Thesis by publication. / Thesis (PhD)--Macquarie University (Division of Information & Communication Sciences, Dept. of Mathematics), 2007. / Bibliography: p. 119-123. / Introduction -- Mackey functors on compact closed categories -- Lax braidings and the lax centre -- On centres and lax centres for promonoidal catagories -- Pullback and finite coproduct preserving functors between categories of permutation representations -- Conclusion. / This thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally promonoidal categories ... The third contribution of this thesis is the study of functors between categories of permutation representations. / x,123 p. ill