Torsors and Stacks

It is shown that the groupoid of G-torsors, suitably defined, can be used to construct a model for the stack associated to G, for arbitrary sheaves of groupoids G. This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Geometric modules and Quinn homology theory

LetR be a ring with unit and invariant basis property. In [1], the authors define a functorK(_;R):TOP/LIP _c-LPEP by combining the open cone construction of [7] with a geometric module construction and show this functor is a homology theory. This paper shows that if attention is restricted to objects TOP/LIP _c with a homotopy colimit structure, then the functorK(_;R) is a Quinn homology theory, In particular, for each having a homotopy colimit structure,K(;R) is a homotopy colimit in the category of -spectra. Furthermore, the constituent spectra of this homotopy colimit are obtained naturally from the fibres of .Partially supported by the National Science Foundation under grant number DMS88-03148.Partially supported by the SNF (Denmark) under grant number 11-7792.     

弱上极限及其性质

系统讨论了弱上极限与弱极限的有关性质，并且利用弱上极限的语言给出了Ｂｒｏｗｎ表示定理的一个新的表述。     

On Full and Faithful Kan Extensions

The paper introduces some mild requirements under which a Kan extension is a full embedding that preserves factorization systems and colimits of -directed diagrams. The existence of such Kan extensions is then used to establish conditions equivalent to algebraic universality of concrete categories such as varieties of algebras of any given similarity type.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

A 2-groupoid Characterisation of the Cubical Homotopy Pushout

We give a 2-track-theoretical characterisation of the homotopy pushout of a 3-corner by recognising the mapping 2-simplex as an initial object in a coherent homotopy category of Hausdorff spaces under a 3-corner with morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.     

Homotopical complexity and good spaces

This paper is an exploration of two ideas in the study of closed classes: the -complexity of a space and the notion of good spaces (spaces for which ). A variety of formulae for the computation of complexity are given, along with some calculations. Good spaces are characterized in terms of the functors and . The main result is a countable upper bound for -complexity when is a good space.

     

Pushouts of Categories,Derived Limits,and Colimits

We provide a counterexample to a theorem of Ford, namely a pushout square of categories with all involved functors injective, such that there is no associated exact “Mayer–Vietoris” sequence of derived limits. Further, we construct a Mayer–Vietoris sequence for derived (co)limits under some additional hypotheses, extending the well-known case of a pushout square of group monomorphisms.     

Covers and Directed Colimits

We find conditions on , a class of objects of a Grothendieck category, sufficient for the existence of -covers. The theory includes the existence of flat covers of modules.