Spahn effect algebra of predicates (changes)

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Predicates

Let $C$ be a category with coproducts (written $+$ ). A predicate is a map $X\stackrel{p}{\to} X+X$ such that $[id_X,id_X]\circ p=id_X$ .

A predicate is hence in particular a coalgebra for the $C$ -endofunctor $X\mapsto X+X$ .

If we denote the coproduct inclusions by $k_1:X\to X+X$ and $k_2:X\to X+X$ , then their coproduct $[k_2,k_1]:X+X\to X+X$ is the swap map (which is a self-inverse equivalence).

For a predicate $p$ the orthocomplement of $p$ is defined to be the composite $p^\perp:=[k_2,k_1]\circ p$ . We have $(p^\perp)^\perp\simeq p$ .

Effect algebra of predicates

Lemma

In a extensive category we have the following diagrams involving coproducts and pullbacks:

(…lots of diagrams…)

Corollary

Predicates with the operations defined by (…) assemble to an effect algebra called the effect algebra of predicates.

Examples

If $X\in Set$ is a set and $p:X\to X+X$ is a predicate on $X$ , then $p(x)=k_1 x$ or $p(x)=k_2 x$ .

In particular we can identify $p$ with a subset $U=\{x|p(x)=k_1 x\}\subseteq X$ . Then $\neg U=\{x|p(x)=k_2 x\}=\{x|p^\perp(x)=k_1 x\}$

Proof

In $Set$ we have disjoint coproducts such that in our case we have

\array{ 0&\to &X\\ \downarrow&&\downarrow^{k_2}\\ X&\stackrel{k_1}{\to}&X+X }

is a pushout square as well as a pullback square. And in $Set$ we have universal coproducts such that in our case we have $X+X\simeq X$ . Now if we assume $[id_X,id_X]\circ p\simeq id_X$ the couniversal property of the coproduct implies that $p$ factors through $k_1$ and $k_2$ followed by an equivalence.

A category having disjoint- and universal coproducts is called an extensive category.

Further examples are listed at

coalgebra of the real interval
some of the examples at coalgebra for an endofunctor

Created on January 8, 2013 at 03:28:48. See the history of this page for a list of all contributions to it.