Two functors are given, and a new functor can be obtained by point-by-point synthesis. It can be verified that this compound operation satisfies the associative law and there is an identity morphism.
So some people may accidentally think that categories and functors can also constitute a new category. However, this statement is inaccurate. The category axiom requires that morphisms between any two objects can form a set, but both categories contain objects of a class, so all functors between them should form a class, not a set.
Therefore, we should always pay attention to distinguish between sets and classes in category theory.
This leads to a definition: a category is called a small category, which is equivalent to a class composed of its objects being a collection. Note that a class is a bigger concept than a set, and a set must be a class, and a class is not necessarily a set.
So it is concluded that small categories and functors can form a new category. Because all the functions between sets still constitute a set.
Some obvious examples of categories
Collection categories: collections and mappings
Category of Topological Space: Topological Space and Continuous Mapping
Group category: group and group homomorphism
Categories of commutative rings: commutative rings and ring homomorphisms.
The Category of Real Vector Space: Real Vector Space and Linear Transformation
Category of real Banach spaces: real Banach spaces and bounded functionals
Set and injectivity
The real Banach space and linear compression have never been seen before, and it is unclear.
Some mathematical structures can also be regarded as categories.
A. The object is a natural number, the arrow is a matrix from n to m, the number of decimal places is n * m, and compounding is the usual multiplication of the matrix. There will be doubt here, because the arrow here is no longer a mapping, but can be understood as the correspondence of order pairs. An ordered pair corresponds to a set of matrices.
B A poset can also be regarded as a category, the object is a set element, and morphism is defined as, so synthesis is transitive and identity is reflexive. These two properties are satisfied in the definition of partial order.
C any set can be regarded as a category, the object is an element of the set, and a morphism is just an identity morphism. Such a category is called discrete category, and the meaning of discrete can consider discrete topology, which is point by point uncorrelated.
D a simple semigroup can be regarded as a category with only one object, the morphism set is m itself, the compound law is the multiplication of the semigroup itself, and the simple semigroup actually functions as a function. Every group is a semigroup, the difference is that the elements in the semigroup do not guarantee the existence of an inverse. Since monosemigroups are all categories, groups are naturally categories, and rings with monosemigroups are also categories. At this time, the synthesis rule is given by the multiplication of rings.
Let's call it a day. There is a lot of content. Although the greatest pleasure of learning category theory is the connection of mathematical objects, it is inevitable that we must understand many mathematical structures before we can truly understand this connection on the basis of full understanding. Like the above structure, one is structure set and structure-preserving mapping, which is rich in content, and the other is a more generalized mathematical structure. These are undeveloped places, which bring new problems and show the vitality of the theory. However, this is not important. Learning mathematics is to seek a kind of fun, a kind of beautiful enjoyment. Whether it is the beauty of mechanical proof, creativity or connection, it is the greatest happiness to realize this beauty through hard study.