Exponentiable morphisms, partial products and pullback complements

Exponentiable morphisms of domains

Given a map f in the category ω-Cpo of ω-complete posets, exponentiability of f in ω-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in ω-Cpo, showing the existence of partial products of the two-point ordered set S = {0 < 1} (Theorem 1.8). Using this ch...

On Exponentiable Morphisms in Classical Algebra

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphi...

Partial Complements and Transposable Dispersions

Suppose A = {a(i, j)}, for i ≥ 0 and j ≥ 0, is the dispersion of a strictly increasing sequence r = (r(0), r(1), r(2), . . .) of integers, where r(0) = 1 and infinitely many postive integers are not terms of r. Let R be the set of such sequences, and define t on R by tr(n) = a(0, n) for n ≥ 0. Let F be the subset of R consisting of sequences r satisfying ttr = r. The set F is characterized in t...

Exponentiable Streams and Prestreams

Inspired by a construction of Escardó, Lawson, and Simpson, we give a general construction of C-generated objects in a topological construct. When C consists of exponentiable objects, the resulting category is Cartesian-closed. This generalizes the familiar construction of compactly-generated spaces, and we apply this to Krishnan’s categories of streams and prestreams, as well as to Haucourt st...

Partial Morphisms in Categories of Effective Objects