Characterizations of some (weak) Grothendieck topologies

Let [Formula: see text] be a small well-powered category which has pullbacks. The aim of this paper is to study and characterize two (weak) topologies on presheaf category [Formula: see text], one of these is constructed by means of a subfunctor of the Yoneda functor which corresponds to an ideal of [Formula: see text], called (weak) ideal topology, and another constructed by a dominion. Naturally, these (weak) topologies of [Formula: see text] introduce two (weak) Grothendieck topologies on [Formula: see text]. To find and study their relations, using the induced presheaf [Formula: see text] given by the fixed dominion [Formula: see text] on [Formula: see text] we construct an action of [Formula: see text] on the subobject classifier [Formula: see text] of [Formula: see text]. Then, we investigate which one of (weak) Grothendieck topologies corresponds one-to-one to the ideals of [Formula: see text] and which one is a sub [Formula: see text]-set of [Formula: see text]. Moreover, among other things, we give some conditions under which the Grothendieck topologies on [Formula: see text] associated to the double negation topology [Formula: see text] and two topologies [Formula: see text] and [Formula: see text] (mentioned above) are sub [Formula: see text]-sets of [Formula: see text].