Topological Hochschild homology and higher characteristics

Campbell, Jonathan A.; Ponto, Kate

doi:10.2140/agt.2019.19.965

Cited by 14 publications

(23 citation statements)

References 31 publications

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“…In [31], Ponto and Shulman showed that classical Hochschild homology extends to a shadow. In [13] Campbell and Ponto proved that topological Hochschild homology of ring spectra and of spectral categories does so as well. In this section, we generalize these result to any nice enough symmetric monoidal, simplicial model category.…”

Section: Hochschild Shadowsmentioning

confidence: 97%

“…naturally, for all A M B and B N A . It is straightforward to check the remaining shadow conditions, by an argument analogous to that in [13].…”

Section: Hochschild Shadowsmentioning

confidence: 99%

“…The notion of a shadow on a bicategory, introduced by Ponto in [30], has proved instrumental in the study of (topological) Hochschild homology, topological restriction homology, algebraic K-theory, and fixed point theory. Both classical Hochschild homology and its homotopical generalization, topological Hochschild homology of ring spectra and of spectral categories, are examples of shadows [31], [13]. An advantage of the shadow approach to studying Hochschild theories is that their Morita invariance is immediate once we have established that they are indeed shadows.…”

Section: Introductionmentioning

confidence: 99%

“…Let SH denote the stable homotopy category. Campbell and Ponto proved in [13] that topological Hochschild homology is an SH-shadow on the bicategory R Sp whose objects are structured ring spectra (i.e., monoids in some nice monoidal model category of spectra, Sp) and whose morphism categories are the homotopy categories of categories of spectral bimodules.…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 2.2.4. [13] Let B be a bicategory equipped with a C-shadow. Let ( A M B , B N A ) be a Morita equivalence.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A shadow framework for equivariant Hochschild homologies

Adamyk¹,

Gerhardt²,

Hess³

et al. 2021

Preprint

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Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category V, as well as for small V-categories and monoids in V equipped with an automorphism. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We demonstrate that Hochschild homology of Green functors and Cn-twisted topological Hochschild homology fit into this framework and therefore extend to bicategorical shadows, whence these theories are Morita invariant. Moreover, the associated linearization maps in each case are lax shadow functors. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic K-theory to twisted topological Hochschild homology.

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Section: Hochschild Shadowsmentioning

confidence: 97%

“…naturally, for all A M B and B N A . It is straightforward to check the remaining shadow conditions, by an argument analogous to that in [13].…”

Section: Hochschild Shadowsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Proposition 2.2.4. [13] Let B be a bicategory equipped with a C-shadow. Let ( A M B , B N A ) be a Morita equivalence.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A shadow framework for equivariant Hochschild homologies

Adamyk¹,

Gerhardt²,

Hess³

et al. 2021

Preprint

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Periodic Points and Topological Restriction Homology

Malkiewich

Ponto

2020

International Mathematics Research Notices

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We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.

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THH and Traces of Enriched Categories

Berman

2020

International Mathematics Research Notices

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We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty $-category. Our results rely crucially on an elementary, model-independent framework for enriched higher-category theory, which may be of independent interest. For those interested only in enriched category theory, read Sections 1.3 and 2.

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Topological Hochschild homology and higher characteristics

Cited by 14 publications

References 31 publications

A shadow framework for equivariant Hochschild homologies

A shadow framework for equivariant Hochschild homologies

Periodic Points and Topological Restriction Homology

THH and Traces of Enriched Categories

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