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561e5817979a03b7ef7ab689faa473aa2bee1573 | subsection | 298 | 1,121 | Global forms of | If V=\alpha ^*({\mathbb {R}}^j) is such a G-representation, then the space
{\mathbf {L}}^G(V,{\mathcal {U}}_G^\perp ) is empty if V has non-trivial G-fixed points,
and contractible otherwise. Moreover, the centralizer C(\alpha )
is precisely the group of G-equivariant linear self-isometries of V,
which acts freely on {... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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50e965056e9999900e8fc959f3bc1a116c84b831 | subsection | 299 | 1,121 | Global forms of | This canonical decomposition provides a homeomorphism{\mathbf {bO}}(V\oplus W)^G \ = \ (G r_{|V|+|W|}(V\oplus W\oplus {\mathbb {R}}^\infty ))^G \ \cong \coprod _{j=0,\dots ,|V|} \left( G r_j(V) \right)^G \times G r_{j+|W|}(W\oplus {\mathbb {R}}^\infty )sending L to the pair (V- L^\perp ,\ L^G).
Every G-invariant subspa... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9ceb6453bcb93f521d87bc16717056cfe8059e4a | subsection | 300 | 1,121 | Global forms of | As we explained in the weak equivalence (REF ),
the set \pi _0(G r_j^{G,\perp } ) bijects with the set of isomorphism classes of
j-dimensional G-representations with trivial fixed points.
Altogether this identifies \pi _0({\mathbf {bO}}({\mathcal {U}}_G)^G) with \mathbf {RO}^\sharp (G),
and unraveling all definitions s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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21435c94d92faf8ab200ff2675032dc4e38e3429 | subsection | 301 | 1,121 | Global forms of | The inclusion of {\mathbf {bO}}_{(m)} into {\mathbf {bO}}_{(m+1)} is a closed embedding,
so the global invariance property of Proposition REF (viii)
entitles us to view the union {\mathbf {bO}} as a global homotopy colimit of
the filtration.The tautological action of O(m) on {\mathbb {R}}^m is faithful,
so the semifre... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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13128203eda7d3b6f6a7ffba605066fdc44710c9 | subsection | 302 | 1,121 | Global forms of | The inclusion {\mathbf {bO}}_{(m)} \longrightarrow {\mathbf {bO}}_{(m+1)}
takes the class u_m to the class\operatorname{res}^{O(m+1)}_{O(m)}(u_{m+1})\ \in \ \pi _0^{O(m)}\left({\mathbf {bO}}_{(m+1)} \right) \ .The morphism \gamma _m factors as the composite of two morphisms of
orthogonal spaces{\mathbf {L}}_{O(m),{\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
5f446480007af81a988c3a3e79a35dcdc0ac6627 | subsection | 303 | 1,121 | Global forms of | So the classes\text{incl}_*( u_m ) \text{\qquad and\qquad }
\operatorname{res}^{O(m)}_{O(m+1)}(u_{m+1}) \text{\quad in\ }
\pi _0^{O(m)}({\mathbf {bO}}_{(m+1)})are represented by the two subspaces({\mathbb {R}}^{m+1}-j({\mathbb {R}}^m))\oplus j({\mathbb {R}}^m) \text{\qquad respectively\qquad }
0\oplus {\mathbb {R}}^{m+... | {
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{
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"raw": "J. P. May, E_{\\infty } ring spaces and E_{\\infty } ring spectra. With contributions by F. Quinn, N. Ray, and J. Tornehave. Lecture Notes in Mathematics, Vol. 77. Springer-Verlag, Berlin-New York,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ad96866c96ffca4f57c5ba4ccc9eec8db165a74b | subsection | 304 | 1,121 | Global forms of | For all n\ge 0 and all inner product spaces V_1,\dots ,V_n
we define a linear isometry\kappa \ : \ (V_1\oplus {\mathbb {R}}^\infty )\oplus \dots \oplus (V_n\oplus {\mathbb {R}}^\infty )\ \cong \ V_1\oplus \dots \oplus V_n\oplus ({\mathbb {R}}^\infty )^nby shuffling the summands, i.e.,\kappa (v_1,x_1,\dots ,v_n,x_n)\ = ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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541059a912d9e1414e1ba31120906ba522018000 | subsection | 305 | 1,121 | Global forms of | We obtain binary pairings\pi _0^G(R)\times \pi _0^G(R)\ \xrightarrow{} \ \pi _0^G(R\boxtimes R)\ \xrightarrow{} \ \pi _0^G(R)\ ,where \varphi \in {\mathcal {L}}(2) is any linear isometric embedding of
({\mathbb {R}}^\infty )^2 into {\mathbb {R}}^\infty .
The second map (and hence the composite) is independent of \varph... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0fc789de44335ddd63f016e89edc57a633d7d96f | subsection | 306 | 1,121 | Global forms of | The zigzag of morphisms passes through the orthogonal space {\mathbf {BO}}^{\prime } with values{\mathbf {BO}}^{\prime }(V) \ = \ Gr_{|V|}(V^2\oplus {\mathbb {R}}^\infty )\ .The structure maps of {\mathbf {BO}}^{\prime } are a mixture of those for {\mathbf {bO}} and {\mathbf {BO}}, i.e.,{\mathbf {BO}}^{\prime }(\varphi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2b5ab60451201b479d0245a999babcc023133fda | subsection | 307 | 1,121 | Global forms of | We define a morphismj \ : \ {\mathbf {BO}}^{\prime }_{(m+1)} \ \longrightarrow \ \operatorname{sh}({\mathbf {BO}}^{\prime }_{(m)})at an inner product space V byj(V)\ : \ {\mathbf {BO}}^{\prime }_{(m+1)}(V)\ &= \ G r_{|V|}(V\oplus V\oplus {\mathbb {R}}^{m+1})\\
&\longrightarrow \ G r_{|V|+1}(V\oplus {\mathbb {R}}\oplus ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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84e8e9686a1a046094906325a023fbcaa2c9826b | subsection | 308 | 1,121 | Global forms of | The inclusion of {\mathbf {BO}}^{\prime }_{(m)} into {\mathbf {BO}}^{\prime }_{(m+1)} is also objectwise a closed embedding,
so the inclusion{\mathbf {BO}}\ =\ {\mathbf {BO}}^{\prime }_{(0)}\ \longrightarrow \ {\bigcup }_{m\ge 0} \, {\mathbf {BO}}^{\prime }_{(m)}\ = \ {\mathbf {BO}}^{\prime }is a global equivalence,
by... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b63b4d6dce513768a04abeaa6f5cb587c2b81978 | subsection | 309 | 1,121 | Global forms of | The abelian \operatorname{Rep}-monoid {\underline{\pi }}_0({\mathbf {bO}})
cannot be extended to a global power monoid.augmentation ideal!of the orthogonal representation ringIf L\subset V\oplus {\mathbb {R}}^\infty is a G-invariant subspace of the same dimension
as V, then[L] - [V] \ &= \ (\dim (L)-\dim (L^\perp ))\cd... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c4b7aa6e342173a949ad2cd3c39c4e45c043ef17 | subsection | 310 | 1,121 | Global forms of | So \pi _0^{\Sigma _3}({\mathbf {bO}}) `is' (via \gamma \circ a_*)
the free abelian submonoid of \mathbf {IO}(\Sigma _3) generated by1 -\sigma \text{\qquad and\qquad } 2 - \nu \ .We abuse notation and also write \sigma for the 1-dimensional
sign representation of the cyclic subgroup of \Sigma _3 generated by
the transpo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e824a748916329d9d024878e3d7c0c8a7acb2577 | subsection | 311 | 1,121 | Global forms of | The orthogonal space {\mathbf {bOP}} is naturally {\mathbb {Z}}-graded: for m\in {\mathbb {Z}} we let{\mathbf {bOP}}^{[m]}(V)\ \subset \ {\mathbf {bOP}}(V)be the path component consisting of all subspaces L\subset V\oplus {\mathbb {R}}^\infty
such that \dim (L)=\dim (V) + m. For fixed m these spaces
form an orthogonal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fab970299dad5bfe92743cb37ab20c2ddbc08f89 | subsection | 312 | 1,121 | Global forms of | Any two linear isometric embeddings from
{\mathbb {R}}^\infty \oplus {\mathbb {R}} to {\mathbb {R}}^\infty are homotopic through
linear isometric embeddings, so the homotopy class of \psi _\sharp
is independent of the choice of \psi .Proposition 4.32
For every linear isometric embedding \psi :{\mathbb {R}}^\infty \op... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5209f30906ab5a3c3ced3536f81710f9a8ab9fea | subsection | 313 | 1,121 | Global forms of | Since both {\mathbf {bOP}}\circ i and \psi _! are global equivalences,
so is the composite \psi _\sharp .Another description of the global homotopy type of {\mathbf {bOP}}
is as a global homotopy colimit of a sequence of self-maps of \mathbf {Gr}:{\mathbf {bOP}}\ \simeq \ \operatorname{hocolim}_{m\ge 1}\, \mathbf {Gr}\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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54798ef718cabc3614dcae833665dc77b186563b | subsection | 314 | 1,121 | Global forms of | The inclusion of {\mathbf {bOP}}_{(m)} into {\mathbf {bOP}}_{(m+1)} is a closed embedding,
so the global invariance property of Proposition REF (ix)
entitles us to view the union {\mathbf {bOP}} as a global homotopy colimit of
the filtration.
This justifies the interpretation of {\mathbf {bOP}} as a global homotopy co... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ae0169a722367d2e4a505a6600eeee7a922daf67 | subsection | 315 | 1,121 | Global forms of | Then the triangle of monoid homomorphisms
on the right of following diagram commutes:{
\pi _0^G({\mathbf {bOP}}) [r]^-{a_*} [dr] & \pi _0^G({\mathbf {BOP}}^{\prime }) [d]_\gamma ^\cong &
\pi _0^G({\mathbf {BOP}}) [l]^-{\cong }_-{b_*} [dl]_(.6){\cong }^{(\ref {eq:pi^G BOP to ROG})} \\
& \mathbf {RO}(G) }The two maps on ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8255a8307b0693258cf19e38cdd22dc561fbfaa7 | subsection | 316 | 1,121 | Global forms of | More explicitly,{\mathbf {BU}}(V) \ = \ Gr_{|V|}^{\mathbb {C}}(V_{\mathbb {C}}^2)\text{\qquad respectively\qquad }
{\mathbf {BSp}}(V) \ = \ Gr_{|V|}^{\mathbb {H}}(V_{\mathbb {H}}^2)\ .The complex and quaternionic analogues of Theorem REF
provide isomorphisms of global power monoids{\underline{\pi }}_0({\mathbf {BUP}})... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0b6829156d0291fd6ba649ccd8feb974d4df58da | subsection | 317 | 1,121 | Global forms of | Periodic versions {\mathbf {bUP}}{\mathbf {bUP}} - periodic global B U
and {\mathbf {bSpP}}{\mathbf {bSpP}} - periodic global B S p
are defined by taking the full Grassmannian
inside V_{\mathbb {C}}\oplus {\mathbb {C}}^\infty respectively V_{\mathbb {H}}\oplus {\mathbb {H}}^\infty ,
as in the real case in Example REF .... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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04d047c2f9c92d75c6f6120f88b180f19fb7fbe0 | subsection | 318 | 1,121 | Global group completion and units | For every orthogonal monoid space R and every compact Lie group G,
the operation (REF )
makes the equivariant homotopy set \pi _0^G(R) into a monoid,
and this multiplication is natural with respect to restriction maps in G.
If the multiplication of R is commutative,
then so is the multiplication of \pi _0^G(R).
In this... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b535d39acfc0538fade7fa7091adbe6c983252f4 | subsection | 319 | 1,121 | Global group completion and units | We write 0 for any morphism that factors through a zero object.We call the category {\mathcal {D}} pre-additivepre-additive category
if `finite products are coproducts';
more precisely, we require that every product A\times B
of two objects A and B is also a co-product, with respect to
the morphismsi_1\ =\ (\operatorna... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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adf349ccd27260c69d662fcdc0432dac6a018493 | subsection | 320 | 1,121 | Global group completion and units | The lower triangle then commutes since
the two morphismsa\bot (b\bot c)\ ,\, ((a\bot b)\bot c)\circ \alpha \ : A\times (A\times A)\ \longrightarrow \ Xhave the same `restrictions', namely a, b respectively c.The commutativity is a consequence of two elementary facts:
first, b\bot a=(a\bot b)\tau
where \tau :A\times A\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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48417b658141bac5ede654dde6c00308105091e5 | subsection | 321 | 1,121 | Global group completion and units | Thus
d(a+ b)=d a+ d b by the definition of `+'.Now we introduce the group-like objects in a pre-additive category.Proposition 5.3
Let {\mathcal {D}} be a pre-additive category.
For every object A of {\mathcal {D}} the following two conditions are equivalent:The shearing morphism
\Delta \bot i_2=(\Delta p_1)+i_2 p_2:A\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1989ceb9783330446923bb526f5e67c030fd3c44 | subsection | 322 | 1,121 | Global group completion and units | A morphism i:R\longrightarrow R^\star in {\mathcal {D}} is a
group completiongroup completion!in a pre-additive category
if for every object T the map{\mathcal {D}}(i,T)\ : \ {\mathcal {D}}(R^\star ,T )\ \longrightarrow \ {\mathcal {D}}(R,T)is injective with image the subgroup {\mathcal {D}}(R,T)^\times of invertible e... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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dfc274028f6195b7488768ac0fa9616d643e37c9 | subsection | 323 | 1,121 | Global group completion and units | The pair (R^\star ,i) is unique up to preferred isomorphism,
and if we choose a group completion i_R:R\longrightarrow R^\star
for every object R, then this extends canonically to a functor(-)^\star \ : \ {\mathcal {D}}\ \longrightarrow \ {\mathcal {D}}and a natural transformation i:\operatorname{Id}\longrightarrow (-)... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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868e063ddbfa00fc99a8e151da3e2536fe3c2673 | subsection | 324 | 1,121 | Global group completion and units | Indeed, given a monoid homomorphism h:M\longrightarrow N that is pointwise invertible,
then we can define f:M^\star \longrightarrow N byf[x,y]\ = \ h(x) - h(y)\ .A routine verification shows that f is indeed a well-defined homomorphism
and that sending h to f is inverse to the restriction map{\mathcal {A}}bMon(i,N) \ :... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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672e9d9940934e2b645f110790f5bd8869e4dfcb | subsection | 325 | 1,121 | Global group completion and units | Dually, g is a cokernel of f if g f=0 and
for every morphism \beta :B\longrightarrow Y such that \beta f=0,
there is a unique morphism \gamma :C\longrightarrow Y such that \gamma g=\beta .Proposition 5.7
Let R be an object of a pre-additive category {\mathcal {D}}.Let e:R^\times \longrightarrow R\times R be a kernel
o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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017aa5d528be4a7cbf5836b05ee270e3bf4b064a | subsection | 326 | 1,121 | Global group completion and units | The relation( d\circ (0,\operatorname{Id}) ) + i \ = \ d\circ ( (0,\operatorname{Id}) + (\operatorname{Id},0)) \ = \ d\circ (\operatorname{Id},\operatorname{Id}) \ = \ 0holds in the monoid {\mathcal {D}}(R,R^\star ), and thus
d\circ (0,\operatorname{Id}) = - i. This shows that d= i\bot (- i).The previous characterizati... | {
"cite_spans": [
{
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"doi": "10.1007/bfb0097438",
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"source_ref_id": "4c0900e5d5eb23d... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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43661220121764dac241930199529fd7d437f73d | subsection | 327 | 1,121 | Global group completion and units | This means that the two binary operations satisfy the interchange law.
Since they also share the same neutral element, they coincide.
Since one of the two operations has inverses, so does the other.The argument that \Sigma R is group-like is dual, using
that \Sigma R is the loop object of R in \operatorname{Ho}({\mathc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0097438",
"end": 1970,
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"source_ref_id": "4c0900e5d5eb23d... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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97c2a272b98c3bfdf07972af744f597e46d5f4be | subsection | 328 | 1,121 | Global group completion and units | So we consider
two morphisms \alpha _1,\alpha _2\in [T,A] such that f\circ \alpha _1=f\circ \alpha _2.
Then by Proposition 4 (ii) of ,
there is an element \lambda \in [A,\Omega D]
such that \alpha _2=\alpha _1\cdot \lambda .
Since the morphism g:B\longrightarrow D has a section,
so does the morphism \Omega g:\Omega B\l... | {
"cite_spans": [
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"doi": "10.1007/bfb0097438",
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"source_ref_id": "4c0900e5d5eb23dd... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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474f659062e4bc066fd601a2400806ca14333da1 | subsection | 329 | 1,121 | Global group completion and units | By
Proposition REF (i) the morphism f
becomes a kernel of q in \operatorname{Ho}({\mathcal {C}}). So the codiagonal morphism
of R has a kernel.(ii) We choose a weak equivalence q:R\longrightarrow \bar{R}
to a fibrant object. A unit morphism \bar{R}^\times \longrightarrow \bar{R} exists in \operatorname{Ho}({\mathcal {... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/surv/099",
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"source_ref_id": "29... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f55e550c3d87061cd352a67f1b8a48c8a2bffbda | subsection | 330 | 1,121 | Global group completion and units | To see this we exploit that both
\operatorname{map}^h(T,R^\times ) and \operatorname{map}^{h,\times }(T,R) are group-like H-spaces,
the multiplication arising from the fact T is a comonoid object up to homotopy.
Moreover, the map u_* is an H-map and bijection on path components
(by the universal property of unit morphi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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25997560fa7d4f2d23c78e84b3d30cd390ff0d15 | subsection | 331 | 1,121 | Global group completion and units | So \eta _R is isomorphic, as an object in the comma category R\downarrow \operatorname{Ho}({\mathcal {C}}),
to i, and hence also a group completion.The previous proposition also has a dual statement
(with the dual proof):
if for every group-like object R of {\mathcal {C}} the
adjunction counit \epsilon :\Sigma (\Omega ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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128f218256057fce9e039d0619a7f1c27ce75055 | subsection | 332 | 1,121 | Global group completion and units | Conversely, if \chi is bijective, then for every x\in M
there is a y\in M such that \chi (x,y)=(x,1), i.e., with x y=1.
Then \chi (x,y x)=(x,x y x)=(x,x)=\chi (x,1), so y x=1
by injectivity of \chi . Thus y is a two-sided inverse for x.For orthogonal monoid spaces R (not necessarily commutative),
the group-like conditi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c821fe075616acc4d59a3a6718655f18d6a6d635 | subsection | 333 | 1,121 | Global group completion and units | We may thus show that for every compact Lie group G the continuous map\chi ^G \ = \ \chi ({\mathcal {U}}_G)^G \ : \ (R\boxtimes R)({\mathcal {U}}_G)^G \ \longrightarrow \ (R\times R)({\mathcal {U}}_G)^G\ = \ R({\mathcal {U}}_G)^G \times R({\mathcal {U}}_G)^Gis a weak equivalence, compare Proposition REF .
Since the mon... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6746828bde9bca76260e5eec1501e990b1ee6849 | subsection | 334 | 1,121 | Global group completion and units | The same argument applies to R\times R instead of R\boxtimes R,
and we obtain a commutative diagram@C=20mm{
\pi _k^G((R\boxtimes R)({\mathcal {U}}_G)^G, 1)
[r]^-{\pi _k(\chi ^G,1)} [d]^\cong _{\pi _k(\varphi _*(-,x)^G)} &
\pi _k^G( (R\times R)({\mathcal {U}}_G)^G, 1) [d]^{\pi _k(\varphi _*(-,\chi ^G(x))^G)}_\cong \\
\p... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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761c5b502fe1cc7458d54b246903baffcee6c95a | subsection | 335 | 1,121 | Global group completion and units | Since \chi ^{R^{\prime }} is a global equivalence by the previous paragraph and
\chi ^R\circ (f\boxtimes f)=(f\times f)\circ \chi ^{R^{\prime }},
the morphism \chi ^R is also a global equivalence.Finally, if R is ultra-commutative, then the point-set level shearing morphism \chi
becomes the shearing morphism in the se... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2fa15a9ce1c77fe59f7d6b56834c771b70273c3c | subsection | 336 | 1,121 | Global group completion and units | Indeed, the commutative square{ R^\times [r]^-p [d]_q & R\boxtimes R [d]^\mu \\
R^{[0,1]}\times _R \lbrace 0\rbrace [r]_-{\operatorname{ev}_0} & R}is a pullback of ultra-commutative monoids,
by definition, and both horizontal morphisms are strong level fibrations,
where q denotes the projection to the second factor.
So... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8a64d2bcb123eeab2f8ce3b5e369b67e78575af4 | subsection | 337 | 1,121 | Global group completion and units | Now we consider the composite\pi _0( T(V)^G)\ &\cong \ \pi _0(\operatorname{map}^{umon}({\mathbb {P}}(B_{\operatorname{gl}}G),T)) \ \xrightarrow{} \\
&\operatorname{Ho}(umon)( {\mathbb {P}}(B_{\operatorname{gl}} G), T ) \ \xrightarrow{} \ \pi _0^G(T)with the adjunction bijection and
the map induced by the localization
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a072554824d277b05cf5726d50624581718f5dc1 | subsection | 338 | 1,121 | Global group completion and units | So the cone is R\rhd [0,1],
the tensor of R with the based space ([0,1], 0),
as defined more generally in (REF ).
Since R is cofibrant and the global model structure is topological,
the left vertical morphism
is an acyclic cofibration, and so the cone C R
is globally equivalent to the zero monoid.We can then construct ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b751c7ee6e4001925e4d500f659d6c65d9e7594e | subsection | 339 | 1,121 | Global group completion and units | So the defining pushout for R^\star can be rewritten
as the realization of a simplicial ultra-commutative monoid,
the two-sided bar construction
with respect to the box product:R^\star \ = \ (R\rhd [0,1])\boxtimes _R (R\times R)\ \cong \ B(R,\Delta [1])\boxtimes _R (R\times R)\ \cong \ B^\boxtimes (\ast , R, R\times R)... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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67491cfc4dbc2fc14e87b7fafdc9a68997ed12a2 | subsection | 340 | 1,121 | Global group completion and units | Since {\mathbf {\Delta }}_{\le n} is contained in {\mathbf {\Delta }}_{\le n+1},
there is a canonical morphism
B^{[n]}\longrightarrow B^{[n+1]} and the realization B^\boxtimes (\ast , R, R\times R)
is the colimit of the sequence of orthogonal spacesR\times R = B^{[0]} \ \longrightarrow \ B^{[1]}\ \longrightarrow \ \cdo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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57796854729cf24eb4af069cc87cc1afaa8e54ec | subsection | 341 | 1,121 | Global group completion and units | So the morphism
B^{[1]}\longrightarrow B^\boxtimes (\ast , R, R\times R) induces a bijection on \pi _0^G.
This proves that the morphism R\longrightarrow B^\boxtimes (\ast , R, R\times R)=R^\star
is a group completion of abelian monoids.The second claim then follows because group completions
of global power monoids are... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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eeb9946dc6e9b9dd6d7d4a68402c27a234982950 | subsection | 342 | 1,121 | Global group completion and units | If f is a global equivalence, so is the morphism B(f):B(R)\longrightarrow B(S).(i) The n-th latching morphism L_n^\Delta (B_\bullet (R))\longrightarrow B_n(R) in the simplicial
direction is the iterated pushout producti^{\Box n} \ : \ Q^n(i)\ \longrightarrow \ R^{\boxtimes n}with respect to the unit morphism \ast \long... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aa88ffb11909c588d951ce70a02e426869165001 | subsection | 343 | 1,121 | Global group completion and units | The induced morphism |\rho _\bullet |:B(M)\longrightarrow \mathbf {B}^\circ M
between the realizations is then a global equivalence
by Proposition REF (ii).The canonical morphismR\times [0,1] \ = \ B_1(R)\times \Delta ^1 \ \longrightarrow \ | B_\bullet (R) |\ = \ B(R)takes R\times \lbrace 0,1\rbrace to the basepoint, ... | {
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"source_ref_id": "2e56d470deef8729f559920366ac... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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61a338d82aa28dc82901b65fa45bbf408fd7188f | subsection | 344 | 1,121 | Global group completion and units | Then
((\Omega B(R))({\mathcal {U}}_G))^G is homeomorphic to
\Omega \left( ( B(R)({\mathcal {U}}_G))^G \right), which is in turn homeomorphic to
the loop space of the geometric realization of the simplicial space[n]\ \longmapsto \ (R^{\boxtimes n}({\mathcal {U}}_G))^G \ .We define i_k:[1]\longrightarrow [n] by i_k(0)=k... | {
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"source_ref_id": "c2eeea34788058e96a06d8882550e0060ad51... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2f319c84a934e798701f1415a3af6958b70bb8c7 | subsection | 345 | 1,121 | Global group completion and units | To see this we consider the `simplicial circle' \mathbf {S}^1,simplicial circle
the simplicial set given by(\mathbf {S}^1)_n \ = \ \lbrace 0,1,\dots ,n\rbrace \ ,with face maps d_i:(\mathbf {S}^1)_n\longrightarrow (\mathbf {S}^1)_{n-1} given byd_i(j) \ = \ {\left\lbrace \begin{array}{ll}
j-1 & \text{ for $i <j$, and}\\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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115738ebf9100acd8c47edf410eb99cc0139ed68 | subsection | 346 | 1,121 | Global group completion and units | Then R has a flat unit by Theorem REF (ii a).
Since ultra-commutative monoids form a topological model category,
R\rhd S^1 is an abstract suspension of R.
The isomorphism (REF )
transforms the adjunction unit R\longrightarrow \Omega (R\rhd S^1)
into the morphism \eta _R:R\longrightarrow \Omega B(R) defined in (REF ).
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a68afcd9d8483fdd5ab5c753b6be4a9161b94ebf | subsection | 347 | 1,121 | Global group completion and units | Indeed, the functor H_*((-)^G;{\mathbb {Z}})
takes strong level equivalences to isomorphisms, which reduces the claim
(by cofibrant approximation in the strong level model structure)
to global equivalences f:X\longrightarrow Y between flat orthogonal spaces. Flat orthogonal
spaces are closed, so the global equivalence ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1354159e1199b3ca8e7806f55c5bcf3dd25e9278 | subsection | 348 | 1,121 | Global group completion and units | Property (i) holds by definition of `group completion'.
We give two alternative proofs for why a global group completion
satisfies property (ii), based on the two different bar construction models
in Construction REF respectively
Corollary REF .The first argument uses the loop space of the bar construction B(R),
which ... | {
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{
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"source_ref_id": "2e56d470deef8729f559920366ac... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fc436576b9d24bd8c1fd50613f583c95f01ac7d5 | subsection | 349 | 1,121 | Global group completion and units | The argument is reproduced in more
detail in the proof of .Now we prove the reverse implication.
We let i:R\longrightarrow R^\star be a morphism of ultra-commutative monoids that
satisfies properties (i) and (ii); we need to show that i is a global group
completion.
We assume first that both R and R^\star are cofibrant... | {
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{
"arxiv_id": "",
"doi": "10.1007/978-1-4471-4393-2",
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"raw": "B. Dundas, T. G. Goodwillie, R. McCarthy, The local structure of algebraic K-theory. Algebra and Applications, 18. Springer-Verlag London, Ltd.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ec746a3de2b213c17866c46fb283048025300064 | subsection | 350 | 1,121 | Global group completion and units | Properties (i) and (ii) are invariant under global equivalences of pairs,
so the morphism i^c:R^c\longrightarrow R^\dagger satisfies (i) and (ii).
Since R^c and R^\dagger are both cofibrant, the morphism i^c
is a global group completion by the special case above.
So the morphism i is also a global group completion.We s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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77fa772b7d6742e114ec880be6c5c4240104d30f | subsection | 351 | 1,121 | Global group completion and units | To this end we define a bi-orthogonal space, i.e., a functor\mathbf {Gr}^\sharp \ : \ {\mathbf {L}}\times {\mathbf {L}}\ \longrightarrow \ {\mathbf {T}}on objects by\mathbf {Gr}^\sharp (U,V)\ = \ \mathbf {Gr}(U\oplus V) \ .For linear isometric embeddings \varphi :U\longrightarrow \bar{U} and \psi :V\longrightarrow \bar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4739377dbd39b0a13b4ef7a8d47f853fe4174bf9 | subsection | 352 | 1,121 | Global group completion and units | Hence for fixed U,\operatorname{colim}_{V\in s({\mathcal {U}}_G)} \mathbf {Gr}^\sharp (U,V)\ = \ \mathbf {Gr}(U\oplus {\mathcal {U}}_G)\ .A colimit over s({\mathcal {U}}_G)\times s({\mathcal {U}}_G) can be calculated in two steps,
first in one variable and then in the other, so we conclude that{\mathbf {BOP}}({\mathcal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3ed77730f9b582c286ec6c36c4917352e61e47f8 | subsection | 353 | 1,121 | Global group completion and units | To see this we observe that all the maps in the colimit system are
closed embeddings; so singular homology commutes with this particular colimit.For U\in s({\mathcal {U}}_G) we denote by j_U:\mathbf {Gr}({\mathcal {U}}_G)^G\longrightarrow \mathbf {Gr}(U\oplus {\mathcal {U}}_G)^G
the map induced by applying the direct s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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26ce5dcada823e4ad2d72c7fb1a60ce4051d2072 | subsection | 354 | 1,121 | Global group completion and units | In the same way as for the homogeneous degree 0 summands in (REF ),
we defined two morphisms
of E_\infty -orthogonal monoid spaces{\mathbf {bOP}}\ \xrightarrow{}\ {\mathbf {BOP}}^{\prime } \ \xleftarrow[\simeq ]{\ b\ } \ {\mathbf {BOP}}\ .The same arguments as in Proposition REF
show that the morphism b is a global eq... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/1970106",
"end": 2167,
"openalex_id": "https://openalex.org/W2331310219",
"raw": "R. Bott, The stable homotopy of the classical groups. Ann. of Math. (2) 70 (1959), 313–337.",
"source_ref_id": "3d572a2df91cfc77d5934bbb659c3ed7... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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53e237c740cb5b6db9335074d450b919bd31f25c | subsection | 355 | 1,121 | Global group completion and units | On the other hand, Harris exhibits an explicit homeomorphism
between the bar construction of \amalg _{n\ge 0}\, Gr _n
and the infinite unitary group, essentially the inverse to the eigenspace decomposition
of a unitary matrix.
Together these two ingredients provide a chain of weak equivalences{\mathbb {Z}}\times B U \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1ab81d936bf005937e732781d28bb6c5b0e28b77 | subsection | 356 | 1,121 | Global group completion and units | Then\beta (V)(0)(x)\ = \ p_{V_{\mathbb {C}}} \ = \ \operatorname{Id}_{V_{\mathbb {C}}} \ ;so \beta (V)(0) is the constant loop at the identity, which is the
unit element of \Omega {\mathbf {U}}(V).
Now we consider subspaces L\in \mathbf {Gr}^{\mathbb {C}}(V) and L^{\prime }\in \mathbf {Gr}^{\mathbb {C}}(W). Then\beta (... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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55c81faf4cddd312fa5946b471b2f43d4650b14c | subsection | 357 | 1,121 | Global group completion and units | The morphism \beta :\mathbf {Gr}^{\mathbb {C}}\longrightarrow \Omega {\mathbf {U}} is a global group completion of
ultra-commutative monoids.We factor \beta ^\flat as a composite
of two morphisms of ultra-commutative monoidsB(\mathbf {Gr}^{\mathbb {C}})\ = \ |B_\bullet (\mathbf {Gr}^{\mathbb {C}})|\ \xrightarrow[\simeq... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.02... |
caf1d7a4df8918da50245be53a6eadc67c145766 | subsection | 358 | 1,121 | Global group completion and units | The object of n-simplices \mathbf {Gr}^{\mathbb {C}}_{\langle n\rangle } is the ultra-commutative monoid
of n-tuples of pairwise orthogonal complex subspaces, i.e.,\mathbf {Gr}^{\mathbb {C}}_{\langle n\rangle }(V)\ = \ \lbrace (L_1,\dots ,L_n)\in ( Gr^{\mathbb {C}}(V_{\mathbb {C}}))^n\ : \ \text{$L_i$ is orthogonal to ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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006475d4bc55769c64b359511b28e73e8218184e | subsection | 359 | 1,121 | Global group completion and units | The universal property of the box product
turns this multi-morphism into a morphism of orthogonal spaces\zeta _n \ : \ B_n(\mathbf {Gr}^{\mathbb {C}})\ = \ (\mathbf {Gr}^{\mathbb {C}})^{\boxtimes n} \ \longrightarrow \ \mathbf {Gr}^{\mathbb {C}}_{\langle n\rangle } \ .The morphisms \zeta _n are compatible with the simp... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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17f53b64c742ab899653160060656b8a88e3af14 | subsection | 360 | 1,121 | Global group completion and units | The value at a euclidean inner product space V is{\mathbf {L}}^{\mathbb {C}}_{G,W}(V) \ = \ \ {\mathbf {L}}^{\mathbb {C}}(W,V_{\mathbb {C}}) / G\ .Here {\mathbf {L}}^{\mathbb {C}} is the space of {\mathbb {C}}-linear maps that preserve
the hermitian inner products.
In the special case of the tautological U(n)-represent... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6265ca6f682dfbfa3d741c5d3777631ba3140471 | subsection | 361 | 1,121 | Global group completion and units | This completes the proof that
the morphism \zeta _n:(\mathbf {Gr}^{\mathbb {C}})^{\boxtimes n}\longrightarrow \mathbf {Gr}^{\mathbb {C}}_{\langle n\rangle } is a global equivalence.Now we observe that the underlying simplicial orthogonal spaces of source and target
of \zeta _\bullet are Reedy flat in the sense of Defin... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2b99c37282491a317bffdc15d95f8b9b0b12a8b5 | subsection | 362 | 1,121 | Global group completion and units | Since source and target of the morphism \zeta _\bullet
are Reedy flat as simplicial orthogonal spaces, and \zeta _\bullet
is a global equivalence in every simplicial dimension,
the induced morphism of realizations\zeta \ = \ |\zeta _\bullet |\ : \ B(\mathbf {Gr}^{\mathbb {C}})\ = \ | B_\bullet (\mathbf {Gr}^{\mathbb ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0021-8693(80)90194-5",
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"raw": "B. Harris, Bott periodicity via simplicial spaces. J. Algebra 62 (1980), no. 2, 450–454.",
"source_ref_id": "1434c53bb10f74c4aa2dc1e... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ed55428e79f461d438e8e2b26be1daf599d0a847 | subsection | 363 | 1,121 | Global group completion and units | We have the relations\epsilon _n(L_1,\dots ,L_n;\, 0, t_1,\dots , t_{n-1})
\ = \ \epsilon _{n-1}(L_2,\dots , L_n;\, t_1,\dots ,t_{n-1})and\epsilon _n(L_1,\dots ,L_n;\, t_1,\dots , t_{n-1},1)\ = \ \epsilon _{n-1}(L_1,\dots , L_{n-1};\, t_1,\dots ,t_{n-1})because \exp (0)=\exp (2\pi i \cdot p_L)=\operatorname{Id}_{V_{\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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07c32db8c5225267d75e3b298dd92e74f7fcd632 | subsection | 364 | 1,121 | Global group completion and units | Unraveling all definitions shows that the composite\mathbf {Gr}^{\mathbb {C}}\wedge S^1\ \longrightarrow \ B(\mathbf {Gr}^{\mathbb {C}}) \ \xrightarrow{}\ |\mathbf {Gr}^{\mathbb {C}}_{\langle \bullet \rangle }|
\ \xrightarrow{}\ {\mathbf {U}}is given at an inner product space V by the mapG r^{\mathbb {C}}(V_{\mathbb {C... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a99995c14945b6c365be9baff318efd041b22834 | subsection | 365 | 1,121 | Global group completion and units | However, we elaborate a bit more and exhibit an explicit chain
of two global equivalences between {\mathbf {BUP}} and \Omega {\mathbf {U}},
see Theorem REF below.We define a morphism of ultra-commutative
monoidsadditive Grassmannian\beta @\bar{\beta } - Bott morphism from {\mathbf {BUP}} to \Omega {\mathbf {U}}\bar{\be... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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30d6c765581e0a5fadcc0d160c8b845c7f2b7b16 | subsection | 366 | 1,121 | Global group completion and units | The map \bar{\beta }(V) is continuous in L.For every inner product space V we have\bar{\beta }(V)(V_{\mathbb {C}}\oplus 0)(x)\ = \ ( (c(x)\cdot p_{V_{\mathbb {C}}\oplus 0})\ +\ p_{0\oplus V_{\mathbb {C}}}) \circ ( (c(-x)\cdot p_{V_{\mathbb {C}}\oplus 0}) + p_{0\oplus V_{\mathbb {C}}})\ =\ \operatorname{Id}_{V_{\mathbb ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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77aaa5965a0aed38e02788de76b9992158fb8248 | subsection | 367 | 1,121 | Global group completion and units | Hence the morphism \bar{\beta } restricts to a morphism of
ultra-commutative monoids\bar{\beta }^{[0]} \ : \ {\mathbf {BU}}\ \longrightarrow \ \Omega ( \operatorname{sh}_\otimes {\mathbf {SU}}) \ .Now we can properly state our global version of complex Bott periodicity.
The embeddings j:V_{\mathbb {C}}\longrightarrow V... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5a95254846e3e49b3ea59637d06c9ce30824406c | subsection | 368 | 1,121 | Global group completion and units | Since the morphism \bar{\beta }^{[0]}:{\mathbf {BU}}\longrightarrow \Omega (\operatorname{sh}_\otimes {\mathbf {SU}}) is
a retract of the global equivalence \bar{\beta }, it is a global equivalence itself.Corollary 5.40
For every compact Lie group G and every finite G-CW-complex A, the map[A,\beta ]^G\ : \ [A,\mathbf ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cb127a045fa5b8f6d0e4d0382f9e07fac81e18f7 | subsection | 369 | 1,121 | Equivariant stable homotopy theory | In this chapter we give a largely self-contained
exposition of many basics about equivariant stable homotopy theory
for a fixed compact Lie group; our model is the category of orthogonal G-spectra.
In Section
we review orthogonal spectra and orthogonal G-spectra;
we define equivariant stable homotopy groups and prove ... | {
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{
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"end": 2584,
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"source_ref_id": "1fe5cf0f78b8a5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5dabdc660c62092717879bbc5d2d98825e62e635 | subsection | 370 | 1,121 | Equivariant orthogonal spectra | In this section we begin to develop some of the basic
features of equivariant stable homotopy theory for compact Lie groups
in the context of equivariant orthogonal spectra.
After introducing orthogonal G-spectra
and equivariant stable homotopy groups,
we discuss shifts by a representation and show that they are
equiva... | {
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{
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"doi": "",
"end": 1172,
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"raw": "J. P. May, E_{\\infty } ring spaces and E_{\\infty } ring spectra. With contributions by F. Quinn, N. Ray, and J. Tornehave. Lecture Notes in Mathematics, Vol. 77. Springer-Verlag, Berlin-New York,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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80648aea59ba982c9dd47aee47dea4724d6ee2cc | subsection | 371 | 1,121 | Equivariant orthogonal spectra | The way to make the continuous dependence
rigorous is to exploit that the complements W-\varphi (V) vary in
a locally trivial way, i.e., they are the fibers of a distinguished vector bundle,
the `orthogonal complement bundle',
over the space of {\mathbf {L}}(V,W) of linear isometric embeddings.
All the structure maps \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8ef3ec93c689f2815dcfd6c59eb2eeb283d03fc5 | subsection | 372 | 1,121 | Equivariant orthogonal spectra | Here, and in the following, we writeG\ltimes _H A - induced based G-spaceG\ltimes _H A \ = \ (G_+)\wedge _H A \ = \ (G_+\wedge A) / \simfor a closed subgroup H of G and a based G-space A;
the equivalence relation is g h\wedge a\sim g \wedge h a
for all (g,h,a)\in G\times H\times A.
Put yet another way: if \dim V=n and ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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40e9875884f1b14403926f556aee01b11c36430e | subsection | 373 | 1,121 | Equivariant orthogonal spectra | We denote the category of orthogonal spectra by {\mathcal {S}}p. {\mathcal {S}}p - category of orthogonal spectraGiven two inner product spaces V and W we define a continuous based mapi_V \ : \ S^V \ \longrightarrow \ {\mathbf {O}}(W,V\oplus W)\text{\qquad by\qquad }
v \ \longmapsto \ ((v,0), (0,-)) \ ,where (0,-):W\lo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d348016219df69f0ec8949e6d1709a91d22ef69f | subsection | 374 | 1,121 | Equivariant orthogonal spectra | Here the group O(m)\times O(n)
acts on the target by restriction, along orthogonal sum, of the O(m+n)-action.
Indeed, the mapO(n)_+\ \longrightarrow \ {\mathbf {O}}({\mathbb {R}}^n,{\mathbb {R}}^n) \ , \quad A\ \longmapsto \ (0,A)is a homeomorphism, so O(n) `is' the endomorphism monoid of {\mathbb {R}}^n
as an object o... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b57fae6848c4bcad80779982141306c86c9b5d6e | subsection | 375 | 1,121 | Equivariant orthogonal spectra | If \varphi :V\longrightarrow W is a linear isometric embedding
and f:S^V\longrightarrow X(V) a continuous based map,
we define \varphi _*f:S^W\longrightarrow X(W) as the compositeS^W \cong \ S^{W-\varphi (V)}\wedge S^V\ &\xrightarrow{} \ S^{W-\varphi (V)}\wedge X(V) \\
&\xrightarrow{} \ X((W-\varphi (V))\oplus V) \ \xr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5118d980a75ceaf4f8e46f3c489544db063e74d6 | subsection | 376 | 1,121 | Equivariant orthogonal spectra | We choose a G-fixed unit vector v_0\in V,
and we let V^\perp denote the orthogonal complement of v_0 in V.
This induces a decomposition{\mathbb {R}}\oplus V^\perp \ \cong \ V \ , \quad (t,v)\ \longmapsto \ t v_0 +vthat extends to a G-equivariant homeomorphism
S^1\wedge S^{V^\perp }\cong S^V on one-point compactificatio... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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36939a6fb39f0f17b6309c1e0650b0299f95d84f | subsection | 377 | 1,121 | Equivariant orthogonal spectra | If k is a positive integer, then we set\pi _k^G (X) \ &= \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, [S^{V\oplus {\mathbb {R}}^k}, X(V)]^G
\text{\qquad and}\\
\pi _{-k}^G (X) \ &= \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, [S^{V}, X(V\oplus {\mathbb {R}}^k)]^G \ .The colimits are taken over the analogo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.04232284426689148,
0... |
122fa0a4efee3a5e72b058b4e4dce468b838be64 | subsection | 378 | 1,121 | Equivariant orthogonal spectra | Then the compositeS^{\bar{V}} \ \xrightarrow[\cong ]{(S^j)^{-1}} \ S^{V\oplus {\mathbb {R}}^{n+k}} \ \xrightarrow{} \ X(V\oplus {\mathbb {R}}^n) \ \xrightarrow[\cong ]{X(j\oplus {\mathbb {R}}^{-k})} \ X(\bar{V}\oplus {\mathbb {R}}^{-k} )represents a class \langle f\rangle \in \pi _k^G(X).We also need a way to recognize... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.016113173216581345,
-0.008315984159708023,
-0.014648339711129665,
0.017135506495833397,
0.02600080333650112,
-0.037688955664634705,
0.045867614448070526,
0.013122471049427986,
0.01447286456823349,
0.026794254779815674,
0.003833745140582323,
-0.012267984449863434,
0.010970995761454105,
0... |
c3e6c62bb4172b19153af1827f190a8c15cd2e22 | subsection | 379 | 1,121 | Equivariant orthogonal spectra | For every G-equivariant linear isometric embedding
\varphi :V\longrightarrow W the relation
\langle \varphi _*f\rangle \ = \ \langle f\rangle \text{\qquad holds in\quad $\pi _k^G(X)$.}Now we let K and G be two compact Lie groups.
Every continuous based functor F:G{\mathbf {T}}_\ast \longrightarrow K{\mathbf {T}}_\ast ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03564462438225746,
-0.00036716554313898087,
-0.01497653964906931,
0.016799969598650932,
0.02685553766787052,
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0.01033785566687584,
0.012313874438405037,
0.016510052606463432,
0.02890022099018097,
-0.01991277001798153,
-0.02931220829486847,
0.002624518470838666,
0.03... |
1bdc9b68cffea98e41f969608de672e24cf917eb | subsection | 380 | 1,121 | Equivariant orthogonal spectra | Then the two isomorphismsc_g^* \ : \ \pi _0^G(X) \ \longrightarrow \ \pi _0^G(c_g^* X) \text{\qquad and\qquad }
(l_g^X)_*\ : \ \pi _0^G(c_g^* X) \ \longrightarrow \ \pi _0^G( X)are inverse to each other.We let V be a G-representation,
and we recall that the G-action on X(V) is diagonally,
from the external G-action on ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06855770945549011,
-0.013186522759497166,
-0.0023408366832882166,
0.01710890233516693,
0.03476724401116371,
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0.02277117222547531,
0.0468548908829689,
0.013964894227683544,
0.034065183252096176,
-0.023625854402780533,
-0.017841488122940063,
0.03662922978401184,
0.0006... |
af583d689091cc42d54ac21d5f75b8c74ca9e1d6 | subsection | 381 | 1,121 | Equivariant orthogonal spectra | So combining the restriction map along c_g
with the effect of l_g^X
gives an isomorphismconjugation homomorphism!on equivariant homotopy groupsg_\star \ : \ \pi _0^{H^g}(X)\ \xrightarrow{}\ \pi _0^H(c_g^*X)\ \xrightarrow{}\ \pi _0^H(X)\ .Moreover,g_\star \circ g^{\prime }_\star \ &= \ (l_g^X)_*\circ (c_g)^* \circ (l_{g... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.027992280200123787,
0.016169928014278412,
-0.0009291034075431526,
0.019144583493471146,
0.004339947365224361,
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0.02221076749265194,
0.026619361713528633,
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0.02160058170557022,
-0.030021147802472115,
-0.047777559608221054,
0.015231766737997532,
-... |
25cbd3536f33b4b0c1f179a80226de65651fee82 | subsection | 382 | 1,121 | Equivariant orthogonal spectra | \Construction 1.17
If A is a pointed G-space, then smashing with A and taking based maps
out of A are two continuous based endofunctors
on the category of based G-spaces. So for every orthogonal G-spectrum X,
we can define two new orthogonal G-spectra X\wedge A and \operatorname{map}_*(A,X)
by smashing with A (and let... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.012851030565798283,
0.017475571483373642,
-0.023931611329317093,
0.00940170418471098,
0.014850419014692307,
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0.017048221081495285,
0.015644069761037827,
0.048565298318862915,
0.006185131147503853,
-0.009081191383302212,
-0.0168955959379673,
0.026083623990416527,
-0.... |
eb32e94823be55011211c2c06292eeb0de856368 | subsection | 383 | 1,121 | Equivariant orthogonal spectra | The W-th loop spectrumloop spectrum
\Omega ^W X=\operatorname{map}_*(S^W,X), defined by(\Omega ^W X)(V) \ = \ \Omega ^W X(V) \ = \ \operatorname{map}_*(S^W,X(V)) \ ,the based mapping space from S^W to the V-th level of X.
We obtain an adjunction between -\wedge S^W and \Omega ^W
as the special case A=S^W of (REF ).Cons... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03433247283101082,
0.023529188707470894,
-0.052185360342264175,
0.01281745731830597,
0.010597290471196175,
-0.02496352232992649,
-0.0008535434608347714,
-0.008575488813221455,
0.06610145419836044,
-0.01625070348381996,
0.0005068808095529675,
-0.027191318571567535,
0.0019359700381755829,
... |
84673929db2dff4092d46be959914d290ec9aa6c | subsection | 384 | 1,121 | Equivariant orthogonal spectra | The values of \operatorname{sh}^V(\operatorname{sh}^W X) and \operatorname{sh}^{V\oplus W}X at an inner product space U
are given by(\operatorname{sh}^V(\operatorname{sh}^W X))(U)\ = \ X((U\oplus V)\oplus W)respectively(\operatorname{sh}^{V\oplus W}X)(U)\ = \ X(U\oplus (V\oplus W)) \ .We use the effect of X on the asso... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.029298238456249237,
0.004787668120115995,
-0.05459848791360855,
-0.009247256442904472,
-0.0071071116253733635,
-0.01983734965324402,
0.006767588201910257,
-0.007042258977890015,
0.0834999829530716,
0.00833168625831604,
-0.0234233308583498,
-0.02551388368010521,
0.011391215957701206,
0.0... |
7cbecfbdae3d260248506958123cd8e14e2722e3 | subsection | 385 | 1,121 | Equivariant orthogonal spectra | Then the orthogonal spectra X\wedge S^V and \operatorname{sh}^V X
become orthogonal G-spectra by letting G act diagonally on X and V.
With respect to these diagonal actions, the morphism \lambda ^V_X:X\wedge S^V\longrightarrow \operatorname{sh}^V X
is a morphism of orthogonal G-spectra.
Our next aim is to show that \la... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02430015057325363,
0.015467892400920391,
-0.023750994354486465,
0.01973910629749298,
0.013652627356350422,
-0.03673243150115013,
0.02054758556187153,
0.008611069992184639,
0.07010891288518906,
0.028403567150235176,
-0.022362850606441498,
-0.002352980663999915,
0.016062811017036438,
0.00... |
86c9e055186321c4f9e7e0e7ce39d972418b4265 | subsection | 386 | 1,121 | Equivariant orthogonal spectra | The morphism
\lambda ^V_X\ :\ X\wedge S^V\ \longrightarrow \ \operatorname{sh}^V X \ ,
\text{\qquad its adjoint\qquad }
\tilde{\lambda }^V_X \ : \ X \ \longrightarrow \ \Omega ^V \operatorname{sh}^V X \ ,
the adjunction unit \eta ^V_X:X\longrightarrow \Omega ^V(X\wedge S^V)
and the adjunction counit \epsilon ^V_X:(\Om... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.035592518746852875,
0.02559547685086727,
-0.029884284362196922,
-0.002144403522834182,
0.000531331286765635,
-0.04212493821978569,
0.01494977343827486,
-0.003823295934125781,
0.06849881261587143,
0.034432556480169296,
-0.013698236085474491,
-0.015735801309347153,
0.022527683526277542,
0... |
c1ae904082d24e673b73bb08db7a6a181a71961a | subsection | 387 | 1,121 | Equivariant orthogonal spectra | We claim that each of the three composites around the triangle{
\pi ^G(A;X\wedge S^V) [rr]^-{(\lambda ^V_X)_*} &&
\pi ^G(A;\operatorname{sh}^V X) [dl]^-{\psi ^V_{X}} \\
& \pi ^G(A;X\wedge S^V) [ul]^-{\varepsilon _V}}is the respective identity.
We consider a based continuous G-map f:S^U\wedge A\longrightarrow X(U)\wedge... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.035101089626550674,
-0.017031658440828323,
-0.055337630212306976,
-0.021381141617894173,
0.012552455067634583,
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0.04273176193237305,
0.0083326930180192,
0.03128575533628464,
0.013498658314347267,
-0.025394875556230545,
0.011087366379797459,
0.05396410822868347,
-0.... |
a275e39df3c644165d9939f11947ee40869bdbfc | subsection | 388 | 1,121 | Equivariant orthogonal spectra | Since the left and right vertical composites differ by conjugation with
an equivariant isometry, they also represent the same class in \pi ^G(A;X\wedge S^V),
by Proposition REF (ii).
Altogether this shows that the composite
\varepsilon _V\circ \psi ^V_X\circ (\lambda ^V_X)_* is the identity.
Since \varepsilon _V^2 is ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07340757548809052,
-0.027733784168958664,
-0.0026715504936873913,
0.013432120904326439,
0.007635180838406086,
-0.047107767313718796,
0.020197760313749313,
0.008542859926819801,
0.03401888906955719,
0.015560208819806576,
-0.04051756113767624,
-0.010487886145710945,
0.02957965061068535,
0... |
c1f3bd5cc64e9564fefc65cc7366f671e6f39ebd | subsection | 389 | 1,121 | Equivariant orthogonal spectra | Then the class (\lambda ^V_X)_*(\varepsilon _V(\psi ^V_{X}\langle g\rangle ))
is represented by the compositeS^U\wedge S^V\wedge A\ &\xrightarrow{}\ S^U\wedge A\wedge S^V \ \xrightarrow{}\ X(U\oplus V)\wedge S^V \\
&\xrightarrow{}\ X(U\oplus V)\wedge S^V\\
&\xrightarrow{}\ X(U\oplus V\oplus V) \ = \ (\operatorname{sh}^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02924288623034954,
-0.015506666153669357,
-0.016620825976133347,
-0.023672087118029594,
0.022588450461626053,
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0.009935866110026836,
0.0296549741178751,
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0.020970629528164864,
-0.01015717163681984,
-0.00017361056234221905,
0.03656887263059616,
-... |
1ea0110bb1e508ffa9b9ea41b5b277bdb04cb6d2 | subsection | 390 | 1,121 | Equivariant orthogonal spectra | Since the left and right vertical composites differ by conjugation with
an equivariant isometry, they represent the same class,
so the composite (\lambda ^V_X)_* \circ \varepsilon _V\circ \psi ^V_X is the identity.Now we prove claim (i) of the proposition.
For k\ge 0, it is the special case A=S^k of the discussion abov... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.08050266653299332,
-0.004314277786761522,
0.004272317513823509,
0.000024854301955201663,
0.015387971885502338,
-0.01444958709180355,
0.037047095596790314,
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0.032011862844228745,
0.040800631046295166,
-0.016326354816555977,
0.006797561887651682,
0.005481535568833351,... |
467fad300b5df9cd57755637ccdde916d8cb55e7 | subsection | 391 | 1,121 | Equivariant orthogonal spectra | The map \kappa sends [g]
to the class represented by the compositeS^{U\oplus V\oplus {\mathbb {R}}^k} \ \xrightarrow{}\ S^{U\oplus {\mathbb {R}}^k\oplus V} \ \xrightarrow{}\ X(U\oplus V) \ ,where g^\flat is the adjoint of g.
This is compatible with stabilization.We claim that the map \kappa is injective.
Indeed, if g:S... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.047212861478328705,
0.005764272529631853,
0.007774710189551115,
0.02629210241138935,
0.017624711617827415,
-0.07422216236591339,
0.0367448516190052,
0.01800619810819626,
0.02481193095445633,
0.02005097083747387,
0.0000856558108353056,
-0.0013352053938433528,
0.01631239429116249,
0.00888... |
db46245734ce56a8e96e177df1420f49be7f1948 | subsection | 392 | 1,121 | Equivariant orthogonal spectra | For k\ge 0 this follows by applying part (i) with A=S^{{\mathbb {R}}^k\oplus W}
and exploiting the natural isomorphism \pi _k^G(\Omega ^W Y)\cong \pi ^G(S^{{\mathbb {R}}^k\oplus W};Y).
To get the same conclusion
for negative dimensional homotopy groups we exploit that
\pi _{-k}^G(Y)=\pi _0^G(\operatorname{sh}^k Y), by ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06733616441488266,
0.01939806528389454,
-0.006039956118911505,
-0.005005951970815659,
-0.006280333269387484,
-0.04126858338713646,
0.02721223421394825,
-0.010698696598410606,
0.04783126339316368,
0.024465065449476242,
-0.034644853323698044,
-0.005211989860981703,
0.006532157305628061,
0... |
dfaabadfac538297c8004766f59fad94fc398248 | subsection | 393 | 1,121 | Equivariant orthogonal spectra | The two homomorphisms of orthogonal G-spectra\Omega ^V (\tilde{\lambda }^V_X)\ , \ \tilde{\lambda }^V_{\Omega ^V X}\ : \ \Omega ^V X \ \longrightarrow \ \Omega ^V(\Omega ^V\operatorname{sh}^V X)are not the same; they differ by the involution on the target
that interchanges the two V-loop coordinates.
An equivariant hom... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07176251709461212,
-0.026880433782935143,
-0.032555531710386276,
0.005408123601227999,
0.019389914348721504,
-0.019496705383062363,
-0.0044203209690749645,
0.01852034404873848,
0.05266246199607849,
-0.0019183964468538761,
-0.019023779779672623,
-0.024439530447125435,
-0.00937458872795105,... |
4b0bf4564b07cbcaa9e067b7026036f6d7bf112c | subsection | 394 | 1,121 | Equivariant orthogonal spectra | Then we set \alpha [f]=[f^\flat ].Next we define the suspension isomorphismsuspension isomorphism-\wedge S^1 \ : \ \pi ^G_k (X)\ \longrightarrow \ \pi ^G_{k+1}(X\wedge S^1) \ .We represent a given class in \pi _k^G(X) by a based G-map
f:S^{V\oplus {\mathbb {R}}^{n+k}}\longrightarrow X(V\oplus {\mathbb {R}}^n);
then f\w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.032984454184770584,
-0.0110838133841753,
-0.03817164897918701,
0.013372281566262245,
0.023021988570690155,
-0.04702039062976837,
0.03826318681240082,
0.042473968118429184,
0.031229961663484573,
0.006007228512316942,
-0.03594420477747917,
-0.010397273115813732,
0.01550055667757988,
0.023... |
05f2e11da74eba97681453449f73318d0be135c1 | subsection | 395 | 1,121 | Equivariant orthogonal spectra | The mapping cone comes with an embedding i:B\longrightarrow C f
and a projection p:C f\longrightarrow A\wedge S^1; the projection
sends B to the basepoint and is given on A\wedge [0,1]
by p(a,x)=a\wedge t(x), wheret\ :\ [0,1]\ \longrightarrow \ S^1\text{\qquad is defined as\qquad }
t(x)\ =\ \frac{2x-1}{x(1-x)}\ .What i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.013547923415899277,
-0.024456443265080452,
-0.053459372371435165,
0.014242101460695267,
0.0487908311188221,
-0.02076432853937149,
0.0418643020093441,
-0.00775801669806242,
0.01588219404220581,
0.021664472296833992,
-0.016477202996611595,
-0.02370886504650116,
0.020535478368401527,
-0.01... |
b9160edb166fdd644695c25b3f024a9480b9fc2e | subsection | 396 | 1,121 | Equivariant orthogonal spectra | The homotopy fiber comes with maps\Omega B\ \xrightarrow{}\ F(f)\ \xrightarrow{}\ A \ .The map p is the projection to the second factor;
the value of the map i on a based loop \omega :S^1\longrightarrow B isi(\omega ) = (\omega \circ t,*) \ ,where t:[0,1]\longrightarrow S^1 was defined in (REF ).The homotopy fiber F(f)... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03287191689014435,
-0.008309544064104557,
-0.024264784529805183,
0.007432044483721256,
0.03711443766951561,
-0.026767568662762642,
0.027545873075723648,
0.009797479026019573,
0.04773600399494171,
0.01973230578005314,
-0.03445904701948166,
-0.025607742369174957,
0.02876674197614193,
-0.0... |
fe7e745a146dfadffae5a3d1a4d064f59a31a292 | subsection | 397 | 1,121 | Equivariant orthogonal spectra | Let f:A\longrightarrow B and \beta :Z\longrightarrow B be morphisms of based G-spaces
such that the composite i\beta :Z\longrightarrow C f is equivariantly null-homotopic.
Then there exists a based G-map h:Z\wedge S^1\longrightarrow A\wedge S^1 such that
(f\wedge S^1)\circ h:Z\wedge S^1\longrightarrow B\wedge S^1
is eq... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.021380821242928505,
0.0011398144997656345,
-0.044104620814323425,
0.006028140429407358,
0.040930312126874924,
-0.03635197505354881,
0.028172018006443977,
0.0057038418017327785,
0.04498976841568947,
0.006997221149504185,
-0.05637455731630325,
-0.024539873003959656,
0.0429447777569294,
-0... |
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