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Adams operations on the twisted K-theory of compact Lie groups J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2024-03-18 Chi-Kwong Fok
In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings.
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On Vietoris–Rips complexes of finite metric spaces with scale 2 J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2024-02-03 Ziqin Feng, Naga Chandra Padmini Nukala
We examine the homotopy types of Vietoris–Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of \([m]=\{1, 2, \ldots , m\}\) equipped with symmetric difference metric d, specifically, \({\mathcal {F}}^m_n\), \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+1}\), \({\mathcal {F}}_n^m\cup {\mathcal {F}}^m_{n+2}\), and \({\mathcal {F}}_{\preceq A}^m\). Here \({\mathcal
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Associative 2-algebras and nonabelian extensions of associative algebras J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2024-02-01 Yunhe Sheng, You Wang
In this paper, we study nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. First we construct an associative 2-algebra using the bimultipliers of an associative algebra. Then we classify nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. Finally we analyze the relation between nonabelian extensions of associative algebras and
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Lambda module structure on higher K-groups J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2024-01-25 Sourayan Banerjee, Vivek Sadhu
In this article, we show that for a quasicompact scheme X and \(n>0,\) the n-th K-group \(K_{n}(X)\) is a \(\lambda \)-module over a \(\lambda \)-ring \(K_{0}(X)\) in the sense of Hesselholt.
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LHS-spectral sequences for regular extensions of categories J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2024-01-20 Ergün Yalçın
In (Xu, J Pure Appl Algebra 212:2555–2569, 2008), a LHS-spectral sequence for target regular extensions of small categories is constructed. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Słomińska’s spectral sequence
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Periodic self maps and thick ideals in the stable motivic homotopy category over $${\mathbb {C}}$$ at odd primes J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-11-23 Sven-Torben Stahn
In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over \({\mathbb {C}}\). In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.
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The homotopy of the $$KU_G$$ -local equivariant sphere spectrum J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-11-20 Tanner N. Carawan, Rebecca Field, Bertrand J. Guillou, David Mehrle, Nathaniel J. Stapleton
We compute the homotopy Mackey functors of the \(KU_G\)-local equivariant sphere spectrum when G is a finite q-group for an odd prime q, building on the degree zero case due to Bonventre and the third and fifth authors.
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Prismatic cohomology and p-adic homotopy theory J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-11-13 Tobias Shin
Historically, it was known by the work of Artin and Mazur that the \(\ell \)-adic homotopy type of a smooth complex variety with good reduction mod p can be recovered from the reduction mod p, where \(\ell \) is not p. This short note removes this last constraint, with an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor
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Weak cartesian properties of simplicial sets J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-11-10 Carmen Constantin, Tobias Fritz, Paolo Perrone, Brandon T. Shapiro
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On the K-theory of $$\mathbb {Z}$$ -categories J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-11-04 Eugenia Ellis, Rafael Parra
We establish connections between the concepts of Noetherian, regular coherent, and regular n-coherent categories for \(\mathbb {Z}\)-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative K-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the K-theory of \(\mathbb {Z}\)-linear
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Self-closeness numbers of rational mapping spaces J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-10-11 Yichen Tong
For a closed connected oriented manifold M of dimension 2n, it was proved by Møller and Raussen that the components of the mapping space from M to \(S^{2n}\) have exactly two different rational homotopy types. However, since this result was proved by the algebraic models for the components, it is unclear whether other homotopy invariants distinguish their rational homotopy types or not. The self-closeness
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Comparison of the colimit and the 2-colimit J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-10-04 Ilia Pirashvili
The 2-colimit (also referred to as a pseudo colimit) is the 2-categorical analogue of the colimit and as such, a very important construction. Calculating it is, however, more involved than calculating the colimit. The aim of this paper is to give a condition under which these two constructions coincide. Tough the setting under which our results are applicable is very specific, it is, in fact, fairly
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The derived Brauer map via twisted sheaves J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-09-22 Guglielmo Nocera, Michele Pernice
Let X be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of X and which we call \(\textsf{Br}^\dagger (X)\) following Lurie. Toën introduced a map \(\phi :\textsf{Br}^\dagger (X)\rightarrow H ^2_{\acute{e }t }(X,{\mathbb {G}}_{\textrm{m}})\) which extends the classical Brauer map, but instead
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Eilenberg–Maclane spaces and stabilisation in homotopy type theory J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-09-21 David Wärn
In this note, we study the delooping of spaces and maps in homotopy type theory. We show that in some cases, spaces have a unique delooping, and give a simple description of the delooping in these cases. We explain why some maps, such as group homomorphisms, have a unique delooping. We discuss some applications to Eilenberg–MacLane spaces and cohomology.
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Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1 J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-08-08 Oleksandra Khokhliuk, Sergiy Maksymenko
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Goodwillie’s cosimplicial model for the space of long knots and its applications J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-08-02 Yuqing Shi
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Centralisers, complex reflection groups and actions in the Weyl group $$E_6$$ J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-06-08 Graham A. Niblo, Roger Plymen, Nick Wright
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A t-structure on the $$\infty $$ -category of mixed graded modules J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-04-27 Emanuele Pavia
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On Kähler differentials of divided power algebras J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-04-10 Ioannis Dokas
The Quillen–Barr–Beck cohomology of augmented algebras with a system of divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the Kähler differentials of an augmented algebra with a system of divided powers in prime characteristic represents Beck derivations. We give a geometrical interpretation of this statement for the sheaf of relative differentials
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Coherent presentations of monoids with a right-noetherian Garside family J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-02-20 Pierre-Louis Curien, Alen Ɖurić, Yves Guiraud
This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for Artin-Tits monoids of spherical type: to general Artin-Tits
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The cohomology of $$C_2$$ -surfaces with $${\underline{{\mathbb {Z}}}}$$ -coefficients J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-01-20 Christy Hazel
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Modules and representations up to homotopy of Lie n-algebroids J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2023-01-05 M. Jotz, R. A. Mehta, T. Papantonis
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Localization $$C^*-$$ algebras and index pairing J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-11-24 Hang Wang, Chaohua Zhang, Dapeng Zhou
Kasparov KK-theory for a pair of \(C^*\)-algebras \((A,\,B)\) can be formulated equivalently in terms of the K-theory of Yu’s localization algebra by Dadarlat-Willett-Wu. We investigate the pairings between K-theory \(K_j(A)\) and the two notions of KK-theory which are Kasparov KK-theory \(KK_i(A,B)\) and the localization algebra description of \(KK_i(A,B)\) and show that the two pairings are compatible
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The $$\mathbb {R}$$ -local homotopy theory of smooth spaces J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-11-11 Severin Bunk
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Multifunctorial K-theory is an equivalence of homotopy theories J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-11-01 Niles Johnson, Donald Yau
We show that each of the three K-theory multifunctors from small permutative categories to \(\mathcal {G}_*\)-categories, \(\mathcal {G}_*\)-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these K-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe
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On the RO(Q)-graded coefficients of Eilenberg–MacLane spectra J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-10-05 Igor Sikora
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Homotopy pro-nilpotent structured ring spectra and topological Quillen localization J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-09-16 Yu Zhang
The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are \({ \mathsf {TQ} }\)-local, where structured ring spectra are described as algebras over a spectral operad \({ \mathcal {O} }\). Here, \({ \mathsf {TQ} }\) is short for topological Quillen homology, which is weakly equivalent to \({ \mathcal {O} }\)-algebra stabilization. An \({ \mathcal {O} }\)-algebra is called
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On families of nilpotent subgroups and associated coset posets J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-09-14 Simon Gritschacher, Bernardo Villarreal
We study some properties of the coset poset associated with the family of subgroups of class \(\le 2\) of a nilpotent group of class \(\le 3\). We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the
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Toward a minimal model for $$H_*(\overline{\mathcal {M}})$$ H ∗ ( M ¯ ) J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-09-08 Benjamin C. Ward
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Unitary calculus: model categories and convergence J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-08-09 Niall Taggart
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Computations of relative topological coHochschild homology J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-08-05 Sarah Klanderman
Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of \(\mathrm {coTHH}\) for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence to study \(\mathrm {coTHH}\) of coalgebra spectra over any commutative
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Smashing localizations in equivariant stable homotopy J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-07-15 Christian Carrick
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Modeling bundle-valued forms on the path space with a curved iterated integral J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-07-13 Cheyne Glass, Corbett Redden
The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that
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An equivalence of profinite completions J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-07-06 Chang-Yeon Chough
The goal of this paper is to establish an equivalence of profinite completions of pro-spaces in model category theory and in \(\infty \)-category theory. As an application, we show that the author’s comparison theorem for algebro-geometric objects in the setting of model categories recovers that of David Carchedi in the setting of \(\infty \)-categories.
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Rational stabilization and maximal ideal spaces of commutative Banach algebras J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-07-01 Kazuhiro Kawamura
For a unital commutative Banach algebra A and its closed ideal I, we study the relative Čech cohomology of the pair \((\mathrm {Max}(A),\mathrm {Max}(A/I))\) of maximal ideal spaces and show a relative version of the main theorem of Lupton et al. (Trans Amer Math Soc 361:267–296, 2009): \(\check{\mathrm {H}}^{j}(\mathrm {Max}(A),\mathrm {Max}(A/I));{\mathbb {Q}}) \cong \pi _{2n-j-1}(Lc_{n}(I))_{{\mathbb
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Cohomology and deformations of twisted Rota–Baxter operators and NS-algebras J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-05-05 Apurba Das
The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an \(L_\infty \)-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology
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On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-04-25 Jesús González, José Luis León-Medina
We compute the Lusternik–Schnirelmann category and all the higher topological complexities of non-k-equal manifolds \(M_d^{(k)}(n)\) for certain values of d, k and n. This includes instances where \(M_d^{(k)}(n)\) is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring \(H^*(M_d^{(k)}(n))\) as described by Dobrinskaya and Turchin. A fine
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$${ \mathsf {TQ} }$$ TQ -completion and the Taylor tower of the identity functor J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-03-30 Nikolas Schonsheck
The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if A is a \((-1)\)-connected \({ \mathcal {O} }\)-algebra with 0-connected \({ \mathsf {TQ} }\)-homology spectrum \({ \mathsf {TQ} }(A)\), then there is a natural weak equivalence \(P_\infty ({ \mathrm {id} })A \simeq A^\wedge
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Resolutions of operads via Koszul (bi)algebras J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-03-03 Pedro Tamaroff
We introduce a construction that produces from each bialgebra H an operad \(\mathsf {Ass}_H\) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and
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On the Euler–Poincaré characteristics of a simply connected rationally elliptic CW-complex J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-02-22 Mahmoud Benkhalifa
For a simply connected rationally elliptic CW-complex X, we show that the cohomology and the homotopy Euler–Poincaré characteristics are related to two new numerical invariants namely \(\eta _{X}\) and \(\rho _{X}\) which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of X.
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Connectedness of graphs arising from the dual Steenrod algebra J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-02-08 Donald M. Larson
We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra \(\mathscr {A}^*\). We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of \(\mathscr {A}^*\) and its structure as a Hopf algebra.
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The completion theorem in twisted equivariant K-theory for proper actions J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-01-31 Noé Bárcenas, Mario Velásquez
We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.
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On graded $${\mathbb {E}}_{\infty }$$ E ∞ -rings and projective schemes in spectral algebraic geometry J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-01-31 Mariko Ohara, Takeshi Torii
We introduce graded \({\mathbb {E}}_{\infty }\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty \)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme
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$$C_2$$ C 2 -equivariant topological modular forms J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2022-01-10 Chua, Dexter
We compute the homotopy groups of the \(C_2\) fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a \({\mathrm {TMF}}\)-module, it is isomorphic to the tensor product of \({\mathrm {TMF}}\) with an explicit finite cell complex.
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Marked colimits and higher cofinality J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-12-16 Abellán García, Fernando
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On the LS-category and topological complexity of projective product spaces J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-11-08 Fişekci, Seher, Vandembroucq, Lucile
We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtain an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.
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Overcategories and undercategories of cofibrantly generated model categories J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-10-13 Hirschhorn, Philip S.
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Sheaves via augmentations of Legendrian surfaces J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-10-09 Rutherford, Dan, Sullivan, Michael
Given an augmentation for a Legendrian surface in a 1-jet space, \(\Lambda \subset J^1(M)\), we explicitly construct an object, \(\mathcal {F} \in \mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})\), of the (derived) category from Viterbo (Shende, V., Treumann, D., Zaslow, E. Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on \(M\times \mathbb {R}\) with singular support
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Rational model for the string coproduct of pure manifolds J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-10-07 Naito, Takahito
The string coproduct is a coproduct on the homology with field coefficients of the free loop space of a closed oriented manifold introduced by Sullivan in string topology. The coproduct and the Chas-Sullivan loop product give an infinitesimal bialgebra structure on the homology if the Euler characteristic is zero. The aim of this paper is to study the string coproduct using Sullivan models in rational
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Equivariant formal group laws and complex-oriented spectra over primary cyclic groups: elliptic curves, Barsotti–Tate groups, and other examples J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-09-27 Hu, Po, Kriz, Igor, Somberg, Petr
We explicitly construct and investigate a number of examples of \({\mathbb {Z}}/p^r\)-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.
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On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$ A ∞ -algebra J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-09-25 de Kleijn, Niek, Wierstra, Felix
In this paper, we develop the \(A_\infty \)-analog of the Maurer-Cartan simplicial set associated to an \(L_\infty \)-algebra and show how we can use this to study the deformation theory of \(\infty \)-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of \(A_\infty \)-algebras like the Maurer-Cartan equation and twist. One of our
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Quasi-categories vs. Segal spaces: Cartesian edition J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-08-20 Rasekh, Nima
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1. On marked simplicial sets (due to Lurie [31]), 2. On bisimplicial spaces (due to deBrito [12]), 3. On bisimplicial sets, 4. On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete
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A cochain level proof of Adem relations in the mod 2 Steenrod algebra J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-08-19 Brumfiel, Greg, Medina-Mardones, Anibal, Morgan, John
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod’s student J. Adem applied the homological
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Relative singularity categories and singular equivalences J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-08-18 Hafezi, Rasool
Let R be a right noetherian ring. We introduce the concept of relative singularity category \(\Delta _{\mathcal {X} }(R)\) of R with respect to a contravariantly finite subcategory \(\mathcal {X} \) of \({\text {{mod{-}}}}R.\) Along with some finiteness conditions on \(\mathcal {X} \), we prove that \(\Delta _{\mathcal {X} }(R)\) is triangle equivalent to a subcategory of the homotopy category \(\mathbb
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Higher order Toda brackets J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-07-27 Aziz Kharoof
We describe two ways to define higher order Toda brackets in a pointed simplicial model category \({\mathcal {D}}\): one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to
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The equivalence between Feynman transform and Verdier duality J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-07-23 Hao Yu
The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and the Feynman transform. Specifically, for each twisted modular operad \(\mathcal {P}\) (taking values in dg-vector spaces over a field k of characteristic 0), there is a certain sheaf
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On the K(1)-local homotopy of $$\mathrm {tmf}\wedge \mathrm {tmf}$$ tmf ∧ tmf J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-07-20 Dominic Leon Culver, Paul VanKoughnett
As a step towards understanding the \(\mathrm {tmf}\)-based Adams spectral sequence, we compute the K(1)-local homotopy of \(\mathrm {tmf}\wedge \mathrm {tmf}\), using a small presentation of \(L_{K(1)}\mathrm {tmf}\) due to Hopkins. We also describe the K(1)-local \(\mathrm {tmf}\)-based Adams spectral sequence.
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The Cantor–Schröder–Bernstein Theorem for $$\infty $$ ∞ -groupoids J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-06-28 Martín Hötzel Escardó
We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or \(\infty \)-groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean
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2-Segal objects and algebras in spans J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-05-17 Walker H. Stern
We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).
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Torsion in the magnitude homology of graphs J. Homotopy Relat. Struct. (IF 0.5) Pub Date : 2021-05-15 Radmila Sazdanovic, Victor Summers
Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular