Journal of Homotopy and Related Structures最新文献

We show that the homotopy groups of a connective (mathbb {E}_k)-ring spectrum with an (mathbb {E}_k)-cell attached along a class (alpha ) in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to (alpha ) through degree 2n. Using this, we prove that the (2n-1)st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that, after localizing at a prime, (X(n+1)) is homotopically unique as the (mathbb {E}_1-X(n))-algebra with homotopy groups in degree (2n-1) killed by an (mathbb {E}_1)-cell. Lastly, we prove analogous theorems for a sequence of (mathbb {E}_k)-ring Thom spectra, for each odd k, which are formally similar to Ravenel’s X(n) spectra and whose colimit is also MU.

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank (ge n), if every normal semisimple reductive R-subgroup of G contains (({{mathrm{{{mathbf {G}}}_m}}}_{,R})^n). We prove that if G has isotropic rank (ge 1) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra (A=R[x_1,ldots ,x_n]/I) over R, the map (H^1_{mathrm {Nis}}(A,G)rightarrow H^1_{mathrm {Nis}}(R,G)) induced by evaluation at (x_1=cdots =x_n=0), is a bijection. If k has characteristic 0, then, moreover, the map (H^1_{acute{mathrm{e}}mathrm {t}}(A,G)rightarrow H^1_{acute{mathrm{e}}mathrm {t}}(R,G)) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is (ge 2), and A is square-free, then (K_1^G(A)=K_1^G(R)), where (K_1^G(R)=G(R)/E(R)) is the corresponding non-stable (K_1)-functor, also called the Whitehead group of G. The corresponding statements for (G={{mathrm{GL}}}_n) were previously proved by Ton Vorst.

Define a (mathcal V^{(d)})-algebra as an associative algebra with a symmetric and invariant co-inner product of degree d. Here, we consider (mathcal V^{(d)}) as a dioperad which includes operations with zero inputs. We show that the quadratic dual of (mathcal V^{(d)}) is ((mathcal V^{(d)})^!=mathcal V^{(-d)}) and prove that (mathcal V^{(d)}) is Koszul. We also show that the corresponding properad is not Koszul contractible.

To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map (eta :Orightarrow O') of operads, the corresponding model ({textit{Bimod}}_{O}^{h}(O,;,O')) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad (mathcal {C}_{1}). We also build an explicit weak equivalence of (mathcal {C}_{1})-algebras from the loop space (Omega {textit{Operad}}^{h}(O,;,O')) to ({textit{Bimod}}_{O}^{h}(O,;,O')).

Building on our previous work, we show that the category of non-negatively graded chain complexes of (mathcal {D}_X)-modules – where X is a smooth affine algebraic variety over an algebraically closed field of characteristic zero – fits into a homotopical algebraic context in the sense of To?n and Vezzosi.

This paper introduces an inductive tree notation for all the faces of polytopes arising from a simplex by truncations, which allows viewing face inclusion as the process of contracting tree edges. These polytopes, known as hypergraph polytopes or nestohedra, fit in the interval from simplices to permutohedra (in any finite dimension). This interval was further stretched by Petri? to allow truncations of faces that are themselves obtained by truncations. Our notation applies to all these polytopes. As an illustration, we detail the case of Petri?’s permutohedron-based associahedra. As an application, we present a criterion for determining whether edges of polytopes associated with the coherences of categorified operads correspond to sequential, or to parallel associativity.

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of (wS_bullet ), which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence (wS_bullet {mathcal {E}}({mathcal {A}},{mathcal {C}},{mathcal {B}}) rightarrow wS_bullet {mathcal {A}}times wS_bullet {mathcal {B}}). A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra ({mathbf {K}}({mathcal {A}}) vee {mathbf {K}}({mathcal {B}}) rightarrow {mathbf {K}}({mathcal {E}}({mathcal {A}},{mathcal {C}},{mathcal {B}}))). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.

We prove that the normalization functor of the Dold-Kan correspondence does not induce a Quillen equivalence between Goerss’ model category of simplicial coalgebras and Getzler–Goerss’ model category of differential graded coalgebras.

The existence of a model structure on the category ({mathcal {D}}) of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category ({mathcal {D}}) whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on ({mathcal {D}}) is to introduce diffeologies on the sets (varDelta ^{p})((p ge 0)) such that (varDelta ^{p}) contains the (kmathrm{th}) horn (varLambda ^{p}_{k}) as a smooth deformation retract.

We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs (({mathcal {X}},omega )) in an abelian category ({mathcal {C}}). We show how to construct from (({mathcal {X}},omega )) a projective exact model structure on ({mathcal {X}}^wedge ), the subcategory of objects in ({mathcal {C}}) with finite ({mathcal {X}})-resolution dimension, via cotorsion pairs relative to a thick subcategory of ({mathcal {C}}). We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander–Buchweitz contexts. Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described.