We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups.

Abstract. We give an algebraicization of rational S 1 -equivariant homotopy theory. There is an algebraic category of "T-systems" which is equivalent to the homotopy category of rational S 1 -simply connected S 1 -spaces. There is also a theory of "minimal models" for T-systems, analogous to Sullivan's minimal algebras. Each S 1 -space has an associated minimal T-system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call 'spider moves.' We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.