Models of intuitionistic set theory in subtoposes of nested realizability toposes

Given a partial combinatory algebra (pca) A (see e.g. [16]) together with a subpca A# of A we will construct the nested realizability topos RT(A,A#) as described in [5] (without giving it a proper name there). It is well known (from e.g. [16]) that RT(A,A#) appears as the exact/regular completion of its subcategory Asm(A,A#) of assemblies. In [5] the authors considered two complementary subtoposes of RT(A,A#), namely the relative realizability topos RTr(A,A#) and the modified relative realizability topos RTm(A,A#), respectively. Within nested realizability toposes we will identify a class of small maps giving rise to a model of intuitionistic set theory IZF (see [6, 13]) as described in [11]. For this purpose we first identify a class of display maps in Asm(A,A#) which using a result of [2] gives rise to the desired class of small maps in the exact/regular completion RT(A,A#) of Asm(A,A#). For showing that the subtoposes RTr(A,A#) and RTm(A,A#) also give rise to models of IZF we will prove the following general result. If E is a topos with a class S of small maps and F is a subtopos of E then there is a class SF of small maps in F which is obtained by closing sheafifications of maps in S under quotients in F . As explained in subsections 1.2.2 and 1.2.3 below this covers also the Modified Realizability topos as studied in [15] and the more recent Herbrand topos of van den Berg.