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Quasicategories vs. Segal spaces: Cartesian edition J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210820
Rasekh, NimaWe 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 quasicategories and complete

A cochain level proof of Adem relations in the mod 2 Steenrod algebra J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210819
Brumfiel, Greg, MedinaMardones, Anibal, Morgan, JohnIn 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cupi 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

Relative singularity categories and singular equivalences J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210818
Hafezi, RasoolLet 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

Higher order Toda brackets J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210727
Aziz KharoofWe 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

The equivalence between Feynman transform and Verdier duality J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210723
Hao YuThe 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 dgvector spaces over a field k of characteristic 0), there is a certain sheaf

On the K(1)local homotopy of $$\mathrm {tmf}\wedge \mathrm {tmf}$$ tmf ∧ tmf J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210720
Dominic Leon Culver, Paul VanKoughnettAs 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.

The Cantor–Schröder–Bernstein Theorem for $$\infty $$ ∞ groupoids J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210628
Martín Hötzel EscardóWe 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

2Segal objects and algebras in spans J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210517
Walker H. SternWe 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: 2Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

Torsion in the magnitude homology of graphs J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210515
Radmila Sazdanovic, Victor SummersMagnitude 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

Leibniz algebras with derivations J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210409
Apurba DasIn this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz

Uniqueness of differential characters and differential Ktheory via homological algebra J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210403
Ishan MataSimons and Sullivan constructed a model of differential Ktheory, and showed that the differential Ktheory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential Ktheory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential

Derived categories of N DG categories J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210331
Junichi Miyachi, Hiroshi NagaseIn this paper we study Ndifferential graded categories and their derived categories. First, we introduce modules over an Ndifferential graded category. Then we show that they form a Frobenius category and that its homotopy category is triangulated. Second, we study the properties of its derived category and give triangle equivalences of Morita type between derived categories of Ndifferential graded

Homotopy theory of monoids and derived localization J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210303
Joe Chuang, Julian Holstein, Andrey LazarevWe use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobarconstruction to the nonsimply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an \(\infty \)category of discrete monoids.

Homotopical perspective on statistical quantities J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210209
Nissim RanadeWe introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with the algebraic product structure. These cumulants have a close relationship with \(A_{\infty }\) and \(C_{\infty }\) morphisms, which are the classical homotopical tools for analyzing deformations of algebraically compatible linear maps. We look at these two different

Smooth functorial field theories from Bfields and Dbranes J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210123
Severin Bunk, Konrad WaldorfIn the Lagrangian approach to 2dimensional sigma models, Bfields and Dbranes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2dimensional, smooth openclosed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition

Homotopy types of gauge groups of $$\mathrm {PU}(p)$$ PU ( p ) bundles over spheres J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210121
Simon ReaWe examine the relation between the gauge groups of \(\mathrm {SU}(n)\) and \(\mathrm {PU}(n)\)bundles over \(S^{2i}\), with \(2\le i\le n\), particularly when n is a prime. As special cases, for \(\mathrm {PU}(5)\)bundles over \(S^4\), we show that there is a rational or plocal equivalence \(\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}\) for any prime p if, and only if, \((120,k)=(120,l)\)

The Segal conjecture for topological Hochschild homology of Ravenel spectra J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210119
Gabriel AngeliniKnoll, J. D. QuigleyIn the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing

The Adams spectral sequence for 3local $$\mathrm {tmf}$$ tmf J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20210106
D. CulverThe purpose of this article is to record the computation of the homotopy groups of 3local \(\mathrm {tmf}\) via the Adams spectral sequence.

Groups up to congruence relation and from categorical groups to ccrossed modules J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20201121
Tamar Datuashvili, Osman Mucuk, Tunçar ŞahanWe introduce a notion of cgroup, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (ccrossed modules) are defined in this category and the semidirect product is constructed. We prove that each categorical group gives rise to a cgroup and to a ccrossed module, which is a connected, special and strict ccrossed module in the

On the relative K group in the ETNC, Part II J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20201106
Oliver BraunlingIn a previous paper we showed that, under some assumptions, the relative Kgroup in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a Kgroup of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and

Homotopy Gerstenhaber algebras are strongly homotopy commutative J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20201101
Matthias FranzWe show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\mathbin {\cup _1}\)product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense

Homotopic distance between functors J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20201013
E. MacíasVirgós, D. MosqueraLoisWe introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LScategory of small categories by Tanaka.

Cohomology and deformations of oriented dialgebras J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200916
Ali N. A. Koam, Ripan SahaWe introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.

Note on Toda brackets J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200828
Samik Basu, David Blanc, Debasis SenWe provide a general definition of Toda brackets in a pointed model category, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.

Cyclic homology for bornological coarse spaces J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200724
Luigi CaputiThe goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent

Bianchi’s additional symmetries J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200720
Alexander D. RahmIn a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of JeanPierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers \(\mathcal {O}\) in an imaginary quadratic number field, and the Borel–Serre

Descent theory and mapping spaces J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200703
Nicholas J. MeadowsThe purpose of this paper is to develop a theory of \((\infty , 1)\)stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasicategory theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty , 1)\)stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition

Higher equivariant and invariant topological complexities J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200621
Marzieh Bayeh, Soumen SarkarIn this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LScategory. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

Transfer ideals and torsion in the Morava E theory of abelian groups J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200523
Tobias Barthel, Nathaniel StapletonLet A be a finite abelian pgroup of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava Ecohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains ptorsion. The proof makes use of transchromatic character theory.

The universal fibration with fibre X in rational homotopy theory J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200402
Gregory Lupton, Samuel Bruce SmithLet X be a simply connected space with finitedimensional rational homotopy groups. Let \(p_\infty :UE \rightarrow B\mathrm {aut}_1(X)\) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map \( \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})\) expressed in terms of derivations of

The unit of the total décalage adjunction J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200319
Viktoriya Ozornova, Martina RovelliWe consider the décalage construction \({{\,\mathrm{Dec}\,}}\) and its right adjoint \(T\). These functors are induced on the category of simplicial objects valued in any bicomplete category \({\mathcal {C}}\) by the ordinal sum. We identify \(T{{\,\mathrm{Dec}\,}}X\) with the path object \(X^{\Delta [1]}\) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy

Verifying the Hilali conjecture up to formal dimension twenty J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200312
Spencer Cattalani, Aleksandar MilivojevićWe prove that in formal dimension \(\le 20\) the Hilali conjecture holds, i.e. that the total dimension of the rational homology bounds from above the total dimension of the rational homotopy for a simply connected rationally elliptic space.

An application of the h principle to manifold calculus J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200311
Apurva NakadeManifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the hprinciple, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor \(\mathrm {Emb}_{\mathrm {Lag}}(,N)\) is the totally

Correction to: Representations are adjoint to endomorphisms J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200306
Gabriel C. DrummondCole, Joseph Hirsh, Damien LejayThe first equation under section “Remark 3” was processed and published incorrectly. The correct equation should read as follows:

On the capacity and depth of compact surfaces J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20200212
Mahboubeh Abbasi, Behrooz MashayekhyK. Borsuk in 1979, at the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus \(g\ge 0\) is equal to \(g+2\). Also, we prove that the capacity and depth of a compact nonorientable surface of genus \(g>0\) is

Representations are adjoint to endomorphisms J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20191230
Gabriel C. DrummondCole, Joseph Hirsh, Damien LejayThe functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the selfenrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism properad

Formulae in noncommutative Hodge theory J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20191121
Nick SheridanWe prove that the cyclic homology of a saturated \(A_\infty \) category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror

The depth of a Riemann surface and of a rightangled Artin group J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20191112
Yves Félix, Steve HalperinWe consider two families of spaces, X: the closed orientable Riemann surfaces of genus \(g>0\) and the classifying spaces of rightangled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that$$\begin{aligned} \text{ depth } \,{\mathbb {Q}}[\pi _1(X)]

Twisting structures and morphisms up to strong homotopy J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20191108
Kathryn Hess, PaulEugène Parent, Jonathan ScottWe define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the barcobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction

Lie theory for symmetric Leibniz algebras J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20191005
Mamuka Jibladze, Teimuraz PirashviliLie algebras and groups equipped with a multiplication \(\mu \) satisfying some compatibility properties are studied. These structures are called symmetric Lie \(\mu \)algebras and symmetric \(\mu \)groups respectively. An equivalence of categories between symmetric Lie \(\mu \)algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of

Another approach to the Kan–Quillen model structure J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190924
Sean MossBy careful analysis of the embedding of a simplicial set into its image under Kan’s \(\mathop {\mathop {\mathsf {Ex}}^\infty }\) functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about \(\mathop {\mathop {\mathsf {Ex}}^\infty }\)

Correction to: Wrong way maps in uniformly finite homology and homology of groups J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190921
Alexander EngelThere is an error in the proof of Theorem 2.16 of Ref. 2. It occured at the end of the secondtolast paragraph of the proof.

Parallel transport of higher flat gerbes as an extended homotopy quantum field theory J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190718
Lukas Müller, Lukas WoikeWe prove that the parallel transport of a flat \(n1\)gerbe on any given target space gives rise to an ndimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimensionindependent

Enhanced $$A_{\infty }$$A∞ obstruction theory J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190716
Fernando MuroAn \(A_n\)algebra \(A= (A,m_1, m_2, \ldots , m_n)\) is a special kind of \(A_\infty \)algebra satisfying the \(A_\infty \)relations involving just the \(m_i\) listed. We consider obstructions to extending an \(A_{n1}\) algebra to an \(A_n\)algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e. \(m_1=0)\)\(A_\infty

On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3planes J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190712
Somnath Basu, Prateep ChakrabortyIn this paper we study the mod 2 cohomology ring of the Grasmannian \(\widetilde{G}_{n,3}\) of oriented 3planes in \({\mathbb {R}}^n\). We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This description allows us to provide lower and upper bounds on the cup length of \(\widetilde{G}_{n,3}\). As another

Weight decompositions of Thom spaces of vector bundles in rational homotopy J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190712
Urtzi Buijs, Federico Cantero Morán, Joana CiriciMotivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of FélixOpreaTanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give

A model structure via orbit spaces for equivariant homotopy J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190626
Mehmet Akif Erdal, Aslı Güçlükan İlhanLet G be discrete group and \(\mathcal F\) be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right Gsimplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on Horbits for all H in \(\mathcal F\). This gives a model categorical criterion for maps that induce weak equivalences

Cohomology of infinite groups realizing fusion systems J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190607
Muhammed Said Gündoğan, Ergün YalçınGiven a fusion system \({\mathcal {F}}\) defined on a pgroup S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize \({\mathcal {F}}\). We study these models when \({\mathcal {F}}\) is a fusion system of a finite group G and prove a theorem which relates the cohomology of an infinite group model \(\pi \) to the cohomology of the group G. We show that for

Dense products in fundamental groupoids J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190603
Jeremy BrazasInfinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally pathconnected metric space, the welldefinedness of countable dense products in the fundamental

Minimality in diagrams of simplicial sets J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190530
Carles Broto, Ramón Flores, Carlos GiraldoWe formulate the concept of minimal fibration in the context of fibrations in the model category \({\mathbf {S}}^{\mathcal {C}}\) of \({\mathcal {C}}\)diagrams of simplicial sets, for a small index category \({\mathcal {C}}\). When \({\mathcal {C}}\) is an EIcategory satisfying some mild finiteness restrictions, we show that every fibration of \({\mathcal {C}}\)diagrams admits a wellbehaved minimal

Hearts and towers in stable $$\infty $$ ∞ categories J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190522
Domenico Fiorenza, Fosco Loregian, Giovanni Luca MarchettiWe exploit the equivalence between tstructures and normal torsion theories on a stable \(\infty \)category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded tstructures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s

Comonad cohomology of track categories J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190514
David Blanc, Simona PaoliWe define a comonad cohomology of track categories, and show that it is related via a long exact sequence to the corresponding \(({\mathcal {S}}\!,\!\mathcal {O})\)cohomology. Under mild hypotheses, the comonad cohomology coincides, up to reindexing, with the \(({\mathcal {S}}\!,\!\mathcal {O})\)cohomology, yielding an algebraic description of the latter. We also specialize to the case where the

Involutions on surfaces J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190514
Daniel DuggerWe use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, with a complete classification finally appearing in the 1990s. In this paper we give a different approach to the classification, using techniques that are more accessible to algebraic topologists as well as a new invariant (which we

Matrix factorizations for quantum complete intersections J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190328
Petter Andreas Bergh, Karin ErdmannWe introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such factorizations.

Characteristic classes as complete obstructions J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190313
Martina RovelliIn the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal

Homotopy types of SU ( n )gauge groups over nonspin 4manifolds J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190312
Tseleung SoLet M be an orientable, simplyconnected, closed, nonspin 4manifold and let \({\mathcal {G}}_k(M)\) be the gauge group of the principal Gbundle over M with second Chern class \(k\in {\mathbb {Z}}\). It is known that the homotopy type of \({\mathcal {G}}_k(M)\) is determined by the homotopy type of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\). In this paper we investigate properties of \({\mathcal

Some characterizations of acyclic maps J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190304
George RaptisWe discuss two categorical characterizations of the class of acyclic maps between spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks along \(\pi _0\)surjective maps define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined

Tate cohomology of connected ktheory for elementary abelian groups revisited J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190110
Po Hu, Igor Kriz, Petr SombergTate cohomology (as well as Borel homology and cohomology) of connective Ktheory for \(G=({\mathbb {Z}}/2)^n\) was completely calculated by Bruner and Greenlees (The connective Ktheory of finite groups, 2003). In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to \(p>2\) prime. We also identify the resulting spectra, which are products of Eilenberg–Mac

Algebraic Hopf invariants and rational models for mapping spaces J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20190103
Felix WierstraThe main goal of this paper is to define an invariant \(mc_{\infty }(f)\) of homotopy classes of maps \(f:X \rightarrow Y_{\mathbb {Q}}\), from a finite CWcomplex X to a rational space \(Y_{\mathbb {Q}}\). We prove that this invariant is complete, i.e. \(mc_{\infty }(f)=mc_{\infty }(g)\) if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure

Computations of orbits for the Lubin–Tate ring J. Homotopy Relat. Struct. (IF 0.382) Pub Date : 20181218
Agnès Beaudry, Naiche Downey, Connor McCranie, Luke Meszar, Andy Riddle, Peter RockWe take a direct approach to computing the orbits for the action of the automorphism group \(\mathbb {G}_2\) of the Honda formal group law of height 2 on the associated Lubin–Tate rings \(R_2\). We prove that \((R_2/p)_{\mathbb {G}_2} \cong \mathbb {F}_p\). The result is new for \(p=2\) and \(p=3\). For primes \(p\ge 5\), the result is a consequence of computations of Shimomura and Yabe and has been