The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current article is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as n-fold loop spaces and pointed G-spaces. This language also includes the diagram categories of spectra as well as spectra in the sense of Lewis, which will be discussed in some detail.
Keywords: loop spaces, spectra, Quillen closed model categories, enriched sheaves.
2000 MSC: 55P42, 18F20, 18G55, 55P35.
Theory and Applications of Categories, Vol. 8, 2001, No. 19, pp 490-508.