Journal of Homotopy and Related Structures最新文献

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).

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, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.

In this paper we study N-differential graded categories and their derived categories. First, we introduce modules over an N-differential 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 N-differential graded categories. Finally, we show that this derived category is triangle equivalent to the derived category of some ordinary differential graded category.

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group 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 one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.

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.