Journal of Homotopy and Related Structures最新文献

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We consider the localisation of the (projective or injective) model category of smooth spaces at the morphisms which become weak equivalences under the singular complex functor. We prove that this localisation agrees with a motivic-style (mathbb {R})-localisation of the model category of smooth spaces. Further, we exhibit the singular complex functor for smooth spaces as one of several Quillen equivalences between model categories for spaces and the above (mathbb {R})-local model category of smooth spaces. In the process, we show that the singular complex functor agrees with the homotopy colimit functor up to a natural zig-zag of weak equivalences. We provide a functorial fibrant replacement in the (mathbb {R})-local model category of smooth spaces and use this to compute mapping spaces in terms of singular complexes. Finally, we explain the relation of our fibrant replacement to the concordance sheaf construction introduced recently by Berwick-Evans, Boavida de Brito and Pavlov.

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 that the right-induced homotopy theory of Bohmann–Osorno (mathcal {E}_*)-categories is equivalent to the homotopy theory of pointed simplicial categories.

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 homotopy pro-nilpotent if it is equivalent to a limit of nilpotent ({ mathcal {O} })-algebras. Our result provides new positive evidence to a conjecture by Francis–Gaisgory on Koszul duality for general operads. As an application, we simultaneously extend the previously known 0-connected and nilpotent ({ mathsf {TQ} })-Whitehead theorems to a homotopy pro-nilpotent ({ mathsf {TQ} })-Whitehead theorem.

The modular operad (H_*(overline{mathcal {M}}_{g,n})) of the homology of Deligne-Mumford compactifications of moduli spaces of pointed Riemann surfaces has a minimal model governed by higher homology operations on the open moduli spaces (H_*(mathcal {M}_{g,n})). Using Getzler’s computation of relations among boundary cycles in (H_4(overline{mathcal {M}}_{1,4})), we give an explicit construction of the first family of such higher operations.

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

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 incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral.

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 can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology.

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 tuning comes from the use of obstruction theory techniques.

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 _{ mathsf {TQ} }) between the limit of the Taylor tower of the identity functor evaluated on A and the ({ mathsf {TQ} })-completion of A. Since, in this context, the identity functor is only known to be 0-analytic, this result extends knowledge of the Taylor tower of the identity beyond its “radius of convergence.”