This thesis is based on three papers which focus on representations of Grothendieck toposes as localic groupoids, localic stacks, and categories of points equipped with convergence data.

In Paper I, we analyse various localic covering theorems and localic groupoid representations for toposes given in the literature. We note that each of these arises from a minimal, concrete object: a locale equipped with a suitable 'amorphous' sheaf. With this definition in hand, we abstract the standard recipe for covering theorems, describe and compare the amorphous sheaves associated to standard covering constructions from the literature, and give a general logical characterisation of amorphous objects.

In Paper II, we show that the 2-category of locales is dense in that of toposes, and that toposes can faithfully be represented as stacks over locales. This is achieved by the combination of an original technical result about the 2-localisation induced by a 2-site, and the key observation that open surjections are of lax descent type in toposes.

In Paper III, we build on the recent development that toposes with enough points can be represented as categories equipped with ultraconvergence data, using it to characterise the surjection–embedding and hyperconnected–localic factorisation systems in terms of points. We introduce analogous classes of functors between abstract categories equipped with ultraconvergence data; provide a new characterisation of separating families of points of a topos; and apply it to show preservation and reflection results between these classes of functors and geometric morphisms.