Slicing Sites and Semireplete Factorization Systems

A factorization system (E ,M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes the P so obtained as follows: their object images have terminal objects, and they admit bicategorically cartesian liftings, up to equivalence, of slice-category projections. It is clear that, and how, (E ,M) can be recovered from such a P . The correspondence thus described is actually the second of three similar ones between certain (E ,M) and certain P that the article presents. In the first one the characterization of the P has all ultimately bicategorical ingredients replaced with their categorical analogues. A category A with such a P is precisely what the author has called a “slicing site”. In the article’s terms the associated (E ,M) are again factorization systems — but the concept conveyed extends the standard one by not obliging isomorphisms to belong to either factor class —, namely those that are “right semireplete” (isomorphisms do belong toM) and “left semistrict” (morphisms inM are monic relative to E). The third correspondence subsumes the other two; here the (E ,M) are all right-semireplete factorization systems.