This note has to do with showing that the “classifying space” of a category, i.e., the geometric realization of the nerve of a category, is homeomorphic to that of the opposite category.
Let be the full inclusion of finite nonempty ordinals into categories. The nerve functor is the “restricted Yoneda embedding”
On there is an involution given by taking opposite categories, and similarly there is an involution on . Denote either involution by . There is an evident isomorphism
i.e., the nerve functor commutes with these involutions. The vertical arrow on the right takes a simplicial set to ; we may call this the opposite simplicial set.
Geometric realization is a functor
where is any convenient category of spaces, such as , and is defined by a coend or tensor product
In this formula, is regarded as an interval consisting of totally ordered elements, and as a topological interval. The hom-object is regarded as living in .
If is the interval with reverse ordering, then there is a homeomorphism
and there is also an interval homeomorphism sending to . Thus we have
Hence the geometric realization of a simplicial set is homeomorphic to the geometric realization of the opposite simplicial set. It follows that the classifying space of a category is homeomorphic to the classifying space of its opposite.