Proposition 8.4.1.1.1 (00KX): Self-Duality for the (2-)Category of Relations

The ($2$-)category of relations is self-dual:

1.
Self-Duality I. We have an isomorphism

\[ \mathrm{Rel}^{\mathsf{op}}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathrm{Rel} \]

of categories.
2.
Self-Duality II. We have a $2$-isomorphism

\[ \boldsymbol {\mathsf{Rel}}^{\mathsf{op}}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\boldsymbol {\mathsf{Rel}} \]

of $2$-categories.

Item 1: Self-Duality I

We claim that the functor

\[ F\colon \mathsf{Rel}^{\mathsf{op}}\to \mathsf{Rel} \]

given by the identity on objects and by $R\mapsto R^{\dagger }$ on morphisms is an isomorphism of categories.

By Chapter 11: Categories, Item 1 of Proposition 11.6.8.1.3, it suffices to show that $F$ is bijective on objects (which is clear) and fully faithful. Indeed, the map

\[ \webleft (-\webright )^{\dagger }\colon \mathrm{Rel}\webleft (A,B\webright )\to \mathrm{Rel}\webleft (B,A\webright ) \]

defined by the assignment $R\mapsto R^{\dagger }$ is a bijection by Chapter 9: Constructions With Relations, of , showing $F$ to be fully faithful.

Item 2: Self-Duality II

We claim that the $2$-functor

\[ F\colon \mathsf{Rel}^{\mathsf{op}}\to \mathsf{Rel} \]

given by the identity on objects, by $R\mapsto R^{\dagger }$ on morphisms, and by preserving inclusions on $2$-morphisms via Chapter 9: Constructions With Relations, of , is an isomorphism of categories.

By , of , it suffices to show that $F$ is:

•
Bijective on objects, which is clear.
•
Bijective on $1$-morphisms, which was shown in Item 1.
•
Bijective on $2$-morphisms, which follows from Chapter 9: Constructions With Relations, of .

Thus $F$ is indeed a $2$-isomorphism of categories.

The Clowder Project

8.4.1 Self-Duality