Journal of Homotopy and Related Structures最新文献

We study nilpotency in the context of exact Mal’tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie’s functor calculus. We show in particular that the reflection into the subcategory of n-nilpotent objects is the universal endofunctor of degree n if and only if every n-nilpotent object is n-folded. In the special context of a semi-abelian category, an object is n-folded precisely when its Higgins commutator of length (n+1) vanishes.

We determine the Artin–Mazur étale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the étale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an étale homotopy point of view.

In previous work of the first author and Jibladze, the (E_3)-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the (E_3)-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms (E_r). In this paper, we introduce 2-track algebras and tertiary chain complexes, and we show that the (E_4)-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.

We use symbolic computation (SC) and homological perturbation (HPT) to compute a resolution of the integers (mathbb {Z}) over the integer group ring of G, the fundamental group of the Klein bottle. From this it is easy to read off the homology of the Klein bottle as well as other information.

This paper studies questions of coherence and strictification related to self-similarity—the identity (Scong Sotimes S) in a semi-monoidal category. Based on Saavedra’s theory of units, we first demonstrate that strict self-similarity cannot simultaneously occur with strict associativity—i.e. no monoid may have a strictly associative (semi-) monoidal tensor, although many monoids have a semi-monoidal tensor associative up to isomorphism. We then give a simple coherence result for the arrows exhibiting self-similarity and use this to describe a ‘strictification procedure’ that gives a semi-monoidal equivalence of categories relating strict and non-strict self-similarity, and hence monoid analogues of many categorical properties. Using this, we characterise a class of diagrams (built from the canonical isomorphisms for the relevant tensors, together with the isomorphisms exhibiting the self-similarity) that are guaranteed to commute, and give a simple intuitive interpretation of this characterisation.

We describe the basic ingredients of a general computational framework for performing machine calculations in the cohomology of groups. This has been implemented in the GAP system for computational algebra and the paper is intended to aid those wishing to extend that implementation to their own needs.

Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the homotopy invariance theorem (by Bousfield and Kan), the homotopy colimit theorem (Thomason), Theorems A and B (Quillen), or the homotopy cofinality theorem (Hirschhorn).

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid. This is the published version of arXiv:1302.7019.

We show that the two models of an extensional version of Martin-L?f type theory, those given by the category of equilogical spaces and by the effective topos, are homotopical quotients of appropriate categories of 2-groupoids.