Translation Groupoids and Orbifold Cohomology

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.

Equivariant shape theory

Equivariant shape theory is an improvement of equivariant homotopy theory which could be regarded as an equivariant version of Borsuk's shape theory. The main result in this paper is a description of the equivariant shape category ShG whose objects are equivariant spaces or G-spaces, i.e. topological spaces endowed with an action of a given topological group G, and whose morphisms are equivariant homotopy or G-homotopy classes of families of multi-valued functions which we call equivariant multi-nets or G-multi-nets. Previously equivariant shape theories have been described only under the assumptions that the group G is either finite or compact. We also study classes of G-spaces on which equivariant shape and equivariant homotopy coincide, look for conditions under which a G-map f: X → Y is an equivariant shape equivalence, and give some characterizations of G-spaces with trivial equivariant shape.