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How to amalgamate a family of structures?

  1. Dec 13, 2012 #1
    How to "amalgamate" a family of structures?

    Given a collection of structures all using the same language, is there a way to come up with a structure C such that each given structure is, in any sense, a "subobject" of C?

    Right now I'm looking at coproducts to see if that will help. Please see the attached pdf.
     

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  2. jcsd
  3. Dec 14, 2012 #2

    Stephen Tashi

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  4. Dec 14, 2012 #3

    chiro

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    Re: How to "amalgamate" a family of structures?

    Hey phoenixthoth.

    For this kind of mathematics, do you have a set theory equivalent of an intersection to extrapolate a subset or "sub-object" of a complete object?
     
  5. Dec 14, 2012 #4
    Re: How to "amalgamate" a family of structures?

    The Rosetta Stone article looks interesting.. Something that unifies certain aspects of Physics, topology, and logic & computation. Should be a fascinating read.

    Yes. (btw, when I say 'structure' I mean this.)

    If you have two structures of the same signature, just intersecting their domains and leaving the signature alone would give rise to the type of object you're referring to. If the signatures are different, the situation is more complicated.

    If coproducts of structures exist, then the coproduct object is exactly what I need. Every time I find a reference that mentions coproducts, the reference says that they don't always exist. So the main question is whether or not coproducts of structures exist in general or perhaps with additional assumptions.
     
  6. Dec 14, 2012 #5

    Hurkyl

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    Re: How to "amalgamate" a family of structures?

    Consider a theory with a constant symbol C and a unary function f.

    Define one structure S to have a single element CS. f(CS) = CS.

    Define another structure T to have two elements CT and *, and f(CT) = f(*) = *.

    Suppose S and T are both substructures of some bigger structure U.

    If i is the embedding S --> U, then [itex] f(C_U) = f(i(C_S)) = i(f(C_s)) = i(C_S) = C_U[/itex]

    If j is the embedding T --> U, then [itex] f(C_U) = f(j(C_T)) = j(f(C_T)) = j(*) \neq j(C_T) = C_U[/itex]

    and thus we have a contradiction.



    I think constant symbols make the difference. If you don't have any, then I believe you can just take the disjoint union of all of your structures, then arbitrarily extend all of the functions to have values on all inputs, but having constant symbols forces relations between the different structures if you try to amalgamate them.
     
  7. Dec 14, 2012 #6
    Re: How to "amalgamate" a family of structures?

    I just have a couple of questions.

    How do we know that [itex] f(i(C_S)) = i(f(C_s)) [/itex] and [itex] f(j(C_T)) = j(f(C_T)) [/itex]?

    What if the word "embedding" were replaced with "injective homomorphism (of structures)?"

    Interestingly enough, the disjoint union is the coproduct of families of sets.

    If the results in my pdf are accurate, then I'm not saying S and T are embedded in U; I'm saying that there is an injective homomorphism from S to U and likewise for T. Is there the same contradiction in that case?
     
  8. Dec 14, 2012 #7

    Hurkyl

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    Re: How to "amalgamate" a family of structures?

    "Injective homomorphism" is usually what one means by embedding; I'd add the caveat that "injective" should actually be "monic", but I believe they mean the same thing here.

    Yes: my proof relied crucially on both i and j being homomorphisms, and j being injective.

    Yep; that's why I took the idea of disjoint union as the starting point of trying to figure something out. It's often a good starting point: take the disjoint union, identify what needs to be identified, then add what you need to make it work for whatever you're doing.
     
  9. Dec 14, 2012 #8
    Re: How to "amalgamate" a family of structures?

    Hey Hurkyl

    How do we know

    [itex]j(*) \neq j(C_T) [/itex]

    ?
     
  10. Dec 14, 2012 #9

    Hurkyl

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    Re: How to "amalgamate" a family of structures?

    Because j is assumed injective, [itex]j(*) = j(c_T)[/itex] would imply [itex]* = C_T[/itex].
     
  11. Dec 15, 2012 #10
    Re: How to "amalgamate" a family of structures?

    Gotchya. Thanks for this counterexample.

    Now I'm wondering if there's any way to salvage any sort of concept of amalgamation.

    Suppose I have a structure in a language with constant symbols, and an interpretation of those constant symbols as elements in the domain of that structure. For example, <N, z, S> the structure whose domain is the set of natural numbers, z is a constant symbol interpreted as 0, and S is the function symbol interpreted as successor.

    What is the relation between that structure, <N, z, S> for instance, and a structure obtained by not assigning an interpretation to constant symbol(s), in this case <N, S>.

    What is the relationship between <N, z, S> and <N,S>?
     
  12. Dec 15, 2012 #11
    Re: How to "amalgamate" a family of structures?

    The aggregate of all structures over a fixed language is a proper class.

    So let's look at your U, the amalgam of S and T.

    You proved that U cannot be a structure in which S and T are embedded.

    But is there another type of object, like a "semi-structure", something that is almost a structure but requirements are weakened and S and T are isomorphic to two sub-objects of U, called S' and T'?

    Something like a category of some kind?
     
  13. Dec 15, 2012 #12

    Hurkyl

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    Re: How to "amalgamate" a family of structures?

    Ignoring size issues, if you have a category C of structures and homomorphisms, you can take it's "cocompletion": the category you get by formally adding all colimits to it. In particular, coproducts are colimits.

    If I haven't made a mistake, in the cocompletion, the map from A to the coproduct of A and B is always monic, so this category would have the property you want.

    (the cocompletion is given by taking the category of presheaves on C. C can be viewed as a good subcategory of the category of presheaves by the "yoneda embedding", which by the yoneda lemma is a full and faithful functor)
     
  14. Dec 15, 2012 #13
    Re: How to "amalgamate" a family of structures?

    This site contains relevant info, yes?

    http://ncatlab.org/nlab/show/free+cocompletion


    I'm having a little trouble since I never studied category theory much.

    How would that statement be formalized in terms of C being the category of structures and homomorphisms, is the category that has the property I want PSh(C)?


    edit: in the definition of the free cocompletion of C, which is PSh(C), why do we need the assumption that C is a small category? http://ncatlab.org/nlab/show/category+of+presheaves
     
    Last edited: Dec 15, 2012
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