Journal of Homotopy and Related Structures最新文献

By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal.

In this paper, we study the (mathbb {Z}/2) action on complex Grassmann manifolds (G_{n}(mathbb {C}^{2n})) given by taking orthogonal complement. We completely compute the associated (mathbb {Z}/2) Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds (G_n(mathbb {R}^{2n})) by Baralić et al. [Forum Math., 30 (2018), pp. 1539–1572].

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring ({mathcal {C}}) over a noncommutative ring A, we show that all A-flat ({mathcal {C}})-comodules are (aleph _1)-directed colimits of A-countably presentable A-flat ({mathcal {C}})-comodules. In the context of a complete, separated topological ring ({mathfrak {R}}) with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat ({mathfrak {R}})-contramodules are (aleph _1)-directed colimits of countably presentable flat ({mathfrak {R}})-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat ({mathcal {C}})-comodules and flat ({mathfrak {R}})-contramodules as (aleph _1)-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion ({mathfrak {R}})-contramodules, all the contramodules of cocycles are cotorsion.

We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher (A_infty ) products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of (A_infty ) algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

We classify all 2-term (L_infty )-algebras up to isomorphism. We show that such (L_infty )-algebras are classified by a Lie algebra, a vector space, a representation (all up to isomorphism) and a cohomology class of the corresponding Lie algebra cohomology.

This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.

By making use of Halperin’s local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré. In the latter half of this article, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with the pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau–de Rham cohomology algebra of the diffeological space. By using the pullback construction, we also discuss a local system model for a topological homotopy pushout.

Let A be the Steenrod algebra over the finite field (k:= {mathbb {F}}_2) and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, (textrm{Ext}^{q, *}_A(k, k),) for all homological degrees (q geqslant 0.) The Singer algebraic transfer of rank q,  formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to (textrm{Ext}^{q, *}_A(k, k).) Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all (qgeqslant 4.) This paper establishes Singer’s conjecture for rank four in the generic degrees (n = 2^{s+t+1} +2^{s+1} - 3) whenever (tne 3) and (sgeqslant 1,) and (n = 2^{s+t} + 2^{s} - 2) whenever (tne 2,, 3,, 4) and (sgeqslant 1.) In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.

We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality.