Origin of Species II

Todd Trimble kindly made the following comment:

I hope I'm not being too obnoxiously anticipatory by also remarking that the usual notion of permutative V-operad can be defined very concisely as a monoid in the monoidal category of V-species, where the monoidal product is species substitution.

This is far from being obnoxious, Todd! Struggling physicists appreciate perceptive comments from knowledgeable category theorists who can see where we are trying to go with M Theory. Indeed, although we tend to talk about operads loosely as one object multicategories, the species definition comes very close to what Rivasseau et al have in mind for redefining quantum field theory.

Note also that from the logos perspective (ie. M theoretic higher topos ideas) the category of finite sets (resp. vector spaces), as it sits in the topos Set (resp. Vect), plays an important role in attempting to define the generalised logic behind the simple operators that we associate to measurement algebras. The one major alteration to these structures that M theory requires, which comes up again and again in many guises, is the relaxation of the monoidal condition. This happens naturally with higher dimensional structures, as illustrated by Batanin's tower of coherence laws. For QFT, this forces a complete change of mathematical language, since too many concepts are only defined in categorical terms. Fortunately, everybody can understand simple diagrams of ribbons and trees.

Aside: Post written with wireless connection from my heavenly warm room amidst the fresh snow at the observatory. Can't seem to find any old skis lying about ...