On the role of description (Q5936011)

The problem treated in this paper is connected with the possibility of an alternative axiomatization of the subobject classifier of a topos. It is proved that, if axiomatizing internal equality through a ``diagonal classifier'' (instead of the subobject classifier), and if an axiom asking for the existence of a ``description operator'', which enables to ``peek'' the sole element of any singleton, is added, then a new definition for the notion of topos is obtained. In this way the role of sentences like ``the unique \(x\) in \(A\) such that \dots'' in ordinary mathematical language is illuminated.