Journal of Homotopy and Related Structures最新文献

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 what seems to be the first published proof of the structure (known to Leray and Koszul) of the cohomology rings

We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra (mathcal{H}_n). More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of (mathcal{H}_n), and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of (mathcal{H}_n) to the Gelfand–Fuks cohomology of the Lie algebra (W_n) of formal vector fields on ({mathbb {R}}^n) respects this multiplicative structure. We then illustrate the machinery for (n=1).

Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.

Let R be any ring with identity and ( Ch (R)) the category of chain complexes of (left) R-modules. We show that the Gorenstein AC-projective chain complexes of?[1] are the cofibrant objects of an abelian model structure on ( Ch (R)). The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when R is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly generated. Constructing this model structure also shows that every chain complex over any ring has a Gorenstein AC-projective precover. These are precisely Gorenstein projective (in the usual sense) precovers whenever R is either a Ding-Chen ring, or, a ring for which all level (left) R-modules have finite projective dimension. For a general (right) coherent ring R, the Gorenstein AC-projective complexes coincide with the Ding projective complexes of [31] and so provide such precovers in this case.

Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent quasi-categories. The method is purely combinatorial and extends to semisimplicial objects in other categories; in particular to semi-simplicial spaces satisfying the Segal condition (semi-Segal spaces).

We generalize Freyd’s well-known result that “homotopy is not concrete”, offering a general method to show that under certain assumptions on a model category (mathcal {M}), its homotopy category (textsc {ho}(mathcal {M})) cannot be concrete. This result is part of an attempt to understand more deeply the relation between set theory and abstract homotopy theory.

The derivatives of the identity functor on spaces in Goodwillie calculus form an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this present paper we carry out similar computations for mod p homology for odd primes p, also for non-connected spectra.

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.

Adapting a result of Félix–Halperin–Lemaire concerning the Lusternik–Schnirelmann category of products, we prove the additivity of a rational approximation for Schwarz’s sectional category with respect to products of certain fibrations.