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Suppose $S$ is a family of $L$-structures where $L$ is some collection of constant symbols, relation symbols, and function symbols. Does the coproduct of elements of $S$ exist?

If not, how does one prove it?

If yes, how is the coproduct defined? Are the maps from elements of S to the coproduct all monic?

Also, any references speaking about this would be appreciated; something that involves mathematical logic and category theory perhaps.

Thanks in advance for your answer.

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If you look at examples it's quite clear that there isn't an easy uniform way of describing the coproduct. For example, in the category of all groups, the coproduct is the so-called "free product", but in the category of all abelian groups, the coproduct is the direct sum. This should be particularly disturbing since the two theories have the same signature and differ in only one equational axiom! –  Zhen Lin May 13 '12 at 18:12
    
So in investigating this question, would it be fair to assume that the domain of the coproduct (if it exists) is going to be the disjoint union of the domains of elements of $S$ (since that is how coproducts work in sets)? If that is the case, I need to formulate how (constant, relation, and function)-symbols are interpreted in the coproduct with that disjoint union as its domain. –  atat May 13 '12 at 18:41
    
That won't work. The underlying set of the coproduct in the examples I mentioned is most certainly not the disjoint union of the underlying sets. –  Zhen Lin May 13 '12 at 18:51
    
Ok, good. Thanks. –  atat May 13 '12 at 19:46
    
Herrlich and Strecker also point out that some classical constructions called products are actually categorical coproducts, and others called sums are really categorical products... Without a database or ontology everyone can agree on, it's difficult to sort it all out. –  alancalvitti May 17 '12 at 3:06

3 Answers 3

In the full-blown generality appearing in the question the answer is that it is very difficult to say much at all. Especially since the formulation of the question mentions (and even that is in a somewhat vague form) the objects but not the morphisms. The notion of coproduct depends crucially on the morphisms. One way to make the question more precise is as follows. Assume that some $L$ structures are given and that some morphisms between these are given so that a category is formed. When are there guaranteed to be coproducts? Well, in this full-blown generality the answer is that it is impossible to know. A more tangible question will thus be: Given some $L$ structure and all of their naturally occurring morphisms, forming a category. Are there coproducts? Even this is too general. In some cases (e.g., groups) coproducts exist. In other cases (e.g., fields) coproducts do not exist.

To really make sense of the situation one needs to delve into the realm of universal algebra, equationally definable theories, operads and other general (but not too general) uniform descriptions of 'algebraic structures'. In such cases much more (but still an absolute answer can't be expected) can be said about when coproducts (and other limits/colimits) exist or not and even obtain formulas when they do exist.

The most relevant reference involving both category theory and logic in the most straightforward manner is universal algebra.

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Thanks for your answer. By morphism, I mean logical homomorphism. I believe I have a candidate for the coproduct tweaked from something someone told me. It is long for a comment so I will give a link to a pdf. In the document, there are some things I know are lacking such as a lack of proof that the "injections" $m_{i}$ are homomorphisms and several instances of needing to prove definitions are well defined. I'm a bit disturbed because of the case of fields as you mentioned. universaldungeon.org/media/4-sci.logic-5-16-2012.pdf –  atat May 16 '12 at 16:05
    
@atat, can you specify what you mean by "language" and the nature of L, I and the morphism between them - Sets and functions? Wyler wrote (NLTQ 1991) "experience tells us that the constructions used in category theory are probably safe, and independent of any particular underlying system of foundations as long as they are true constructions, expressible in a first-order language." –  alancalvitti May 17 '12 at 3:09
    
Sure. By language I mean a collection of constant symbols, n-ary relation symbols (n>0), and n-ary function symbols (n>0). I is some non-empty set used to index the set of L-structures. And by morphism, I mean homomorphism in the sense given in the following article: en.wikipedia.org/wiki/… (The notation there is a bit different.) One problem with the pdf document I linked to is that the $m_{i}$ don't seem to be homomorphisms. –  atat May 17 '12 at 15:44
    
Let me know if this is valid reasoning: universaldungeon.org/media/coproduct_of_structures.pdf –  atat May 18 '12 at 6:08

You definitely have a coproduct if only 1-ary function symbols belong to $L$. The domain is the coproduct of domains (which is a disjoint union of sets, of course) of elements of $S$. The function of the coproduct modeling a symbol $f$ is the parallel coproduct of functions modeling the symbol $f$ in all elements of $S$.

A parallel coproduct (this is a homemade name) is the map of the coproduct functor on morphisms in the category of sets. I hope you can find its concrete definition somewhere.

Example of such $L$: a multirelation between $U_0, U_1, \ldots$ defined as functions $R\to U_0, R\to U_1, \ldots$. I cannot elaborate on this but can redirect you to [1]. The author talks about coproducts of relational systems, but I believe that relations and multirelations are quite similar in the respect.

  1. Foniok. Homomorphisms and structural properties of relational systems.
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One thing you can do is to take the atomic diagrams of the two models and then consider its Henkin model. It can only be done if they don't contradict each other. For example, if $c$ and $d$ are two constants in the language, and $M_1 \vDash c=d$ and $M_2 \nvDash c=d$, is there a coproduct? How should $c$ and $d$ be interpreted in the coproduct? Usual homomorphism will not work here.

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This is probably going to see like a very naive question. If c and d are different constant symbols then how can $M_{1}\models c=d$?I wish I had more room to comment right here. In the document universaldungeon.org/media/4-sci.logic-5-16-2012.pdf the interpretation of relation symbols is given and I'm interpreting equality as a binary relation symbol. The rest of the comment can be found here: universaldungeon.org/media/2-math.stackexchange.5-17-2012.pdf –  atat May 17 '12 at 16:00
    
@atat, constants are symbols of language, I mean these are two different symbols. A model defines their interpretation by mapping them to members of the model's universe, if the model maps them to the same member then they are equal in the model. Two different constant symbols might be equal in a model while unequal in another one. If you want to know more about model theory I would suggest David Marker's book, the first few chapters should give you a good understanding of basic model theory. –  Kaveh May 17 '12 at 20:04
    
The question about the categorical structure of category of models (of a logic) is also studied by model theorist though I don't know any accessible reference, maybe someone else can direct you to one if you ask a new question. –  Kaveh May 17 '12 at 20:10
    
I didn't realize that "interpretation" does not need to be 1-1. Basic error on my part. –  atat May 17 '12 at 21:10

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