Foams and Surface Braids as Higher Categories and Generalizations

Five Categorical Principles are articulated: (1) Different things are not equal. (2) Arrows are used to compare things. (3) Doing followed by un-doing may, or may not, be the same as not doing. (4) Simultaneity is illusory, (5) Change followed by exchange is comparable to exchange followed by change. Then starting from a modification of algebraic axioms (multiplication, comultiplication, (co)associativity, and pairing) embedded foams are formulated as quadruple arrows in a certain multi-category. Briefly, quintuple arrows will be discussed in this context, and then we'll move to surface braids, braided 3-manifolds and illustrate a few braided 4-manifolds.