Journal of Homotopy and Related Structures最新文献

Let X be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of X and which we call (textsf{Br}^dagger (X)) following Lurie. Toën introduced a map (phi :textsf{Br}^dagger (X)rightarrow H ^2_{acute{e }t }(X,{mathbb {G}}_{textrm{m}})) which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of (phi ) to a subgroup (textsf{Br}(X)subset textsf{Br}^dagger (X)), which we call the derived Brauer group, on which (phi ) becomes an isomorphism (textsf{Br}(X)simeq H ^2_{acute{e }t }(X,{mathbb {G}}_{textrm{m}})). This map may be interpreted as a derived version of the classical Brauer map which offers a way to “fill the gap” between the classical Brauer group and the cohomogical Brauer group. The group (textsf{Br}(X)) was introduced by Lurie by making use of the theory of prestable (infty )-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of (infty )-categories between the Brauer space of invertible presentable prestable ({{mathcal {O}}}_X)-linear categories, and the space (Map (X,K ({mathbb {G}}_{textrm{m}},2))). We offer an alternative proof of this equivalence of (infty )-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.

In this note, we study the delooping of spaces and maps in homotopy type theory. We show that in some cases, spaces have a unique delooping, and give a simple description of the delooping in these cases. We explain why some maps, such as group homomorphisms, have a unique delooping. We discuss some applications to Eilenberg–MacLane spaces and cohomology.

Let (T= S^1times D^2) be the solid torus, (mathcal {F}) the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle (S^1times 0), which is the central circle of the torus T, and (mathcal {D}(mathcal {F},partial T)) the group of diffeomorphisms of T fixed on (partial T) and leaving each leaf of the foliation (mathcal {F}) invariant. We prove that (mathcal {D}(mathcal {F},partial T)) is contractible. Gluing two copies of T by some diffeomorphism between their boundaries, we will get a lens space (L_{p,q}) with a Morse–Bott foliation (mathcal {F}_{p,q}) obtained from (mathcal {F}) on each copy of T. We also compute the homotopy type of the group (mathcal {D}(mathcal {F}_{p,q})) of diffeomorphisms of (L_{p,q}) leaving invariant each leaf of (mathcal {F}_{p,q}).

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield–Kan homotopy spectral sequence of the tower of fibrations, given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in Conant (Am J Math 130(2):341–357. https://doi.org/10.1353/ajm.2008.0020, 2008), we give a combinatorial interpretation of the differentials (d^1) mapping into the diagonal terms, by introducing the notion of (i, n)-marked unitrivalent graphs.

The compact, connected Lie group (E_6) admits two forms: simply connected and adjoint type. As we previously established, the Baum–Connes isomorphism relates the two Langlands dual forms, giving a duality between the equivariant K-theory of the Weyl group acting on the corresponding maximal tori. Our study of the (A_n) case showed that this duality persists at the level of homotopy, not just homology. In this paper we compute the extended quotients of maximal tori for the two forms of (E_6), showing that the homotopy equivalences of sectors established in the (A_n) case also exist here, leading to a conjecture that the homotopy equivalences always exist for Langlands dual pairs. In computing these sectors we show that centralisers in the (E_6) Weyl group decompose as direct products of reflection groups, generalising Springer’s results for regular elements, and we develop a pairing between the component groups of fixed sets generalising Reeder’s results. As a further application we compute the K-theory of the reduced Iwahori-spherical (C^*)-algebra of the p-adic group (E_6), which may be of adjoint type or simply connected.

In this work, we shall study in a purely model-independent fashion the (infty )-category of mixed graded modules over a ring of characteristic 0, as defined by D. Calaque, T. Pantev, M. Vaquié, B. Toën and G. Vezzosi. First, we collect some basic results about its main formal properties, clarifying foundational questions in a systematic manner, to serve as a reference for future work. Finally, we shall endow such (infty )-category with a both left and right complete accessible t-structure, showing how this identifies the (infty )-category of mixed graded modules with the left completion of the Beilinson t-structure on the (infty )-category of filtered modules.

The Quillen–Barr–Beck cohomology of augmented algebras with a system of divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the Kähler differentials of an augmented algebra with a system of divided powers in prime characteristic represents Beck derivations. We give a geometrical interpretation of this statement for the sheaf of relative differentials. As an application in the theory of modular Lie algebras we prove that any special derivation of a divided power algebra is a Beck derivation and we apply the theorem to Witt algebras.

This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for Artin-Tits monoids of spherical type: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.

Let (C_2) denote the cyclic group of order 2. We compute the (RO(C_2))-graded cohomology of all (C_2)-surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether or not the action preserves the orientation, and one other invariant.

This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general (nin {mathbb {N}}). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.