  Multi-valued functions are common in computable analysis (built upon the Type
2 Theory of Effectivity), and have made an appearance in complexity theory
under the moniker search problems leading to complexity classes such as PPAD
and PLS being studied. However, a systematic investigation of the resulting
degree structures has only been initiated in the former situation so far (the
Weihrauch-degrees).
  A more general understanding is possible, if the category-theoretic
properties of multi-valued functions are taken into account. In the present
paper, the category-theoretic framework is established, and it is demonstrated
that many-one degrees of multi-valued functions form a distributive lattice
under very general conditions, regardless of the actual reducibility notions
used (e.g. Cook, Karp, Weihrauch).
  Beyond this, an abundance of open questions arises. Some classic results for
reductions between functions carry over to multi-valued functions, but others
do not. The basic theme here again depends on category-theoretic differences
between functions and multi-valued functions.
