Homotopy theory using higher dimensional categories

Orbifolds form a mathematical tool which can be used to study certain types of symmetry. Tilings of the plane such as created by the Dutch artist M.C. Escher and crystals are two well-known examples of this kind of symmetry. These examples have in common that the local symmetry is finite. There are finitely many crystals grouped together around any corner of the crystal and there are finitely many tiles with the motive coming together in any center of symmetry of the tiling. In two dimensions there are two basic types of symmetry patterns: purely rotational or a combination of rotations and reflections as in a snowflake. Several types of symmetry may be combined as one can see in the case of crystals. In order to study this type of symmetry it is useful to consider the so-called quotient of the pattern. This quotient is obtained by 'folding' the pattern so that corresponding motives come on top of one another. E.g., for a snowflake the quotient is a thirty degree circle sector. One could think of this circle segment as generating the symmetry; if one wants to cut a snow-flake pattern out of paper, one needs to fold the paper to obtain a thirty degree circle sector and the cut pattern in the folded paper will generate the symmetric pattern of the snowflake. The quotient of a crystal consists of the asymmetric unit of the crystal where one has to identify sides which correspond to each other under the symmetry. Dr. Satake introduced the concept of orbifolds in 1956. They have since been used in mathematics, physics (string theory) and crystallography. My research is focussed on the foundational questions: what properties do orbifolds have and how can we characterize different types of orbifolds? The questions I ask are influenced by the applications in physics and crystallography. The methods I use to describe orbifolds and other objects with symmetry come from category theory. Where mathematics is used to abstract some of the properties of the physical world to obtain models that can be used in a variety of situations, category theory is used to abstract even further and develop theories and constructions that can be applied in several areas of mathematics.