Todd Trimble Nerve of opposite category

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 $i: \Delta \to Cat$ be the full inclusion of finite nonempty ordinals into categories. The nerve functor $N$ is the “restricted Yoneda embedding”

On $Cat$ there is an involution given by taking opposite categories; by restriction along $i$, there is an involution on $\Delta$. Denote either involution by $\sigma$. We have an isomorphism

natural in categories $C$, so there is an evident isomorphism

i.e., the nerve functor commutes with these involutions. The vertical arrow on the right takes a simplicial set $X$ to $X \circ \sigma$; we may call this the opposite simplicial set.

Geometric realization is a functor

where $Top$ is any convenient category of spaces, such as $CG Haus$, and $R$ is defined by a coend or tensor product

In this formula, $[n] \in Ob(\Delta^{op})$ is regarded as an interval consisting of $n+1$ totally ordered elements, and $I$ as a topological interval. The hom-object $Int([n], I)$ is regarded as living in $Top$.

If $\sigma I$ is the interval $I$ with reverse ordering, then there is a homeomorphism

and there is also an interval homeomorphism $I \to \sigma I$ sending $t$ to $1-t$. Thus we have

Hence the geometric realization of a simplicial set $X$ 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.