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    • CommentRowNumber1.
    • CommentAuthorThomas Holder
    • CommentTimeOct 5th 2020

    Added a reference to

    • Christian Maurer, Universes in Topoi , pp.285-296 in Lawvere, Maurer, Wraith (eds.), Model Theory and Topoi , LNM 445 Springer Heidelberg 1975.

    diff, v19, current

    • CommentRowNumber2.
    • CommentAuthorkrinsman
    • CommentTime14 hours ago
    In the conditions 1-4 defining "universe", should 1. be "every monomorphism a: A -> I, for which I is U-small, is also U-small"?

    Otherwise, why wouldn't the identity morphism of every object, include "U-large" objects, be "U-small"? E.g. the identity morphisms for E and U from el: E -> U?

    (I guess the identity morphism of a "U-large" object V being small doesn't preclude the morphism V -> 1 being U-large, but if so obviously my intuition for this is bad.)

    E.g. if the "ambient topos" is the category of Z(F)C sets, E is the disjoint union of all hereditarily finite sets, and U is a natural numbers object, then shouldn't monomorphisms (i.e. injective functions) whose domain is an infinite set be "U-large"?

    P.S. Even dumber question: what is the notation "el" for the morphism "el: E -> U" intended to suggest? "Elevate"? "Elementary"?
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTime9 hours ago
    • (edited 9 hours ago)

    While I didn’t write this:

    The point is that morphisms are regarded as stand-ins for (the families of) their fibers: In saying that such a morphism “is UU-small” one means to say that all its fibers are UU-small, hence that it represents a family of UU-small sets.

    Here the index-set of these UU-small sets itself need not be UU-small. In this sense every identity morphism counts as being UU-small, since it represents a family of singleton sets.

    Finally, the notation “elel” is meant to refer to the “elements” of the universe, I think.

    (All of this would be good to further clarify in the entry. But I won’t edit at the moment.)