Australian Category Seminar

Any geometric subject should begin with the correct abstract operations. To begin, we should have a category with finite limits, subobject classifiers, exponentiation. For homotopy theory we should have an object I to parametrize paths; ie paths shoud be a representble functor into $\omega$-categories. The idea is to first study this structure and later study homotopy classes of functions. In fact, work with Michael Johnson has lead to considering many sorted co-$\omega$-category objects (where the sorts form an $\omega$-category) to parametrize paths. The Moore-path category is obtained from a co-category with sorts being real intervals. The free monoid on a set arises from a co-monoid with sorts being natural numbers. In simplicial sets, Street's orientals yield a many-sorted co-$\omega$-category to parametrize paths (Johnson).