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nLab > Latest Changes: ETCS with elements

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  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2022

    I may not agree that “concrete” is the right word for this.

    I added a query box requesting the anonymous page creator to come discuss at the nForum.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    Author of this page: I’m not entirely sure what a good name for such a model would be, I don’t really like the current name myself. I chose “concrete” in part because in the categorical semantics of the existence of collections of elements in the model category CC is basically the same as a faithful functor U:C→SetU:C \to Set, which is the definition of a concrete category, but mostly because I am terrible at coming up with a better name for this model. So if you have any other suggestions for names feel free to list them out.

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    To elaborate on the “concrete” part, the sets of morphisms of any category are already defined as a functor Mor:C×C→SetMor:C \times C \to Set, function composition ∘ A,B,C:Mor(B,C)×Mor(A,B)→Mor(A,C)\circ_{A,B,C}:Mor(B,C) \times Mor(A, B) \to Mor(A,C) is already defined as a family of functions in SetSet, and a concrete category is defined as a category with a functor U:C→SetU:C \to Set, so one could append to the concrete category the structure of function evaluation defined as a family of functions ε A,B:Mor(A,B)×U(A)→U(B)\epsilon_{A,B}:Mor(A, B) \times U(A) \to U(B) in SetSet satisfying certain axioms.

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022

    Clarifying the last sentence in my previous comment: many (but not all) concrete categories (Set being one of them, but also Ab, CRing, Top, Met, Diff, et cetera) have a function evaluation structure ε A,B:Mor(A,B)×U(A)→U(B)\epsilon_{A,B}:Mor(A,B) \times U(A) \to U(B) as described in the last sentence of the previous comment, which comes from the fact that group homomorphisms/ring homomorphisms/continuous functions/isometries/differentiable functions are functions of sets which could evaluate an element of a domain to return an element in the codomain.

    Of course, in other concrete categories (such as Rel), such a structure doesn’t exist, since relations aren’t functions.

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeMay 18th 2022

    I don’t understand post #5; the faithfulness of U:C→SetU : C \to Set means that you lose nothing by reinterpreting morphisms A→BA \to B of CC as functions U(A)→U(B)U(A) \to U(B).

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022
    Ah, you're right, sometimes I keep on forgetting that the functor is faithful.
    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeMay 18th 2022
    Anyhow, in line with SEAR I suppose I could call the model "SEAF", but it has choice and doesn't have replacement or unbounded separation, so maybe a more proper name for it could be "Bounded SEAFC", but apart from the explicit inclusion of elements and function evaluation, it is pretty much ETCS.

  9. new temporary name, surely there’s a better name out there.

    Anonymous

    diff, v4, current

  10. somehow this more obvious name never crossed my mind until now.

    Anonymous

    diff, v4, current

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