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in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes hom(X, Y) be disjoint. In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms, (say, as the second and third components of an ordered triple).)
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319:). As far as I know, the majority of recent works in the subject use Hom to denote the abelian group of morphisms, and Mor for the underlying set. Maybe I'm wrong on this point, but certainly Osborne's "Basic Homological Algebra" uses this convention, Lang's "Algebra" certainly uses the Mor notation. Another common notation which is not even mentioned in the article is
1639:, i.e. it is "possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions", but some are not. E.g. every (pre-)ordered set can be seen as a category, with the elements of the set being the objects of the category and a morphism/arrow between to objects
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Your phrase "general relation between two arbitrary objects" implies that one could have a morphism from a number to morphism. This is not that you intend to say, but it is indeed confusing also a function from a set to another set preserves the empty structure. So it is not wrong to say that it is a
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Thanks for the discussion, and no apology called for -- my fault for not checking back. I have gone boldly ahead to revise the intro so that it combines my suggested first sentence with the original para. I've not added your link to homomorphism, as I'm not sure where to work it in, so please do so.
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I don't know much category theory, but having been given a few diagrams to chase, I've found out that monomorphisms are written as arrows with a hook (βͺ), epimorphisms are written with double-headed arrows (β ), and isomorphisms are written with double arrows (β). Furthermore, the algebraists who use
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I can't make sense of that statement (and the sentance that follows it). In the category of sets, the codomain is going to be the set of all of the second elements of the pairs. So if two sets of ordered pairs are identical, then thier codomains are identical, (as are thier domains). So this sentance
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Thanks James and Paul. I guess the issue here was that order of sentences gets in the way of exposition for us readers who don't already know the material. Within the
Definition section, introducing the domain and codomain operations before introducing the domain and codomain features of a morphism
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Another example of a bimorphism that is not an isomorphism: Consider the free monoid on one generator, i.e. the natural numbers with 0 under addition, as a category, in the standard way by having one object and each number corresponding to an arrow. Then every arrow is epi and mono, essentially by
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merely a graphic showing which morphisms imply which others in a 'normal' context. Iso points to epi and mono, which point to surjection and injection, respectively, endo points to homomorphism (and nothing else), etc. I believe the table would be accurate, and give a feel for the interdependence
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which probably don't even use the same wording in the definition. Every proposition in here that isn't trivial needs to be justified by a source. If someone doesn't explain why there aren't some I'm going to start removing non-trivial bits of information (such as the following passage: For example,
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I'm thinking of scattering some pointers to this request around, unless there is a better way of accomplishing it. I also don't know how this is best accomplished in text (as above), where the use of various browsers that don't support
Unicode must be taken into account. I didn't see anything in
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If someone could give me examples, and functions or equivalent objects therein, of a category in which there exists a so-called 'bimorphism' which is NOT an isomorphism, by all means discuss it here. As it stands, with my undergraduate background, I see no reason why a bimorphism does not always
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Reading what I wrote again I am now afraid I may have stifled the discussion (it's a knack, sorry). That was of course not my intention. There is always room for improvement, and if you believe that the lead section may be incomprehensible to the lay reader, by all means, go on and improve it. β
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In the definition section the last paragraph give a quick blurb on how two functions on sets when viewed as functions on sets are the same, but when viewed as morphisms on a category are different (distinct). I don't understand the mechanism that it's talking about, perhaps that section could be
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In a concrete category, every surjective morphism has a set-theoretic right-inverse function. But does it always have a right-inverse morphism? I know for example that this is the case in any category of algebraic objects (in the sense of universal algebra). Is it true more generally? I know
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I agree, these edits must be reverted, and I'll do it. In fact the article defines automorphisms for any structure, and there is no reason to choose, as a prominent example, a very specialized structure, much less known than many much more common examples (such as topological spaces or vector
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Thanks Mark for the comment that there might be two distinct meanings of "morphism" in play here, very useful. However, your comment about "synonym" doesn't seem to be describing a synonym, but rather an alternative definition. And having decided to use "map" as the relationship in your first
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It is difficult to get closer to the notion of isomorphism while remaining strictly weaker than with the concept of a bimorphism -- it is not possible to have an arrow with left-inverse and right-inverse which are not the same. Indeed, any monomorphism with right inverse, and similarly any
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we have an obligation to make the lead as accessible as possible. I used "map" as term somewhat less concrete than "function", but still familiar enough to someone with a little math training to give them an intuitive idea of what role a morphism plays in category theory
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I think the challenge here is that there are at least two uses of the word morphism in mathematics: (1) an arrow in category theory and (2) the general concept of a structure preserving mapping. The lead has to discuss both these uses without confusing the reader. How
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Does the intro sentence need to be so hedgey and ethereal-sounding? Isn't the following more straightforward statement true, or at least close enough for a good start? "In mathematics, a morphism is a structure-preserving mapping between two mathematical structures."
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that suggests ways of accomplishing this. Still, I'm mostly concerned with the commutative diagrams, where I've had one of those "working mathematician disappointed by wikipedia" moments. Perhaps if I were less averse to m*thw*rld I would have found it easier.
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The entire point of the notion of morphisms as "structure-preserving mappings" is to be as general as possible. While this may cause some confusion or ambiguity at first, the abundance of examples on this page should clarify the power of such a general notion.
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The edit that tried to remove the paragraph, and just leave the (slightly modified) more correct paragraph beneath it, was reverted. It was reverted on grounds that it was "too vague", despite the fact that the paragraph is still on the page currently.
1657:, i.e. the morphisms are not functions, and there is no structure to preserve. The nomenclature is certainly derived from morphisms as "structure-preserving mappings between two mathematical structures" but it is more abstract and general than that. β
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A holomorphism, as opposed to a holomorphic function, generally means a holomorphic map between complex manifolds, in other words a morphism in the category of complex manifolds. Of course, a holomorphic function is just a holomorphic map to
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these things consider them so standard that the usage isn't glossed in a typical article. It would be really nice if this usage (and any other ways that convey the same information) were described on the morphism page, and were used in
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Better to state only those implications which are true in all categories. It is too problematic to qualify what constitutes a "normal" category. A quick glance at such at qualified implication table is likely to confuse readers. --
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Given a morphism f which has domain X and codomain Y, we write fΒ : X β Y. Thus a morphism is represented by an arrow from its domain to its codomain. We can also write formal statements about the domain and codomain of f, using
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I am not following your line of reasoning. Your concluding sentence appears to say "The nomenclature is certainly derived from morphisms, but is more abstract and general than ". Ie: morphisms are more abstract than
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The definition of "morphism" is going to be "vague" (general), since it's just an abstract mathematical object as part of the definition of another abstract mathematical object (which that paragraph below links to.
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775:. It has set-theoretic right-inverse function (the logarithm), but this cannot be continuous on the whole circle, I think. If it were, then the right-inverse would be a homeomorphism onto a subset of the line? -
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Therefore surjectivity is weaker than split epimorphism, as the above example shows. And epimorphism is weaker than surjectivity, as the dense image example shows. Does this same heirarchy hold for injections?
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is a group homomorphism mapping from one group to another. In this sense, the morphism in category theory is a useful abstraction of the universal properties that all these structure-preserving mappings
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the meaning of "two operations defined on every morphism" is that there are two operations, and that each of these two can be applied to any morphism. However, if anyone knows better please correct it.
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331:). I might find time to hunt this question down in the next few days & see if I should change the article. Anyone with strong views on this subject, please chip in. Ben 11:05, 11 August 2006 (UTC)
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respectively. The term "arrow" is simply a synonym for "morphism". It is used to emphasize the abstract nature of morphisms, as distinct from the more concrete and prototypical example of functions.
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In what sense can a morphism actually BE an arrow? Surely this is confusing the concept "morphism" with the symbol used to represent it? Or is arrow written in italics some special technical term?
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Isn't that the point? That the objects of a category need not be sets and the morphisms need not be maps/mappings/functions? Or perhaps I am missing some subtlety in your use of the word "map". β
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There should be more examples on this topic that can elaborate how to find a specific function or map that is actually a morphism because every function on some objects is not a morphism e.g.
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somewhere.) I still personally prefer the intro as it is currently worded. But I agree that the lead section should be as accessible as possible ("as simple as possible, but not simpler"). β
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In what sense of the word "operation" are domain and codomain "operations" on a morphism? In lay terms they are properties or characteristics or just parts of a morphism, but "operations"?
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JA: No, you are confusing codomain and range. But some of these articles have been worked over in a way that encourages that confusion, so it might help to rewrite things more clearly.
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In most algebraic categories, like groups, rings, vector spaces, bimorphisms are always isomorphisms. One generally needs to go to a nonalgebraic category for a counterexample. Take the
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is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.
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saysΒ : "In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures...". No suggestion that a homomorphism is "an abstraction derived from...".
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The first paragraph describes a morphism as being a "structure preserving map", however, this is a completely unsourced statement, and one that I cannot find used anywhere else.
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In the first sentence of the intro we read: "In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures."
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examples of epics which are not surjective, but none of them are split epic. So for example, is there a continuous surjective map who has no continuous right-inverse? -
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That would seem a tall order to fill. Your discussion of morphisms as functions, and the example where they are not, doesn't seem to be more abstract than "map".
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Damn, that was fast, thanks. One more thing: how does the addition of an implication table (limited to certain categories) at the top of the page grab folks?
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imply an isomorphism, both being precisely bijective homomorphisms. The more general information on morphisms, on which I am unaware, would be helpful. -Cory
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For example, in the category of sets, where morphisms are functions, two functions may be identical as set of ordered pairs, but have different codomains.
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As long as the lead section makes it clear from the start that morphisms need not be functions at all I should not protest too loudly if something like
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844:." thus eminding sloppy readers such as I that range and codomain are not the same. I'll make the change if you don't like it, revert, I'm not picky.
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There is no subtlety here; the idea is to give a context for the term morphism and to explain to the non-specialist reader what morphisms are. From
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Nevertheless, I agree that the previous version of the lead was very far to be optimal. For this reason, I have completely rewritten the lead.
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To me, for morphisms to be "an abstraction derived from structure-preserving mappings between two mathematical structures", as in the intro,
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Thus in this category, every arrow is a bimorphism, but only n = 0, the identity, has an inverse and thus only n = 0 is an isomorphism.
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creates a mental hiccup. I think the section would be more helpful if presented the same info in a slightly different order:
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Y, where X and Y are some mathematical objects but it is a map that brings us from (X,*) into (X',*') i.e. f:(X,*)-: -->
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I think the article is currently incorrect in stating that the set of morphisms in a category is denoted Hom
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That's incorrect. 'Morphism' is used in a more general sense than the one you describe. See the page on
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Dohh. Of course. Thanks. I'll try to tweak that sentance to say "..ordered pairs, having the same
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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is the right word to use here. Range and codomain are usually used as synonyms for each other. --
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of all these various named morphisms, if we could qualify which categories it holds in. -Cory
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necessarily a homeomorphism; the inverse map need not be continuous. Take for example the map
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spaces). However, it would be useful to quote that the automorphisms form always a group.
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to another. The notion of morphism recurs in much of contemporary mathematics. In
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the freeness assumption, however, only the identity arrow has an inverse.
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Holomorphism exits? And I'm not talking about "holomorphic functions"!
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is a map from one object to another. In other fields of mathematics,
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whatever the map preserves has to be more abstract than structure.
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and one isn't! Therefore, an alternative definition of a function
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from the half-open unit interval to the unit circle given by
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epimorphism with a left inverse, must be an isomorphism. --
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I have slightly reworded this in the hope of clarifying. I
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instead of range, this might prevent the confusion above.
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i.e sin is not a morphism of multiplication over addition.
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This map is bijective and continuous, but the inverse is
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My god, almost none of this is sourced save for notes in
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Disambiguation needed for structure-preserving mapping?
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expanded on a little and possibly include an example?
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I'm not sure what you mean by an implication table. --
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This is either incorrect, or technical yet unexplained
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For example the logarithm to any base is a morphism
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741:Hhmmm very nice. Thanks! =)
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2077:to the sentence
2065:
2037:
1955:Tobias Bergemann
1928:Tobias Bergemann
1834:Tobias Bergemann
1752:, morphisms are
1659:Tobias Bergemann
1656:
1570:, and codomain,
1522:, and codomain,
1376:
1374:
1373:
1368:
1338:
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169:Citation Needed!
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1719:category theory
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1396:Category theory
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257:category theory
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1865:
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1758:linear algebra
1713:
1708:
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1249:of three sets
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700:
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369:homeomorphisms
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232:203.175.98.124
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699:Holomorphism?
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148:High-priority
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2070:called the
2054:G-structure
2035:two changes
1853:Mark viking
1820:mathematics
1796:Mark viking
1723:mathematics
1536:Paul August
123:Mathematics
114:mathematics
70:Mathematics
2223:Categories
2099:idempotent
2063:the change
1750:set theory
1676:morphisms.
1554:Definition
1402:articles.
1211:surjective
825:Jon Awbrey
197:Definition
1888:from one
1754:functions
1744:from one
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693:Render787
248:Dysprosia
39:is rated
2206:D.Lazard
2157:D.Lazard
2050:contribs
2015:Daviddwd
1986:Gwideman
1920:Gwideman
1898:Gwideman
1894:morphism
1879:morphism
1824:morphism
1786:morphism
1774:topology
1735:morphism
1727:morphism
1695:Gwideman
1637:concrete
1622:Gwideman
1589:Gwideman
1444:Gwideman
1417:Dan Hoey
722:contribs
709:unsigned
658:implies
594:implies
528:unsigned
500:unsigned
344:unsigned
290:contribs
228:unsigned
2139:Tea2min
1885:mapping
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517:Fropuff
490:Fropuff
244:Be bold
150:on the
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