(∞, n )-Limits I: Definition and first consistency results

We give a model-independent definition of limits for diagrams valued in an (, )category.We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of (, 1)-limits for (, 1)-categories, and with itself across different values of .Contents 1 Limits in an (, )-category 8 1.1 Limits in an (, )-category . . . . . . . . . . . . . . . . . . . . . . .8 1.2 Self-consistency with respect to varying . . . . . . . . . . . . . . . . 10 2 Homotopy limits in a 2-category 14 2.1 Homotopy limits in a 2-category . . . . . . . . . . . . . . . . . . . . .15 2.2 Consistency with respect to strict context . . . . . . . . . . . . . . . .16 3 Auxiliary results 19 3.1 Model structure for weakly horizontally invariant double categories . .19 3.2 Characterization of homotopy 2-limits . . . . . . . . .