Exponentiable morphisms, partial products and pullback complements

In a category K with finite limits, the exponentiability of a morphisms s is (rather easily) characterised in terms of K admitting partial products (essentially those of Pasynkov) over s; and that of a monomorphism is characterised in terms of the new concept of a pullback complement (a universal construction of a pullback diagram whose top and right sides are given). Then, characterisations, previously given by the first author for the category Sp of topological spaces, of the notions of totally reflective subcategory and of hereditary factorisation system are shown to be instances of simple results on adjointness and factorisations.

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