Restriction presheaves and restriction colimits

Restriction categories, as defined by Cockett and Lack, are an abstraction of the notion of partial functions between sets, and therefore, are important in furthering our understanding of what it means to be partial. This thesis builds upon the work of Cockett and Lack, by providing restriction analogues of notions from ordinary category theory. One such notion is that of free cocompletion. We show that every restriction category may be freely completed to a cocomplete restriction category, and that this free cocompletion can be described in terms of a restriction category of restriction presheaves. Indeed, a restriction presheaf is defined precisely so that this is the case. We then generalise free cocompletion to join restriction categories, which are categories whose compatible maps may be combined in some way. To do this, we introduce the notion of join restriction presheaf, and show that for any join restriction category, its join restriction category of join restriction presheaves is its free cocompletion. The second half of this thesis explores the notion of restriction colimit. More precisely,we define the restriction colimit of a restriction functor weighted by a restriction presheaf. We also show that cocomplete restriction categories may be characterised as those having all such restriction colimits. Finally, we give applications of restriction colimits. Some examples of restriction colimits are gluings of atlases in a restriction category, and composition o frestriction profunctors. We conclude this thesis with notions in category theory that have no analogue in the restriction setting.