Journal of Homotopy and Related Structures最新文献

We 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 of Munkholm.

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.

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.

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 (Gmathbf {BornCoarse}) of equivariant bornological coarse spaces to the cocomplete stable (infty )-category (mathbf {Ch}_infty ) of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory (mathcal {X}K^G_{}) and to coarse ordinary homology?({{,mathrm{mathcal {X}H},}}^G) by constructing a trace-like natural transformation (mathcal {X}K_{}^Grightarrow {{,mathrm{mathcal {X}H},}}^G) that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for ({{,mathrm{mathcal {X}HH},}}_{}^G) with the associated generalized assembly map.

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 compactification of the quotient of hyperbolic 3–space by (mathrm {SL_2}(mathcal {O})). Consider the map?(alpha ) induced on homology when attaching the boundary into the Borel–Serre compactification. How can one determine the kernel of?(alpha ) (in degree 1) ? Serre used a global topological argument and obtained the rank of the kernel of?(alpha ). He added the question what submodule precisely this kernel is.

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 for the presheaf of quasi-categories associated to a presheaf of model categories to be a higher stack. In the final section, we apply this result to construct the higher stack of unbounded complexes associated to a ringed site.

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.

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.

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. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.