chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32cf6a3a02362843e5919199e50646d0dab107ce | subsection | 498 | 1,121 | The double coset formula | Since \psi ^M_G(\operatorname{tr}_{H_j}^G(y)) and |W_G H_j|\cdot x both belong to I_j
and agree at the component of H_j, we can thus conclude that\psi ^M_G(\operatorname{tr}_{H_j}^G(y)) - |W_G H_j|\cdot x \ \in \ I_{j-1} \ .This proves (REF ) and finishes the proof
that the cokernel of \psi ^M_G is annihilated by the n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/memo/0543",
"end": 1787,
"openalex_id": "https://openalex.org/W2049930703",
"raw": "J. P. C. Greenlees, J. P. May, Generalized Tate cohomology. Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178 pp.",
"source_ref_id": "5736f7... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.024978792294859886,
0.014747710898518562,
-0.011077949777245522,
-0.012855609878897667,
0.010513370856642723,
-0.06323281675577164,
0.016388041898608208,
-0.018173329532146454,
0.02403274178504944,
0.04714994877576828,
-0.05056793987751007,
-0.0395205095410347,
0.017639270052313805,
0.0... |
593f70187500f44388baaaa939c8e1f4bbc575d5 | subsection | 499 | 1,121 | The double coset formula | One could deduce part (ii) of the following theorem
from the results in
by using the action of the Burnside ring A(H) on the value
M(H) of any G-Mackey functor M, and showing that after inverting the group order,
the functor \tau _H:G\operatorname{-{\mathcal {M}}ack}\longrightarrow W_G H \operatorname{-mod} becomes is... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/memo/0543",
"end": 407,
"openalex_id": "https://openalex.org/W2049930703",
"raw": "J. P. C. Greenlees, J. P. May, Generalized Tate cohomology. Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178 pp.",
"source_ref_id": "5736f74... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01349275279790163,
0.022424358874559402,
0.02938048727810383,
0.0029689394868910313,
0.0003825584426522255,
-0.06208649277687073,
0.010701148770749569,
0.014613970182836056,
0.02095991186797619,
0.05439814180135727,
-0.04222491756081581,
-0.024880360811948776,
0.03166868910193443,
0.006... |
968e47f4aed9039a2bfc5e530527d3ed6609838d | subsection | 500 | 1,121 | The double coset formula | The conjugation map
\gamma _\star :(\rho _H N)(K^\gamma )\longrightarrow (\rho _H N)(K),
for \gamma \in G, is precomposition with the W_G H-map(G/K)^H \ \longrightarrow \ (G/K^\gamma )^H \ , \quad g\cdot K \ \longmapsto \ g \gamma \cdot K^\gamma \ .For L\le K, the restriction map
\operatorname{res}^K_L:(\rho _H N)(K)\l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03255912661552429,
0.024259144440293312,
-0.039424922317266464,
-0.0020349454134702682,
0.009917260147631168,
-0.040737051516771317,
0.030362073332071304,
0.025052525103092194,
-0.019559888169169426,
0.039424922317266464,
-0.01473857369273901,
-0.009551084600389004,
0.009772315621376038,
... |
adea9a3b147c08783e4006d999c508cb93817d41 | subsection | 501 | 1,121 | The double coset formula | Hence for every W_G H-map f:(G/L)^H\longrightarrow N all g J\in (G/J)^H we obtain the relation(\operatorname{res}^K_J(\operatorname{tr}_L^K (f)))(g J)\ &= \ \sum _{\gamma L\in (G/L)^H\ :\ \gamma K= g K} f(\gamma L)\\
&= \ \sum _{k\in R} \ \sum _{ j\in S_k}\, f(j k L)\\
&= \ \sum _{k\in R} \ \sum _{ j\in S_k}\, k_\star ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01980704441666603,
-0.008507262915372849,
-0.029222259297966957,
0.0003071007668040693,
0.0047495705075562,
-0.00558885233476758,
0.004478711634874344,
-0.0003144921502098441,
-0.01518336683511734,
0.02745213732123375,
-0.04001083970069885,
-0.033082954585552216,
0.037721890956163406,
0... |
8b098fbbd8fb47b29a4943a7bafb247039f1b610 | subsection | 502 | 1,121 | The double coset formula | For a subgroup K of G
we define\eta _H^M(K)\ : \ M(K)\ \longrightarrow \ \operatorname{map}^{W_G H}((G/K)^H,\tau _H M) \ = \ \rho _H(\tau _H M)(K)by\eta _H^M(K)(x)(g K)\ = \ [ g_\star ( \operatorname{res}^K_{H^g}(x) ) ] \ .We show that these additive maps indeed define a morphism of G-Mackey functors.
For L\le K\le G, ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
0.013022505678236485,
0.009314867667853832,
-0.022291600704193115,
-0.007674657739698887,
-0.020369121804833412,
-0.05837622284889221,
-0.012732608243823051,
-0.0058971275575459,
0.029096564278006554,
0.017546433955430984,
-0.020491182804107666,
-0.014906840398907661,
0.01306827925145626,
... |
769316a5911ff122945f64edaaeef07f580d508a | subsection | 503 | 1,121 | The double coset formula | Similarly, for \gamma \in G and x\in M(K^\gamma ) we have\gamma _\star (\eta _H^M(K^\gamma )(x))(g\cdot K)\ &= \ \eta _H^M(K^\gamma )(x)(g\gamma \cdot K^\gamma )\ = \ [ (g\gamma )_\star (\operatorname{res}^{K^\gamma }_{H^{g\gamma }}(x)) ] \\
&= \ [ g_\star (\operatorname{res}^K_{H^g}(\gamma _\star (x))) ] \ = \ \eta _H... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.048612695187330246,
0.027113860473036766,
-0.042570438235998154,
-0.00841490924358368,
-0.004421069752424955,
-0.016402587294578552,
0.025847427546977997,
-0.00021611840929836035,
-0.008010566234588623,
-0.008979463018476963,
-0.015960099175572395,
0.002542401198297739,
0.0229788813740015... |
d9eb7bc726fa5b318d5d0afa3a4673838165d9c4 | subsection | 504 | 1,121 | The double coset formula | So for all x\in M(L) we obtain the relation\operatorname{tr}^K_L(\eta _H^M(L)(x))(g K)\ &= \ \sum \, \eta _H^M(L)(x)(\gamma L)\ = \ \sum \, [ \gamma _\star (\operatorname{res}^L_{H^{\gamma }}(x)) ] \\
_(\ref {eq:double_coset_rho_H})
&= \ [ g_\star (\operatorname{res}^K_{H^g}(\operatorname{tr}^K_L(x)))] \ = \ \eta _H^M(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0063544525764882565,
0.023510711267590523,
-0.028957383707165718,
-0.020261015743017197,
-0.024532916024327278,
-0.032619014382362366,
0.01679772324860096,
-0.004519823472946882,
0.005275034811347723,
0.0037588912528008223,
-0.03252747282385826,
0.0025211842730641365,
0.019803311675786972... |
6f2505a9523f9fbe36ec5be6e55ed2638440fb41 | subsection | 505 | 1,121 | The double coset formula | Since (\tau _H,\rho _H)
is an adjoint pair for every subgroup H of G,
a right adjoint to the product functor \tau is given by\rho \ : \ {\prod }_{(H)}\, W_G H\operatorname{-mod}\longrightarrow G\operatorname{-{\mathcal {M}}ack}\ , \quad \rho ( (N_H)_{(H)})\ = \ {\prod }_{(H)} \, \rho _H(N_H)\ ,the product of the G-Mack... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02047779969871044,
0.006618662737309933,
-0.002296504331752658,
-0.00792713649570942,
-0.01878403127193451,
-0.02659672498703003,
-0.024338368326425552,
-0.0062486277893185616,
0.01236755307763815,
0.03530970588326454,
-0.03152543306350708,
-0.05142338573932648,
0.0013008437817916274,
0... |
8de54c7a998809512972acee07e88a1d2cd5cb13 | subsection | 506 | 1,121 | The double coset formula | So taking W_G H-equivariant maps into \tau _H M
provides an isomorphism of abelian groups\rho _H(\tau _H M)(K)\ = \ \operatorname{map}^{W_G H}( (G/K)^H, \tau _H M) \ &\longrightarrow \ {\prod }_{L\in R_{H,K}}\, (\tau _H M)^{W_{^{g_L} K} H} \\
f \qquad &\longmapsto \quad ( f(g_L K) )_{L} \ .Because H^{g_L}=L, the map (g... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.004902278073132038,
0.032900579273700714,
-0.01876981183886528,
-0.021394534036517143,
-0.01573307067155838,
-0.019548071548342705,
0.02147083356976509,
0.01077356655150652,
0.01690809056162834,
0.03872990608215332,
-0.02636929601430893,
0.005218923091888428,
0.03418242186307907,
0.0295... |
360c63dd842a9ed310be35ab989554edc095ac55 | subsection | 507 | 1,121 | The double coset formula | Since the functor \tau is additive, it commutes with finite products,
so it suffices to show that for every individual subgroup H of G
and every Q-local W_G H-module N, there is an
Q-local G-Mackey functor M such that \tau M is isomorphic to N.
We let e:\tau (\rho _H N)\longrightarrow \tau (\rho _H N) be the idempotent... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.021218793466687202,
0.0085195517167449,
0.020059462636709213,
-0.0042559620924293995,
0.005327580031007528,
-0.05869871750473976,
0.029639191925525665,
0.010754313319921494,
0.020089972764253616,
0.05686819553375244,
-0.010533125139772892,
-0.015650957822799683,
0.0005014865892007947,
0... |
c07a9c2834825b24cd9b37017080f4e989f186ab | subsection | 508 | 1,121 | The double coset formula | The following proposition is well known,
and closely related statements appear in Appendix A of ;
however, I am not aware of a reference for the following statement in this form.Proposition 4.23
For every finite group G, every orthogonal G-spectrum X
and every integer k the reduced geometric fixed point map\bar{\Phi }... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02904415875673294,
0.02803737483918667,
-0.02878483571112156,
0.024086516350507736,
-0.006803411059081554,
-0.05930865928530693,
0.05485440790653229,
-0.005403830204159021,
0.03627469018101692,
0.014918690547347069,
-0.004561031237244606,
-0.024345839396119118,
0.015635641291737556,
0.0... |
7fef7494727365d631f9518cbf38485da5c3bbdd | subsection | 509 | 1,121 | The double coset formula | Equivariant homotopy groups commute with wedges, so we can identify
the homotopy group Mackey functor of X\wedge (A/B) as{\underline{\pi }}_k(X\wedge A/B)\ \cong \ {\bigoplus }_{i\in I}\ {\underline{\pi }}_k(X\wedge S^n\wedge (G/H_i)_+)\ .Since A has no G-fixed points, the groups H_i are all proper subgroups of G,
so t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.023406153544783592,
-0.030562272295355797,
-0.041471920907497406,
0.027831045910716057,
0.01924065127968788,
-0.022322818636894226,
0.026930809020996094,
-0.0016068842960521579,
0.019408492371439934,
0.018004734069108963,
-0.005988862831145525,
-0.04186863452196121,
0.0408005565404892,
... |
ecd3ab7fcfaa495c7b43cf8994cefe6d6c8c7e04 | subsection | 510 | 1,121 | The double coset formula | The inclusion i:S^0\longrightarrow \tilde{E}{\mathcal {P}}_G gives rise to a commutative square:\begin{aligned}
@C=15mm{
\tau _G( {\underline{\pi }}_k(X))[r]^-{\tau _G({\underline{\pi }}_k(X\wedge i))}[d]_{\bar{\Phi }} &
\tau _G( {\underline{\pi }}_k(X\wedge \tilde{E}{\mathcal {P}}_G)) [d]^{\bar{\Phi }} \\
\Phi ^G_k(X)... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06287136673927307,
-0.009651975706219673,
-0.033205848187208176,
0.020890995860099792,
0.03775333985686302,
-0.01710650511085987,
0.03729553893208504,
0.0309931430965662,
0.0412936694920063,
0.03601369634270668,
-0.02032637409865856,
-0.015336341224610806,
0.004867952782660723,
0.031115... |
48cba2cfdbe3b797eecd9df87071c388657fa380 | subsection | 511 | 1,121 | The double coset formula | Under this identification, the map \bar{\psi }^M_G
becomes the product of the maps\pi _k^G(X) \ \xrightarrow{}\ \pi _k^H(X)\ \xrightarrow{} \ \Phi _k^H(X)\ .So Propositions REF
and REF together prove:Corollary 4.25
For every finite group G, every orthogonal G-spectrum X
and every integer k the map(\Phi ^H\circ \opera... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
0.003267943160608411,
-0.0023865713737905025,
-0.00006998988101258874,
0.007360025774687529,
-0.009279203601181507,
-0.06138315051794052,
0.011324291117489338,
-0.03204987943172455,
0.038490377366542816,
0.04178693890571594,
-0.03574324771761894,
0.0008718330063857138,
0.043557312339544296,
... |
680419b3d3ecd64dabeda9288bc66c0060c8ec37 | subsection | 512 | 1,121 | Products | In this section we recall the smash product of orthogonal spectra
and orthogonal G-spectra and study its formal and homotopical properties.
Like the box product of orthogonal spaces, the smash product
of orthogonal spectra is a special case of Day's convolution product on categories
of enriched functors, compare Append... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.037564244121313095,
0.01702749729156494,
-0.021299628540873528,
0.002092200331389904,
0.025388669222593307,
-0.016096781939268112,
0.009810646064579487,
0.013281753286719322,
0.039425674825906754,
0.03048471361398697,
-0.023786621168255806,
0.017195330932736397,
0.007933959364891052,
0.... |
7a2ba6c4dc4323d40050b8553acb2b42c8e92ba3 | subsection | 513 | 1,121 | Products | A smash product of two orthogonal spectra is now a universal example
of a bimorphism from (X,Y).Definition 5.1 A smash productsmash product\wedge - smash product of orthogonal spectra
of two orthogonal spectra X and Y is a pair (X\wedge Y,i)
consisting of an orthogonal spectrum X\wedge Y
and a universal bimorphism i:(X... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.030374575406312943,
0.010892740450799465,
-0.048025697469711304,
-0.02073282189667225,
0.025950351729989052,
-0.020702309906482697,
-0.02448578178882599,
0.020259886980056763,
0.04244202375411987,
0.013516762293875217,
-0.022868653759360313,
0.006327400915324688,
0.006663031410425901,
-... |
7f84ffdc7fbadbfb095b64d7500d08ca5ba1bd4e | subsection | 514 | 1,121 | Products | In the case at hand this specializes to a preferred isomorphism{\mathbf {O}}(V,-)\wedge {\mathbf {O}}(W,-)\ \cong \ {\mathbf {O}}(V\oplus W,-)specified, via the universal property (REF ),
by the bimorphism with (U,U^{\prime })-component\oplus \ : \ {\mathbf {O}}(V,U)\wedge {\mathbf {O}}(W,U^{\prime })\ \longrightarrow ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.019311822950839996,
0.013347429223358631,
-0.04451176896691322,
0.004450414329767227,
0.021721987053751945,
-0.00942709855735302,
0.03932534158229828,
-0.0034646112471818924,
0.02564231865108013,
0.026313502341508865,
0.0013576241908594966,
0.02384232170879841,
0.016749117523431778,
0.0... |
d9dbf2bbcbca24bddd25a95ec99c17ff52210988 | subsection | 515 | 1,121 | Products | Since an orthogonal G-spectrum is the same data as an orthogonal
spectrum with continuous G-action, and since skeleta are functorial,
the skeleta of an orthogonal G-spectrum automatically come as orthogonal G-spectra.Construction 5.6 (Skeleton filtration of orthogonal spectra)
As in the unstable situation in Section ,... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.007810243871062994,
-0.005768973380327225,
-0.047403257340192795,
0.016192806884646416,
0.029424816370010376,
-0.03668944910168648,
-0.03047788329422474,
-0.003161107422783971,
0.04773901775479317,
0.010767224244773388,
0.0008684938075020909,
-0.004834186285734177,
-0.0011007598368451,
... |
7da1e03f49ce53248dac109c1e08bb9cc03fdc07 | subsection | 516 | 1,121 | Products | As we remarked earlier,
the skeleta of (the underlying orthogonal spectrum of)
an orthogonal G-spectrum inherit a continuous G-action by functoriality.
In other words, the skeleta and the various morphisms between them
lift to endofunctors and natural transformations on the category of
orthogonal G-spectra.
If X is an ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1257,
"openalex_id": "https://openalex.org/W2183692372",
"raw": "M. Stolz, Equivariant structure on smash powers of commutative ring spectra. PhD thesis, University of Bergen, 2011.",
"source_ref_id": "26ca4953153b3ac27dc6db... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.030692672356963158,
0.008111962117254734,
-0.04688607156276703,
0.000001609705577720888,
0.02348882332444191,
-0.055036190897226334,
-0.002081408863887191,
0.014445855282247066,
0.012835673056542873,
0.011324696242809296,
-0.0070970128290355206,
-0.02562555857002735,
-0.010141860693693161... |
c6c6b7ca8fb28c6475b81b25ce4fe8ff1e0741cd | subsection | 517 | 1,121 | Products | Indeed, if H
is a closed subgroup of G and Y an orthogonal H-spectrum, thenL_m (G\ltimes _H Y) \ &\cong \ (G\times O(m))\ltimes _{H\times O(m)} (L_m Y)
\text{\quad and}\\
(G\ltimes _H Y)({\mathbb {R}}^m)\ &\cong \ (G\times O(m))\ltimes _{H\times O(m)} Y({\mathbb {R}}^m)\ .Since induction from H\times O(m) to G\times O(... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1839,
"openalex_id": "https://openalex.org/W2183692372",
"raw": "M. Stolz, Equivariant structure on smash powers of commutative ring spectra. PhD thesis, University of Bergen, 2011.",
"source_ref_id": "26ca4953153b3ac27dc6db... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02660767361521721,
-0.007555877789855003,
-0.036951713263988495,
0.008864139206707478,
0.028957204893231392,
-0.026455108076334,
-0.013372492045164108,
-0.013471661135554314,
-0.00048392327153123915,
-0.014059043489396572,
-0.023922495543956757,
0.0022141276858747005,
0.0255702193826437,
... |
952b4b775e1f97da20a8709812c08ef4b7097750 | subsection | 518 | 1,121 | Products | The theorem is stronger than the earlier result
of Mandell and May
because the class of G-flat of orthogonal G-spectra
is strictly larger than the cofibrant G-spectra
in the sense of .Since Stolz' thesis is not published, the notation and
level of generality in
is different from ours,
and the characterization of flat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 185,
"openalex_id": "https://openalex.org/W2075488415",
"raw": "M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.",
"source_ref_id": "1fe5cf0f78b8a5a... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.009300421923398972,
0.024078860878944397,
-0.03335639461874962,
0.0063477857038378716,
0.03775101527571678,
-0.050416067242622375,
0.011711359024047852,
0.01256586890667677,
0.004798986949026585,
0.01915017142891884,
0.004955392796546221,
-0.045899372547864914,
-0.003933032974600792,
0.... |
2ff20c5c0abf5847da27edba567460627b9ccf07 | subsection | 519 | 1,121 | Products | Since the K-action on B is free, the stabilizer group \Gamma
is the graph of a continuous homomorphism
\alpha :H\longrightarrow K from some closed subgroup H of G (namely the projection
of \Gamma to G). So we can rewrite( (G\times K)/\Gamma )_+ \triangleright _{K,W} C \ \cong \ G\ltimes _H ( \operatorname{sh}^{\alpha ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03850265592336655,
0.0424993634223938,
-0.021295445039868355,
-0.0009162304340861738,
0.020837808027863503,
-0.07895790040493011,
0.0026771854609251022,
0.029182083904743195,
0.020044567063450813,
0.04307904094457626,
-0.014148658141493797,
-0.030509235337376595,
0.04039422795176506,
0.... |
143808b8d59a665823f180b56b74d80ab3368d28 | subsection | 520 | 1,121 | Products | Since the input data is stable under passage to closed subgroups of G
(just restrict \alpha to such a subgroup), it is no loss of generality
to assume H=G.We can represent every class of \pi _k^G(C\wedge \alpha ^*({\mathbf {O}}({\mathbb {R}}^m,-)/K ))
by a based G-mapf \ : \ S^{V\oplus {\mathbb {R}}^{m+k}}\ \longrighta... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04418547451496124,
-0.0006598832551389933,
-0.044032901525497437,
0.018964968621730804,
0.01563885062932968,
-0.028577139601111412,
0.024137232452630997,
-0.003240293590351939,
0.010947195813059807,
0.0175307709723711,
-0.015745652839541435,
-0.030545346438884735,
0.03362734615802765,
0... |
da3f23d11ab4842a042b1620b62a027f434712c3 | subsection | 521 | 1,121 | Products | This space also has a commuting free right
action of O(V) by right translation.
Since O(V\oplus {\mathbb {R}}^m)/K is a smooth manifold
and the (G\times O(V))-action is smooth,
Illman's theorem
provides a finite (G\times O(V))-CW-structure on O(V\oplus {\mathbb {R}}^m)/K;
so O(V\oplus {\mathbb {R}}^m)/K_+ is cofibrant... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01456063",
"end": 382,
"openalex_id": "https://openalex.org/W2156908593",
"raw": "S. Illman, The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann. 262 (1983), 487–501.",
"source_ref_id": "1095cd... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.039060078561306,
0.028730519115924835,
-0.04452238604426384,
-0.01908756233751774,
0.029401864856481552,
-0.023619141429662704,
-0.00027249555569142103,
0.024732964113354683,
0.04333227500319481,
0.017668582499027252,
-0.012686897069215775,
-0.02070489339530468,
0.027647212147712708,
0.... |
59d2b358eb54a5e8937562a1320e49e214e58e64 | subsection | 522 | 1,121 | Products | Since the stabilization represents the same element as f, this shows that
\pi _k^G(C\wedge \alpha ^*({\mathbf {O}}({\mathbb {R}}^m,-)/K))=0.Step 3:
We let \Gamma be a closed subgroup of G\times O(m).
We claim that smashing with the orthogonal G-spectrumG_m\left( (G\times O(m))/\Gamma _+\right) \ = \ {\mathbf {O}}({\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.029955502599477768,
0.02029588259756565,
-0.03793651610612869,
-0.020494263619184494,
0.019593920558691025,
-0.05289137735962868,
-0.00602390943095088,
0.018510455265641212,
0.013680646196007729,
0.03589166700839996,
-0.01855623535811901,
-0.028933078050613403,
0.04303337633609772,
0.01... |
3cfecb0825d43e5446c67d7217ae2f51ad5d7f76 | subsection | 523 | 1,121 | Products | We claim that smashing with the orthogonal G-spectrum G_m A
preserves G-stably contractible orthogonal G-spectra.
A cofibrant based (G\times O(m))-space is equivariantly homotopy
equivalent to a based (G\times O(m))-CW-complex, so it is no loss of generality
to assume an equivariant CW-structure with skeleton filtratio... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.031334567815065384,
0.024805260822176933,
-0.03398900479078293,
-0.011113550513982773,
0.03984707593917847,
-0.05507194995880127,
-0.009374435991048813,
-0.0029175931122153997,
0.0022787735797464848,
0.014057840220630169,
-0.0016904877265915275,
-0.03139558807015419,
0.01671990565955639,
... |
941a82412b33530574698e35b7981f4dd99df034 | subsection | 524 | 1,121 | Products | X) is G-stably contractible,
where \operatorname{sk}^m \! X is the m-skeleton
in the sense of Construction REF .
The induction starts with m=-1, where there is nothing to show.
For m\ge 0 the morphism
j_m:\operatorname{sk}^{m-1}\! X\longrightarrow \operatorname{sk}^m\! X is an h-cofibration of orthogonal G-spectra,
hen... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01759050227701664,
-0.002559242770075798,
-0.07060611248016357,
0.027140939608216286,
0.031092319637537003,
-0.02070278488099575,
0.013593354262411594,
0.026393381878733635,
0.01272374577820301,
0.002835762919858098,
-0.013578098267316818,
-0.03682868555188179,
0.008757110685110092,
-0.... |
bf8ba26e54bf28c6bbdfeabf3ddd9b3a65bd3e29 | subsection | 525 | 1,121 | Products | We letf\ :\ S^{U\oplus {\mathbb {R}}^{m+k}}\ \longrightarrow \ X(U\oplus {\mathbb {R}}^m) \text{\quad and\quad }
g\ :\ S^{V\oplus {\mathbb {R}}^{n+l}}\ \longrightarrow \ Y(V\oplus {\mathbb {R}}^n)represent classes in \pi ^G_k(X) respectively \pi ^G_l(Y),
for suitable G-representations U and V.
The class [f]\times [g] i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04886506497859955,
-0.0038876875769346952,
-0.034550439566373825,
0.022372325882315636,
0.02840033918619156,
-0.03226131945848465,
0.0202663354575634,
0.014375668950378895,
0.016359571367502213,
0.020006902515888214,
-0.01252911239862442,
0.003345929319038987,
-0.009454062208533287,
0.0... |
5d6b6faee5de862cbaf33652302e9e03236ab929 | subsection | 526 | 1,121 | Products | (Restriction) For all x\in \pi _k^G ( X ) and y\in \pi _l^G( Y )
and all continuous homomorphisms \alpha :K\longrightarrow G the relation
\alpha ^*(x)\times \alpha ^*(y)\ = \ \alpha ^*(x\times y)
holds in \pi _{k+l}^K(\alpha ^*(X\wedge Y)).
(Transfer) Let H be a closed subgroup of G.
For all x\in \pi _k^G(X) and z\i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05996986851096153,
0.02250777557492256,
-0.01847926527261734,
0.04248247295618057,
-0.01134544424712658,
-0.04510710760951042,
0.03640918806195259,
0.011681153438985348,
0.04409997910261154,
0.009178594686090946,
-0.03836240619421005,
-0.02096656523644924,
0.010513801127672195,
0.075137... |
0e8136e1de6f4455b5b51ccc24fc124f8fbccda9 | subsection | 527 | 1,121 | Products | Indeed, iff \ : \ S^{U\oplus {\mathbb {R}}^{m+k}}\ \longrightarrow \ X(U\oplus {\mathbb {R}}^m) \text{\quad and\quad }
g \ : \ S^{V\oplus {\mathbb {R}}^{n+l}}\ \longrightarrow \ Y(V\oplus {\mathbb {R}}^n)represent classes in \pi _k^G(X) respectively \pi _l^G(Y), then
the left and right vertical composites in the diagra... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.028086891397833824,
-0.023494737222790718,
-0.028224198147654533,
-0.02607305534183979,
0.005145194940268993,
-0.009054627269506454,
-0.0004708005872089416,
0.03179417550563812,
-0.0029749830719083548,
0.008993602357804775,
0.009382638148963451,
0.015546193346381187,
0.020184114575386047,... |
62a5eda3e00c0200669b6b958a616368af26353e | subsection | 528 | 1,121 | Products | Since the two composites differ by conjugation by a G-equivariant linear isometry,
they represent the same class
by Proposition REF (ii).(vi) The following diagram of abelian groups
commutes by naturality of the pairing (REF ), and because
restriction from G to H is multiplicative:\hspace*{-19.91684pt}@C=6mm@R=6mm{
\p... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06585107743740082,
-0.0008477410883642733,
-0.006625540088862181,
-0.018308918923139572,
0.001138585852459073,
-0.01603556051850319,
0.018598809838294983,
0.02177235670387745,
0.022718315944075584,
0.032528843730688095,
-0.05041055753827095,
-0.008109325543045998,
0.019483741372823715,
... |
980c82af8855cf0765a4ed5edfbf47722a74604a | subsection | 529 | 1,121 | Products | Since the external transfer is inverse to the Wirthmüller isomorphism
(up to the effect of the involution S^{-\operatorname{Id}}:S^L\longrightarrow S^L),
we can read the diagram backwards and conclude that the upper part of the
following diagram commutes:@C=10mm@R=6mm{
\pi _k^G(X) \times \pi _l^H(Y\wedge S^L) [r]^-{\op... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1912,
"openalex_id": "",
"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 | [
-0.08667232096195221,
0.03555396571755409,
-0.02450629509985447,
0.022110598161816597,
0.0173344649374485,
-0.027634432539343834,
-0.01273381244391203,
0.013824845664203167,
0.032776787877082825,
0.027359766885638237,
-0.03994861617684364,
0.012482034973800182,
-0.010193153284490108,
0.037... |
012b43e5e8516fbc9cad9493f6b6c18ad096c95a | subsection | 530 | 1,121 | Products | Via the universal property of the smash product the data contained in the
multiplication morphism can be made more explicit: \mu :R\wedge R\longrightarrow R corresponds to
a collection of based continuous maps \mu _{V,W}:R(V)\wedge R(W)\longrightarrow R(V\oplus W)
that together form a bimorphism. The associativity and ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1368,
"openalex_id": "",
"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 | [
-0.06340816617012024,
0.017896480858325958,
-0.028622165322303772,
0.006949571426957846,
0.029781699180603027,
-0.05275876820087433,
-0.007918392308056355,
0.03065134771168232,
0.018094822764396667,
0.03518268093466759,
-0.031398940831422806,
-0.005057701375335455,
-0.004584733862429857,
0... |
3d44026a7ed084cc14d41ea8f32778f3125f5644 | subsection | 531 | 1,121 | Products | The multiplicative unit is the class of the unit map S^0\longrightarrow R(0).
If the multiplication of R is commutative, then the relation
x\cdot y\ = \ (-1)^{kl}\cdot y \cdot x
holds for all classes x\in \pi _k^G(R) and y\in \pi ^G_l(R).
(Restriction) The restriction maps
\operatorname{res}^G_H:\pi _*^G(R)\longrig... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1255,
"openalex_id": "",
"raw": "J. P. C. Greenlees, J. P. May, Localization and completion theorems for M{\\rm U}-module spectra. Ann. of Math. (2) 146 (1997), 509–544.",
"source_ref_id": "e5b776bc28c1d76efd5f4dda378412379c... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07440914958715439,
0.011742789298295975,
-0.03754030168056488,
0.04019559174776077,
0.021578043699264526,
-0.06500881910324097,
0.023394016548991203,
0.01457357220351696,
0.02000623382627964,
0.004417852498590946,
-0.047306884080171585,
-0.020601384341716766,
0.0031073756981641054,
0.00... |
c164cff6fec57d65cf5791be208d00168bff7adb | subsection | 532 | 1,121 | Global stable homotopy theory | In this chapter we embark on the investigation of
global stable homotopy theory.
In Section we specialize the equivariant
theory of the previous chapter to global stable homotopy types,
which we model by orthogonal spectra (with no additional action of any groups).
Section introduces the category of global functors,
th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05131997540593147,
-0.008169395849108696,
-0.03145713359117508,
0.012913900427520275,
0.00668959878385067,
-0.05601871386170387,
-0.002736480673775077,
0.02582780085504055,
0.04048847034573555,
0.01286050584167242,
-0.030816394835710526,
0.01771179959177971,
-0.01655237004160881,
0.0009... |
030784488863b48fafa3a64c19b2261ca2e6c137 | subsection | 533 | 1,121 | Orthogonal spectra as global homotopy types | In this section we specialize the equivariant stable homotopy
theory of Chapter to global stable homotopy types,
which we model by orthogonal spectra (with no additional group action).
Given an orthogonal spectrum X and a compact Lie group G,
we obtain an orthogonal G-spectrum by letting G act trivially
on the values o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04227130115032196,
0.0035518575459718704,
-0.040592655539512634,
0.02756027691066265,
0.025851113721728325,
-0.08478676527738571,
0.028445379808545113,
0.03287089616060257,
0.030642878264188766,
0.02069309912621975,
-0.04025692865252495,
-0.01587081328034401,
0.0008445676066912711,
-0.0... |
ec65d21db84ca9646b9f240e264e219573a11687 | subsection | 534 | 1,121 | Orthogonal spectra as global homotopy types | If Y=X_G arises from
an orthogonal spectrum, then this G-action is trivial.Remark 1.2 (Global homotopy types are split G-spectra)
Obviously, only very special orthogonal G-spectra Y are part of
a `global family', i.e., arise as X_G for an orthogonal spectrum X.
However, it is not a priori clear what the homotopical
si... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02707573026418686,
-0.007070367224514484,
-0.01987181417644024,
0.016544584184885025,
0.03803422674536705,
-0.07960935682058334,
-0.006746038794517517,
0.00943224411457777,
0.05442618578672409,
0.018849225714802742,
-0.044291865080595016,
-0.03299759328365326,
0.0051625510677695274,
-0.... |
79f16747c4b97a387d80e248167081b767d9bb2e | subsection | 535 | 1,121 | Orthogonal spectra as global homotopy types | In later sections we will also consider a relative notion of
global equivalence, the `{\mathcal {F}}-equivalences',
based on a global family {\mathcal {F}}global family
of compact Lie groups. There we require that
the induced map \pi _k^G (f): \pi _k^G (X)\longrightarrow \pi _k^G(Y) is an isomorphism
for all integers k... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04367998242378235,
0.018283933401107788,
-0.035224806517362595,
0.022847287356853485,
0.010759743861854076,
-0.05918622389435768,
0.01916913315653801,
-0.006967124994844198,
0.023778270930051804,
0.02733432874083519,
-0.03604895994067192,
-0.0007235165103338659,
0.022374162450432777,
0.... |
c97a7c5a10de65f51f3d2b95d6d5f8822974bb68 | subsection | 536 | 1,121 | Orthogonal spectra as global homotopy types | The cobase change of an h-cofibration that is also a global equivalence is
another global equivalence
by Corollary REF (i).
Every h-cofibration of orthogonal spectra is in particular levelwise
a closed embedding. So the class of h-cofibrations that are also global equivalences
is closed under sequential
composition by... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.038277845829725266,
0.00885213352739811,
-0.028784697875380516,
0.011576454155147076,
0.009050543420016766,
-0.05317385122179985,
0.03009725548326969,
-0.010584404692053795,
0.04783204570412636,
-0.00005115255044074729,
-0.02725846692919731,
-0.02116880938410759,
0.037942077964544296,
0... |
bc97441dc22616bb726e4fed69da96fcb8a7b848 | subsection | 537 | 1,121 | Orthogonal spectra as global homotopy types | The construction is clearly functorial in the orthogonal spectrum X;
moreover, \Omega ^\bullet has a left adjoint `unreduced suspension spectrum'
functor \Sigma ^\infty _+ that we discuss
in Construction REF below.If G acts on V by linear isometries, then the
G-fixed subspace of (\Omega ^\bullet X)(V) is the space
of G... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06866874545812607,
0.008759080432355404,
-0.004036196507513523,
0.02411036007106304,
0.033052556216716766,
-0.061191484332084656,
0.021897699683904648,
0.03534151613712311,
0.06579992175102234,
-0.0012760942336171865,
-0.031465545296669006,
-0.013413295149803162,
0.019150950014591217,
0... |
25ef63a9700d5b9c048d1410a5d8acb633cab84c | subsection | 538 | 1,121 | Orthogonal spectra as global homotopy types | Given an orthogonal space Y, we denote by
\operatorname{tel}_i Y(V_i) the mapping telescope of the sequence of G-spacesY(V_1) \ \longrightarrow \ Y(V_2) \ \longrightarrow \ \cdots \ \longrightarrow \ Y(V_i) \ \longrightarrow \ \cdots \ ;the maps in the sequence are induced by the inclusions,
so they are G-equivariant, ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07190226763486862,
0.008980153128504753,
-0.018616460263729095,
0.024720218032598495,
0.05233972147107124,
-0.022156639024615288,
-0.015396728180348873,
-0.0054094549268484116,
0.04141399636864662,
0.035096604377031326,
-0.026688680052757263,
-0.021759895607829094,
0.005382751114666462,
... |
f8357c8b2fa7a5e88fee01afcaf497319f3b9121 | subsection | 539 | 1,121 | Orthogonal spectra as global homotopy types | For every compact based G-space A the canonical map\operatorname{colim}_{n\ge 1}\, [A, S^{V_j}\wedge \operatorname{tel}_{[0,n]} Y(V_i)_+ ]^G \ \longrightarrow \ [A, S^{V_j}\wedge \operatorname{tel}_i Y(V_i)_+ ]^Gis thus bijective. Indeed, every continuous map from the compact spaces A
and A\wedge [0,1]_+ to S^{V_j}\wed... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06648489832878113,
0.00863876286894083,
-0.0173843652009964,
-0.026572590693831444,
0.03757709264755249,
-0.049604203552007675,
-0.010966344736516476,
-0.013568657450377941,
0.006368416827172041,
0.03602028265595436,
-0.0017208843491971493,
-0.025809448212385178,
-0.026847321540117264,
... |
7a44dc02cc0a481e323908395a7dcdcd10145334 | subsection | 540 | 1,121 | Orthogonal spectra as global homotopy types | For negative values of k, the argument is similar:
we insert {\mathbb {R}}^{-k} into the second variable
of the sets of equivariant homotopy classes.Corollary 1.9
The unreduced suspension spectrum functor
\Sigma ^\infty _+ takes global equivalences of orthogonal spaces
to global equivalences of orthogonal spectra.Let ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.08076601475477219,
0.004681590013206005,
-0.002226330805569887,
0.00010748914064606652,
0.03867367282509804,
-0.07618743926286697,
0.03903995826840401,
0.020328860729932785,
0.06867858022451401,
0.05070005729794502,
-0.019840478897094727,
0.0014784972881898284,
-0.009752359241247177,
0.... |
ba3324a2cfcd0cdfcab3209b13e6f3719490b4ad | subsection | 541 | 1,121 | Orthogonal spectra as global homotopy types | Then the suspension spectrum \Sigma ^\infty _+ Y is globally connective.
Moreover, for every compact Lie group K the equivariant homotopy group
\pi _0^K(\Sigma ^\infty _+ Y) is a free abelian group with a basis
given by the elements\operatorname{tr}_L^K(\sigma ^L(x)) \ ,where L runs through all conjugacy classes of clo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06390541046857834,
0.01306947972625494,
-0.006744553800672293,
0.009292835369706154,
0.04269516095519066,
-0.04760861396789551,
0.04907349497079849,
-0.006958181969821453,
0.03640838339924812,
0.04568595811724663,
-0.009048688225448132,
0.02322445809841156,
0.0063783335499465466,
0.0466... |
03e8b87e8f4a6e9fad9e37b6705286ade5af4ca1 | subsection | 542 | 1,121 | Orthogonal spectra as global homotopy types | The stable tautological classstable tautological classtautological class!stable|seestable tautological classe_{G,V} - stable tautological class in \pi _0^G(\Sigma ^\infty _+ {\mathbf {L}}_{G,V}) ise_{G,V}\ = \ \sigma ^G(u_{G,V})\ \in \ \pi _0^G(\Sigma ^\infty _+ {\mathbf {L}}_{G,V}) \ .Explicitly, e_{G,V} is the homoto... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05772082135081291,
0.0032088321167975664,
-0.003029503859579563,
-0.00439545139670372,
0.0613531768321991,
-0.04636590555310249,
0.016055606305599213,
0.0369950495660305,
0.02916565164923668,
0.015239090658724308,
0.015994559973478317,
-0.011003890074789524,
-0.005116580054163933,
0.006... |
25e69786d6a5841b61a463b86fe4ff410e18416d | subsection | 543 | 1,121 | Orthogonal spectra as global homotopy types | Proposition REF thus says that
\pi _0^K(\Sigma ^\infty _+ {\mathbf {L}}_{G,V})
is a free abelian group with a basis given by the elements\operatorname{tr}_L^K(\sigma ^L(\alpha ^*(u_{G,V})))\ = \ \operatorname{tr}_L^K(\alpha ^*(\sigma ^G(u_{G,V})))\ = \ \operatorname{tr}_L^K(\alpha ^*(e_{G,V}))where L runs through all c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03973936662077904,
0.011262538842856884,
-0.0071840654127299786,
-0.01352115161716938,
0.030857525765895844,
-0.04691198468208313,
0.06055522337555885,
0.027652738615870476,
0.006836879998445511,
-0.0038934352342039347,
-0.03726710006594658,
0.02391381934285164,
-0.0037465491332113743,
... |
0cd0ee92570dd3b617f9bafa27341ca290891e12 | subsection | 544 | 1,121 | Orthogonal spectra as global homotopy types | Moreover p_H:H\longrightarrow e denotes the unique homomorphism to the trivial group
and 1=\operatorname{res}^{A_3}_e(e_{A_3}) is the restriction of the class e_{A_3} to the
trivial group.Now we discuss how multiplicative features related to the smash
product and the homotopy group pairings work out for global homotopy... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06703450530767441,
0.012675809673964977,
0.006486718077212572,
-0.013133696280419827,
0.02240588702261448,
-0.02123064547777176,
0.011744774878025055,
0.058700982481241226,
0.03983607888221741,
0.015018659643828869,
-0.07112495601177216,
0.014858399517834187,
-0.01921594701707363,
-0.00... |
3448e8968d4eba7cb61d3917c00fe8435cb76d43 | subsection | 545 | 1,121 | Orthogonal spectra as global homotopy types | The bimorphism corresponding to the induced product on \Omega ^\bullet R
thus has as (V,W)-component the composite\operatorname{map}_*(S^V, R(V)) \times \operatorname{map}_*(S^W, R(W)) \ \xrightarrow{} \ &\operatorname{map}_*(S^{V\oplus W}, R(V)\wedge R(W)) \\
\xrightarrow{} \ &\operatorname{map}_*(S^{V\oplus W}, R(V\o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.054043788462877274,
0.006179484538733959,
-0.007587033789604902,
0.0019778164569288492,
0.014144154265522957,
-0.03375066816806793,
0.008163022808730602,
0.036558136343955994,
0.07165150344371796,
0.031202582642436028,
-0.027739020064473152,
-0.00780827458947897,
-0.0025633417535573244,
... |
ea888372e923de5ad8446f4f6151af755784dff0 | subsection | 546 | 1,121 | Orthogonal spectra as global homotopy types | We omit the verification that this morphism is indeed inverse to
the morphism (REF ).Construction 1.19 (Orthogonal ring spectra from orthogonal monoid spaces)
The suspension spectrum \Sigma ^\infty _+ M
of an orthogonal monoid space M
becomes an orthogonal ring spectrum via the multiplication map(\Sigma ^\infty _+ M )... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06514047086238861,
0.010042743757367134,
-0.017185606062412262,
0.022206978872418404,
0.033577561378479004,
-0.07338223606348038,
-0.005906598176807165,
0.042857177555561066,
0.03449331223964691,
0.007459560409188271,
-0.05161786824464798,
-0.028052525594830513,
-0.015361734665930271,
0... |
4025ea41e3e103379231f68fa5d71edaa3532e41 | subsection | 547 | 1,121 | Orthogonal spectra as global homotopy types | We refer
to Remark REF below for more details.Theorem 1.22
Let G,K and L be compact Lie groups and X, Y and Z orthogonal spectra.(Biadditivity)
The product \boxtimes :\pi _k^G(X) \times \pi _l^K (Y) \longrightarrow \pi _{k+l}^{G\times K}(X\wedge Y)
is biadditive.
(Unitality)
Let 1\in \pi _0^e({\mathbb {S}}) denote th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06242155283689499,
0.01206633634865284,
-0.017390169203281403,
0.0029555659275501966,
-0.0037468948867172003,
-0.053695958107709885,
0.008321348577737808,
0.01587997004389763,
0.011852772906422615,
0.043292365968227386,
-0.03362099081277847,
-0.031912483274936676,
-0.018625786527991295,
... |
9d4cd872466a07efdd622f414484f740fa151977 | subsection | 548 | 1,121 | Orthogonal spectra as global homotopy types | For part (vi) we start with two special cases, namely L=K respectively H=G.
The two proofs are analogous, so we only treat the case L=K:\operatorname{tr}_{H\times K}^{G\times K}(x\boxtimes y)\ &= \ \operatorname{tr}_{H\times K}^{G\times K}( p_H^*(x)\times p_K^*(y))\ = \ \operatorname{tr}_{H\times K}^{G\times K}(p_H^*(x... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.027609912678599358,
0.014751233160495758,
-0.009958799928426743,
0.014964908361434937,
0.015354103408753872,
-0.04282665252685547,
0.043467678129673004,
0.01858975924551487,
0.022893792018294334,
0.028601977974176407,
-0.01388890016824007,
-0.01295788586139679,
0.01057693175971508,
0.04... |
7e302eabe12259fbe9e1262a634bdfc3351c8c11 | subsection | 549 | 1,121 | Orthogonal spectra as global homotopy types | When X is an orthogonal spectrum, representing a global homotopy type,
then \alpha ^*(X_G)=X_K, and the inflation maps become homomorphismsinflation map!of geometric fixed point homotopy groups\alpha ^* \ : \ \Phi ^G_k (X) \ \longrightarrow \ \Phi ^K_k (X) \ .These inflation maps between the geometric fixed point homot... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1723,
"openalex_id": "https://openalex.org/W1507692549",
"raw": "G. M. Kelly, Basic concepts of enriched category theory. Reprint of the 1982 original. Repr. Theory Appl. Categ. No. 10 (2005), vi+137 pp.",
"source_ref_id": "... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03995485603809357,
-0.004494539462029934,
-0.03096577525138855,
0.014925839379429817,
0.037085674703121185,
-0.09913931787014008,
0.006528147030621767,
0.03265981376171112,
0.056589990854263306,
0.016116242855787277,
-0.02002321183681488,
-0.012896046973764896,
0.011675119400024414,
-0.... |
0b2ee4718be32204ee4a5bfbaaedfc01ea4aa720 | subsection | 550 | 1,121 | Orthogonal spectra as global homotopy types | So one should think of the semifree orthogonal spectrum F_{G,V}=F_{G,V} S^0
as the `global Thom spectrum' associated to a `virtual global vector bundle',
namely the negative of the vector bundle over B_{\operatorname{gl}}G associated to the
G-representation V.The special case G=O(m) of the orthogonal group
with V=\nu _... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s11511-009-0036-9",
"end": 1069,
"openalex_id": "https://openalex.org/W2019538266",
"raw": "S. Galatius, U. Tillmann, I. Madsen, M. Weiss, The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195–239.",
"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03637247532606125,
0.000985977821983397,
-0.017148766666650772,
-0.0008019413216970861,
0.05037831887602806,
-0.07774921506643295,
-0.015638332813978195,
0.03463318571448326,
0.04427555203437805,
0.04442812129855156,
-0.022534456104040146,
0.0035853739827871323,
0.026836905628442764,
-0... |
85d0615895540ff7db2dae7c37910fef2970d77a | subsection | 551 | 1,121 | Orthogonal spectra as global homotopy types | Indeed, a morphism(F_{G,V}A)\wedge (F_{K,W}B)\ \longrightarrow \ F_{G\times K,V\oplus W}(A\wedge B)is obtained by the universal property (REF )
from the bimorphism with (U,U^{\prime })-component(F_{G,V}A)(U)\wedge (F_{K,W}B)(U^{\prime }) \ = \ ({\mathbf {O}}(V,U)&\wedge _G A)\wedge ({\mathbf {O}}(W,U^{\prime })\wedge _... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03368379920721054,
0.020411651581525803,
-0.013218755833804607,
-0.014195098541676998,
0.010693996213376522,
-0.05138000100851059,
0.006964064668864012,
0.03392788767814636,
0.04326415807008743,
0.0214642696082592,
-0.020518438890576363,
0.000500089256092906,
0.029793689027428627,
0.024... |
04adcd10dc139704c519fc8bd5cc0c8b419a1a17 | subsection | 552 | 1,121 | Orthogonal spectra as global homotopy types | We will now prove a generalization of the fact that \Sigma ^\infty _+ \rho _{V,W}/G
is a global equivalence for these global Thom spectra.
Given G-representations V and W,
we define a restriction morphism of orthogonal spectra\index {symbol}{\lambda _{G,V,W} - {fundamental global equivalence of orthogonal spectra}}
\la... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.008596057072281837,
0.023324526846408844,
0.005670956801623106,
0.007741790264844894,
0.002949889050796628,
-0.0816434696316719,
-0.013843693770468235,
-0.0023587672039866447,
0.0513170100748539,
0.028480635955929756,
-0.004175990354269743,
0.01623106375336647,
0.026711083948612213,
0.0... |
5037bc4acf43e49aad21d54a39e943106ac480dd | subsection | 553 | 1,121 | Orthogonal spectra as global homotopy types | In a first step we produce a K-representation U^{\prime }\in s({\mathcal {U}}_K)
with U\subseteq U^{\prime } and
a continuous (K\times G)-equivariant maph\ :\ {\mathbf {L}}(W,U) \ \longrightarrow \ {\mathbf {L}}(V\oplus W,U^{\prime })such that the lower right triangle in the diagram\begin{aligned}
@C=15mm{
{\mathbf {L}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.059690602123737335,
0.03481951728463173,
-0.026366401463747025,
-0.025557709857821465,
-0.009574605152010918,
-0.029418068006634712,
0.029784267768263817,
-0.0041807834059000015,
0.007770306896418333,
0.016417967155575752,
0.0006837641121819615,
0.009544088505208492,
0.045988619327545166,... |
5e8b5443ecc244b29a82971ca8c539562bbfdeaa | subsection | 554 | 1,121 | Orthogonal spectra as global homotopy types | We conclude that the restriction map
\rho ({\mathcal {U}}_K):{\mathbf {L}}(V\oplus W,{\mathcal {U}}_K) \longrightarrow {\mathbf {L}}(W,{\mathcal {U}}_K) is both
a (K\times G)-weak equivalence and a (K\times G)-fibration.Since {\mathbf {L}}(W,U) is cofibrant as a (K\times G)-space
(by Proposition REF (ii)),
the (K\time... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.010026096366345882,
0.005489936098456383,
-0.027926567941904068,
-0.017824171110987663,
-0.0072868503630161285,
-0.025484900921583176,
0.0192586500197649,
0.02788078784942627,
-0.0026858339551836252,
0.05469334498047829,
-0.002661035628989339,
-0.011735263280570507,
0.030765006318688393,
... |
cf7fb8021a67a502dd8d0e91ab17e5ecf48b6e51 | subsection | 555 | 1,121 | Orthogonal spectra as global homotopy types | So after increasing U^{\prime }, if necessary, we have proved the claim
subsumed in the diagram (REF ).Now we lift the data produced in the first step to the Thom spaces
of the orthogonal complement bundles. The diagram (REF )
is covered by morphisms of (K\times G)-vector bundles:\begin{aligned}
@C=12mm{
(U^{\prime }-U... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.029122743755578995,
0.002408461645245552,
-0.024896971881389618,
-0.009740634821355343,
-0.020259300246834755,
-0.04469860717654228,
0.021937405690550804,
0.0001259770360775292,
0.013226516544818878,
0.02122039720416069,
-0.020945798605680466,
0.009565196931362152,
0.022715436294674873,
... |
d61d9f53fc7fa6d4df0a731305fce7c75c9189c2 | subsection | 556 | 1,121 | Orthogonal spectra as global homotopy types | In particular, the square\begin{aligned}
{
\xi (V\oplus W, U^{\prime })\times V [r]^-{\bar{\rho }(U^{\prime })} [d] & \xi (W,U^{\prime }) [d]\\
{\mathbf {L}}(V\oplus W, U^{\prime }) [r]_-{\rho (U^{\prime })} & {\mathbf {L}}(W,U^{\prime })}
\end{aligned}is a pullback; so the composite(U^{\prime }-U)\times \xi (W,U) \ \l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0308561772108078,
0.007015889044851065,
-0.049260009080171585,
0.006779355462640524,
-0.018861640244722366,
0.006111720576882362,
0.041202612221241,
0.0012656450271606445,
0.007668263744562864,
0.01851065456867218,
-0.021455878391861916,
-0.007221902254968882,
0.043369561433792114,
0.01... |
fee088f8ca39b2209277f24c67d42a8d2e0a4398 | subsection | 557 | 1,121 | Orthogonal spectra as global homotopy types | Again because the square (REF ) is a pullback, the composite\xi (V\oplus W,U)\times V \times (U^{\prime }-U) \times [0,1]\ \longrightarrow \ {\mathbf {L}}(V\oplus W,U) \times [0,1]\ \xrightarrow{} \ {\mathbf {L}}(V\oplus W, U^{\prime })and the map of total spaces(U^{\prime }-U) \times \xi (V\oplus W,U)\times V\times [0... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01600978896021843,
0.005772834178060293,
-0.03641502186655998,
0.011843276210129261,
-0.006692366674542427,
-0.0034282163251191378,
0.027517301961779594,
0.01997789740562439,
0.025090651586651802,
0.019947374239563942,
-0.03128700703382492,
-0.013392364606261253,
0.02161092683672905,
0.... |
86f98d0d2c8ef2128f5762a1944606d7ff986901 | subsection | 558 | 1,121 | Orthogonal spectra as global homotopy types | We conclude that \bar{H} makes the upper left triangle in (REF )
commute up to equivariant homotopy of vector bundle maps.Passing to Thom spaces in (REF )
gives a diagram of (K\times G)-equivariant based maps:@C=20mm{
S^{U^{\prime }-U}\wedge {\mathbf {O}}(V\oplus W,U)\wedge S^V
[r]^{\sigma _{U,U^{\prime }-U}\wedge S^V}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05765362083911896,
0.01837346702814102,
-0.025316929444670677,
-0.009224885143339634,
-0.01671008951961994,
-0.0283384807407856,
0.013200207613408566,
-0.012086201459169388,
-0.0038856680039316416,
0.03168049827218056,
-0.012711876071989536,
0.025347450748085976,
0.0377541184425354,
0.0... |
9ce5b2135e0cc8baacb1ff411adc76f15acae468 | subsection | 559 | 1,121 | Orthogonal spectra as global homotopy types | For A=S^0 and \bar{U}={\mathbb {R}}^k this shows that \pi _{-k}^G(\lambda _{G,V,W})
is an isomorphism.
So \lambda _{G,V,W} is a global equivalence. | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03332488611340523,
0.035430580377578735,
-0.002626394620165229,
0.008964454755187035,
-0.0256040096282959,
-0.03964196518063545,
0.01580795831978321,
-0.011611830443143845,
0.03973351791501045,
0.02297952212393284,
-0.001576408976688981,
0.019439516589045525,
0.03417937085032463,
0.0439... |
4bed69e78064f52ec6451fed06da84ad63449db0 | subsection | 560 | 1,121 | Global functors | This section is devoted to the category of global functors,
the natural home of the collection of equivariant
homotopy groups of a global stable homotopy type.
The category of global functors is a symmetric monoidal abelian category
with enough injectives and projectives that plays the same role
for global homotopy the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.048663511872291565,
-0.007417752873152494,
-0.026620008051395416,
0.017116131260991096,
0.012005699798464775,
-0.03252370283007622,
-0.020701061934232712,
0.019816270098090172,
0.029472697526216507,
0.038259588181972504,
-0.035818785429000854,
0.02823704108595848,
-0.008641968481242657,
... |
c5b37e00c9428386dae58e1eec6cea533ee01d36 | subsection | 561 | 1,121 | Global functors | The functor \pi _0^K is abelian group valued,
so the set {\mathbf {A}}(G,K) is an abelian group under objectwise addition of transformations.
Proposition REF ,
applied to the category of orthogonal spectra,
shows that set valued natural transformations between
the two reduced additive functors \pi _0^G and \pi _0^K are... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2071,
"openalex_id": "",
"raw": "B. Stenström, Rings of quotients. An introduction to methods of ring theory. Die Grundlehren der Mathematischen Wissenschaften, Band 217. Springer-Verlag, New York-Heidelberg, 1975. viii+309 pp.",
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.049845144152641296,
0.013453304767608643,
-0.020679783076047897,
0.02054242603480816,
0.016101231798529625,
-0.05787286534905434,
0.004727352410554886,
0.004109248053282499,
0.037971436977386475,
0.033331841230392456,
-0.03482750058174133,
0.02902800403535366,
0.005578199401497841,
0.02... |
c02df4474ce52d320ce9b47a92d8c3c1c37de09b | subsection | 562 | 1,121 | Global functors | This calculation has two ingredients:
We identify natural transformations
from \pi _0^G to \pi _0^K with the group \pi _0^K(\Sigma ^\infty _+ B_{\operatorname{gl}}G),
and then we exploit the explicit calculation of the latter group
in Corollary REF .Proposition 2.5 global classifying space
Let G and K be compact Lie gr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07884924113750458,
0.012732138857245445,
0.015165663324296474,
0.008429606445133686,
0.026928970590233803,
-0.0560549758374691,
0.011000446043908596,
0.01507411990314722,
0.04134703055024147,
0.038417644798755646,
-0.012732138857245445,
-0.011145389638841152,
0.011236933059990406,
0.016... |
d12a65b4e97c2b7362a49c641aa59863e4261d51 | subsection | 563 | 1,121 | Global functors | For a pair (L,\alpha ) consisting of a closed subgroup L of K
and a continuous group homomorphism \alpha :L\longrightarrow G we define[L,\alpha ]\ = \ \operatorname{tr}_L^K\circ \alpha ^*\ \in \ {\mathbf {A}}(G,K) \ ,the natural transformation whose value at X is the composite\pi _0^G (X) \ \xrightarrow{}\ \pi _0^L(X) ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06675057858228683,
0.04234451800584793,
-0.012775047682225704,
-0.0004995615454390645,
-0.012500479817390442,
-0.014674144797027111,
0.01729779690504074,
0.012317433953285217,
0.007043436635285616,
0.0045074946247041225,
-0.014300426468253136,
-0.018884189426898956,
0.014346187934279442,
... |
390a49487378f65ca33ca17344cc456616a29ea2 | subsection | 564 | 1,121 | Global functors | By Proposition REF the composite{\mathbb {Z}}\lbrace [L,\alpha ]\ |\ \ |W_KL| <\infty ,\, \alpha :L\longrightarrow G\rbrace
\ \longrightarrow \ \operatorname{Nat}(\pi _0^G,\pi _0^K)
\ \xrightarrow{} \ \pi _0^K(\Sigma ^\infty _+ {\mathbf {L}}_{G,V})is an isomorphism, where the first map takes [L,\alpha ] to \operatorna... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.052467599511146545,
-0.022342223674058914,
-0.00620363000780344,
0.04367721825838089,
0.016192007809877396,
-0.0058259181678295135,
-0.007680139504373074,
0.016054658219218254,
0.027943041175603867,
0.02354785054922104,
-0.055855561047792435,
0.004475312307476997,
0.0021670737769454718,
... |
79cdc16fceb75db5a6792552403c1877cd916ea5 | subsection | 565 | 1,121 | Global functors | Then \alpha and \alpha ^{\prime } belong to the same path component of the space \hom (K,G)
of continuous homomorphisms, so they are conjugate by an element of G,
compare Proposition REF .This explicit description allows us to relate our notion of global functor
to other `global' versions of Mackey functors.
For exampl... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02566643",
"end": 453,
"openalex_id": "https://openalex.org/W2043632576",
"raw": "P. Symonds, A splitting principle for group representations. Comment. Math. Helv. 66 (1991), no. 2, 169–184.",
"source_ref_id": "091a2ace1d82f... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.059593841433525085,
-0.0010546400444582105,
0.005511591210961342,
-0.014433121308684349,
0.03860021010041237,
-0.05724425986409187,
0.012236113660037518,
-0.007708598859608173,
0.025738557800650597,
0.042017776519060135,
0.0021779367234557867,
0.00856680516153574,
-0.02808813564479351,
... |
d984e9acc56734691f622cd0f0baf2be7f2fa852 | subsection | 566 | 1,121 | Global functors | By Proposition REF ,
the action on the unit 1\in \pi _0({\mathbb {S}}) is an isomorphism of global functors{\mathbb {A}}\ = \ {\mathbf {A}}(e,-) \ \longrightarrow \ {\underline{\pi }}_0 ( {\mathbb {S}})from the Burnside ring global functor {\mathbb {A}} to the 0-th homotopy
global functor of the sphere spectrum.Example... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0085965",
"end": 1585,
"openalex_id": "https://openalex.org/W205144100",
"raw": "T. tom Dieck, Transformation groups and representation theory. Lecture Notes in Mathematics, Vol. 766. Springer-Verlag, Berlin, 1979. viii+309 pp.",... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.062004003673791885,
0.0014370083808898926,
0.003758916398510337,
0.005465780850499868,
0.009588955901563168,
-0.05098852887749672,
0.008582000620663166,
0.03487725183367729,
-0.003888599807396531,
0.04711327701807022,
-0.01690463349223137,
0.02471616305410862,
-0.00972626730799675,
0.00... |
ff6a781191ac5657742715314072cca795e079db | subsection | 567 | 1,121 | Global functors | Moreover, the class of a general compact G-ENR X
is expressed in terms of this basis by the formula[X]\ = \ {\sum }_{(H)}\ \chi ^{\text{AS}}(G\backslash X_{(H)})\cdot [G/H]\ ;the sum is over conjugacy classes of closed subgroups,
X_{(H)} is the orbit type subspace, and \chi ^{\text{AS}}
is the Euler characteristic base... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0624820776283741,
-0.03444446623325348,
-0.018640894442796707,
0.011738271452486515,
0.0008637811988592148,
-0.033895306289196014,
-0.013057778589427471,
0.017435794696211815,
-0.005285654217004776,
0.01122724823653698,
-0.0346275195479393,
-0.001662731054238975,
0.025688432157039642,
0... |
c364f4cbc777b095e09924a3eb7f5cf6e174f8cb | subsection | 568 | 1,121 | Global functors | The transfer maps \operatorname{tr}_H^G:\mathbf {RU}(H)\longrightarrow \mathbf {RU}(G) along a closed subgroup inclusion
H\le G are given by the smooth induction
of Segal .
If H is a subgroup of finite index of G, then this induction sends the class
of an H-representation to the induced G-representation \operatorname{m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01394272",
"end": 519,
"openalex_id": "https://openalex.org/W4230690795",
"raw": "V. P. Snaith, Explicit Brauer induction. Invent. Math. 94 (1988), no. 3, 455–478.",
"source_ref_id": "979e93cae9b68dba5d84e67fa6e77495efb7b0fa... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0989249125123024,
-0.02326657809317112,
-0.005702218506485224,
0.03185613825917244,
0.03612803667783737,
-0.034907493740320206,
-0.014097257517278194,
0.023876847699284554,
0.014623615890741348,
0.03080342337489128,
-0.014776183292269707,
0.014402393251657486,
0.0476621575653553,
0.0223... |
5bf014fdb475010e0dcf70fc82199194fc813136 | subsection | 569 | 1,121 | Global functors | Generating operations can be displayed as
follows:@R=1mm@!C=4mm{
&&&&&&&\\
F(C_3) [rr]_{\operatorname{res}} &&
F(e) @<.9ex>@/^1pc/[ll]^{p^*} @/_1pc/[ll]_{\operatorname{tr}}
&& F(C_3)[rr]_{\operatorname{res}} &&
F(e) @/_1pc/[ll]_{\operatorname{tr}} \\
@<2ex>@(dl,ul)[uu]^{\alpha ^*} &&&&&& @<-2ex>@(dr,ur)[uu]_{\tau } \\
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04365750402212143,
0.010960138402879238,
-0.04695240780711174,
0.030432116240262985,
0.004656342789530754,
-0.03008127026259899,
-0.0049042231403291225,
0.04384055361151695,
-0.008946584537625313,
0.05180324241518974,
-0.03651854023337364,
0.007489809300750494,
0.015986395999789238,
0.0... |
9c39bab47337a3f6405c2cbe62a2ad65431a6790 | subsection | 570 | 1,121 | Global functors | Given operations \tau \in {\mathbf {A}}(G,K) and \psi \in {\mathbf {A}}(G^{\prime },K^{\prime }),
there is a unique operation\tau \times \psi \ \in \ {\mathbf {A}}(G\times G^{\prime },\, K\times K^{\prime })with the following property:
for all orthogonal spectra X and Y and all classes x\in \pi _0^G(X)
and y\in \pi _0^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.029765909537672997,
0.004535067360848188,
-0.013433563522994518,
-0.0037359953857958317,
0.015836501494050026,
-0.0006336316582746804,
0.008978594094514847,
0.01733166165649891,
0.05931822210550308,
0.05187293142080307,
-0.008292040787637234,
-0.02189342863857746,
0.04207810387015343,
0... |
80a5c606df2e1c5a1e0e3b5d74ad722f776fcc1e | subsection | 571 | 1,121 | Global functors | There is thus a unique operation \tau \times \psi \in {\mathbf {A}}(G\times G^{\prime },K\times K^{\prime })
that satisfies(\tau \times \psi )(e_{G,V}\boxtimes e_{G^{\prime },V^{\prime }})\ = \ \tau (e_{G,V})\boxtimes \psi (e_{G^{\prime },V^{\prime }})in \pi _0^{K\times K^{\prime }}(\Sigma ^\infty _+ {\mathbf {L}}_{G,V... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.005926952231675386,
0.020519306883215904,
-0.012830288149416447,
0.006926219444721937,
-0.013318479061126709,
0.01155641209334135,
0.017635924741625786,
0.020763402804732323,
0.04408981278538704,
0.06419720500707626,
0.010038441978394985,
-0.03151887655258179,
0.04933787137269974,
0.024... |
f2f4a3d0924ec2d71d9457a4c6b71e0f9ee1439d | subsection | 572 | 1,121 | Global functors | Naturality then yields(\Sigma ^\infty _+\rho _{G,V,W}\wedge \Sigma ^\infty _+&\rho _{G^{\prime },V^{\prime },W^{\prime }})_*( (\tau \times \psi )(e_{G,V\oplus W}\boxtimes e_{G^{\prime },V^{\prime }\oplus W^{\prime }}))\\
&= \ (\tau \times \psi )( (\Sigma ^\infty _+\rho _{G,V,W})_*(e_{G,V\oplus W})\boxtimes (\Sigma ^\in... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.017321400344371796,
0.01837441883981228,
-0.007035388145595789,
-0.021747132763266563,
-0.027714241296052933,
-0.033818699419498444,
0.01961057260632515,
0.029545579105615616,
0.04721798375248909,
0.017779234796762466,
-0.014337847009301186,
0.0019820414017885923,
-0.010263120755553246,
... |
25aff8a4f7c1ca247bcf759471c41da915734fd0 | subsection | 573 | 1,121 | Global functors | So in particular it induces an isomorphism on \pi _0^{K\times K^{\prime }},
and we can conclude that(\tau \times \psi )(e_{G,V\oplus W}\boxtimes e_{G^{\prime },V^{\prime }\oplus W^{\prime }})\ = \ \tau (e_{G,V\oplus W})\boxtimes \psi (e_{G^{\prime },V^{\prime }\oplus W^{\prime }}) \ .Now the relation (REF ) follows by ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0160067081451416,
0.029587233439087868,
-0.02284274809062481,
0.008758675307035446,
-0.024887455627322197,
-0.0024643312208354473,
0.005981534253805876,
0.02601662278175354,
0.04193177819252014,
0.05221635475754738,
-0.04278628155589104,
-0.014824134297668934,
0.028015553951263428,
0.01... |
666cfc717b15c3bf7e30423c884d920769128808 | subsection | 574 | 1,121 | Global functors | Using parts (v) and (vi) of Theorem REF we deduce[L\times L^{\prime },\alpha \times \alpha ^{\prime }](x\boxtimes y)\ &= \ \operatorname{tr}_{L\times L^{\prime }}^{K\times K^{\prime }}( (\alpha \times \alpha ^{\prime })^*(x\boxtimes y))\\
&= \ \operatorname{tr}_{L\times L^{\prime }}^{K\times K^{\prime }}( \alpha ^*(x)\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03298100829124451,
0.019449947401881218,
-0.009709718637168407,
-0.001451118616387248,
0.00945038627833128,
-0.01946520246565342,
-0.023202642798423767,
0.05278182029724121,
0.012768318876624107,
0.036154020577669144,
-0.05015798285603523,
-0.04674089327454567,
-0.011677595786750317,
0.... |
976550945425064b824308f0cfc43cee70994aca | subsection | 575 | 1,121 | Global functors | The relation((\tau \times \psi )+(\tau ^{\prime }\times \psi ))(x\boxtimes y)
&= \ (\tau \times \psi )(x\boxtimes y)+(\tau ^{\prime }\times \psi )(x\boxtimes y)\\
&= \ (\tau (x)\boxtimes \psi (y))+(\tau ^{\prime }(x)\boxtimes \psi (y)) \\
&= \ (\tau (x)+\tau ^{\prime }(x))\boxtimes \psi (y) \ = \ (\tau +\tau ^{\prime }... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01646655797958374,
0.022876152768731117,
-0.035252779722213745,
0.02023600973188877,
-0.02983514405786991,
0.016405513510107994,
0.006558390334248543,
0.0326278954744339,
0.04578283056616783,
0.07136017084121704,
-0.032536331564188004,
-0.07972316443920135,
0.017595866695046425,
0.03085... |
1794511933151d2b88d4d1e7fba89fb7bbf78ee0 | subsection | 576 | 1,121 | Global functors | The relation(\tau \times (\psi \times \kappa ))(a_{G,G^{\prime },G^{\prime \prime }}^*((x\boxtimes y)\boxtimes z)) \ &= \ ((\tau \times (\psi \times \kappa ))(x\boxtimes (y\boxtimes z))) \\
&= \ \tau (x)\boxtimes (\psi (y)\boxtimes \kappa (z))\\
&= \ a_{K,K^{\prime },K^{\prime \prime }}^*( (\tau (x)\boxtimes \psi (y))\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0022-4049(93)90030-w",
"end": 1914,
"openalex_id": "https://openalex.org/W2008207368",
"raw": "P. Webb, Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01872318610548973,
0.01110878586769104,
-0.030076121911406517,
0.010940933600068092,
-0.0016947383992373943,
-0.052461545914411545,
0.03134264424443245,
0.00235756509937346,
0.02285846322774887,
0.06186128780245781,
-0.046296779066324234,
-0.015808656811714172,
0.02975567616522312,
0.04... |
d54a5d78a68e5ba6476112999efbad5755fa17bb | subsection | 577 | 1,121 | Global functors | Composition\circ \ : \ {\mathbb {A}}^\text{c}(K,L) \times {\mathbb {A}}^\text{c}(G,K) \ \longrightarrow \ {\mathbb {A}}^\text{c}(G,L)is induced by the balanced product over K, i.e., it is the
biadditive extension of(S,T) \ \longmapsto \ S\times _K T \ .Here S has a left L-action and a commuting free right K-action,
whe... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.01804649457335472,
0.01999911479651928,
-0.026726506650447845,
0.040150780230760574,
0.011601317673921585,
-0.04350684955716133,
0.00032678761635906994,
0.008702894672751427,
-0.008016427047550678,
0.048144325613975525,
-0.010083459317684174,
-0.014774328097701073,
-0.0024712865706533194,... |
5ca5be04de7e30c59f3002ecb8ff3530e91c5d99 | subsection | 578 | 1,121 | Global functors | We omit the details.The restriction of the monoidal structure on the Burnside category
to finite groups has an interpretation in terms of the cartesian product of
bisets: under the equivalence of categories \Psi :{\mathbf {A}}_{{\mathcal {F}}in}\cong {\mathbb {A}}^c,
it corresponds to the monoidal structure{\mathbb {A}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0060438",
"end": 1661,
"openalex_id": "https://openalex.org/W1859542606",
"raw": "B. Day, On closed categories of functors. Reports of the Midwest Category Seminar, IV pp. 1–38. Lecture Notes in Mathematics, Vol. 137. Springer-Ve... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02563563734292984,
0.009109807200729847,
-0.018967321142554283,
0.002227859105914831,
-0.002235488733276725,
-0.06616435945034027,
-0.015617902390658855,
0.04049820452928543,
0.009987217374145985,
0.050111569464206696,
-0.05999960005283356,
-0.015419530682265759,
-0.005699351895600557,
... |
bc7254791b5be4b305f1b83c95639dbd97a220bb | subsection | 579 | 1,121 | Global functors | We denote by F\otimes F^{\prime }:{\mathbf {A}}\otimes {\mathbf {A}}\longrightarrow {\mathcal {A}}b the objectwise
tensor product given on objects by(F\otimes F^{\prime })(G,G^{\prime })\ = \ F(G)\otimes F^{\prime }(G^{\prime }) \ .A bimorphism is a natural transformationF\otimes F^{\prime }\ \longrightarrow \ F^{\prim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03514982759952545,
-0.005534114316105843,
-0.009893517009913921,
0.008528104983270168,
-0.006285472307354212,
-0.03734669089317322,
-0.017285048961639404,
0.04482213035225868,
0.01199121680110693,
0.03353269025683403,
-0.0675535723567009,
-0.00781870074570179,
0.004180144518613815,
0.01... |
f12d8d762fcdb27f87cacf32bc11ab2c9dc43d2a | subsection | 580 | 1,121 | Global functors | Equivalently: for every compact Lie group G the maps \lbrace b_{G,G^{\prime }}\rbrace _{G^{\prime }}
form a morphism of global functors F(G)\otimes F^{\prime }(-)\longrightarrow F^{\prime \prime }(G\times -)
and for every compact Lie group G^{\prime } the maps \lbrace b_{G,G^{\prime }}\rbrace _G
form a morphism of glob... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05659407749772072,
-0.024407150223851204,
-0.033132705837488174,
-0.002015496604144573,
0.018488416448235512,
-0.03938703611493111,
-0.020151153206825256,
0.030386900529265404,
-0.009038272313773632,
0.02250034175813198,
-0.036732759326696396,
-0.004885243717581034,
-0.0096179423853755,
... |
3e0228d6b6a68bbbfa348c585179757b183b8737 | subsection | 581 | 1,121 | Global functors | The map d is the difference of two homomorphisms; one of them
sums the tensor products of{\mathbf {A}}(H\times H^{\prime },-)\otimes {\mathbf {A}}(G,H)\otimes {\mathbf {A}}(G^{\prime },H^{\prime })\ \longrightarrow \ {\mathbf {A}}(G\times G^{\prime },-)\ , \ \varphi \otimes \tau \otimes \tau ^{\prime } \ \longmapsto \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.026309188455343246,
-0.01703077368438244,
-0.03720521926879883,
-0.00797363743185997,
-0.017976928502321243,
-0.02269243821501732,
-0.005085578188300133,
0.024859435856342316,
-0.008874010294675827,
0.0335121676325798,
-0.041661303490400314,
-0.02122742496430874,
0.013688713312149048,
-... |
9ba03327b4fe7fbe2eace925ca2f1cbba6f2c707 | subsection | 582 | 1,121 | Global functors | Theorem REF
describes explicit free generators for the morphism groups
in the Burnside category; using this, the value (F\Box M)(K)
can be expanded into a cokernel of a morphism between two huge sums
of tensor products of values of F and M.Now we consider a short exact sequence of global functors0 \longrightarrow M \l... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2192,
"openalex_id": "",
"raw": "L. G. Lewis, Jr., When projective does not imply flat, and other homological anomalies. Theory Appl. Categ. 5 (1999), no. 9, 202–250.",
"source_ref_id": "a04e77bacbb8b1d0a9f861e7f197c4b7e9869... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04613664746284485,
-0.026470597833395004,
-0.01849127560853958,
-0.004130786284804344,
0.013212413527071476,
-0.011404478922486305,
-0.03304629027843475,
0.036280736327171326,
0.02259536273777485,
0.023815909400582314,
-0.04033905267715454,
0.02601289190351963,
0.0001380266185151413,
0.... |
c26a8fdc9d414123511712c34e3f23d348b8a7ad | subsection | 583 | 1,121 | Global functors | Theorem 6.10 of shows that
the representable functor {\mathbf {A}}_{C_p} is not flat,
where C_p is a cyclic group of prime order p.box product!of global functors|)We now remark that bimorphisms of global functors can be identified
with another kind of structure that we call `diagonal products'.Definition 2.19
Let X,Y ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 296,
"openalex_id": "",
"raw": "L. G. Lewis, Jr., When projective does not imply flat, and other homological anomalies. Theory Appl. Categ. 5 (1999), no. 9, 202–250.",
"source_ref_id": "a04e77bacbb8b1d0a9f861e7f197c4b7e98691... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0502554252743721,
-0.009680348448455334,
-0.010626261122524738,
0.008841232396662235,
-0.017774004489183426,
-0.030192920938134193,
0.01386830024421215,
0.022381514310836792,
0.0010326849296689034,
0.03582262620329857,
-0.014493823051452637,
-0.02045917510986328,
-0.0072240266017615795,
... |
e4c90e23316a32b5a4fe99f983490d0b27e43b7c | subsection | 584 | 1,121 | Global functors | For a group homomorphism
\alpha :K\longrightarrow G we have \Delta _G\circ \alpha =(\alpha \times \alpha )\circ \Delta _K,
so the following diagram commutes:@C=12mm{
X(G)\otimes Y(G) [r]^-{\mu _{G,G}} [d]_{\alpha ^*\otimes \alpha ^*} &
Z(G\times G)[r]^-{\Delta _G^*} [d]^{(\alpha \times \alpha )^*} &
Z(G)[d]^{\alpha ^*}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06415800750255585,
-0.006402065511792898,
-0.008218145929276943,
0.0011999104171991348,
-0.004719520453363657,
-0.009614543989300728,
-0.007237615063786507,
0.01945800706744194,
-0.0067912256345152855,
0.030674975365400314,
-0.028813110664486885,
-0.03226213902235031,
0.027546431869268417... |
352954db893869d2ed89dadb3b8e17f0cd742804 | subsection | 585 | 1,121 | Global functors | If the diagonal product \nu was defined from an external product \mu as above, then\nu _{G\times K}\circ (p_G^*\otimes p_K^*)\ &= \ \Delta _{G\times K}^* \circ \mu _{G\times K,G\times K}\circ (p_G^*\otimes p_K^*)\\
&= \ \Delta _{G\times K}^*\circ (p_G\times p_K)^*\circ \mu _{G,K}\ = \ \mu _{G,K}because the composite (p... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04209515079855919,
0.019796013832092285,
-0.015491869300603867,
-0.01532397698611021,
-0.026603279635310173,
0.010600102134048939,
-0.010600102134048939,
0.013736632652580738,
0.007974878884851933,
0.06862211227416992,
-0.032021619379520416,
-0.017094476148486137,
-0.01707921363413334,
... |
5733493b13c2b614a9d32a485911327649b5eb0c | subsection | 586 | 1,121 | Global functors | Then\operatorname{tr}_{H\times K}^{G\times K}(\mu _{H,K}(x\otimes y)) \ &= \ \operatorname{tr}_{H\times K}^{G\times K}(\nu _{H\times K}(p_H^*(x)\otimes \operatorname{res}^{G\times K}_{H\times K}(p_K^*(y)))) \\
&=\ \nu _{G\times K}(\operatorname{tr}_{H\times K}^{G\times K}(p_H^*(x))\otimes p_K^*(y)) \\
&=\ \nu _{G\times... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.031057709828019142,
0.03285771235823631,
-0.020822089165449142,
0.006151287350803614,
-0.03139330446720123,
-0.020913613960146904,
0.012752575799822807,
-0.00003390501660760492,
0.025520406663417816,
0.02635939233005047,
-0.02866278775036335,
-0.03874586522579193,
0.03053906373679638,
0... |
d8c408e0b66d4c7daf8bd5facb2ed9523d8a0649 | subsection | 587 | 1,121 | Global model structures for orthogonal spectra | In this section we establish the strong level and global model structures
on the category of orthogonal spectra. Many arguments are parallel
to the unstable analogs in Section ,
so there is a certain amount of repetition.
The main model structure of interest for us is
the global model structure,
see Theorem REF .
The w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0098370760679245,
-0.005364983808249235,
-0.036967258900403976,
-0.014416009187698364,
0.03669252246618271,
-0.0461861789226532,
-0.0020357174798846245,
0.02197888121008873,
0.022284144535660744,
0.014568640850484371,
-0.03260200843214989,
-0.017079422250390053,
-0.003945514559745789,
0... |
5309b3fc4b728b3de69a3c0ef3bfbbe28a3ad12e | subsection | 588 | 1,121 | Global model structures for orthogonal spectra | We specialize the general recipe to the category of orthogonal spectra.
For a morphism f:X\longrightarrow Y of orthogonal spectra and m\ge 0
we have a commutative square of O(m)-spaces:@C=12mm{ L_m X[r]^{L_m f}[d]_{\nu _m^X} & L_m Y [d]^{\nu _m^Y}\\
X({\mathbb {R}}^m)[r]_{f({\mathbb {R}}^m)} & Y({\mathbb {R}}^m)}We thu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.013506930321455002,
-0.006036147940903902,
-0.04780079796910286,
-0.042672742158174515,
0.05445506051182747,
-0.027563296258449554,
-0.0008976004319265485,
0.01555204764008522,
0.04404633119702339,
0.021488992497324944,
-0.01871129684150219,
0.013384833931922913,
0.0029226860497146845,
... |
4b6a4a872fc36c7b764018ec78bd6832af8c2750 | subsection | 589 | 1,121 | Global model structures for orthogonal spectra | A morphism f:X\longrightarrow Y of orthogonal spectra isan {\mathcal {F}}-level equivalenceF-level equivalence@{\mathcal {F}}-level equivalence!of orthogonal spectra
if the map f({\mathbb {R}}^m):X({\mathbb {R}}^m)\longrightarrow Y({\mathbb {R}}^m) is an {\mathcal {F}}(m)-equivalence for all m\ge 0;
an {\mathcal {F}}-... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
0.017256122082471848,
-0.013800321146845818,
-0.04134756326675415,
-0.02776847407221794,
0.026242733001708984,
-0.04244609549641609,
0.007567672058939934,
0.022825075313448906,
0.02746332623064518,
0.011870259419083595,
-0.03597695752978325,
-0.021451909095048904,
0.032894961535930634,
-0.... |
41f7809eb755f94b74347717ab8f6ca4f0af690e | subsection | 590 | 1,121 | Global model structures for orthogonal spectra | The morphism f is an {\mathcal {F}}-level fibration
if and only if for every compact Lie group G
and every faithful G-representation V the map
f(V):X(V)\longrightarrow Y(V) is an ({\mathcal {F}}\cap G)-fibration.Now we are ready to establish the {\mathcal {F}}-level model structure.Proposition 3.5
Let {\mathcal {F}} b... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.013681049458682537,
-0.023974265903234482,
-0.023318063467741013,
-0.006428490858525038,
0.030154773965477943,
-0.00488336430862546,
-0.010354258120059967,
0.029010234400629997,
0.016420312225818634,
0.020433826372027397,
-0.02055591158568859,
-0.003414539620280266,
0.022234566509723663,
... |
2465858527ce2b5e53f13e13880d1b86feb3de76 | subsection | 591 | 1,121 | Global model structures for orthogonal spectra | Since the functor under consideration
is a left adjoint, it suffices to prove the claim for the generating
acyclic cofibrations, i.e., the maps( O(m)/H \times j_k)_+for all H\in {\mathcal {F}}(m) and all k\ge 0,
where j_k:D^k\times \lbrace 0\rbrace \longrightarrow D^k\times [0,1] is the inclusion.
Up to isomorphism, th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.044184595346450806,
-0.016828594729304314,
-0.03353513404726982,
0.00793369859457016,
0.012236203998327255,
-0.014013657346367836,
-0.012732059694826603,
0.019361276179552078,
0.006831372156739235,
0.022595783695578575,
-0.03060576692223549,
-0.009978150948882103,
-0.003207807894796133,
... |
d6e9891c297242435c5b1e069061fa5bb5ffc1af | subsection | 592 | 1,121 | Global model structures for orthogonal spectra | The {\mathcal {F}}-level model structure is topological
by Proposition REF ,
where we take {\mathcal {G}} as the set of semifree orthogonal spectra F_{H,{\mathbb {R}}^m}
for all m\ge 0 and all H\in {\mathcal {F}}(m), and {\mathcal {Z}}=\emptyset
as the empty set.When {\mathcal {F}}=\langle e\rangle is the minimal glob... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1112/s0024611501012692",
"end": 577,
"openalex_id": "https://openalex.org/W2086997195",
"raw": "M. A. Mandell, J. P. May, S. Schwede, B. Shipley, Model categories of diagram spectra. Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0270619485527277,
-0.018316039815545082,
-0.029458297416567802,
-0.01825498603284359,
0.03296887129545212,
-0.04459955543279648,
0.02237609401345253,
0.012485433369874954,
-0.01353097427636385,
0.022208197042346,
-0.019827112555503845,
0.016438644379377365,
0.0058153425343334675,
0.0263... |
7632201884c238562a8cffa8567d0a3bfc5b088a | subsection | 593 | 1,121 | Global model structures for orthogonal spectra | This means that global \Omega -spectra abound, because every
orthogonal spectrum admits a global equivalence to a global \Omega -spectrum.Remark 3.9 Global \Omega -spectra are a very rich kind of structure,
because they encode compatible equivariant
infinite loop spaces for all compact Lie groups at once.
For a global ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 569,
"openalex_id": "https://openalex.org/W2075488415",
"raw": "M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.",
"source_ref_id": "1fe5cf0f78b8a5a... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06781035661697388,
0.029342779889702797,
-0.009552043862640858,
-0.0139389643445611,
-0.0025444142520427704,
-0.08862343430519104,
0.01722724735736847,
0.003845231607556343,
0.04388446733355522,
-0.0011653951369225979,
0.020141689106822014,
-0.036224521696567535,
-0.0010900545166805387,
... |
e5a2c57fd1c1a4316672e53f776d75d8b9cdc9fb | subsection | 594 | 1,121 | Global model structures for orthogonal spectra | Specialized to the trivial group, the condition in Definition REF
says that X is in particular a non-equivariant \Omega -spectrum
in the sense that the adjoint structure map
\tilde{\sigma }_{{\mathbb {R}},W}:X(W)\longrightarrow \Omega X({\mathbb {R}}\oplus W) is a weak equivalence of
(non-equivariant) spaces for every... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05748138949275017,
0.02106328122317791,
-0.001976590370759368,
-0.03409809246659279,
0.02237592078745365,
-0.028969641774892807,
0.028237007558345795,
-0.0051856879144907,
0.06178562343120575,
0.03000754304230213,
-0.010554535314440727,
-0.01298139151185751,
0.014767191372811794,
0.0214... |
a16c0613d0a681f120bb3450994fca6c64ac3cff | subsection | 595 | 1,121 | Global model structures for orthogonal spectra | We denote by \xi (f) the resulting compositeY \ \xrightarrow{} \ \Omega ( \operatorname{sh}Y)\ \xrightarrow{}\ F(\operatorname{sh}f) \ .We note that the orthogonal spectrum F(\operatorname{Id}_{\operatorname{sh}X}) has a preferred
contraction, so part (ii) of the next proposition is a way to make precise that
the seque... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.02532065100967884,
-0.011927721090614796,
-0.01689569652080536,
-0.03824807330965996,
0.04441416263580322,
-0.0368744395673275,
0.010126734152436256,
0.01619361713528633,
0.055830586701631546,
0.021749204024672508,
-0.01392712164670229,
0.004910741001367569,
0.0006477065617218614,
0.023... |
37db7fabadd1d540b1bbae0d033339e6fad17690 | subsection | 596 | 1,121 | Global model structures for orthogonal spectra | We consider the commutative diagram@C=10mm{ X(W)^G [d]_{f(W)^G} [r]^-{(\tilde{\sigma }_{V,W})^G} &
\operatorname{map}_*^G(S^V, X(V\oplus W)) [d]^{\operatorname{map}_*^G(S^V,f(V\oplus W))} \\
Y(W)^G [d]_{(\xi (f)(W))^G}[r]_-{(\tilde{\sigma }_{V,W})^G} &
\operatorname{map}_*^G(S^V, Y(V\oplus W)) [d]^{\operatorname{map}_*... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.026043739169836044,
0.0026680720038712025,
-0.008643103763461113,
-0.027966123074293137,
0.03268054500222206,
-0.043604571372270584,
0.024518035352230072,
0.019818872213363647,
0.03747124969959259,
0.046808548271656036,
-0.006690204609185457,
-0.018293170258402824,
0.001637269277125597,
... |
009695c6f2b7bccdc84252d3267d26ec9dddc5f4 | subsection | 597 | 1,121 | Global model structures for orthogonal spectra | An orthogonal spectrum X is an
{\mathcal {F}}-\Omega -spectrumF-Omega-spectrum@{\mathcal {F}}-\Omega -spectrum
if for every compact Lie group G in {\mathcal {F}}, every G-representation V
and every faithful G-representation W the adjoint structure map
\tilde{\sigma }_{V,W}\ :\ X(W)\ \longrightarrow \ \operatorname{map... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03892919048666954,
0.016139138489961624,
-0.024010401219129562,
-0.010853494517505169,
0.008061942644417286,
-0.05903447046875954,
0.05613613501191139,
0.009091612882912159,
0.027625691145658493,
0.017817120999097824,
-0.02509346231818199,
-0.006422095932066441,
0.017984919250011444,
0.... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.