Boolean algebra as a fragment of the theory of Boolean toposes (Q5951347)

As is well known, the classical presentation of the theory of Boolean functions is based on the understanding of Boolean functions as some relations over the set \(\{0,1\}\). In this paper, the presentation relies on the notion of an arrow, i.e., a mapping abstracted from data. It is shown that many relations of Boolean algebra also take place in arbitrary toposes. However, the so-called ``fundamental'' set of relations, i.e., the set of relations generating all other relations of Boolean algebra, is fulfilled only in Boolean toposes. In other words, the relations of Boolean algebra are fulfilled within the framework of a very narrow class of toposes, namely Boolean toposes. In particular, an example of a Boolean topos can be the category of sets or, what is the same, set-mathematics whose modern building is constructed from ``blocks'' called sets.