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Tuesday, August 27, 2013

**Abstract:** Homotopy limits are well-modeled by mapping spaces; a map of diagrams induces a map the other direction on homotopy limits. To map from larger to smaller homotopy limits is very natural, but to build a "wrong way" map, and to get that map for all homotopy functors applied to your diagram -- this is where always cartesian cubes come in. In this talk I will give a classification of square diagrams which are homotopy pullbacks (i.e. cartesian) and which retain this property when any equivalence-preserving functor is applied, as well as discuss work towards a classification for higher dimensional cubes.