Explicit Morphisms in the Galois-Tukey Category

If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these "cardinal characteristics of the continuum", it was discovered that many of the associated inequalities can be interpreted as morphisms within the "Galois-Tukey" category. This thesis aims to reformulate traditional direct proofs of cardinal characteristic inequalities by making the underlying morphisms explicit. New, purely categorical results are also discussed.