A monoid, as I understood it a few weeks ago, was a rather simple algebraic structure. Consider a set M of elements with a binary operation * that is associative along with an element which acts as the identity with respect to the operation. It’s like a group but the elements needn’t all have inverses in M. The usual example is under addition. While I tend to like things with less and less structure these seemed a bit barren to me.
They show up in MacLane’s text quickly as a Monoid has just enough structure to be thought of as a category all by itself. It’s the category with a single formal object and the morphisms are the elements. Composition of morphisms is given by the binary operation on the monoid and the identity morphism is the identity element. It would look like a point with lots of arrows (as many as there are elements) leaving and arriving at the point.
Well that’s neat but perhaps contrived. I’m reading “Frobenius Algebras and 2D Topological Quantum Field Theories” by Joachim Kock and I’ve come across it again. The idea of a monoid can be abstracted so we don’t have to “look inside” the object (like we do with sets and their elements). An object M is a monoid if it has an morphism where I left “*” to make the product ambiguous because it needn’t always be the cartesian product as we’ll see. We also need some notion of an identity. I’m fairly sure there’s more to it than that, but that’s why this post is “Monoids – 1” and not “Monoids: the definitive post”
Consider the category 2Cob of 2-cobordisms. This consist of oriented compact 1-manifolds as objects and oriented compact 2-manifolds as morphisms. Given two manifolds A and B the 2-manifold acts as morphism by having as its boundary the disjoint union of A and B. The orientation is what gives it ‘direction’ from A to B or vice versa.
Now consider which is an object of 2Cob. This is a monoid. The product on 2Cob (and in fact nCob) is the disjoint union so we would need a morphism (The means ‘disjoint union’). This morphism is shown below,
The identity is just the cylinder for some interval I. It’s fun to draw the picture that indicates associativity of this operation. I haven’t used the disjoint union much before and I can see it’s importance here. set-theoretically but is a pair of circles that are distinguishable from one another. The wikipedia page on disjoint union gives the details of the construction.
This is a nice example because we’re not multiplying any elements of . This isn’t as a subset of . This is a rather abstract sense and there’s no peeking inside because there isn’t anything inside of except for open and closed sets.
Categories can, themselves, be monoidal. In a naive sense (the only sense I have at the moment) they are categories that possess a product that is associative and an identity element relative to that product. So for nCob this would be the disjoint union and the cylinder while for the category of vector spaces (also monoidal) this would be the tensor product with its base field as the identity. These will turn out to be important since Topological Quantum Field Theories are functors between nCob and Vect. That’s not quite right, they are structure preserving functors and one of the structures is that both categories are monoidal.