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# Translation Groupoids and Orbifold Cohomology

Published:2009-12-04
Printed: Jun 2010
• Dorette Pronk
• Laura Scull
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## Abstract

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.
 Keywords: orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions
 MSC Classifications: 57S15 - Compact Lie groups of differentiable transformations 55N91 - Equivariant homology and cohomology [See also 19L47] 19L47 - Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 18D05 - Double categories, $2$-categories, bicategories and generalizations 18D35 - Structured objects in a category (group objects, etc.)