Global Homotopy Theory

Schwede, Stefan

doi:10.1017/9781108349161

Cited by 116 publications

(403 citation statements)

References 132 publications

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“…After taking fixed points for a graph subgroup, these two maps become inverse homotopy equivalences. The proof for this claim is completely analogous to the one for the G − Γ-space ku G modeling connective equivariant K-theory, which is discussed in detail in [34,Theorem 6.3.19], so we refrain from giving it here. It reduces to the fact that the spaces…”

Section: Proofs Of the Main Resultsmentioning

confidence: 98%

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Commuting matrices and Atiyah's real K‐theory

Gritschacher

Hausmann

2019

Journal of Topology

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Section: Proofs Of the Main Resultsmentioning

confidence: 98%

“…In particular, there is a G-equivalence Ω ∞ (X(S) ∧ A) Ω ∞ (X A (S)). Now, if A is G-homeomorphic to the geometric realization of a finite based G-simplicial set, then by [34,Propositions B.37(ii) and B.54(iii)], the prolonged G − Γ-space X A is again G-cofibrant and special. Thus, in that case, we have…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Commuting matrices and Atiyah's real K‐theory

Gritschacher

Hausmann

2019

Journal of Topology

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“…Remark The flat model structure on orthogonal spectra and the previous lemma also appear in more general equivariant contexts in [, Proposition 2.10.1; , Theorem III.5.10].…”

Section: Diagram Spectra and Bispectramentioning

confidence: 87%

“…Remark This use of the term flat follows Schwede's terminology . Our flat orthogonal spectra are called

double-struckS

‐cofibrant in and should not be confused with the more general flat objects of [, Definition B.15].…”

Section: Diagram Spectra and Bispectramentioning

confidence: 99%

“…For closed

h

‐cofibrations in the category of all topological spaces, this is shown in [, Application (vi)]. The desired statement in the category of compactly generated weak Hausdorff spaces

scriptT

we are working in follows because

h

‐cofibrations in

scriptT

are automatically closed (see, for example, [, Proposition A.31]).

□

…”

Section: Topological Hochschild Homologymentioning

confidence: 99%

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Comparing cyclotomic structures on different models for topological Hochschild homology

Dotto

Malkiewich

Patchkoria

et al. 2019

Journal of Topology

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The topological Hochschild homology THH(A) of an orthogonal ring spectrum A can be defined by evaluating the cyclic bar construction on A or by applying Bökstedt's original definition of prefixTHH to A. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for THH(A). This implies that the two versions of topological cyclic homology resulting from these variants of THH(A) are equivalent.

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Hermitian K-theory for stable $$\infty $$-categories I: Foundations

Calmès

Dotto

Harpaz

et al. 2022

Sel. Math. New Ser.

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This paper is the first in a series in which we offer a new framework for hermitian $${\text {K}}$$ K -theory in the realm of stable $$\infty $$ ∞ -categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré $$\infty $$ ∞ -category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived $$\infty $$ ∞ -categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on $$\infty $$ ∞ -categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré $$\infty $$ ∞ -categories, showing in particular that they form a bicomplete, closed symmetric monoidal $$\infty $$ ∞ -category. We also study the process of tensoring and cotensoring a Poincaré $$\infty $$ ∞ -category over a finite simplicial complex, a construction featuring prominently in the definition of the $${\text {L}}$$ L - and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré $$\infty $$ ∞ -category using generators and relations. We extract its basic properties, relating it in particular to the 0th $${\text {L}}$$ L - and algebraic $${\text {K}}$$ K -groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.

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Global Homotopy Theory

Cited by 116 publications

References 132 publications

Commuting matrices and Atiyah's real K‐theory

Commuting matrices and Atiyah's real K‐theory

Comparing cyclotomic structures on different models for topological Hochschild homology

Hermitian K-theory for stable $$\infty $$-categories I: Foundations

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