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

CatySet Cat opSet i opSet Δ op.Cat \stackrel{y}{\to} Set^{Cat^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta^{op}}.

On CatCat there is an involution given by taking opposite categories; by restriction along ii, there is an involution on Δ\Delta. Denote either involution by σ\sigma. We have an isomorphism

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

natural in categories CC, so there is an evident isomorphism

Cat N Set Δ op σ Set σ op Cat N Set Δ op,\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 XX to XσX \circ \sigma; we may call this the opposite simplicial set.

Geometric realization is a functor

R:Set Δ opTopR: Set^{\Delta^{op}} \to Top

where TopTop is any convenient category of spaces, such as CGHausCG Haus, and RR is defined by a coend or tensor product

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

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

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

Int([n],I)Int(σ[n],σI)Int([n], I) \cong Int(\sigma [n], \sigma I)

and there is also an interval homeomorphism IσII \to \sigma I sending tt to 1t1-t. Thus we have

nX(n)Int([n],I) nX(n)Int(σ[n],σI) nX(n)Int(σ[n],I) nXσ(n)Int([n],I)\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 XX 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