Journal of Homotopy and Related Structures最新文献

We introduce a construction that produces from each bialgebra H an operad (mathsf {Ass}_H) controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of H and the Koszul model of H. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take H to be the mod-2 Steenrod algebra ({mathscr {A}}), then this notion of an associative H-algebra coincides with the usual notion of an (mathscr {A})-algebra considered by homotopy theorists. This makes available to us an operad (mathsf {Ass}_{{mathscr {A}}}) along with its minimal model that controls the category of associative ({mathscr {A}})-algebras, and the notion of strong homotopy associative ({mathscr {A}})-algebras.

For a simply connected rationally elliptic CW-complex X, we show that the cohomology and the homotopy Euler–Poincaré characteristics are related to two new numerical invariants namely (eta _{X}) and (rho _{X}) which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of X.

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra (mathscr {A}^*). We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of (mathscr {A}^*) and its structure as a Hopf algebra.

We introduce graded ({mathbb {E}}_{infty })-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective ({mathbb {N}})-graded ({mathbb {E}}_{infty })-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the (infty )-category of almost perfect quasi-coherent sheaves over a spectral projective scheme (text { {Proj}},(A)) associated to a connective ({mathbb {N}})-graded ({mathbb {E}}_{infty })-ring A can be described in terms of ({{mathbb {Z}}})-graded A-modules.

We compare different algebraic structures in twisted equivariant K-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.

We compute the homotopy groups of the (C_2) fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a ({mathrm {TMF}})-module, it is isomorphic to the tensor product of ({mathrm {TMF}}) with an explicit finite cell complex.

Given a marked (infty )-category (mathcal {D}^{dagger }) (i.e. an (infty )-category equipped with a specified collection of morphisms) and a functor (F: mathcal {D}rightarrow {mathbb {B}}) with values in an (infty )-bicategory, we define , the marked colimit of F. We provide a definition of weighted colimits in (infty )-bicategories when the indexing diagram is an (infty )-category and show that they can be computed in terms of marked colimits. In the maximally marked case (mathcal {D}^{sharp }), our construction retrieves the (infty )-categorical colimit of F in the underlying (infty )-category (mathcal {B}subseteq {mathbb {B}}). In the specific case when , the (infty )-bicategory of (infty )-categories and (mathcal {D}^{flat }) is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable (infty )-localization of the associated coCartesian fibration ({text {Un}}_{mathcal {D}}(F)) computes . Our main theorem is a characterization of those functors of marked (infty )-categories ({f:mathcal {C}^{dagger } rightarrow mathcal {D}^{dagger }}) which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits

We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtain an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.

Let (mathcal {M}) be a model category and let Z be an object of (mathcal {M}). We show that if (mathcal {M}) is cofibrantly generated, cellular, left proper, or right proper, then both the model category of objects of (mathcal {M}) over Z and the model category of objects of (mathcal {M}) under Z are as well.

The string coproduct is a coproduct on the homology with field coefficients of the free loop space of a closed oriented manifold introduced by Sullivan in string topology. The coproduct and the Chas-Sullivan loop product give an infinitesimal bialgebra structure on the homology if the Euler characteristic is zero. The aim of this paper is to study the string coproduct using Sullivan models in rational homotopy theory. In particular, we give a rational model for the string coproduct of pure manifolds. Moreover, we study the behavior of the string coproduct in terms of the Hodge decomposition of the rational cohomology of the free loop space. We also give computational examples of the coproduct rationally.