Restriction categories and their free cocompletion

The universal property of the Yoneda embedding which exhibits the presheaf category Ĉ as the free cocompletion of C is well-known to category theorists. On the other hand, restriction categories are less well-studied (having only been introduced since the early 1990's). In this thesis, we describe free cocompletion within the restriction setting by introducing the notion of restriction presheaf. We also motivate and give a difinition of co-complete M-category and cocompletion restriction category.