This is a cobordism between 2 copies of a circle and three copies.
A monoid is a set with an associative multiplication. A monoidal category is a category with a `multiplication' that is associative `up to isomorphism'. The classical example is the category, Vect, of finite dimensional vector spaces with the tensor product as the multiplication. This is also the easiest example to study in detail as we have the numerous tools and algorithms of linear algebra available in it.
A beautiful geometric example of a monoidal category comes from the study of closed manifolds of some fixed dimension d say, with disjoint union as the `multiplication', but what should be the morphisms in the category. These are the cobordisms.
A topological quantum field theory is a way of comparing a cobordism
category with the category of vector spaces, in other words a representation
of Cobord(d) on Vect. Of course that representation should preserve the
`tensor' so should send a disjoint union of manifolds to a tensor product of
the corresponding vector spaces.
Homotopy Quantum Field Theories were introduced by Turaev to allow cobordisms with extra structure to be handled.He classified those 1+1 dimensional HQFTs with background space a K(G, 1). Brightwell and Turner have classified those with background space a K(G, 2). Turaev and Porter have recently extended these two cases to handle an arbitrary 2-type as background. This is the most general case for this dimension. The key idea is that of a formal map with coefficients in a crossed module or, more generally still, a crossed complex. These correspond to various types of higher dimensional bundles akin to gerbes. (Two preprints, the first with Turaev on Hqfts the second on the interpretation, together with prepint on Yetter's invariant with Faria Martins, are available plus preprint versions of earlier papers (on TFQTs defined using crossed modules and homotopy n-types resp.). I also have available some slides from my talks at the Streetfest July 2005 and the extended version given at the Cambridge Category Theory Seminar, November 2005. For these just click here for the Canberra slides and here for the Cambridge version.)