Monday, 10 June 2013

Bijection Categories

Some thoughts on bijections. One can define a category $\mathsf{Bij}(\kappa)$ whose objects are sets $X$ of some fixed cardinality, and whose morphisms are bijections $X \rightarrow Y$. Let $\kappa$ be a cardinal and consider the class:
$\mathsf{Bij}(\kappa) := \{f: X \rightarrow Y \mid |X| = \kappa \mbox{ and } f : X \rightarrow Y \mbox{ is a bijection}\}$
That is, $\mathsf{Bij}(\kappa)$ is the class of bijections between sets of cardinality $\kappa$.

$\mathsf{Bij}(\kappa)$ is a groupoid, and therefore a category, such that :
i. the objects of $\mathsf{Bij}(\kappa)$ are sets $X$ of cardinality $\kappa$;
ii. the morphisms of $\mathsf{Bij}(\kappa)$ are the bijections $X \rightarrow Y$.
iii. for each $X$, the identity morphism $1_X$ is $i: X \rightarrow X$.
where, for any object $X$, for any $x \in X$, $i(x) = x$.