Homomorphisms as structure-preserving maps

  1. Fredrik

    Fredrik 10,263
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    Homomorphisms as "structure-preserving" maps

    A function f between groups is said to be a homomorphism if it "preserves" the product in the sense that f(xy)=f(x)f(y). A function f between fields is said to be a homomorphism if it "preserves" both addition and multiplication in the sense that f(xy)=f(x)f(y), f(x+y)=f(x)+f(y). Homomorphisms are often described as "structure-preserving maps". We should be able to define homomorphisms as structure-preserving maps, if we define "structure" first, and also what it means for the structure to be "preserved". I suggest the following definition:

    We define a structure as a 4-tuple (S,R,O,C), where S is a set, R is a set of relations on S, O is a set of operations on S, and C is a subset of S. The members of C are called constants. If (S,R,O,C) and (S',R',O',C') are structures, a map f:S→S' is said to be structure preserving if the following holds for all x,y,x1,...,xn in S:

    a) If r is a relation in R such that (x,y) is in r, then there's a r' in R' such that (f(x),f(y)) is in r'.
    b) For each n=1,2,..., if m is an operation of arity n in O, then f(m(x1,...,xn))=m(f(x1),...,f(n))
    c) If x is in C, then f(x) is in C'.

    (A relation of arity n is a subset of Sn. An operation of arity n is a function from Sn into S).

    Unless I missed something essential (and I might have), this should take care of the definition of "homomorphism" for groups, rings, fields, topological spaces and a lot more, but not for metric spaces, vector spaces, manifolds and a lot more. A metric space involves a function that isn't an operation. A vector space also involves a function (scalar multiplication) that isn't an operation, and it also involves a previously defined structure (a field). And manifolds...uh...I don't even want to think about it.

    What I'd like to know if there is a more general definition that covers these other things as well. I know the term homomorphism is used for vector spaces, but I don't think I've heard it used for metric spaces. I've heard people say that diffeomorphisms are isomorphisms between manifolds, but perhaps they just meant isomorphisms as they are defined in category theory (where they don't have to be structure preserving).

    By the way, I have studied some mathematical logic since the other discussion, so I'm now familiar with how structures/algebras are defined in model theory/universal algebra. I understand the reason why texts on those subjects talk about relation symbols instead of relations and so on, and I feel that we don't need to do that here, as long as we're just looking for a single definition of homomorphism.
     
    Last edited: Jul 2, 2010
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  3. Office_Shredder

    Office_Shredder 4,500
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    Re: Homomorphisms as "structure-preserving" maps

    You should look into category theory
     
  4. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    I'm familiar with the basic definitions of category theory, so I know that it doesn't answer my question. It can't tell us which functions are the homomorphisms of a given class of structures (e.g. groups). Instead, the choice of what functions to call homomorphims (or just morphisms) is part of what defines a category. For example, category theory allows us to define a category that consists of the class of vector spaces and all functions between vector spaces that satisfy f(x+y)=f(x)+f(y) for all x,y. This defines a category of vector spaces, just not the standard one, and its isomorphisms are group isomorphisms, not what we'd normally consider vector space isomorphisms (i.e. linear bijections).
     
  5. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    If you have any prior notion of "object with structure" and "structure-preserving map", you can collect all instances into a category. Then your prior notion of "object with structure" is now the same as the new notion of "object in that category", and similarly for maps.

    You can accelerate thing by skipping the first step of trying to come up with a prior notion of structure. :wink:


    A constant is a nullary operation -- i.e. an operation of arity 0. You don't need to treat them separately.


    m is not an element of O'; it is not an operation on S'.


    Merely mapping constants to constants is inadequate. e.g. in a variety of universal algebras with two constant symbols 0 and 1, the function associated to a morphism must actually satisfy f(0)=0 and f(1)=1.



    Fields: Their characterization is essentially non-algebraic. I suppose you luck out because you don't care about characterizing them, and the notions of ring homomorphism between fields and field homomorphism between fields are equivalent.

    Topological spaces: You have types of stuff: points and open sets. Morphisms are awkward because the induced functions work in both directions: points are mapped forwards, but open sets are mapped backwards. You could switch to the notion of Locale so that you have only one type of stuff, but the function associated to a morphism still works backwards.

    Homotopy types: The homotopy category is not concrete -- roughly speaking its objects cannot be represented as "sets with structure", no matter clever a notion of structure you might come up with.


    A diffeomorphism is indeed an isomorphism in the category of smooth manifolds. (Or, more verbosely, the category of smooth manifolds and smooth maps) The reason we have many words like "diffeomorphism", "isometry", "homeomorphism" and so forth is that we are frequently interested in several categories at once. e.g. when studying Riemann manifolds, we might be interested both in smooth metric-preserving maps as well as plain ordinary smooth maps, and from time to time even merely twice-differentiable maps!
     
  6. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    I know. Also, every operation of arity n is a relation of arity n+1, so we don't even have to treat operations separately. I chose to treat them separately to make the definition look more like the definitions in the mathematical logic books I'm reading and the definition in this book on universal algebra that I downloaded. (Except that they don't include any relations).

    D'oh. That first one was a major brain fart by me, and the second was was pretty silly too. I'm glad you caught them, because now I finally see why these books talk about relation symbols instead of relations and so on. It's not just because it has to be that way when we're dealing with first-order formal languages. It's also needed for the definition of homomorphisms that I'm trying to make. New attempt:

    I define a structure to be a triple (S,R,I) where S is a set, R is a set of relation symbols such that the arity of the symbol is explicitly included in the notation (e.g. r(n) is a relation symbol with arity n), and I is a function that assigns a relation on S of the appropriate arity to each relation symbol.

    Suppose that (S,R,I) and (S',R,I') are two structures with the same relation symbols. A function f:S→S' is said to be a homomorphism if

    [tex](f(x_1),...,f(x_n))\in I'r^{(n)} \Leftrightarrow (x_1,...,x_n)\in Ir^{(n)}[/tex]

    for each relation symbol r(n) of arity n and for each n=1,2,... (I think this would be a bit clearer if we treat operations and constants separately).

    That's another good catch. You're quite useful. :smile: Now I'm going to have to think about whether my (new) definition allows e.g. a non-abelian division ring to be isomorpic to a field (i.e. abelian division ring).


    I was thinking that S is the set and that each subset of S is a unary relation on S. You're right about the homomorphisms. My improved definition doesn't work. This makes me wonder why homeomorphic topological spaces are considered equivalent, instead of topological spaces that would be isomorphic according to my definition.

    I guess what we really want is a notion of equivalence that ensures that every theorem that we derive using the properties of a specific "structure" (a concept that I still haven't defined satisfactorily) will be guaranteed to hold for any structure that's "equivalent" to the one we used.

    If "structure" doesn't apply to all these things that are defined by "a set and then some", like groups, fields, topological spaces, Hilbert spaces and manifolds, then I wonder, isn't there some other term that can be used? Aren't they all special cases of a single more general concept?
     
    Last edited: Jul 3, 2010
  7. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    It's OK for other people to contribute here too. :smile: (Not that I don't appreciate Hurkyl's contributions. I always do). Let's focus on one specific detail to make things easier. Why is the category of topological spaces that has continous maps as morphisms more interesting than the category of topological spaces that has open maps as morphisms? (f:X→Y is said to be open if f(E) is open for every open E).

    I've been thinking that we want the morphisms to have the property that the corresponding isomorphisms tell us when two structures are equivalent in the sense that any theorem that holds for a specific structure also holds for all isomorphic structures. Does that hold for topological spaces when the isomorphisms are continuous bijections but not when they're bijective open maps?
     
  8. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    I assume you meant "homeomorphism" and not "continuous bijection"?


    Incidentally, all isomorphisms in the category of topological spaces and continuous maps are also open.
    Additionally, all isomorphisms in the category of topological spaces and open maps are also continuous.
     
  9. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    Yes. I assumed those two were the same, but I guess I should have thought it through.

    So...the isomorphisms of the first category are the same as the isomorphisms of the second category. I guess that solves that problem.

    I'll try to prove the things you wrote here as an exercise. Thank you again for your comments.

    Edit: I'm too lazy to try hard to find an example of a continuous bijection with an inverse that isn't continuous, but unless I made a mistake, I have verified the other details.
     
    Last edited: Jul 4, 2010
  10. dx

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    Re: Homomorphisms as "structure-preserving" maps

    I think the only way we can have a useful general theory of 'structure preserving maps' is if that theory talks only about properties that all structure preserving maps possess, and this is the case with category theory as far as I can tell.

    To me, this seems to imply that this general theory cannot define a structure preserving map as a specific type of map between sets, but only characterize them by their structure-independent features (and therefore the theorems of category theory apply to all 'structures'). For example, we require that composition of structure preserving maps φ : A → B and χ : B → C must be a structure preserving map ψ : A → C and that there exists an 'identity structure preserving map' and so on.

    The reason this is powerful is kinda the same as the reason the notion of 'topological space' is powerful. It isolates and characterizes the idea of 'continuity' in a terminology which ensures that no irrelevant features of specific examples enter the description. To study continuous functions on Euclidean space, we don't need much of the information contained in the euclidean metric. So the old definition of continuity (For every epsilon, there exits a delta such that...) is 'redundant'. The modern definition of continuity (If Δ is open, so is φ-1[Δ]) uses a terminology which does not refer to the metric at all, and applies to any topological space.
     
    Last edited: Jul 4, 2010
  11. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    But category theory doesn't have a concept of "structure preserving" as far as I know. For example, we can define a category of groups with morphisms that don't "preserve" group multiplication.

    I would like these things to follow logically from a general definition of "structure preserving". And I think my second attempt in this thread works (although it would have been clearer if I had mentioned operation symbols and constant symbols explicitly). It seems to work at least for groups, rings (and therefore also fields), modules (and therefore also vector spaces), and topological spaces. I wonder if we can also make it work for metric spaces, manifolds, fiber bundles... I have to get some sleep, but I'll think about that tomorrow.
     
  12. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    It doesn't. So that's one less thing to worry about.
     
  13. dx

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    Re: Homomorphisms as "structure-preserving" maps

    In category theory, the 'structure' of the objects in a category is completely encoded in the morphisms and their compositions. So you can't say that the morphisms 'don't preserve the structure', since they define what the 'structure' is in that particular category. For example, consider a topological group. If this is viewed as an object in the category of topological spaces, there is no need for the morphisms to preserve the group structure.

    Consider the definition of a topoligical space. The axioms that characterize open sets do not follow from a general definition of 'open set'. This reminds me of something that Niels Bohr said: 'A mutually exclusive relationship always exists between the practical use of any word and attempts at its strict definition'.
     
  14. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    Then the structure that is being preserved by the morphisms isn't the group structure. :wink:


    One nice concrete1 example of a category is the category of natural numbers and matrices. The objects of this category are natural numbers, and they are only really there to serve as a reminder of when we're allowed to multiply two matrices.

    But if you really want to, you can think of the arrows as structure-preserving maps. What structure? The one implicitly defined by matrix arithmetic, of course! And we can use matrices to describe "elements" and even talk about subobjects (via the rowspace of a matrix).

    Of course, this category is a concrete category; it's equivalent to the category of finite-dimensional vector spaces and linear transformations. The forgetful functor sends a natural number n to the set Rn, and it sends matrices to the appropriate set functions.


    1: Meant in the natural language sense.
     
  15. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    OK, that makes sense. But it makes me wonder if this is a useful point of view. The reason why the explicitly defined "structure-preserving maps" are significant is that they give us a concept of "isomorphism" that ensures that theorems derived using the properties of a specific structure (e.g. SO(3)) holds for all structures that are isomorphic to it (e.g. SU(2)/Z2). That can be proved as a metatheorem. Is there a corresponding metatheorem about the isomorphisms we end up with when we define the structure implicitly by a choice of arrows, and what exactly does it say?
    I don't really understand this comment. I haven't suggested that they should. What I have suggested is that the appropriate concept of "isomorphism" should come from taking the morphisms to be (explicitly) structure-preserving, i.e. in this case they should be open maps, not continuous maps. However, it also turns out that both choices give us the same isomorphisms.
     
    Last edited: Jul 5, 2010
  16. Landau

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    Re: Homomorphisms as "structure-preserving" maps

    The standard example is
    [tex][0,2\pi )\to S^1[/tex]
    [tex]t\mapsto e^{it}.[/tex]
     
  17. dx

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    Re: Homomorphisms as "structure-preserving" maps

    A topological space is defined as a set X together with a collection of subsets of X called its 'topology', such that

    i.) If A and B are in the topology, then so is their intersection.

    ii.) If Aλ (λ in Λ) are in the topology, so is their union.

    iii.) X and {} are in the topology.

    These properties of open sets are not derived from a definition of 'open set', but merely postulated. We have an intuitive idea of what an open set is: "a set which contains all points sufficiently close to to each of its points", and such intuitve ideas are always at the back of ones mind when the language of topological spaces is being used. Otherwise it would just be a formal play with symbols.

    Similarly, we have an intuitive idea of what a 'structure preserving map' is, and from that idea we abstract the essential features so that they can be discussed independently of specific examples. An isomoprhism is a morphism A → B such that there exists a morphism B → A which composes with the original map to give the identity (both ways). This is a 'categorical definition' of an isomorphism, and does not require any concrete definition of what a 'morphism' is. It is taken to be a primitive notion, and enters only as an element of the Hom(_,_) sets.

    Such categorical definitions are useful because they apply to all categories. For example, if you are able to find a categorical definition of the 'direct product' or 'direct sum' of two objects, that definition can be applied to any category, and therefore the language of category theory helps us find appropriate definitions. If you want to consider objects of the form of triples (S, R, I), then category theory will help you find a natural definition of a 'product' of such objects, etc.
     
    Last edited: Jul 5, 2010
  18. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    In the end, I bet it simply boils down to two things (which I'm not sure are actually different):
    • Theorem: If F is a functor from C to D, and f is an isomorphism in C, then F(f) is an isomorphism in D
    • For any diagram drawn in a category, we can create an 'isomorphic' diagram by replacing its objects with isomorphic ones (choosing a specific isomorphism), and composing the arrows with the isomorphisms and their inverses as appropriate.
    (A functor is a "structure-preserving map" between categories, if you will. It only preserves structure up to natural isomorphism, though)

    Example 1: "Cartesian product" is a functor from Set x Set to Set. It immediately follows that if X and X' are bijective sets, and Y and Y' are bijective sets, then X x Y and X' x Y' are bijective sets.

    Example 2:: The "singular homology" construction defines functors on the category Top of topological spaces and continuous maps. It immediately follows that homeomorphic topological spaces have isomorphic singular homology groups.


    Following up on that latter example: there is a relation on continuous maps called "homotopy". This relation is an equivalence relation. It is even a "congruence" on Top: f~g implies fh~gh and kf~kg. It turns out that that homotopic maps turn into equal maps after taking singular homology. There are a variety of other purposes for which the difference between homotopy and equality is insignificant.

    Therefore, it is desirable to construct a new category hTop -- the homotopy category -- by modding out by this equivalence relation. Its objects are again topological spaces, but the arrows are now equivalence classes of continuous maps. Isomorphims in this category are homotopy equivalences. An isomorphism class of objects is a homotopy type.

    So to reorganize the above data, homology is a property of homotopy types; the structure of "topological space" contains a lot of irrelevant information. Homology defines a functor on hTop, and the various other purposes I mentioned can be carried out in hTop rather than in Top.

    So, in these cases, the topic of interest really is this category hTop (or other related constructions). Whatever 'structure' a homotopy type consists of, that's the structure we want to study.

    The problem... hTop is not concrete. This is the example I gave earlier of a category that cannot be viewed as "sets with structure". There is a theorem that any functor U:hTop->Set must not be faithful. That means there are homotopy types X and Y, and a pair of distinct arrows f,g from X to Y with the property that U(f) = U(g).
     
  19. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    I should also add you can build a formal language out of diagrams rather than logical connectives as is usually done. I imagine that in such a language, preservation of logic by isomorphism is self-evident.

    I didn't pay too much attention to it, but the one construction I saw was used to prove a cute little theorem: a property (described in this language) of categories is preserved and reflected by equivalence of categories if and only if the property can be described on a blackboard. (i.e. which amounts to never asserting as a conclusion that two objects must be equal. Isomorphic is still okay)

    ("preserved" means that if X has the property, then so does F(X). "reflected" means that if F(X) has the property, then so does X)
     
    Last edited: Jul 5, 2010
  20. Hurkyl

    Hurkyl 16,090
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    Re: Homomorphisms as "structure-preserving" maps

    I've attached a diagram that expresses the proposition "A pair of objects has a product".

    A model in a category C of this diagram is a functor from the 2-point category into C (i.e. a pair of objects) with the following property:

    There exists a functor from the category drawn in the second diagram that is compatable with the first one (i.e. every pair of arrows X <-- Z --> Y such that X and Y were the objects selected by the first functor) with the property that:

    For every functor from the third diagram compatible with the second functor:

    There exists a unique functor from the fourth diagram compatible with the third. (the triangles commute)



    In this language, it's immediate that the property is invariant under isomorphisms; if I replace the two objects with isomorphic ones (and choose a specific isomorphisms), I can produce compatible natural isomorphisms on all of the relevant functors by swapping those two objects for their isomorphic copy, and composing the arrows appropriately.

    It's also immediately evident that if XxY is a product of X and Y, then XxY is also a product of X' and Y'. (If X' and Y' are isomorphic to X and Y)
     

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  21. Fredrik

    Fredrik 10,263
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    Re: Homomorphisms as "structure-preserving" maps

    I understood this part at least, but I had difficulties after that. I'll try again tomorrow.
     
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