The concept of duality can be a powerful way of obtaining new results which come easily within category theory, but which are not obvious in the theory to which category theory is being applied. As an advanced example, the category of Boolean Algebras is dual to the category of Stone Spaces. See, Stone Duality on Wikipedia for the motivation.
Morphisms are the central objects of study in category theory. For this reason, properties of morphisms can be very important.
Depending on its properties aA morphism f:X→Y may be any ofis called the following:following if it satisfies the given property:
Intuitively, an isomorphism is a way of transforming from one object to another in a way that makes them indistinguishable using the information of the category.
Intuitively, f being a monomorphism indicates that all of the information given by morphisms into X is preserved when composing by f. It generalizes the notion of an injective function, since in most concrete categories (like sets, groups, and topological spaces) every injective map is a monomorphism. However, even in concrete categories (and certainly more generally), monomorphisms need not be injective.
For any notion in a category, its dual is obtained by `reversing all the arrows' and 'reversing the order of composition'. If a statement is true in any category, then its dual is true in any category. As a corollary, if a statement is true in some categories, its dual is true in the duals of those categories.
As an example, consider the definition of a terminal object given above. A statement about terminal object is that any two terminal objects are isomorphic. Let's examine the exact statement. Assume T is terminal. Then for any X there is unique f:X→T. If we reverse the arrows, we get that for every X there is unique f:X←T. This is the definition of an initial object. Consider another terminal object T′. The statement that T′ is isomorphic to T is means that there is some f:T→T′ and g:T′→T such that gf=1T and fg=1T′. The dual of this is just the statement that there is some f:T←T′ and g:T′←T such that fg=1T and gf=1T′, this is exactly the same property! (The morphisms f and g have just been renamed). Hence, the dual of the statement that a terminal object is unique up to isomorphism is the statement that every initial object is unique up to isomorphism.
Similarly, if something is true for every category with an initial object, its dual will be true for every category with a terminal object.
The concept of duality can be a powerful way of obtaining new results which come easily within category theory, but which are not obvious in the theory to which category theory is being applied.
Morphisms are the central objects of study in category theory. For this reason, properties of morphisms can be very important. Depending on its properties a morphism f:X→Y may be any of the following:
Many mathematical constructions (such as products) appear across different fields of mathematics, consisting of different ingredients but nevertheless capturing a similar idea (and often even under the same name). Category theory allows one to precisely describe the property that these different constructions allcapall at once. This allows one to prove theorems about all these structures at once. Hence, once you prove that a specific mathematical structure is, say, a product, then all the category-theoretic theorems about products are true for that structure. In fact, sometimes there are structures which non-obviously satisfy a category-theoretic property. Especially when category-theoretic duality is involved.
Intuitively, f being a monomorphism indicates that all of the information givencaptured by the collection of morphisms into X is preserved when composing by f. It generalizes the notion of an injective function, since in most concrete categories (like sets, groups, and topological spaces) every injective map is a monomorphism. However, even in concrete categories (and certainly more generally), monomorphisms need not be injective.
Intuitively, f being an epimorphism indicates that all the information captured by the collection of morphisms out of Y is preserved when composing by f.. It generalizes the notion of a surjective function. However, in an even stronger sense than for monomorphisms, a function being epimorphic and a function being surjective are far from equivalent.
Properties that more closely match surjectivity include Section / Split Epimorphism, and regular epimorphism. strict epimorphism, strong epimorphism, and extremal epimorphism. Note that despite the names, not all of these are necessarily epimorphisms, but are epimorphisms in "nice" categories.
A category consists of a collection of objects and a collection of morphisms. A morphism f goes from one object, say X, to another, say Y, and is drawn as an arrow from X to Y. Note that X may equal Y (in(in which case f is referred to as an endomorphism). The object X is called the source or domain of f and Y is called the target or codomain of f. This is written as f:X→Y.
An endomorphism is a morphism from an object to itself.
An automorphism is a morphism from a structure to itself that preserves all the information of the structure that is distinguishable by the category. Intuitively, it gives "another view" of a structure (say, by moving around its elements) that leaves it essentially unchanged.
A morphism is a retraction if its effect can be "reversed" or inverted by another morphism applied after it. For example, every injective map is a retraction. The morphism which inverts the retraction is a section.
A morphism is a section if it "reverses" the effect of some other morphism applied before it. The morphism which is inverted is a retraction.
Many mathematical constructions (such as products) appear across different fields of mathematics, consisting of different ingredients but nevertheless capturing a similar idea (and often even under the same name). Category theory allows one to precisely describe the property that these different constructions capallall at once. This allows one to prove theorems about all these structures at once. Hence, once you prove that a specific mathematical structure is, say, a product, then all the category-theoretic theorems about products are true for that structure. In fact, sometimes there are structures which non-obviously satisfy a category-theoretic property. Especially when category-theoretic duality is involved.
Doing something in category theory which relies on a specific construction (insteadconstruction, that is, requiring objects to be equal instead of being up to isomorphism)merely isomorphic, is colloquially referred to as evil.
Doing something in category theory which relies on a specific construction, that is, requiring objectsconstruction (instead of being up to be equal instead of merely isomorphic,isomorphism) is colloquially referred to as evil.
Many mathematical constructions (such as products) appear across different fields of mathematics, consisting of different ingredients but nevertheless capturing a similar idea (and often even under the same name). Category theory allows one to precisely describe the property that these different constructions all at once. This allows one to prove theorems about all these structures at once. Hence, once you prove that a specific mathematical structure is, say, a product, then all the category-theoretic theorems about products are true for that structure. In fact, sometimes there are structures which non-obviously satisfy a category-theoretic property. Especially when category-theoretic duality is involved.
In addition, category theory allows the simple description of functors, natural transformations and adjunctions. These are mathematically powerful concepts which are very difficult to describe without the language of category theory. In fact, one of the founders of category theory, Saunders Mac Lane, has remarked that category theory was initially developed in order to provide a language in which to speak about natural transformations.
Powerfully, functors and adjunctions between categories allow one to translate concepts from one mathematical theory to another. They provide a "translation" (either full or partial) that allows one type of object to be viewed as another, and theorems to be translated across domains. In fact, using duality, very non-obvious translations can be found because a theorem in one category can be translated to its "opposite theory" in the other category. Connections which are not obvious in the language of the mathematical theories themselves, become clear in the language of category theory.