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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2014
    • (edited Mar 3rd 2014)

    I gather the following is true and is shown in Battenfield-Schröder-Simpson (pdf), but I haven’t really fully absorbed yet how AdmRepAdmRep is actually embedded in RT(𝒦 2)RT(\mathcal{K}_2).

    The subcategory on the effectively computable morphisms of the function realizability topos RT(𝒦 2)RT(\mathcal{K}_2) is the Kleene-Vesley topos KVKV. The category of “admissible representations” AdmRepAdmRep (whose morphisms are computable functions (analysis), see there) is a reflective subcategory of RT(𝒦 2)RT(\mathcal{K}_2) (BSS) and the restriction of that to KVKV is AdmRep effAdmRep_{eff}

    AdmRep eff KV AdmRep RT(𝒦 2) \array{ AdmRep_{eff} &\hookrightarrow& KV \\ \downarrow && \downarrow \\ AdmRep &\hookrightarrow& RT(\mathcal{K}_2) }

    This is currently stated this way in the entry function ralizability and computable function (analysis), but please criticize/handle with care, I’ll try to further fine-tune as need be.