Advanced examples of categories

Question

I'm looking for some example of categories which requires some effort to prove that it is a category (For example it is straightforward to prove that $\mathbf{Set}$ is a category, I don't want that sort of example).

Answer 1

For various flavors of cobordism categories it can be difficult to prove that composition is well-defined. A typical example is Segal's conformal cobordism category used in conformal field theory, where, loosely speaking,

• the objects are finite disjoint unions of circles, and
• the morphisms between unions of circles are (isomorphism classes of) Riemann surfaces having those circles as boundary.

Composition is given by gluing together Riemann surfaces along common boundaries; it's clear that this can be done topologically, but there's some difficulty in showing that one can compatibly glue smooth and holomorphic structures along the boundaries too. The proof that this is possible is nontrivial; see conformal welding for details.

(Also, as described here, this category doesn't have identity morphisms; those need to be added in by hand.)