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cfea11441c5bc82456a0464194ef0fd1ffecfb7e | subsection | 998 | 1,121 | Compactly generated spaces | Hence C(K,G)=\operatorname{map}(K,G),
i.e., the compact-open topology is the topology of the internal mapping
space in the category {\mathbf {T}} of compactly generated spaces.We let \hom (K,G) denote the set of continuous homomorphisms
with the subspace topology of \operatorname{map}(K,G).
Since \operatorname{map}(K,G... | {
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"Stefan Schwede"
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54d35e8bd7df94479d58bae0a9212ca8efde5991 | subsection | 999 | 1,121 | Compactly generated spaces | Since O is open and (k,\beta (k))\in O,
there are open subsets U_k\subset K and V_k\subset G with(k,\beta (k))\ \subseteq \ U_k\times V_k \ \subseteq \ O \ .Since \beta is continuous, the set \beta ^{-1}(V_k) is open in K,
hence so is U_k\cap \beta ^{-1}(V_k); moreover, k\in U_k\cap \beta ^{-1}(V_k).
Since K is compact... | {
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"Stefan Schwede"
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f44870e520dd581f2c324c70ee560987808e2f48 | subsection | 1,000 | 1,121 | Compactly generated spaces | So the G^\circ -orbits are open, closed and path connected;
hence they coincide with the path components and the connected components.For a closed subgroup H of a compact Lie group G,
we let C_G H and N_G H denote the centralizer respectively
normalizer of H in G, and we write W_G H=(N_G H)/H for the Weyl group.Proposi... | {
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b29b43cc984237d37b5967c8235fff6f365652ad | subsection | 1,001 | 1,121 | Compactly generated spaces | We write `\times ' for the pairing
and X^K for the cotensor of an object X with a compactly generated space K.
A homotopy is then a morphism H:A\times [0,1]\longrightarrow X
defined on the pairing of a {\mathcal {C}}-object with the unit interval.
For a homotopy and any t\in [0,1] we denote by H_t:A\longrightarrow X th... | {
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c5910b058dbbb0587833d3676c2fce8ea85e48b5 | subsection | 1,002 | 1,121 | Compactly generated spaces | Let {\mathcal {C}}^{\prime } be another category tensored over the category {\mathbf {T}},
and F:{\mathcal {C}}\longrightarrow {\mathcal {C}}^{\prime } a continuous functor that commutes with colimits and
tensors with [0,1]. Then F takes h-cofibrations
in {\mathcal {C}} to h-cofibrations in {\mathcal {C}}^{\prime }.
I... | {
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e72635629a95de192d9dadc8bcc472e93d97a584 | subsection | 1,003 | 1,121 | Compactly generated spaces | So the map(-,1)\ : \ A\ \longrightarrow \ B\cup _f (A\times [0,1]) \ , \quad a \ \longmapsto \ (a,1)is a closed embedding by direct inspection of the topology on the pushout.The universal example of the homotopy extension property provides
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21b267fc3a15e957370bf15ca8344cd9217f119f | subsection | 1,004 | 1,121 | Compactly generated spaces | We let {\mathbf {\Delta }} denote
the simplicial indexing category, with objects the finite
totally ordered sets [n]=\lbrace 0\le 1\le \dots \le n\rbrace
for n\ge 0. Morphisms in {\mathbf {\Delta }} are all weakly monotone maps.
We let\Delta ^n \ = \ \lbrace (t_1,\dots ,t_n)\in [0,1]^n \ |\ t_1\le t_2\le \dots \le t_n... | {
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33cd8238d606cab65ffdcac01bb8578e18a4126e | subsection | 1,005 | 1,121 | Compactly generated spaces | A more categorical way to say this is that |X| is a coend of the functor{\mathbf {\Delta }}^{\operatorname{op}}\times {\mathbf {\Delta }}\ \longrightarrow \ \mathbf {Spc}\ , \quad ([m],[n])\ \longmapsto \ X_m\times _0\Delta ^n \ .We recall that the equivalence relation generated by (REF )
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a8f0a56732723830c7f889845aef8ce44c870928 | subsection | 1,006 | 1,121 | Compactly generated spaces | Then the inclusion induces a closed embedding |Y|\longrightarrow |X|.(i) Since \Delta ^n is compact and X_n is a k-space,
X_n\times _0\Delta ^n=X\times \Delta ^n is a k-space in the product topology
by Proposition REF (vi),
and so the disjoint union \coprod _{n\ge 0}\, X_n\times \Delta ^n
is a k-space.
As a quotient s... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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41a12705074a9e446c3c1906ffa81e7fdde96fab | subsection | 1,007 | 1,121 | Compactly generated spaces | So there are surjective morphisms
\sigma :[k]\longrightarrow [l] and \bar{\sigma }:[\bar{k}]\longrightarrow [l],
injective morphisms \delta :[k]\longrightarrow [m] and \bar{\delta }:[\bar{k}]\longrightarrow [n],
and u\in \Delta ^k, \bar{u}\in \Delta ^{\bar{k}} such that\delta ^*(y)\ = \ \sigma ^*(x)\ , \quad s\ = \ \de... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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970645612814f6f320b05aa03be71576017f979e | subsection | 1,008 | 1,121 | Compactly generated spaces | So the set(\sigma ^*\times \delta _*\times \bar{\sigma }^*\times \bar{\delta }_*)
\left((X_l\times \sigma _*\times X_l\times \bar{\sigma }_*)^{-1}
( \Delta _{X_l\times \Delta ^l})\right)is closed in X_k\times \Delta ^m\times X_{\bar{k}}\times \Delta ^n.
Since E_{m,n} is the inverse image of this latter closed set
under... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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f4b62a50e4dffaaaaf8a82bf1fcdfa7eb373efba | subsection | 1,009 | 1,121 | Compactly generated spaces | We let (y,s)\in X_m\times \Delta ^m be a point whose equivalence class
lies in the image of |\iota |:|Y|\longrightarrow |X|.
As we showed in the previous paragraph, the representative
(x,t)\in X_l\times \Delta ^l
of minimal dimension in the equivalence class of (y,s) must then
lie in the simplicial subspace Y, i.e., we... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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dfbc164fa5814e69e13c2b887b2b1c4a9f05cdb0 | subsection | 1,010 | 1,121 | Compactly generated spaces | We conclude that ( X_m\times \Delta ^m) \cap q^{-1}( |\iota |(A) )
is closed in X_m\times \Delta ^m for every m\ge 0,
hence the set q^{-1}(|\iota |(A)) is closed.
Since q is a quotient map, this shows that |\iota |(A) is closed in |X|.The next proposition is about the interaction of geometric realization
and products f... | {
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64c61479af8505dc4ddc1646992ce10268ca5b67 | subsection | 1,011 | 1,121 | Compactly generated spaces | May gives a proof
in the category of Hausdorff k-spaces (which he calls
`compactly generated Hausdorff spaces'), but the proof does not use
the Hausdorff property.Proposition 4.70For every simplicial k-space Y:{\mathbf {\Delta }}^{\operatorname{op}}\longrightarrow {\mathbf {K}} and every k-space K,
the canonical map
|... | {
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8384867c4d5ea59999f3ada9a2ea0249c9ae3eba | subsection | 1,012 | 1,121 | Compactly generated spaces | Moreover, colimits commute with coends, so the canonical maps\operatorname{colim}_I &\left( |-|\circ F\right)\ = \ \operatorname{colim}_I \left( \int ^{[n]\in {\mathbf {\Delta }}} F_n\times \Delta ^n\right)\ \xrightarrow{}\ \int ^{[n]\in {\mathbf {\Delta }}} \operatorname{colim}_I (F_n \times \Delta ^n) \\
&\xrightarro... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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5f601b3b0f25db729b8887ac440674dd46cbdc6b | subsection | 1,013 | 1,121 | Compactly generated spaces | Moreover, under these identifications, the map \delta for Z=\Delta [m,n] specializes
to the canonical map |\Delta [m]\times \Delta [n]|\longrightarrow |\Delta [m]|\times |\Delta [n]|.
It is a classical fact,
already observed by Milnor ,
that this canonical map is a homeomorphism;
other references are
and .We still nee... | {
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"Stefan Schwede"
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5efe806f3b21a247727abf96709bbfcff1c8731d | subsection | 1,014 | 1,121 | Compactly generated spaces | A specific example
is given by taking both X and Y as wedges of the simplicial 1-sphere,
where X has countably infinitely many copies,
and Y has uncountably many copies,
compare .Now we discuss the latching spaceslatching space!of a simplicial space
of a simplicial space X, of which there are competing definitions in t... | {
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... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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7ea054723286b5f4b28218535148418952343c8e | subsection | 1,015 | 1,121 | Compactly generated spaces | We let {\mathbf {\Delta }}(n)_\circ denote the full subcategory with all objects
except the identity of [n].A simplicial topological space X:{\mathbf {\Delta }}^{\operatorname{op}}\longrightarrow \mathbf {Spc}
can be restricted along the forgetful functor{\mathbf {\Delta }}(n)_\circ ^{\operatorname{op}}\ \xrightarrow{}... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b51100be2824d1716c480a424d236d34f975ee80 | subsection | 1,016 | 1,121 | Compactly generated spaces | Indeed, an isomorphism \kappa :{\mathbf {\Delta }}(n)^{\operatorname{op}}\longrightarrow {\mathcal {P}}(n) is given on objects by\kappa (\sigma :[n]\longrightarrow [k]) \ = \ \lbrace i\in \lbrace 1,\dots , n\rbrace \ : \ \sigma (i) > \sigma (i-1)\rbrace \ .In the other direction, a subset U\subset \lbrace 1,\dots ,n\rb... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2e1f3ce5c22dcd843b5449ab5c5b4c15a58dc521 | subsection | 1,017 | 1,121 | Compactly generated spaces | The category {\mathbf {\Delta }}(2)_\circ has three objects and two non-identity morphisms,
and L_2(X) is a pushout of the diagramX_1 \ \xleftarrow{}\ X_0 \ \xrightarrow{} X_1 \ .Proposition 4.75
Let X:{\mathbf {\Delta }}^{\operatorname{op}}\longrightarrow \mathbf {Spc} be a simplicial topological space and n\ge 0.The... | {
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{
"arxiv_id": "",
"doi": "10.2307/1969364",
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"raw": "S. Eilenberg, J. A. Zilber, Semi-Simplicial Complexes and Singular Homology. Ann. of Math. (2) 51 (1950), 499–513.",
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
a8b3c9e716c5a6e68573ba5fca38d099bb762886 | subsection | 1,018 | 1,121 | Compactly generated spaces | Because i\le j, the second relation means that\bar{\sigma }(a)\ = \ {\left\lbrace \begin{array}{ll}
\ \sigma (a) & \text{for $0\le a\le i$,}\\
\sigma (a-1) & \text{for $i+1\le a\le j$, and}\\
\ \sigma (a) & \text{for $j+1\le a\le n-1$.}
\end{array}\right.}We define \tau :[n-2]\longrightarrow [k] by\tau (a)\ = \ {\left\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c0ec73459da5aca447ddbc4f1d465f8f52d9c400 | subsection | 1,019 | 1,121 | Compactly generated spaces | Since X_n is compactly generated, so is L_n X,
by Proposition REF (i).The proof of part (iii) of the previous proposition
makes critical use of the fact that all degeneracy maps
in a simplicial compactly generated space are closed embeddings.
This is not the case more generally for simplicial k-spaces
or simplicial to... | {
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{
"arxiv_id": "",
"doi": "10.1007/bfb0097438",
"end": 1658,
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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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ddd97001205028210ee5f1e773ffeb5eb1d04ff1 | subsection | 1,020 | 1,121 | Compactly generated spaces | The argument makes essential use of Lillig's `union theorem' ,
so it is not of a formal, model category theoretical nature.Since cofibrations of compactly generated spaces are in particular
closed h-cofibrations,
every Reedy cofibrant simplicial compactly generated space
is in particular proper and good.
Whenever we wa... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01228231",
"end": 123,
"openalex_id": "https://openalex.org/W2155673231",
"raw": "J. Lillig, A union theorem for cofibrations. Arch. Math. (Basel) 24 (1973), 410–415.",
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5b88e0d7c0e283dd9d3afa6af0f755c42a79ea3f | subsection | 1,021 | 1,121 | Compactly generated spaces | If the spaces X_0,\dots , X_n consist only of a single point each,
then the realization |X| is n-connected.(i) We claim that this simplicial space |-|\circ \operatorname{Sing}\circ X
sending [m] to the realization of the simplicial set \operatorname{Sing}(X_m)
is automatically Reedy cofibrant.
Indeed, for every simplic... | {
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"doi": "",
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"raw": "A. K. Bousfield, E. M. Friedlander, Homotopy theory of \\Gamma -spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1f6419901b301620e710bdd24190adf2ce2cc451 | subsection | 1,022 | 1,121 | Compactly generated spaces | Then X satisfies the \pi _t-Kan condition at a
if the following condition holds:
for all tuples of elements x_i\in \pi _t(X_{m-1},d_i^*(a))
for i\in \lbrace 0,1,\dots ,k-1,k+1,\dots ,m\rbrace satisfyingd_i^*(x_j)\ = \ d_{j-1}^*(x_i) \text{\quad in\quad $\pi _t(X_{m-2},(d_j d_i)^*(a))$}for all 0\le i<j\le m with i\ne k,... | {
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"raw": "P. G. Goerss, J. F. Jardine, Simplicial homotopy theory. Progress in Mathematics, 174. Birkhäuser Verlag, Basel, 1999. xvi+510 pp.",
"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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157eef1bd59ecd46902c3667522af77fbd2f874d | subsection | 1,023 | 1,121 | Compactly generated spaces | The simplicial spaces X and Y satisfy the \pi _*-Kan condition.
The morphism of simplicial sets \pi _0(f):\pi _0(X)\longrightarrow \pi _0(Y)
is a Kan fibration.Then the sequence (|i|,|f|) is a homotopy fiber sequence.We transfer the question into the context of simplicial sets by use of the
singular complex functor, a... | {
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{
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"raw": "A. K. Bousfield, E. M. Friedlander, Homotopy theory of \\Gamma -spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9f6fa46ff22f4617352fa4e0c2d73b2008d102aa | subsection | 1,024 | 1,121 | Compactly generated spaces | After geometric realization we can identify the diagonal realization
with the iterated realization
as in Proposition REF (iii),
so this proves the claim.At some later stage we will need to know that the product of two
Reedy cofibrant simplicial spaces is again Reedy cofibrant.
The proof of this fact is rather formal a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6d4de4cdf78cf4b8fd094a0df135d0ecee111395 | subsection | 1,025 | 1,121 | Compactly generated spaces | The upper horizontal latching morphism is a cofibration by hypothesis;
so the lower horizontal morphism is a cofibration.In the general case we choose a chain of intermediate subsets{\mathcal {Y}}\ = \ {\mathcal {Y}}_0 \ \subset \ {\mathcal {Y}}_1 \ \subset \ \dots \ \subset {\mathcal {Y}}_k \ = \ {\mathcal {Z}}such th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
7f76cbf57a689c3ac03bfadbcb1401f18d32f2b1 | subsection | 1,026 | 1,121 | Compactly generated spaces | This latter morphism is a cofibration by the pushout product property.Now we take m=n and let {\mathcal {X}} be the subposet of {\mathcal {P}}(n+n)
of proper diagonal elements, i.e., the sets U+U
for a proper subset U of \lbrace 1,\dots ,n\rbrace .
The latching object L_n(X\boxtimes Y) is then a colimit
of the functor ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bb13180a0e67ccf7dd6ba3e69f54af00e3fe48f4 | subsection | 1,027 | 1,121 | Equivariant spaces | In this appendix we collect basic results about the equivariant homotopy
theory of G-spaces. Initially G can be any topological group, but we will
eventually specialize to compact Lie groups.
We start out by checking that taking fixed points commutes with certain kinds of
colimits, namely pushouts and sequential colimi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
9f0654f125ef742599bf8a5ed7cf7c33c1c26e6a | subsection | 1,028 | 1,121 | Equivariant spaces | When restricted to finite groups, these theorems show that
evaluation of a G-cofibrant {\mathbf {\Gamma }}-G-space F on spheres provides
a positive G-\Omega -spectrum if F is `special'
(compare Definition REF ),
and a full fledged G-\Omega -spectrum if F is `very special'
(compare Definition REF ).
As we explain in Rem... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
284f5213faa3dfca0fd9e4f8b2a4f6d5725293ad | subsection | 1,029 | 1,121 | Equivariant spaces | This colimit comes with a preferred G-action making
it a colimit in the category of G-spaces.
One has to beware that whenever the colimit in \mathbf {Spc} is not weak Hausdorff,
the functor w changes the underlying set; in particular, the forgetful functor to sets
does not preserve such colimits.Now we consider a close... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
9feb230a76a706915329e539bb47a278bc167c5c | subsection | 1,030 | 1,121 | Equivariant spaces | In particular, D is the set-theoretic
disjoint union of the images of B-i(A) and C, which are both G-invariant.
So D^G is the set-theoretic disjoint union of the images of
(B-i(A))^G=B^G-(i^G)(A^G) and C^G. The canonical map
B^G\cup _{A^G}C^G\longrightarrow D^G is thus a continuous bijection.Now we show that the canoni... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2bb3a1434f44799fb716d9b6fa058bfc33c2f11f | subsection | 1,031 | 1,121 | Equivariant spaces | This shows that \kappa is a closed map.(iii)
Points of compactly generated spaces are closed
(Proposition REF (iii)),
so the subspace X\times \lbrace y_0\rbrace \cup \lbrace x_0\rbrace \times Y is closed in X\times Y.
The claim then follows by applying part (i) to the pushout of the diagram\ast \ \longleftarrow \ X\ti... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a0a54e515f129b733eb4ed1cd27b95e670e2c2d4 | subsection | 1,032 | 1,121 | Equivariant spaces | Let
X_0 \ \longrightarrow \ X_1 \ \longrightarrow \ \dots \ \longrightarrow \ X_n \ \longrightarrow \ \dots
be a sequence of closed embeddings of G-spaces
and X_\infty a colimit of the sequence in the category of G-spaces.
Then for every compact space K,
every continuous G-map from G/H\times K to X_\infty
factors th... | {
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{
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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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073e6f43234b91b6398d57d71c08e42af747af07 | subsection | 1,033 | 1,121 | Equivariant spaces | The cofibrations are the retracts of
generalized CW-complexes, i.e., relative cell complexes in which cells
can be attached in any order and not necessarily to cells of lower dimensions.We consider a model category {\mathcal {M}} that is also enriched, tensored and cotensored
over the category {\mathbf {T}} of compactl... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/surv/099",
"end": 1183,
"openalex_id": "https://openalex.org/W1583122470",
"raw": "M. Hovey, Model categories. Mathematical Surveys and Monographs, 63. Amer. Math. Soc., Providence, RI, 1999, xii+209 pp.",
"source_ref_id": "29... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
415fae9ccafd584d4dc26106a04a517039d3bb77 | subsection | 1,034 | 1,121 | Equivariant spaces | So pushout product with i_l preserves the set of
generating cofibrations \lbrace K\times i_k\rbrace _{K\in {\mathcal {G}},k\ge 0} (up to isomorphism).
This takes care of the part
of the pushout product property that involves the cofibrations only.Similarly, the pushout product of j_k with i_l is isomorphic to j_{k+l}.
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
227180839a003b8821f7450904d8dd1956f62a35 | subsection | 1,035 | 1,121 | Equivariant spaces | Then the induced map of pushouts\gamma \cup \beta \ : \ C\cup _A B \ \longrightarrow \ \bar{C}\cup _{\bar{A}} \bar{B}is a {\mathcal {C}}-equivalence.We let H be a closed subgroup from the set {\mathcal {C}},
and we contemplate the commutative diagram of fixed points:@C=10mm{
C^H [d]_{\gamma ^H} & A^H [l]_-{g^H} [d]^{\a... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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56b696ae9e55ce1a191a3ae04c5a1dd58c0d3047 | subsection | 1,036 | 1,121 | Equivariant spaces | In the category of (non-equivariant) spaces,
the set \lbrace i_k: \partial D^k \longrightarrow D^k\rbrace _{k\ge 0} of inclusions of spheres into discs
detects Serre fibrations that are simultaneously weak equivalences.
By adjointness, the setI_{\mathcal {C}}\ = \ \lbrace G/H\times i_k \ : \ G/H \times \partial D^k\ \l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8a5434d576a2d8770f917202cd1f57b6b5ffdc4f | subsection | 1,037 | 1,121 | Equivariant spaces | To this end we observe that the morphisms in J_{\mathcal {C}} are inclusions of deformation
retracts internal to the category of G-spaces.
This property is inherited by coproducts and cobase changes,
so every morphism obtained by cobase changes of coproducts
of morphisms in J_{\mathcal {C}} is a homotopy equivalence of... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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beffdf173c1194abe0bfe06a71ddb01caf3c23aa | subsection | 1,038 | 1,121 | Equivariant spaces | If E^{\prime } is another universal G-space for {\mathcal {C}}, then every continuous G-map
from E^{\prime } to E is a G-equivariant homotopy equivalence.(i) This all follows from the existence of the {\mathcal {C}}-projective model
structure described in Proposition REF .
We let A be a {\mathcal {C}}-cofibrant G-space... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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718b189fba489c9aeecd44403f41001f966a3ad1 | subsection | 1,039 | 1,121 | Equivariant spaces | We consider a pushout square of G-spaces on the left{A [r]^-i[d]&B[d] &&
A^N[r]^-{i^N}[d]&B^N[d]\\
C [r]_j& D && C^N [r]_{j^N}& D^N }such that i is a G-cofibration for which the claim holds.
Then i is a closed embedding,
and the square on the right is also a pushout of G-spaces
(by Proposition REF (i)).
In particular,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1848,
"openalex_id": "https://openalex.org/W324411233",
"raw": "R. S. Palais, The classification of G-spaces. Mem. Amer. Math. Soc. 36 (1960), iv+72 pp.",
"source_ref_id": "7df74b609a79fe221dcafd5e7b7c46c0284c6a5c",
"s... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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38d4a875447f7775c9bb25462c3bd325ad19ecc0 | subsection | 1,040 | 1,121 | Equivariant spaces | Since X is weak Hausdorff, the diagonal \Delta _X
is closed in X\times X. So G\times \Delta _X is closed in
G\times X\times X.
The orbit relation E is the image of G\times \Delta _X under the above composite,
so E is closed in X\times X.(ii)
If O is an open subset of X, then g O is open for every g\in G,
since left tra... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.18910/10274",
"end": 1663,
"openalex_id": "https://openalex.org/W1569755823",
"raw": "S. Illman, Restricting the transformation group in equivariant CW complexes. Osaka J. Math. 27 (1990), no. 1, 191–206.",
"source_ref_id": "766694... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
541e58ccf6ede674da16f5967bc3de8287235a87 | subsection | 1,041 | 1,121 | Equivariant spaces | For every closed normal subgroup N of G
the orbit functor N\backslash -:G{\mathbf {T}}\longrightarrow (G/N){\mathbf {T}}
takes G-cofibrations to G/N-cofibrations.(i) The restriction functor \alpha ^* preserves colimits,
so we may show that it takes the generating G-cofibrations
G/H\times i_k:G/H\times \partial D^k\long... | {
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{
"arxiv_id": "",
"doi": "10.1007/978-3-662-12918-0",
"end": 597,
"openalex_id": "https://openalex.org/W1597292016",
"raw": "T. Bröcker, T. tom Dieck, Representations of compact Lie groups. Graduate Texts in Math., Vol. 98, Springer-Verlag, New York, 1985. x+313 pp.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2a7b97f574b3946920610072df9fd1c7823a7b38 | subsection | 1,042 | 1,121 | Equivariant spaces | Since G is compact Lie,
restriction along the diagonal embedding G\longrightarrow G\times G
preserves cofibrations by Proposition REF (i).Now we prove a decomposition result for certain kinds of fixed points.
We let G and K be topological groups and X a (K\times G)-space.
We want to describe
the K-fixed points (G\back... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
9667912d7e04d76220832588da51bd94489f4a10 | subsection | 1,043 | 1,121 | Equivariant spaces | Since (G\backslash X)^K a closed subset of the orbit space,
\bar{X} is a (K\times G)-invariant closed subspace of X.
In particular, \bar{X} is compactly generated in the subspace topology.For a given continuous homomorphism \alpha :K\longrightarrow G, we setX^{(\alpha )}\ = \ G\cdot X^{\alpha }\ ,the smallest G-subspac... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b93f3954699b2de575a216bc8b8ff4c3a87927a3 | subsection | 1,044 | 1,121 | Equivariant spaces | The graph of \bar{\beta } is the inverse image of the diagonal under the
continuous map\bar{X}\times K\times G \ \longrightarrow \ \bar{X}\times \bar{X}\ , \quad (x,k,g)\ \longmapsto \ (k x, g x)\ .So the graph of \bar{\beta } is closed, hence \bar{\beta } and \beta are
continuous.By Proposition REF the space \hom (K,G... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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17e5b300d8bfb8597b69e9c325d076bade48d863 | subsection | 1,045 | 1,121 | Equivariant spaces | We start with the special case where A is a G-CW-complex with skeleton filtration\emptyset = A^{-1} \ \subset \ A^0 \ \subset \ A^1 \ \subset \ \dots \ \subset \ A^n
\ \subset \ \dots \ .We show by induction over n that the map A^n\times _G f:A^n\times _G X\longrightarrow A^n\times _G Y
is a weak equivalence. The induc... | {
"cite_spans": [
{
"arxiv_id": "",
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"raw": "J. M. Boardman, R. M. Vogt, Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, Vol. 347. Springer-Verlag,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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270ec09ca21af69929c250775da76533fb346b07 | subsection | 1,046 | 1,121 | Equivariant spaces | In the body of this book, we always work in the category {\mathbf {T}} of compactly
generated spaces. However, this full subcategory is not closed under
quotient spaces nor coends inside the ambient category
of all topological spaces; since the construction of the prolongation
involves a coend (quotient space), some ca... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.... |
a100d2d69ad5dbb08fb2ba45775afa28e5cfc36f | subsection | 1,047 | 1,121 | Equivariant spaces | A more categorical way to describe F(K) is as a coend of the functor{\mathbf {\Gamma }}\times {\mathbf {\Gamma }}^{\operatorname{op}}\ \longrightarrow \ {\mathbf {K}}\ , \quad (m^+,n^+)\ \longmapsto \ F(m^+)\times K^n \ .Remark 4.106
We want to justify the abuse of notation of not distinguishing
between the original {... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00209-010-0770-x",
"end": 2016,
"openalex_id": "https://openalex.org/W3100544416",
"raw": "C. Berger, I. Moerdijk, On an extension of the notion of Reedy category. Math. Z. 269 (2011), no. 3-4, 977–1004.",
"source_ref_id": "1... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.... |
858debe7621bfbda457f607bec652ad440178d4a | subsection | 1,048 | 1,121 | Equivariant spaces | The next proposition in particular implies that
the reduction map (REF ) defined in the proof of part (ii)
is a bijection from F(K) to the set-theoretic disjoint union,
for m\ge 0, of the sets F(m_+)^{\text{nd}}\times _{\Sigma _m}C_m(K),
where F(m_+)^{\text{nd}} is the set of non-generate elements of F(m_+).Proposition... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/1969364",
"end": 1694,
"openalex_id": "https://openalex.org/W2327936544",
"raw": "S. Eilenberg, J. A. Zilber, Semi-Simplicial Complexes and Singular Homology. Ann. of Math. (2) 51 (1950), 499–513.",
"source_ref_id": "3989abb5f... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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950b2215d85d0c303a0593601869ae745325f9a6 | subsection | 1,049 | 1,121 | Equivariant spaces | We must have k<m, so the inductive
hypothesis provides an injective morphism
\delta :l_+\longrightarrow k_+ and a non-degenerate element x\in F(l_+)^{\text{nd}}
such that F(\beta )(z)=F(\delta )(x). But then \delta ^{\prime }\delta is also injective andy \ = \ F(\alpha )(z)\ = \ F(\delta ^{\prime })(F(\beta )(z))\ = \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
0c421aedefa3b2439449d886570d583ddd645f31 | subsection | 1,050 | 1,121 | Equivariant spaces | HenceF(\lambda )(x)\ = \ F(\bar{\sigma }\bar{\delta }\lambda )(x)\ = \ F(\bar{\sigma })(F(\delta )(x))\ = \ F(\bar{\sigma })(F(\bar{\delta })(\bar{x}))\ = \ \bar{x}\ .So \lambda is the bijection with the desired properties.(ii)
We call a quadruple (\sigma ,\delta ,u,x) consisting of
a surjective morphism \sigma :m_+\lo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.008979955688118935,
0.00033498401171527803,
0.006... |
75c67c40d007a297d845b96995a31ed29ee73e81 | subsection | 1,051 | 1,121 | Equivariant spaces | If(\sigma :m_+\longrightarrow k_+,\delta :l_+\longrightarrow k_+,u,x)\text{\quad and\quad }
(\bar{\sigma }:m_+\longrightarrow \bar{k}_+,\bar{\delta }:\bar{l}_+\longrightarrow \bar{k}_+,\bar{u},\bar{x})are reduction data for the same element (y,s),
then k=\bar{k}, l=\bar{l} and there are bijective morphisms
\lambda :l_+... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.012062439695000648,
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0.01663258485496044,
-0.023529767990112305,
0.03... |
bd3a5ddfd4f30d653c759d201f02e03abafaa95a | subsection | 1,052 | 1,121 | Equivariant spaces | Using that both quadruples are reduction data, we know thatF(\bar{\delta })(\bar{x})\ = \ F(\bar{\sigma })(y)\ = \ F(\beta )(F(\sigma )(y)) \ = \ F(\beta )(F(\delta )(x))\ = \ F(\beta \delta )(x)\ .Since \bar{\delta } and \beta \delta are both injective and
x and \bar{x} are both non-degenerate, then essential uniquene... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.012860950082540512,
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0.0006645968533121049,
0.01312030479311943,
0.... |
802e46345da4606a93923d673fc34ff7ca95124d | subsection | 1,053 | 1,121 | Equivariant spaces | We choose a factorization\sigma \circ \alpha \ = \ \bar{\delta }\circ \bar{\sigma }as a surjective morphism \bar{\sigma }:\bar{m}_+\longrightarrow \bar{k}_+
followed by an injective morphism \bar{\delta }:\bar{k}_+\longrightarrow k_+.
Using part (i) we writeF(\bar{\sigma })(\bar{y}) \ = \ F(\delta ^{\prime })(\bar{x})f... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02286040596663952,
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0.008881679736077785,
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0.0132... |
c3748fe41ead32ee9f9cedc0a8db8ee2fb2d6eba | subsection | 1,054 | 1,121 | Equivariant spaces | If (x,t)\in F(l_+)\times K^l is of minimal dimension in its equivalence class, then x
is non-degenerate and t\in C_l(K), for otherwise (x,t) would be equivalent
to an element of smaller dimension. So (\operatorname{Id},\operatorname{Id},t,x) is a reduction datum
for (x,t), and hence \rho (x,t)=[x,t].
Now we let (y,s) b... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7537ec1529d05e8f23e92f97a6a9e87260ffe429 | subsection | 1,055 | 1,121 | Equivariant spaces | Then the inclusions E\longrightarrow F and L\longrightarrow K
induce a closed embedding \iota :E(L)\longrightarrow F(K).(i)
The category {\mathbf {T}} is closed under products inside the category {\mathbf {K}},
so the functor (REF ) takes values in {\mathbf {T}};
the issue is that a priori, the quotient topology need n... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bd4ce0d1c7ddc12eab9c107ae3d2fa8facf13385 | subsection | 1,056 | 1,121 | Equivariant spaces | Proposition REF (ii) provides
surjective morphisms
\sigma :m_+\longrightarrow k_+ and \bar{\sigma }:n_+\longrightarrow \bar{k}_+,
injective morphisms \delta :l_+\longrightarrow k_+ and \bar{\delta }:l_+\longrightarrow \bar{k}_+
and u\in K^k, \bar{u}\in K^{\bar{k}} and (x,t)\in F(l_+)\times K^l
such thatF(\sigma )(y)\ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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551674fdc028446196370337ed1e38577aa5635b | subsection | 1,057 | 1,121 | Equivariant spaces | Since E_{m,n} is the inverse image of this latter closed set
under a continuous map, this show the claim that E_{m,n}
is a closed subset of F(m_+)\times K^m\times F(n_+)\times K^n.(ii)
Our first claim is that the map \iota :E(L)\longrightarrow F(K) is injective.
We let (x;\,l_1,\dots ,l_m)\in E(m_+)\times L^m
be a mini... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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535d3f2d41ca07a152910f7d0a1886e27c42c809 | subsection | 1,058 | 1,121 | Equivariant spaces | As we argued in the injectivity statement, (x,t) is then also
a minimal representative in its (F,K)-equivalence class.Proposition REF (ii)
provides a surjective morphism \sigma :m_+\longrightarrow k_+,
an injective morphism \delta :l_+\longrightarrow k_+ and u\in K^k such thatF(\sigma )(y)\ = \ F(\delta )(x)\ , \quad ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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076ac3c7359bc23c72ca1e0d050aa15f6899a6b8 | subsection | 1,059 | 1,121 | Equivariant spaces | The prolongation comes with a continuous,
based assembly mapassembly map!of a {\mathbf {\Gamma }}-space\alpha \ : \ K\wedge F(L) \ \longrightarrow \ F(K\wedge L) \ , \ \alpha (k\wedge [x;\, l_1,\dots ,l_n])\ = \ [x;\,k\wedge l_1,\dots ,k\wedge l_n]\ .The assembly map is natural in all three variables and associative an... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bcf1a0a76167f339f5104e46993864dc48f0280d | subsection | 1,060 | 1,121 | Equivariant spaces | In the category {\mathbf {K}}, product with any space preserves proclusions
(Proposition REF ), so the mapF(m_+)\times q^m\ : \ F(m_+)\times (K\times L)^m \ \longrightarrow \ F(m_+)\times (K\wedge L)^mis a proclusion for every m\ge 0.
The continuous map\psi _m \ : \ F(m_+)\times (K\times L)^m \ \longrightarrow \ F_K(L)... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bbb54e5c6921f079317eae6f8b28c35457e3e4d7 | subsection | 1,061 | 1,121 | Equivariant spaces | So the maps define a continuous map\kappa \ : \ |F\circ A|\ \longrightarrow \ F(|A|)\ .In terms of elements, \kappa is thus given by\kappa [[x;\,a_1,\dots ,a_m],t] \ = \ [x;\,[a_1,t],\dots ,[a_m,t]]\ ,for x\in F(m_+), a_1,\dots ,a_m\in A_n and t\in \Delta ^n.The following proposition is .Proposition 4.112
For every si... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 290,
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"raw": "R. Woolfson, Hyper-\\Gamma -spaces and hyperspectra. Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 229–255.",
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"start": ... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4fff64b2d3121f960e771356d52c624e2688824b | subsection | 1,062 | 1,121 | Equivariant spaces | So taking coends over {\mathbf {\Delta }} gives a homeomorphism\int ^{[m]\in {\mathbf {\Delta }}} \int ^{k_+\in {\mathbf {\Gamma }}} F(k_+)\times A_m^k\times \Delta ^m\ &\cong \ \int ^{[m]\in {\mathbf {\Delta }}} F(A_m)\times \Delta ^m\ = \ |F\circ A|\ .On the other hand, if we fix k and l and exploit that geometric
re... | {
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"raw": "S. Mac Lane, Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp.",
"source_ref_id": "05a93e42555651e93a155... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e34a21f0e133ed946c57c92469dc717d4875f27d | subsection | 1,063 | 1,121 | Equivariant spaces | This explains why we look
for a practical condition to ensure that simplicial G-spaces of the form F\circ A
are Reedy cofibrant; the concept of `G-cofibrancy' introduced in
Definition REF below does the job.Construction 4.113 We let {\mathcal {P}}(n) denote the power set of \lbrace 1,\dots ,n\rbrace ,
i.e., the set of ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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36e7dd376e21c8e5701276e49709542a3d19e1b8 | subsection | 1,064 | 1,121 | Equivariant spaces | We recall that the `simplicial circle' \mathbf {S}^1:{\mathbf {\Delta }}^{\operatorname{op}}\longrightarrow {\mathbf {\Gamma }}\mathbf {S}^1 - simplicial circlesimplicial circle
is given on objects by \mathbf {S}^1_n = n_+,
with face maps d_i:n_+\longrightarrow (n-1)_+ given byd_i(j) \ = \ {\left\lbrace \begin{array}{l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e5c95e13e03c120d00983b7424c6070c8a851f9a | subsection | 1,065 | 1,121 | Equivariant spaces | Moreover {\mathbf {\Delta }}(n)_\circ is the full subcategory with all objects except
the identity of [n].As we recalled in Remark REF ,
the category {\mathbf {\Delta }}(n)^{\operatorname{op}} is isomorphic to the
poset category {\mathcal {P}}(n).
Indeed, an isomorphism is given on objects by\kappa (\sigma :[n]\longrig... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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695806eb15cb2ab04756e02821904ee8e8d9f00a | subsection | 1,066 | 1,121 | Equivariant spaces | Claim (ii) is then a special case of
Proposition REF (iii).(iii) We contemplate the commutative square:{
L_n\left( (F\circ \mathbf {S}^1)^G\right)[d]_\cong [r]
& (L_n(F\circ \mathbf {S}^1))^G[d]^\cong \\
\operatorname{colim}_{U\subsetneq \lbrace 1,\dots ,n\rbrace } \, ( F(U_+))^G[r] &
\left( \operatorname{colim}_{U\su... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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80c62df1c7d3c07183a0b3200894fb6ec5729b7c | subsection | 1,067 | 1,121 | Equivariant spaces | So the map |l_n|
is a (G\times \Sigma _n)-cofibration without any hypotheses on E.Our notion of `G-cofibrant' should not be confused with
cofibrancy in the strict model structure that Bousfield and Friedlander
introduce for non-equivariant {\mathbf {\Gamma }}-simplicial sets
in ,
and generalized to {\mathbf {\Gamma }}-... | {
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"raw": "A. K. Bousfield, E. M. Friedlander, Homotopy theory of \\Gamma -spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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27e14b811e8941a8521e65bdf6a59dbb53af8301 | subsection | 1,068 | 1,121 | Equivariant spaces | Then the canonical map\operatorname{colim}_{A\in {\mathcal {Y}}} \, F(A_+)\ \longrightarrow \ F(T_+)is a (G\times K)-cofibration.We prove the following more general statement.
We let {\mathcal {Y}}\subset {\mathcal {Z}}\subset {\mathcal {P}}(T) be two K-invariant sets
that are both closed under passage to subsets.
We s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ca0950e46e667bc3105e03ef7d281c8b7835e2f7 | subsection | 1,069 | 1,121 | Equivariant spaces | The claim then holds for each pair ({\mathcal {Y}}_i,{\mathcal {Y}}_{i-1}).
Since (G\times K)-cofibrations are stable under composition,
this proves the general case.Proposition 4.120
Let G be a finite group, F a G-cofibrant {\mathbf {\Gamma }}-G-space,
and A a simplicial finite based G-set.For all m,n\ge 0 the `doubl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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77a6a7171cd9909c883f6e49360d2bb971f0a2d9 | subsection | 1,070 | 1,121 | Equivariant spaces | We let {\mathcal {Y}} denote the subposet of
{\mathcal {P}}(S\times \lbrace 1,\dots ,n\rbrace ) consisting of all subsets that are
contained in I_U\times \lbrace 1,\dots ,n\rbrace
for some (U,V)\in ({\mathcal {P}}(m)\times {\mathcal {P}}(n))^\circ .
Then the functor (REF ) factors as the composite( {\mathcal {P}}(m)\t... | {
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{
"arxiv_id": "",
"doi": "10.1007/978-3-642-85844-4",
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"raw": "P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6d1f2eba2012072d767a28c3c8f710955cc408a2 | subsection | 1,071 | 1,121 | Equivariant spaces | Since \tau ^*(A_l)=\sigma ^*(\alpha ^*(A_l)), the set \tau ^*(A_l)
is contained in \sigma ^*(A_k),
and similarly for (\sigma ^{\prime })^*(A_{k^{\prime }}).
Conversely, we let a\in A_m be a simplex such that
a=\sigma ^*(x)=(\sigma ^{\prime })^*(y) for some x\in A_k and y\in A_{k^{\prime }}.
We write x=\beta ^*(z) and y... | {
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"arxiv_id": "",
"doi": "10.2307/1969364",
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"raw": "S. Eilenberg, J. A. Zilber, Semi-Simplicial Complexes and Singular Homology. Ann. of Math. (2) 51 (1950), 499–513.",
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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369362de51b92f6c2a6fb946ae803e768a61a3a4 | subsection | 1,072 | 1,121 | Equivariant spaces | So the comma category B\downarrow \varphi is connected.
Since the functor \varphi is final, we can conclude that
the canonical (G\times \Sigma _n)-equivariant map\operatorname{colim}_{ (U,V)\in ( {\mathcal {P}}(m)\times {\mathcal {P}}(n) )^\circ } \,
F( (I_U\times V)_+)\ \longrightarrow \ \operatorname{colim}_{B\in {\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0c956bd986b4aa39efe15864f2d87f0dfe9855b8 | subsection | 1,073 | 1,121 | Equivariant spaces | This proves the claim.Part (iii) is the special case of part (i) for n=1.(iv) The restriction functor from G-spaces to H-spaces
preserves colimits, and hence latching objects,
and takes G-cofibrations to H-cofibrations
by Proposition REF (i); so the latching map
l_n:L_n(F\circ A)\longrightarrow F(A_n) is an H-cofibrat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ba11e942be54f20e935aac23483b79f670098ad0 | subsection | 1,074 | 1,121 | Equivariant spaces | In more detail, we let G be a compact Lie group
and F a {\mathbf {\Gamma }}-G-space.
We define an orthogonal G-spectrum F({\mathbb {S}}) byF({\mathbb {S}})(V) \ = \ F(S^V) \ .The structure map \sigma _{V,W}:S^V\wedge F({\mathbb {S}})(W)\longrightarrow F({\mathbb {S}})(V\oplus W)
is the assembly map for K=S^V and L=S^W,... | {
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"doi": "10.1007/bf01405351",
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"raw": "S. Illman, Smooth equivariant triangulations of G-manifolds for G a finite group. Math. Ann. 233 (1978), no. 3, 199–220.",
"source_ref_id": "... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2a4711921d851a7cf7be70c89ada4a7aeb055ae0 | subsection | 1,075 | 1,121 | Equivariant spaces | Altogether, S^V is G-homeomorphic to the geometric realization
of the G-simplicial set A=(\Delta [d]/\partial \Delta [d])\wedge B.Proposition REF
provides a G-equivariant homeomorphismF(S^V)\ \cong \ F(|A|) \ \cong \ |F\circ A | \ .Taking G-fixed points commutes with realization
by Proposition REF (iv),
so F(S^V)^G i... | {
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"doi": "10.1016/s0079-8169(08)x6007-6",
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"raw": "T. tom Dieck, Transformation Groups. De Gruyter Studies in Mathematics, 8. Walter de Gruyter & Co., Berlin, 1987. x+312 pp.",
"sou... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4f19bec65eac34a9bb3a8845ee376e23c3b12980 | subsection | 1,076 | 1,121 | Equivariant spaces | So every based continuous H-map S^V\longrightarrow F(S^{{\mathbb {R}}^k\oplus V}) is equivariantly
null-homotopic by ,
and the set [S^V,F(S^{{\mathbb {R}}^k\oplus V})]^H has only one element.
Passage to the colimit over V\in s({\mathcal {U}}_H) proves the claim.We can also show that prolongation of G-cofibrant {\mathbf... | {
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"doi": "10.1016/s0079-8169(08)x6007-6",
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"sour... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b3c32c118ee534355fd6f67f55c4c862032637f4 | subsection | 1,077 | 1,121 | Equivariant spaces | Every finite based G-CW-complex is based G-homotopy equivalent
to the realization of a finite based G-simplicial set.(i)
We let c Y= (Y\times [0,1]) / (Y\times \lbrace 1\rbrace ) denote
the unreduced cone of a space Y.
We start with a very special case, namely when there is
a pushout square of G-spaces{
G/H\times |\par... | {
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{
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"doi": "",
"end": 1298,
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"raw": "D. M. Kan, On c. s. s. complexes. Amer. J. Math. 79 (1957), 449–476.",
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... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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b29c9603d51c125b0948a27d13226d57c817a493 | subsection | 1,078 | 1,121 | Equivariant spaces | We absorb the homotopy into the mapping cylinder of the map
|\tilde{\Phi }|:G/H\times |D|\longrightarrow |A|, and obtain a commutative diagram of G-spaces:{
|A|\cup _{ |\tilde{\Phi }|} (G/H\times |D|\times [0,1])
[d]_{\operatorname{Id}_{|A|}\cup K} & G/H\times |D| [r]^-{\text{incl}}
[l]_-{(-,1)} [d]^{G/H\times \varphi ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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05f545fff8a03d4200512b663cf3afa1e26e2bb7 | subsection | 1,079 | 1,121 | Equivariant spaces | So the triple (B,i,p\circ h) has the desired properties.
If the map f is arbitrary, we use the equivariant cellular approximation theorem
(see for example ,
or )
and the equivariant homotopy extension property of the G-map |i|:|A|\longrightarrow |B|
to reduce to the cellular case.(ii) We let X be a finite based G-CW-c... | {
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"arxiv_id": "",
"doi": "10.1090/s0002-9904-1967-11712-9",
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"raw": "G. E. Bredon, Equivariant cohomology theories. Lecture Notes in Mathematics, Vol. 34, Springer-Verlag, Berlin-New York, 1967. vi+64 pp.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b60a14a5ea41016fe2883b7b49a113071458ae69 | subsection | 1,080 | 1,121 | Equivariant spaces | The final aim is to show that for finite G and very special (respectively special) F,
the evaluation on spheres F({\mathbb {S}}) is a G-\Omega -spectrum
(respectively `positive' G-\Omega -spectrum),
see Theorem REF respectively
Theorem REF below.If F is any {\mathbf {\Gamma }}-space and S a finite set, then we define t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4cf1f436af8677eb014c94803fa02c2ea1fc17ea | subsection | 1,081 | 1,121 | Equivariant spaces | We recall that l_Z:G\ltimes _H Z\longrightarrow Z denotes the H-equivariant
projection to the wedge summand indexed by the preferred coset e H, i.e.,l_Z[g,z]\ = \ {\left\lbrace \begin{array}{ll}
g\cdot z & \text{ if $g\in H$, and}\\
\ \ast & \text{else.}
\end{array}\right.}The H-equivariant map F(l_Z): F(G\ltimes _H Z)... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 684,
"openalex_id": "",
"raw": "K. Shimakawa, A note on \\Gamma _G-spaces. Osaka J. Math. 28 (1991), no. 2, 223–228.",
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"start": 620
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]
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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763eca22e10db6e996319b71c2612821f87e4d3f | subsection | 1,082 | 1,121 | Equivariant spaces | So the map (P_n)^\Gamma is a weak equivalence if and only if
the map (P_{\alpha ^*\lbrace 1,\dots ,n\rbrace })^H is a weak equivalence.(i)\Longrightarrow (iii)
In the commutative square\begin{aligned}
{
F( (T\amalg U)_+)[r]^-{(F(p_T),F(p_U))}[d]_{P_{T\amalg U}}^\simeq &
F(T_+)\times F(U_+) [d]^{p_T\times p_U}_\simeq \\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6abae9c7b40aa79c6ec651535d8fadce62fe7e7c | subsection | 1,083 | 1,121 | Equivariant spaces | For every finite index subgroup H of G, there is a commutative square
of G-maps{
F( (G/H)_+)[r]^-{P_{G/H}}[d]_\cong & \operatorname{map}(G/H,F(1_+)) [d]^\cong \\
F(G\ltimes _H 1_+)[r]_-{\omega _{1_+}} &\operatorname{map}^H(G,F(1_+))
}where the right vertical map is adjoint to the H-map\operatorname{map}(G/H,F(1_+)) \ \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bd42325699c1f1e77c7882db9e893f6ee41068b5 | subsection | 1,084 | 1,121 | Equivariant spaces | For every finite based G-CW-complex X,
the shifted {\mathbf {\Gamma }}-G-space F_X is special.(i)
We wish to show that for every closed subgroup H of G the map(F(p_X)^H,F(p_Y)^H)\ : \ F(X\vee Y)^H\ \longrightarrow \ F(X)^H\times F(Y)^His a weak equivalence.
The underlying {\mathbf {\Gamma }}-H-space of F is H-cofibrant... | {
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"source_ref_id": "7666947... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2d3e112a7c3fa2e32fe072430d9bb8432d0a99cf | subsection | 1,085 | 1,121 | Equivariant spaces | We let G^\circ be the identity component of G
and we write \bar{G}=G/G^\circ for the finite group of path components.
The {\mathbf {\Gamma }}-\bar{G}-space F^{G^\circ } is \bar{G}-cofibrant
by Proposition REF (ii)
and special by Proposition REF (ii).
Since X and Y are finite G-CW-complexes, X^{G^\circ }
and Y^{G^\cir... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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142f9cdc650b3ad8d46b9c0275d246f13e9a403b | subsection | 1,086 | 1,121 | Equivariant spaces | The morphism of simplicial spaces (REF )
is a weak equivalence in every simplicial dimension
by Proposition REF (iii);
moreover, source and target are Reedy cofibrant
by Proposition REF (iv).
As a levelwise weak equivalence between Reedy cofibrant simplicial spaces,
the morphism (REF ) induces
a weak equivalence on g... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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34cfec4ce10a359372e178a567f71778fb1c9b0a | subsection | 1,087 | 1,121 | Equivariant spaces | Proposition REF lets us rewrite
source and target of this map as( F^{G^\circ }( \bar{G}\ltimes _{\bar{H}} Z^{H^\circ }))^{\bar{G}}
\ &\cong \ ( F^{G^\circ }( (G\ltimes _H Z)^{G^\circ }))^{\bar{G}} \\
\ &\cong \ ( F( G\ltimes _H Z)^{G^\circ })^{\bar{G}} \ = \ F(G\ltimes _H Z)^Grespectively( F^{G^\circ }(Z^{H^\circ }))^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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84d8e40f5dd7b2b28a97d8eba6246fd153b476e9 | subsection | 1,088 | 1,121 | Equivariant spaces | The latter is a weak equivalence by the additivity property of part (i),
applied to the underlying {\mathbf {\Gamma }}-K-space of F,
and the spaces K\ltimes _{K\cap {^{\gamma _i} H}} c_{\gamma _i}^*(Z);
there is a slight caveat, namely that the underlying (K^{\gamma _i}\cap H)-space
of Z need not admit the structure of... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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55ce19373b78234ede2c59f32634de2a5d415baa | subsection | 1,089 | 1,121 | Equivariant spaces | Proposition REF now applies and shows that
the shifted {\mathbf {\Gamma }}-G-space F_X is special.We still have to recall the notion of a `very special' {\mathbf {\Gamma }}-G-space.
We let G be a compact Lie group and F a special {\mathbf {\Gamma }}-G-space.
We let p_1,p_2:2_+\longrightarrow 1_+ denote the two projecti... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1935d071395b166fac8f4a149ae70ed4a923ace1 | subsection | 1,090 | 1,121 | Equivariant spaces | The map(\pi _0(F(p_1)^H),\pi _0(F(p_2)^H))\ : \ \pi _0(F((S\amalg S)_+)^H)
\ \longrightarrow \ \pi _0(F(S_+)^H)\times \pi _0(F(S_+)^H)is bijective by Proposition REF (iii);
inverting this map and composing with the effect of
the fold map \nabla :(S\amalg S)_+\longrightarrow S_+ on \pi _0(F(-)^H)
yields a binary operat... | {
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"raw": "A. K. Bousfield, E. M. Friedlander, Homotopy theory of \\Gamma -spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1729ff9575a8f4b599a9743fce6a27c57d37fc5f | subsection | 1,091 | 1,121 | Equivariant spaces | Moreover, the map of simplicial sets
\beta :\pi _t((F\circ A)^H)_{\text{free}}\longrightarrow \pi _0( (F\circ A)^H)
discussed in B.3 of
is underlying a surjective morphism of simplicial groups,
and is thus a Kan fibration by .
So proves that
(F\circ A)^H satisfies the \pi _*-Kan condition.Proposition 4.141
Let H be a... | {
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"raw": "A. K. Bousfield, E. M. Friedlander, Homotopy theory of \\Gamma -spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130,... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cb7259292165c50d1d22e7e3cc5ec2638d18d03b | subsection | 1,092 | 1,121 | Equivariant spaces | In various respects, goes further than we do here;
for example, it contains detailed comparisons of prolongation
(the `conceptual Segal machine'),
bar construction (the `homotopical Segal machine')
and the operadic approach to equivariant delooping (via a `generalized Segal machine').
Moreover, for special {\mathbf {\G... | {
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{
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"doi": "",
"end": 286,
"openalex_id": "https://openalex.org/W2609689518",
"raw": "J. P. May, M. Merling, A. M. Osorno, Equivariant infinite loop space theory, I. The space level story. arXiv:1704.03413",
"source_ref_id": "34149da69cd32c6d7851... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a886d5af4fc1b67d4f8211f06e03016b37e7f46b | subsection | 1,093 | 1,121 | Equivariant spaces | For fixed n\ge 0 the lower row in the commutative diagram@C=15mm{
F(A_n)^H[r]^-{F(i_n)^H} @{=}[d] &
F(B_n)^H[r]^-{F(q_n)^H} [d]^{(F(r)^H,F(q_n)^H)}_\simeq &
F(B_n/A_n)^H @{=}[d]\\
F(A_n)^H [r] & F(A_n)^H\times F( B_n/A_n)^H[r]_-{\text{proj}}& F( B_n/A_n )^H}is a homotopy fiber sequence.
The middle vertical map is a wea... | {
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"arxiv_id": "",
"doi": "10.1007/978-3-0346-0189-4",
"end": 1148,
"openalex_id": "https://openalex.org/W1530632394",
"raw": "E. B. Curtis, Simplicial homotopy theory. Advances in Math. 6 (1971), 107–209.",
"source_ref_id": "8d947d2c59ce44306ec55e1cb80ea5784fa... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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3d66a2803bfca27c9ec9a17d1fb91756e1b5340f | subsection | 1,094 | 1,121 | Equivariant spaces | Source and target of h/k are cofibrant as G-spaces,
so this G-weak equivalence is even a G-homotopy equivalence.
The upper row in the commutative diagram of G-spaces@C=18mm{
F(|A|)[r]^-{F(|i|)} [d]_{F(k)}^\simeq &
F(|B|)[r]^-{F(|q|)} [d]_{F(h)}^\simeq &
F(|B/A|) [d]^{F(h/k)}_\simeq \\
F(X) [r]_-{F(j)} & F(Y)[r]_-{F(p)}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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5a294abed0dfc31a645736b12f39efd777187d9c | subsection | 1,095 | 1,121 | Equivariant spaces | This completes the proof that the sequence (REF )
is a G-homotopy fiber sequence.Now we treat a general homotopy cocartesian square
of finite based G-CW-complexes:{ X[r]^-j[d]_k & Y[d]^h \\ Z [r]_i & P}By replacing Y and P by the reduced mapping cylinders
of j respectively i,
we may assume that the horizontal maps are ... | {
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"doi": "10.2140/agt.2006.6.2257",
"end": 1165,
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"raw": "A. J. Blumberg, Continuous functors as a model for the equivariant stable homotopy category. Algebr. Geom. Topol. 6 (2006), 2257–2295.",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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035c63b5fafd892316beda1c40e9376bdeb889d4 | subsection | 1,096 | 1,121 | Equivariant spaces | The previous paragraph shows that the map\tilde{\alpha }\ : \ F^{G^\circ }(X^{G^\circ })\ \longrightarrow \ \operatorname{map}_*(S^V,F^{G^\circ }(X^{G^\circ }\wedge S^V))induces a weak equivalence on \bar{G}-fixed points.
Proposition REF provides a homeomorphism( F^{G^\circ }( X^{G^\circ }) )^{\bar{G}}\ \cong \ (F(X)^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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4b1afbfc7aa542d3ef8a4e071163932000539d42 | subsection | 1,097 | 1,121 | Equivariant spaces | Moreover, for every subgroup H of G the space( F_{S^1}(1_+))^H\ \cong \ F(S^1)^His path connected by Proposition REF (i).
So the abelian monoid \pi _0( F_{S^1}(1_+)^H) has only one element,
and is thus an abelian group. Hence F_{S^1} is very special.So Theorem REF applies to the
G-cofibrant and very special {\mathbf {... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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