braided category

In an ordinary symmetric setting, swapping two objects and swapping them back returns you exactly where you started — like commuting numbers. A braided category relaxes this: you may still swap two tensor factors, but the swap is a genuine braiding, like crossing one strand over another. Doing it twice need not undo it, just as two strands crossed twice stay linked. The picture is literally that of braids, and that is no coincidence.

Formally, a braided monoidal category is a monoidal category (C, ⊗, 1) equipped with a natural family of isomorphisms c_{X,Y} : X ⊗ Y -> Y ⊗ X, the braiding, satisfying the two hexagon axioms relating c to the associativity constraint. When in addition c_{Y,X} ∘ c_{X,Y} = id_{X⊗Y} for all X, Y, the braiding is a symmetry and one has a symmetric monoidal category; a genuinely braided category is one where this fails.

The hexagon axioms are exactly the Yang-Baxter / braid relations in categorical form, so the braiding gives, for each object X, a representation of the braid group on the tensor powers of X. This is why braided categories are the natural target for knot and tangle invariants: a tangle diagram is read as a composite of braidings, cups, and caps, and a braided (ribbon) category turns it into a number.