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”

$Cat \stackrel{y}{\to} Set^{Cat^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta^{op}}.$

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

$N(C^{op}) = Cat(i-, C^{op}) \cong Cat(i(-^{op}), (C^{op})^{op}) = Cat(i(-^{op}), C)$

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

$\array{
Cat & \stackrel{N}{\to} & Set^{\Delta^{op}} \\
^\mathllap{\sigma} \downarrow & \cong & \downarrow^\mathrlap{Set^{\sigma^{op}}} \\
Cat & \underset{N}{\to} & Set^{\Delta^{op}},
}$

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

$R: Set^{\Delta^{op}} \to Top$

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

$R(X) = \int^n X(n) \cdot Int([n], I).$

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

$Int([n], I) \cong Int(\sigma [n], \sigma I)$

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

$\array{
\int^n X(n) \cdot Int([n], I) & \cong & \int^n X(n) \cdot Int(\sigma [n], \sigma I) \\
& \cong & \int^n X(n) \cdot Int(\sigma [n], I) \\
& \cong & \int^n X \sigma (n) \cdot Int([n], I)
}$

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.

Revised on August 15, 2018 at 10:55:03
by
Todd Trimble