Prior work towards the subject of higher-dimensional categories gives rise to several examples of a category over $Cat$ to which the slice-category construction can be lifted universally. The present paper starts by supplying this last clause with a precise meaning. It goes on to establish for any such category a certain embedding in a presheaf category, to describe the image, and hence to derive conditions collectively sufficient for that functor to be an equivalence. These conditions are met in the foremost of the examples: the category of dendrotopic sets.
Keywords: slice categories, presheaf category, polytopic sets
2000 MSC: 18D99
Theory and Applications of Categories,
Vol. 22, 2009,
No. 5, pp 97-135.