词条 | Hopfian object |
释义 |
In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object, which is an object B such that every monomorphism from B into B is necessarily an automorphism. The two conditions have been studied in the categories of groups, rings, modules, and topological spaces. The terms "hopfian" and "cohopfian" have arisen since the 1960s, and are said to be in honor of Heinz Hopf and his use of the concept of the hopfian group in his work on fundamental groups of surfaces. {{harv|Hazewinkel|2001|p=63}} PropertiesBoth conditions may be viewed as types of finiteness conditions in their category. For example, assuming Zermelo–Fraenkel set theory with the axiom of choice and working in the category of sets, the hopfian and cohopfian objects are precisely the finite sets. From this it is easy to see that all finite groups, finite modules and finite rings are hopfian and cohopfian in their categories. Hopfian objects and cohopfian objects have an elementary interaction with projective objects and injective objects. The two results are:
The proof for the first statement is short: Let A be an injective hopfian object, and let f be an injective morphism from A to A. By injectivity, f factors through the identity map IA on A, yielding a morphism g such that gf=IA. As a result, g is a surjective morphism and hence an automorphism, and then f is necessarily the inverse automorphism to g. This proof can be dualized to prove the second statement. Hopfian and cohopfian groups{{main|hopfian group}}Hopfian and cohopfian modulesHere are several basic results in the category of modules. It is especially important to remember that RR being hopfian or cohopfian as a module is different from R being hopfian or cohopfian as a ring.
Hopfian and cohopfian ringsThe situation in the category of rings is quite different from the category of modules. The reader should note that the morphisms in the category of rings with unity are required to preserve the identity, that is, to send 1 to 1.
Hopfian and cohopfian topological spaces
References
|author=Baumslag, Gilbert |chapter=Hopficity and abelian groups |title=Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962) |publisher=Scott, Foresman and Co. | location=Chicago, Ill. |year=1963 |pages=331–335 |mr=0169896 }}
|title=Encyclopaedia of mathematics. Supplement. Vol. III |editor=Hazewinkel, M. |publisher=Kluwer Academic Publishers |place=Dordrecht |year=2001 |pages=viii+557 |isbn=1-4020-0198-3 |mr=1935796 }}
|author=Varadarajan, K. |title=Hopfian and co-Hopfian objects |journal=Publ. Mat. |volume=36 |year=1992 |number=1 |pages=293–317 |issn=0214-1493 |mr=1179618 |doi=10.5565/PUBLMAT_36192_21}}
|author=Varadarajan, K. |title=Some recent results on Hopficity, co-Hopficity and related properties |conference=International Symposium on Ring Theory |series=Trends Math. |publisher=Birkhäuser Boston |year=2001 |pages=371–392 |mr=1851216 }} External links
4 : Category theory|Group theory|Module theory|Ring theory |
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