A morphism is a map between objects that preserves the structure under consideration. Its precise compatibility conditions depend on the category of objects.
The identity morphism on an object is the neutral element for composition in a category: and whenever the composites are defined.
An endomorphism is a morphism : its source and target are the same object, with the same specified structure. For a group it is an operation-preserving group homomorphism back into that group; for a vector space it is a linear endomorphism; for a ring it is a ring endomorphism. A self-map must satisfy the chosen compatibility conditions to qualify. The notation records the category in which these conditions are imposed. See Riehl, Category Theory in Context, Definition 1.1.10 and the following paragraph.
An endomorphism can lose information or miss part of its target. On the additive integers, preserves addition and is injective, but is not surjective: no integer maps to . On the vector space over the real numbers, the linear endomorphism loses the second coordinate, so it is neither injective nor surjective. Neither is an automorphism. The same doubling rule becomes a group isomorphism when its target is the additive even integers, as explained under isomorphism.
The finite cyclic group offers a small example of information loss that can be drawn completely. With addition in modular arithmetic, the endomorphism sends the inputs to . Its image of a group homomorphism is and its kernel of a group homomorphism is . Each reached output has two inputs; the other three outputs are missed.
For an endomorphism of a group or ring with a finite underlying set, injectivity is equivalent to surjectivity by counting; either condition makes it an automorphism. The additive integers example shows why the finiteness assumption matters. A linear endomorphism of a finite-dimensional vector space also has equivalent injectivity and surjectivity, this time by the rank-nullity theorem even when its underlying set is infinite.
Self-maps can be iterated without changing the object: , , and so on. Composition in a category and the identity morphism make the endomorphisms of a fixed object in a locally small category into a monoid. This packages repeated compatible operations, including irreversible ones, into a single structure.
The general language has a documented historical milestone in the 1945 paper of Samuel Eilenberg and Saunders Mac Lane. They defined categories of objects and compatible maps, and distinguished maps admitting inverses within that category. Their examples included groups, vector spaces, and topological spaces. This gave a common framework for compatible self-maps and their invertible cases. See Eilenberg and Mac Lane, General Theory of Natural Equivalences, pp. 234 and 237–239.
Every endomorphism of the additive finite cyclic group , for a positive integer , has the form , determined by . This is the self-map case of homomorphism from a finite cyclic group: repeated addition forces the value at every element. Conversely, each choice of respects addition and gives a well-defined map on congruence classes.
Let , the greatest common divisor. The image of a group homomorphism is the cyclic subgroup generated by , with size by order of an element of a finite cyclic group. The kernel of a group homomorphism isIndeed, is equivalent to dividing , after cancelling and using coprimality. Consequently is a group automorphism exactly when and are coprime. In that case a modular inverse of gives . This connects reversible self-maps directly to the generators of a finite cyclic group.[ref]
A linear endomorphism is a linear map from a vector space to itself. In a finite-dimensional basis it is represented by a square matrix. It need not be invertible; an invertible linear endomorphism is an automorphism of the vector space.
Let be a singular endomorphism of a finite-dimensional vector space. Choose and a nonzero linear functional vanishing on , possible because that image of a linear map is proper. Then is a nonzero rank-one endomorphism with .
An automorphism is an isomorphism : a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.
For example, is an automorphism of the vector space over the real numbers, with inverse . Both are linear maps. The linear endomorphism from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.
The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.
A group automorphism is a bijective group homomorphism . It preserves the group operation, the identity element, and inverse elements; its inverse is another group homomorphism. Thus it is a group isomorphism with identical source and target. These maps form the automorphism group under function composition.[ref]
For the additive finite cyclic group , take . Its output list on is , a permutation of all six elements. It preserves addition because , and applying it twice gives . Hence it is a group automorphism and its own inverse function. Compare the collapsing in the figure under endomorphism. The general criterion under endomorphism of a finite cyclic group says exactly when such a map is reversible: its multiplier must be coprime to the size of the finite cyclic group.
A permutation need not be a group automorphism, even if it fixes the identity element. Let exchange and in and fix . It is a bijection, and , butThus it fails to preserve addition and is not even a group homomorphism. Reversibility alone does not preserve the group operation.
A group automorphism carries each cyclic subgroup onto the cyclic subgroup generated by the image of its defining element, preserving the order of a group element. In particular, it carries a generator of a group to another generator of a group. For , the existing calculation of generators of a finite cyclic group leaves only the images and for , so its automorphism group consists exactly of the identity function and . This turns the problem of compatible relabeling into a much smaller choice than enumerating arbitrary permutations.
A subgroup is characteristic when every automorphism of the ambient group maps it to itself. A uniquely determined subgroup of a given order or structural type is characteristic.
A nontrivial group with no proper nontrivial characteristic subgroup. A minimal nontrivial normal subgroup of a finite group has this property: every characteristic subgroup of it is normal in the ambient group.
An isomorphism is a morphism with a two-sided inverse that is also a morphism. Two objects are isomorphic when such a map exists between them. For groups, rings, and vector spaces, a bijective structure-preserving map has a structure-preserving inverse; in other settings the inverse condition needs separate attention, as for a homeomorphism between topological spaces.
For example, is a group isomorphism from the additive integers to the additive even integers. It is bijective onto that target, satisfies , and has inverse . The two groups have the same addition pattern even though they are different sets. It is not a ring isomorphism for their inherited multiplications, since while .
A group isomorphism is a bijective group homomorphism . It matches the entire group operation: , with each product taken in its own group. Its inverse is automatically a group homomorphism. A group automorphism is the special case .[ref]
The target is part of the claim. The doubling example under isomorphism is a group isomorphism , between the additive integers and the additive even integers. Exactly the same rule , declared as , is an injective endomorphism that misses and every other odd integer, so is not a group automorphism. Changing the declared target changes surjectivity, even though it does not change the formula or its compatibility with addition.
An isomorphism lets a calculation in one model solve the same problem in another. The example under group identifies addition modulo four with function composition of rotations of a square via . Through this group isomorphism, adding exponents corresponds exactly to composing motions. More generally, a group isomorphism preserves order of a group element, carries subgroups to subgroups, and carries a generating set of a group to a generating set of a group. Having equally many elements is insufficient: the permutation in group automorphism already fails compatibility on two inputs.
Why does the inverse preserve the group operation? Write and , possible because of surjectivity. ThenThis proves the inverse condition directly. Analogous calculations work for each operation of a ring and for addition and scalar multiplication in a vector space. The argument relies on preserving those operations; it supplies no general continuity claim for the inverse of a continuous bijection between topological spaces.
A ring isomorphism is a bijective ring homomorphism. It preserves all ring operations and has a ring-homomorphic inverse.
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In mathematics, particularly in the field of category theory, a **morphism** is a structure-preserving map between two objects in a category. The concept of a morphism helps to generalize mathematical concepts by focusing on the relationships and transformations between objects rather than just the objects themselves. A morphism typically has the following characteristics: 1. **Objects**: In a category, you have objects which can be anything: sets, topological spaces, vector spaces, etc.