Internal categories as models of homotopy types

[摘要] A homotopy n-type is a topological space which has trivial homotopy groups abovedegree n. Every space can be constructed from a sequence of such homotopy types, in asense made precise by the theory of Postnikov towers, yielding improving `approximations'to the space by encoding information about the first n homotopy groups for increasing n.Thus the study of homotopy types, and the search for models of such spaces that can befruitfully investigated, has been a central problem in homotopy theory.Of course, a homotopy 0-type is, up to weak homotopy equivalence (isomorphism ofhomotopy groups), a discrete set. It is well-known that a connected 1-type can be represented,again up to weak homotopy equivalence, as the classifying space of its fundamentalgroup: this is the geometric realization of the simplicial set that is the nerve of the groupregarded as a category with one object. Another way to phrase this is that the homotopycategory of 1-types obtained by localizing at maps which are weak homotopy equivalences| formally adding inverses for these | is equivalent to the skeleton of the category ofgroups.In [Mac Lane and Whitehead] it was proved that connected homotopy 2-types canbe modeled, in the sense described above, by crossed modules of groups. A crossed moduleis equivalently what in [Loday] is called a 1-cat-group, but now often referred to as acat1

[发布机构] University:University of Glasgow;Department:School of Mathematics and Statistics

[关键词] Abstract homotopy theory, model categories, simplicial objects