Hopfian object
Encyclopedia
In the branch of mathematics called category theory
, a hopfian object is an object A such that any surjective morphism
of A onto A is necessarily an automorphism
. The dual notion
is that of a cohopfian object, which is an B such that every injective morphism from B into B is necessarily an automorphism. The two conditions have been studied in the categories of group
s, rings
, module
s, and topological spaces.
The terms "hopfian" and "cohopfian" have arisen since the 1960's, 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 group
s of surfaces.
, 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 object
s. The two results are:
The proof for the first statement is short: Let A be and 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.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a hopfian object is an object A such that any surjective morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
of A onto A is necessarily an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
. The dual notion
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
is that of a cohopfian object, which is an B such that every injective morphism from B into B is necessarily an automorphism. The two conditions have been studied in the categories of group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s, rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s, and topological spaces.
The terms "hopfian" and "cohopfian" have arisen since the 1960's, and are said to be in honor of Heinz Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
and his use of the concept of the hopfian group in his work on fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s of surfaces.
Properties
Both 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 setsCategory of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
, 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 object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...
s. The two results are:
- An injective hopfian object is cohopfian.
- A projective cohopfian object is hopfian.
The proof for the first statement is short: Let A be and 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 modules
Here 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.- A Noetherian moduleNoetherian moduleIn abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....
is hopfian, and an Artinian moduleArtinian moduleIn abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...
is cohopfian. - The module RR is hopfian if and only if R is a directly finite ring. Symmetrically, these two are also equivalent to the module RR being hopfian.
- In contrast with the above, the modules RR or RR can be cohopfian or not in any combination. An example of a ring cohopfian on one side but not the other side was given in . However if either of these two modules is cohopfian, R is hopfian on both sides (since R is projective as a left or right module) and directly finite.
Hopfian and cohopfian rings
The 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.- If R satisfies the ascending chain conditionAscending chain conditionThe ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
on ideals, then R is hopfian. This can be proven by analogy with the fact for Noetherian modules. The counterpart idea for "cohopfian" does not exist however, since if f is a ring homomorphism from R into R preserving identity, and the image of f is not R, then the image is certainly not an ideal of R. In any case, this shows that a one sided Noetherian or Artinian ring is always hopfian. - Any simple ring is hopfian, since the kernel of any endomorphism is an ideal, which is necessarily zero in a simple ring. In contrast, in , and example of a non-cohopfian fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
was given. - The full linear ring EndD(V) of a countable dimensional vector space is a hopfian ring which is not hopfian as a module, since it only has three ideals, but it is not directly finite. The paper also gives an example of a cohopfian ring which is not cohopfian as a module.
- Also in , it is shown that for a Boolean ringBoolean ringIn mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
R and its associated Stone space X, the ring R is hopfian in the category of rings if and only if X is cohopfian in the category of topological spaces, and R is cohopfian as a ring if and only if X is hopfian as a topological space.
Hopfian and cohopfian topological spaces
- In , a series of results on compact manifolds are included. Firstly, the only compact manifolds which are hopfian are finite discrete spaceDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
s. Secondly, compact manifolds without boundary are always cohopfian. Lastly, compact manifolds with nonempty boundary are not cohopfian.