Homotopy Theories of Algebras over Operads

Homotopy theories of algebras over operads, including operads over “little n-cubes,” are defined. Spectral sequences are constructed and the corresponding homotopy groups are calculated.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 278–285.Original Russian Text Copyright © 2005 by V. A. Smirnov.     

一类局部定向完备集及其范畴的性质

本文给出了局部定向完备集的概念及其在此结构下的一种新的双小于关系，从而进一步给出了一种新的连续性概念，接着讨论了局部定向完备集，连续的局部定向完备集等对象的一些性质，最后考察了三种范畴的笛卡儿闭性，并证明了范畴LDCPO是范畴ALG的反射满子范畴．     

Double Short Exact Sequences and K1 of an Exact Category

We introduce a modification of the Bass group K ₁ ^det . Our group is defined by generators and relations, in terms of double short exact Sequences instead of automorphisms. We establish its relation to Quillen's K₁ by dealing with loops in the G-construction.     

Geometrical Decomposition of the Free Loop Space on a Manifold with Finitely Many Closed Geodesics

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.     

范畴 RMRl上的一个函子及范畴A Grn0

本文定义了一个由范畴 _RM_R^l到范畴Ａ G_rn⁰ 的函子Ｇ，并证明了函子Ｇ保持分量正合及全正合，关于范畴ＡＧG_rn⁰ 证明了定理：     

L-fuzzy拓扑空间的层浓度

在Ｌ－ｆｕｚｙ拓扑学中引入了层稠密集、层第一纲集、层Ｂａｉｒｅ空间、层浓度、及层可分空间等概念，研究了它们的基本性质，讨论了它们与分明拓扑学中相应概念的关系．     

Group of Self-Equivalences of Sum Object

In general category, the group of self-equivalences of sum object can decompose as the product of its two subgroups under certain conditions and also several split short exact sequences are obtained. Subsequently, we apply the above results to the category of space under a space and get some results which are dual to those in fibrewise category.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

Meet continuous lattices, limit spaces, and L-topological spaces

In this paper, a systematic investigation of the relationship between meet continuous lattices, limit spaces, and L-topological spaces is given. It is a continuation of the investigation on this topic by Höhle (2000, 2001). The relationship between the Lowen functors and the functors introduced by Höhle (2000, 2001) is made clear.     

The p-chain Spectral Sequence

We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah–Hirzebruch-type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum and Connes and Farrell and Jones to compute algebraic K- and L-groups of group rings and topological K-groups of reduced group C*-algebras.