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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
    • CommentTimeMar 29th 2024
    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
    • CommentTimeMar 29th 2024
    • (edited Mar 29th 2024)

    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.)

    • CommentRowNumber4.
    • CommentAuthorkrinsman
    • CommentTimeMar 30th 2024
    Perfect, this is a really good explanation! The article makes a lot more sense to me now. Unsurprisingly it turns out the heuristics I was using were wrong.

    For future reference of myself or other readers, a useful reference for the intuition of the ideas being discussed is the second paragraph of p. 86 (section 4.4) of Lawvere and Rosebrugh's "Sets for Mathematics", i.e. how in the category Set the monomorphisms are exactly those functions whose non-empty fibers are singletons, and how for general functions the non-empty fibers can be larger. (Cf. also exercise 4.17 therein.)

    In particular this explains how the "size" (in the "fiber-wise" sense above) of terminal maps X -> 1 also classifies the "size" of the object X itself, because (in Set) such maps have exactly one non-empty fiber, namely X itself. So in fact unless X is a subterminal object ("the empty set or a singleton"), the morphism X -> 1 is always "larger" than the identity id: X -> X (at least in the category Set). (Which I definitely didn't realize before your explanation.)
    • CommentRowNumber5.
    • CommentAuthorkrinsman
    • CommentTimeMar 30th 2024
    Would it be OK / make sense to add a reference to Thomas Streicher's "Fibered Categories a la Jean Benabo" for this article? Thomas Streicher lists it as a good introductory reference in his "Universes in Topoi", and it seems to explain what the term "closed under reindexing" means in section 6.

    The explanation in terms of fibers also seems to suggest that a reader should have more experience / better intuition for fibered categories and / or indexed categories before expecting to be able to appreciate the most important high-level ideas of this article. (E.g. why, as mentioned in section 6, internal categories in a locally cartesian closed category represent indexed (sub)categories is unclear to me as a category theory dilettante.)
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2024
    • (edited Mar 30th 2024)

    Sure, if you have a reference that is useful to you, then it is probably useful to others, too, and worth adding.

    And yes, the idea of universes and of object classifiers is a generalization of that of subobject classifiers: the latter are universes of (just) (-1)-truncated objects.

    This is a rather general theme that is touched on in many entries, but I am not sure if we have one that serves as a stand-alone introduction for readers completely new to the perspective.

    The entry relative point of view could have been such an entry point, but it remains a stub.

    The entry categorical models of dependent type theory, which is all about this theme, too, is more extensive, but it hardly serves as an introduction.