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  • Homotopic distance between functors
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-10-13
    E. Macías-Virgós, D. Mosquera-Lois

    We 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 LS-category of small categories by Tanaka.

    更新日期:2020-10-13
  • Cohomology and deformations of oriented dialgebras
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-09-16
    Ali N. A. Koam, Ripan Saha

    We 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.

    更新日期:2020-09-16
  • Note on Toda brackets
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-08-28
    Samik Basu, David Blanc, Debasis Sen

    We 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.

    更新日期:2020-08-28
  • Cyclic homology for bornological coarse spaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-24
    Luigi Caputi

    The 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

    更新日期:2020-07-24
  • Bianchi’s additional symmetries
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-20
    Alexander D. Rahm

    In a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of Jean-Pierre 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

    更新日期:2020-07-21
  • Descent theory and mapping spaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-03
    Nicholas J. Meadows

    The 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 quasi-category 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

    更新日期:2020-07-03
  • Higher equivariant and invariant topological complexities
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-06-21
    Marzieh Bayeh, Soumen Sarkar

    In this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.

    更新日期:2020-06-23
  • Transfer ideals and torsion in the Morava E -theory of abelian groups
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-05-23
    Tobias Barthel; Nathaniel Stapleton

    Let A be a finite abelian p-group of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava E-cohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains p-torsion. The proof makes use of transchromatic character theory.

    更新日期:2020-05-23
  • The universal fibration with fibre X in rational homotopy theory
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-04-02
    Gregory Lupton; Samuel Bruce Smith

    Let X be a simply connected space with finite-dimensional 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

    更新日期:2020-04-02
  • The unit of the total décalage adjunction
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-19
    Viktoriya Ozornova; Martina Rovelli

    We 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

    更新日期:2020-03-19
  • Verifying the Hilali conjecture up to formal dimension twenty
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-12
    Spencer Cattalani; Aleksandar Milivojević

    We 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.

    更新日期:2020-03-12
  • An application of the h -principle to manifold calculus
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-11
    Apurva Nakade

    Manifold 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 h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor \(\mathrm {Emb}_{\mathrm {Lag}}(-,N)\) is the totally

    更新日期:2020-03-11
  • Correction to: Representations are adjoint to endomorphisms
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-06
    Gabriel C. Drummond-Cole, Joseph Hirsh, Damien Lejay

    The first equation under section “Remark 3” was processed and published incorrectly. The correct equation should read as follows:

    更新日期:2020-03-06
  • On the capacity and depth of compact surfaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-02-12
    Mahboubeh Abbasi; Behrooz Mashayekhy

    K. 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 non-orientable surface of genus \(g>0\) is

    更新日期:2020-02-12
  • Representations are adjoint to endomorphisms
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-12-30
    Gabriel C. Drummond-Cole; Joseph Hirsh; Damien Lejay

    The 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 self-enrichment 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

    更新日期:2019-12-30
  • Formulae in noncommutative Hodge theory
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-21
    Nick Sheridan

    We 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

    更新日期:2019-11-21
  • The depth of a Riemann surface and of a right-angled Artin group
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-12
    Yves Félix; Steve Halperin

    We consider two families of spaces, X: the closed orientable Riemann surfaces of genus \(g>0\) and the classifying spaces of right-angled 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)]

    更新日期:2019-11-12
  • Twisting structures and morphisms up to strong homotopy
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-08
    Kathryn Hess; Paul-Eugène Parent; Jonathan Scott

    We 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 bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction

    更新日期:2019-11-08
  • Lie theory for symmetric Leibniz algebras
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-10-05
    Mamuka Jibladze; Teimuraz Pirashvili

    Lie 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

    更新日期:2019-10-05
  • Another approach to the Kan–Quillen model structure
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-09-24
    Sean Moss

    By 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 \(\mathop {\mathop {\mathsf {Ex}}^\infty }\)

    更新日期:2019-09-24
  • Correction to: Wrong way maps in uniformly finite homology and homology of groups
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-09-21
    Alexander Engel

    There is an error in the proof of Theorem 2.16 of Ref. 2. It occured at the end of the second-to-last paragraph of the proof.

    更新日期:2019-09-21
  • Parallel transport of higher flat gerbes as an extended homotopy quantum field theory
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-18
    Lukas Müller; Lukas Woike

    We prove that the parallel transport of a flat \(n-1\)-gerbe on any given target space gives rise to an n-dimensional 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 dimension-independent

    更新日期:2019-07-18
  • Enhanced $$A_{\infty }$$A∞ -obstruction theory
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-16
    Fernando Muro

    An \(A_n\)-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 \(A_{n-1}\) 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

    更新日期:2019-07-16
  • On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-12
    Somnath Basu; Prateep Chakraborty

    In this paper we study the mod 2 cohomology ring of the Grasmannian \(\widetilde{G}_{n,3}\) of oriented 3-planes 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

    更新日期:2019-07-12
  • Weight decompositions of Thom spaces of vector bundles in rational homotopy
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-12
    Urtzi Buijs; Federico Cantero Morán; Joana Cirici

    Motivated 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 Félix-Oprea-Tanré 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

    更新日期:2019-07-12
  • A model structure via orbit spaces for equivariant homotopy
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-26
    Mehmet Akif Erdal; Aslı Güçlükan İlhan

    Let 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 G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in \(\mathcal F\). This gives a model categorical criterion for maps that induce weak equivalences

    更新日期:2019-06-26
  • Cohomology of infinite groups realizing fusion systems
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-07
    Muhammed Said Gündoğan; Ergün Yalçın

    Given a fusion system \({\mathcal {F}}\) defined on a p-group 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

    更新日期:2019-06-07
  • Dense products in fundamental groupoids
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-03
    Jeremy Brazas

    Infinitary 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 path-connected metric space, the well-definedness of countable dense products in the fundamental

    更新日期:2019-06-03
  • Minimality in diagrams of simplicial sets
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-30
    Carles Broto; Ramón Flores; Carlos Giraldo

    We 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 EI-category satisfying some mild finiteness restrictions, we show that every fibration of \({\mathcal {C}}\)-diagrams admits a well-behaved minimal

    更新日期:2019-05-30
  • Hearts and towers in stable $$\infty $$ ∞ -categories
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-22
    Domenico Fiorenza; Fosco Loregian; Giovanni Luca Marchetti

    We exploit the equivalence between t-structures 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 t-structures 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

    更新日期:2019-05-22
  • Comonad cohomology of track categories
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-14
    David Blanc; Simona Paoli

    We 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

    更新日期:2019-05-14
  • Involutions on surfaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-14
    Daniel Dugger

    We 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

    更新日期:2019-05-14
  • Matrix factorizations for quantum complete intersections
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-28
    Petter Andreas Bergh; Karin Erdmann

    We 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.

    更新日期:2019-03-28
  • Characteristic classes as complete obstructions
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-13
    Martina Rovelli

    In 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

    更新日期:2019-03-13
  • Homotopy types of SU ( n )-gauge groups over non-spin 4-manifolds
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-12
    Tseleung So

    Let M be an orientable, simply-connected, closed, non-spin 4-manifold and let \({\mathcal {G}}_k(M)\) be the gauge group of the principal G-bundle 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

    更新日期:2019-03-12
  • Some characterizations of acyclic maps
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-04
    George Raptis

    We 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

    更新日期:2019-03-04
  • Tate cohomology of connected k-theory for elementary abelian groups revisited
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-01-10
    Po Hu; Igor Kriz; Petr Somberg

    Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for \(G=({\mathbb {Z}}/2)^n\) was completely calculated by Bruner and Greenlees (The connective K-theory 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

    更新日期:2019-01-10
  • Algebraic Hopf invariants and rational models for mapping spaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-01-03
    Felix Wierstra

    The 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 CW-complex 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

    更新日期:2019-01-03
  • Computations of orbits for the Lubin–Tate ring
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-12-18
    Agnès Beaudry; Naiche Downey; Connor McCranie; Luke Meszar; Andy Riddle; Peter Rock

    We 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

    更新日期:2018-12-18
  • A multiplicative K -theoretic model of Voevodsky’s motivic K -theory spectrum
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-28
    Youngsoo Kim

    Voevodsky defined a motivic spectrum representing algebraic K-theory, and Panin, Pimenov, and Röndigs described its ring structure up to homotopy. We construct a motivic symmetric spectrum with a strict ring structure. Then we show that these spectra are stably equivalent and that their ring structures are compatible up to homotopy.

    更新日期:2018-11-28
  • A comonadic interpretation of Baues–Ellis homology of crossed modules
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-27
    Guram Donadze; Tim Van der Linden

    We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an

    更新日期:2018-11-27
  • Equivariant chromatic localizations and commutativity
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-27
    Michael A. Hill

    In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed

    更新日期:2018-11-27
  • Yoga of commutators in DSER elementary orthogonal group
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-15
    A. A. Ambily

    In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an

    更新日期:2018-11-15
  • A theorem on multiplicative cell attachments with an application to Ravenel’s X ( n ) spectra
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-15
    Jonathan Beardsley

    We show that the homotopy groups of a connective \(\mathbb {E}_k\)-ring spectrum with an \(\mathbb {E}_k\)-cell attached along a class \(\alpha \) in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to \(\alpha \) through degree 2n. Using this, we prove that the \(2n-1\)st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that

    更新日期:2018-11-15
  • The Dold–Thom theorem via factorization homology
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10
    Lauren Bandklayder

    We give a new proof of the classical Dold–Thom theorem using factorization homology. Our method is direct and conceptual, avoiding the Eilenberg–Steenrod axioms entirely in favor of a more general geometric argument.

    更新日期:2018-11-10
  • Isotropic reductive groups over discrete Hodge algebras
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10
    Anastasia Stavrova

    Let G be a reductive group over a commutative ring R. We say that G has isotropic rank \(\ge n\), if every normal semisimple reductive R-subgroup of G contains \(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\). We prove that if G has isotropic rank \(\ge 1\) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra \(A=R[x_1,\ldots ,x_n]/I\) over R, the map \(H^1_{\mathrm

    更新日期:2018-11-10
  • Stabilization of derivators revisited
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10
    Ian Coley

    We revisit and improve Alex Heller’s results on the stabilization of derivators in Heller (J Pure Appl Algebra 115(2):113–130, 1997), recovering his results entirely. Along the way we give some details of the localization theory of derivators and prove some new results in that vein.

    更新日期:2018-11-10
  • Koszuality of the $$\mathcal V^{(d)}$$ V ( d ) dioperad
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-30
    Kate Poirier; Thomas Tradler

    Define a \(\mathcal V^{(d)}\)-algebra as an associative algebra with a symmetric and invariant co-inner product of degree d. Here, we consider \(\mathcal V^{(d)}\) as a dioperad which includes operations with zero inputs. We show that the quadratic dual of \(\mathcal V^{(d)}\) is \((\mathcal V^{(d)})^!=\mathcal V^{(-d)}\) and prove that \(\mathcal V^{(d)}\) is Koszul. We also show that the corresponding

    更新日期:2018-10-30
  • The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-25
    Jeehoon Park; Donggeon Yhee

    Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant \(L_\infty \)-algebras. More precisely, for a given GBV algebra \((\mathcal

    更新日期:2018-10-25
  • Delooping derived mapping spaces of bimodules over an operad
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-17
    Julien Ducoulombier

    To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map \(\eta :O\rightarrow O'\) of operads, the corresponding model \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad \(\mathcal {C}_{1}\). We also build an explicit weak equivalence of \(\mathcal

    更新日期:2018-10-17
  • A combinatorial model for the path fibration
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-29
    Manuel Rivera; Samson Saneblidze

    We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set \({\widehat{{\varvec{\Omega }}}}X\) such that its geometric realization \(|{\widehat{{\varvec{\Omega }}}}X|\), a space built

    更新日期:2018-09-29
  • Unstable splittings in Hodge filtered Brown–Peterson cohomology
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-27
    Gereon Quick

    We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown–Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson’s unstable splitting. As a consequence, we obtain an analog of Quillen’s theorem for Hodge filtered Brown–Peterson cohomology for complex manifolds.

    更新日期:2018-09-27
  • Topology of scrambled simplices
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-27
    Dmitry N. Kozlov

    In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a \(\Delta \)-complex

    更新日期:2018-09-27
  • Homotopical algebraic context over differential operators
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-18
    Gennaro Di Brino; Damjan Pištalo; Norbert Poncin

    Building on our previous work, we show that the category of non-negatively graded chain complexes of \(\mathcal {D}_X\)-modules – where X is a smooth affine algebraic variety over an algebraically closed field of characteristic zero – fits into a homotopical algebraic context in the sense of Toën and Vezzosi.

    更新日期:2018-09-18
  • On the topological computation of $$K_4$$ K 4 of the Gaussian and Eisenstein integers
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-08-18
    Mathieu Dutour Sikirić; Herbert Gangl; Paul E. Gunnells; Jonathan Hanke; Achill Schürmann; Dan Yasaki

    In this paper we use topological tools to investigate the structure of the algebraic K-groups \(K_4(R)\) for \(R=Z[i]\) and \(R=Z[\rho ]\) where \(i := \sqrt{-1}\) and \(\rho := (1+\sqrt{-3})/2\). We exploit the close connection between homology groups of \(\mathrm {GL}_n(R)\) for \(n\le 5\) and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive

    更新日期:2018-08-18
  • Syntactic aspects of hypergraph polytopes
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-08-13
    Pierre-Louis Curien; Jovana Obradović; Jelena Ivanović

    This paper introduces an inductive tree notation for all the faces of polytopes arising from a simplex by truncations, which allows viewing face inclusion as the process of contracting tree edges. These polytopes, known as hypergraph polytopes or nestohedra, fit in the interval from simplices to permutohedra (in any finite dimension). This interval was further stretched by Petrić to allow truncations

    更新日期:2018-08-13
  • Equivariant formality of isotropic torus actions
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-07-24
    Jeffrey D. Carlson

    Considering the potential equivariant formality of the left action of a connected Lie group K on the homogeneous space G / K, we arrive through a sequence of reductions at the case G is compact and simply-connected and K is a torus. We then classify all pairs (G, S) such that G is compact connected Lie and the embedded circular subgroup S acts equivariantly formally on G / S. In the process we provide

    更新日期:2018-07-24
  • Waldhausen Additivity: classical and quasicategorical
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-07-12
    Thomas M. Fiore; Malte Pieper

    We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\). A consequence is Additivity for

    更新日期:2018-07-12
  • On a Quillen adjunction between the categories of differential graded and simplicial coalgebras
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-06-30
    W. Hermann B. Sore

    We prove that the normalization functor of the Dold-Kan correspondence does not induce a Quillen equivalence between Goerss’ model category of simplicial coalgebras and Getzler–Goerss’ model category of differential graded coalgebras.

    更新日期:2018-06-30
  • Model category of diffeological spaces
    J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-06-20
    Hiroshi Kihara

    The existence of a model structure on the category \({\mathcal {D}}\) of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category \({\mathcal {D}}\) whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on \({\mathcal {D}}\)

    更新日期:2018-06-20
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