e7cbbc | subsection | 76 | 1,121 | Global model structure for orthogonal spaces | A simplicial object X can be restricted along the forgetful functor{\mathbf {\Delta }}(n)^{\operatorname{op}}_\circ \ \xrightarrow{} \ {\mathbf {\Delta }}^{\operatorname{op}} \ ,\quad (\sigma :[n]\longrightarrow [k])\ \longmapsto \ [k]\ .The n-th latching object of X is the colimit over {\mathbf {\Delta }}(n)^{\operato... | {
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"Stefan Schwede"
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165ba247079eb1ade69192a32553ec804603714b | subsection | 77 | 1,121 | Global model structure for orthogonal spaces | Reedy's paper – albeit highly influential – remains unpublished, but an account of
the Reedy model structure can
for example be found in .
If we form the Reedy model structure starting
with the global model structure of orthogonal spaces,
then the cofibrant objects are precisely the
Reedy flat simplicial orthogonal spa... | {
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825c6bbe7310953814fb367745b7a843ca4069aa | subsection | 78 | 1,121 | Monoidal structures | This section is devoted to monoidal products on the category of orthogonal spaces,
with emphasis on global homotopical features.
Our main focus is the box product of orthogonal spaces,
a special case of a Day type convolution product,
and the `good' monoidal structure for orthogonal spaces.
We prove in Theorem REF that... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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b24508c2439287f58ce06448fde723a1aaa5e662 | subsection | 79 | 1,121 | Monoidal structures | We will often refer to this bijection as the
universal property of the box product of orthogonal spaces.
universal property!of the box product
Very often only the object X\boxtimes Y will be referred to
as the box product, but one should keep in mind that it comes equipped
with a specific, universal bimorphism.The exis... | {
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"Stefan Schwede"
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451e840effd151fadd8acdd4f97a4f6e5f9061b1 | subsection | 80 | 1,121 | Monoidal structures | The functor X\boxtimes - preserves global equivalences.(i) For an orthogonal space Z we denote by \operatorname{sh}Z the orthogonal space defined by(\operatorname{sh}Z)(V) \ = \ Z( V\oplus V) \text{\qquad and\qquad }
(\operatorname{sh}Z)(\varphi )\ = \ Z(\varphi \oplus \varphi )\ ;thus \operatorname{sh}Z is isomorphic ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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c5c56755ef2fe4279f8b66c211ce04aee09a1cb7 | subsection | 81 | 1,121 | Monoidal structures | Moreover, f_0=\lambda \circ \rho _{X,Y} and f_1=(X\boxtimes Y)\circ i_1, so this is
the desired homotopy.
The morphism (X\boxtimes Y)\circ i_1 is a global equivalence by
Theorem REF , hence so is the morphism \lambda \circ \rho _{X,Y}.The shift functor preserves products, and under the canonical isomorphism
\operatorna... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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b88e3d7aea4e484fc16a5854108fde803198cd20 | subsection | 82 | 1,121 | Monoidal structures | The mapA\times B \ \xrightarrow{} \ &( {\mathbf {L}}_{G,V} A)(V)\times ({\mathbf {L}}_{K,W} B) (W)\\
\xrightarrow{}\quad &( ( {\mathbf {L}}_{G,V} A)\boxtimes ({\mathbf {L}}_{K,W} B) )(V\oplus W)is (G\times K)-equivariant, so it extends freely to a morphism of orthogonal spaces{\mathbf {L}}_{G\times K,V\oplus W}(A\times... | {
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"Stefan Schwede"
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8e48032498e58b3ebe1cd9c86b6ce6f58150bcff | subsection | 83 | 1,121 | Monoidal structures | A special case of this shows that the box product of two global classifying
spaces is another global classifying space.global classifying space
Indeed, if G acts faithfully on V and K acts faithfully on W, then
the (G\times K)-action on V\oplus W is also faithful, and hence( B_{\operatorname{gl}}G) \boxtimes (B_{\opera... | {
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f4436d13eba013a937adff375e622ee9d5e3f124 | subsection | 84 | 1,121 | Monoidal structures | The first map in (REF )
is a quotient map, and its source is compact by hypothesis (a)
and because the spaces {\mathbf {L}}({\mathbb {R}}^i,V) are all compact.
So the space (\operatorname{sk}^m Y)(V) is quasi-compact.
Since Y(V) is Hausdorff, the map i_m(V) is a closed map.
So all that remains is to show that i_m(V) is... | {
"cite_spans": []
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"Stefan Schwede"
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82342dc1a9a7f47aa96e3940b11baedd6a2d933c | subsection | 85 | 1,121 | Monoidal structures | An element of y\in Y({\mathbb {R}}^m) is in the image of the latching map
if an only if it is in the image of the map Y(U)\longrightarrow Y({\mathbb {R}}^m)
for some (m-1)-dimensional subspace U of {\mathbb {R}}^m.
Then the orthogonal reflection in the hyperplane U fixes y.
Conversely, if y is fixed by a reflection,
th... | {
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a0e32f09c3e721d50fc89d13465ab96015e18f22 | subsection | 86 | 1,121 | Monoidal structures | Finally, if a pair(\varphi ,\psi ) \ \in \ {\mathbf {L}}(V,U)\times {\mathbf {L}}(W,U)is fixed by the reflection in some hyperplane U^{\prime } of U,
then the images of \varphi and \psi are contained in U^{\prime },
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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e62a45bcb1e789a284727788815499cbdf33fc34 | subsection | 87 | 1,121 | Monoidal structures | The next construction introduces the orthogonal space
{\mathbf {L}}^{\mathbb {C}}_{G,W} for a unitary G-representation W.
In contrast to their orthogonal cousins {\mathbf {L}}_{G,V},
the unitary analogs are not representable nor semifree in any sense.
However, the unitary versions also enjoy various useful properties,
... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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24cf197dae171d3265ef303076d75725874584b4 | subsection | 88 | 1,121 | Monoidal structures | We define the orthogonal space {\mathbf {L}}^{\mathbb {C}}_{G,W} by{\mathbf {L}}^{\mathbb {C}}_{G,W}(V) \ = \ \ {\mathbf {L}}^{\mathbb {C}}(W,V_{\mathbb {C}}) / G\ .We define a morphism of orthogonal spacesf_{G,W} \ : \ {\mathbf {L}}_{G, u W} \ \longrightarrow \ {\mathbf {L}}^{\mathbb {C}}_{G, W}as follows.
The mapj_W ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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97ba56c352cee1915a7edc330b3eb0bb1f829dbc | subsection | 89 | 1,121 | Monoidal structures | Since {\mathbb {C}}\otimes _{\mathbb {R}}{\mathcal {U}}_K is a complete complex G-universe, the complex analog
of Proposition REF (i), proved in much the same way,
shows that the target of \tilde{f} is also such a universal space for the
same family of subgroups of K\times G.
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0.012054443359375,
0.13623046875,
0.038238525390625... |
8f1045198d4d8fbc6bd25dbac6658a6eb9e142e2 | subsection | 90 | 1,121 | Monoidal structures | The maps{\mathbf {L}}^{\mathbb {C}}(W,V_{\mathbb {C}})/G \times {\mathbf {L}}^{\mathbb {C}}(U,V^{\prime }_{\mathbb {C}})/K \ &\longrightarrow \ {\mathbf {L}}^{\mathbb {C}}(W\oplus U,(V\oplus V^{\prime })_{\mathbb {C}})/ ( G\times K )\\
( \varphi \cdot G,\, \psi \cdot K ) \qquad \quad &\longmapsto \qquad (\varphi \oplus... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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47845fc2a1b1c9c5be6421f1a691c6b0e6b380e6 | subsection | 91 | 1,121 | Global families | In this section we explain a variant of unstable global homotopy
theory based on a global family, i.e., a class {\mathcal {F}} of
compact Lie groups with certain closure properties.
We introduce {\mathcal {F}}-equivalences, a relative version of global equivalences,
and establish an {\mathcal {F}}-relative version of t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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027e1ec22fbe84d53f191aa6c1e4c519e5ea15cf | subsection | 92 | 1,121 | Global families | A morphism f:X\longrightarrow Y of orthogonal spaces isan {\mathcal {F}}-level equivalenceF-level equivalence@{\mathcal {F}}-level equivalence!of orthogonal spaces
if for every compact Lie group G in {\mathcal {F}} and every G-representation V the map
f(V)^G:X(V)^G\longrightarrow Y(V)^G is a weak equivalence;
an {\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
62,
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d8f472e8fed79d5100bc8003a9598e84c2a7d8ed | subsection | 93 | 1,121 | Global families | The {\mathcal {F}}-level equivalences, {\mathcal {F}}-level fibrations
and {\mathcal {F}}-cofibrations form a topological and cofibrantly generated model structure,
the {\mathcal {F}}-level model structure,F-level model structure@{\mathcal {F}}-level model structure!for orthogonal spaces
on the category of orthogonal s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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41e48e1c9784e2cec555b23dd8f145b24a7fc453 | subsection | 94 | 1,121 | Global families | A morphism f:X\longrightarrow Y of orthogonal spaces
is an {\mathcal {F}}-equivalenceF-equivalence@{\mathcal {F}}-equivalence!of orthogonal spaces
if the following condition holds: for every compact Lie group G in {\mathcal {F}},
every G-representation V, every k\ge 0
and all maps \alpha :\partial D^k\longrightarrow X(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
62,
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2c33a2287cb36f4d4c2a3b03ee52aee20532a30b | subsection | 95 | 1,121 | Global families | For every compact Lie group G in the family {\mathcal {F}}
and every exhaustive sequence \lbrace V_i\rbrace _{i\ge 1}
of G-representations the induced map
\operatorname{tel}_i f(V_i)\ : \ \operatorname{tel}_i X(V_i)\ \longrightarrow \ \operatorname{tel}_i Y(V_i)
is a G-weak equivalence.Definition 4.6 F-global fibrati... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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56a15be741d722e4a508badd639696bee066fd25 | subsection | 96 | 1,121 | Global families | Then the canonical morphism
f_\infty :Y_0\longrightarrow Y_\infty to the colimit of the sequence \lbrace f_n\rbrace _{n\ge 0}
is an {\mathcal {F}}-equivalence.
Let
{
C [d]_\gamma & A [l]_-g [d]^\alpha [r]^-f & B [d]^\beta \\
C^{\prime } & A^{\prime } [l]^-{g^{\prime }}[r]_-{f^{\prime }} & B^{\prime } }
be a commutat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5ffdced40847199593ad20e84f94d71472620a57 | subsection | 97 | 1,121 | Global families | The set I_{{\mathcal {F}}} detects the acyclic fibrations
in the {\mathcal {F}}-level model structure,
which coincide with the acyclic fibrations
in the {\mathcal {F}}-global model structure.Also in Proposition REF
we defined J_{{\mathcal {F}}} as the set of all morphisms G_m j
for m\ge 0 and for j in the set
of gener... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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