guest post by Mike Stay
Thanks to primates’ propensity for trade and because our planet rotates, everyone is familiar with abelian groups. We can add, we’ve got a zero element, addition is commutative and associative, and we can negate any element—or, using multiplicative language: we can multiply, we’ve got a 1 element, multiplication is commutative and associative, and we can divide 1 by any element to get its inverse.
Thanks to the fact that for most practical purposes we live in ℝ 3, everyone’s familiar with at least one vector space. Peano defined them formally in 1888. Vector spaces are “categorified” abelian groups: instead of being elements of a set, vector spaces are objects in a compact closed category. But we can “multiply” them using the tensor product; we have the 1-dimensional vector space that plays the role of 1 up to an isomorphism called the unitor; the tensor product is associative up to an isomorphism called the associator, and is commutative up to an isomorphism called the braiding; and every object A has a “weak inverse”, or dual, an object equipped with morphisms for “cancelling”, e A:A⊗A *→1 and i A:1→A *⊗A and some “yanking” equations. As always in categorification, when we weaken equations to isomorphisms, we have to add new equations: a pentagon equation for the associator, triangle equations for the unitors, hexagon equations for the braiding, and braiding twice is the identity.
Thanks to the fact that everyone has family and friends, everyone is (ahem) familiar with relations. Sets, relations, and implications form a compact closed bicategory. In a compact closed bicategory, we weaken the equations above to 2-isomorphisms and add new equations: an associahedron with 14 vertices, 7-vertex prisms for the unitors, shuffle polytopes and a map of permutahedra for the braiding, and an equation governing the syllepsis. The syllepsis is what happens when we weaken the symmetry equation: braiding twice is merely isomorphic to the identity.