Category theory and category theory

Category (category) is not only a mathematical language, but also a philosophical point of view. The essence of algebraic thought is: abstract, but concise. The generalization is very high. 0. Preliminary concept: morphism, the most common example of such a process is a function or mapping that maintains a structure in a sense. In set theory, for example, the state ejection is a function, in group theory, they are group homomorphism, and in topology they are continuous functions. In the range of a universal algebra (Universal algebra), a state shot is usually a homomorphism.
Abstract studies of the state-firing and the structures (or objects) they define are part of the category theory. In categorical theory, a state-shot does not have to be a function, and is usually considered an arrow between two objects (not necessarily collections). Unlike mapping an element of a set to another set, they are just a relationship between the domain and the codomain. Although the nature of the state-shot is abstract, most people's intuitive (in fact, most of the terms) comes from a specific category, where the object is a set of additional structures and the state-shot is the function that keeps the structure. 2. Basic ideas

Algebra thought is different from analytic thought, its original intention is to use the most concise language to unify as many concepts as possible, that is, the pursuit of universality. (That is, the algebra idea is a bottom-up summary, the analysis is Top-down refinement).

Most mathematical objects are abstract to the point where they cannot be abstracted, and they must be described in terms of category language. In short, category is the ultimate abstraction (Ultimate abstraction) of most mathematical objects. The "abstraction" here refers to the process of describing different concepts in the same way. For example, all sets form a category, all linear spaces form a category, all groups form a category, and all manifolds form a category, so categories are abstractions of sets, linear spaces, groups, and manifolds.

Category is the description of commonality.

But abstraction cannot be blind, as a mathematical object (object) must have its intrinsic operating rules (the rules define objects), otherwise the concept of category is too broad. In order to give the operational rules within the category, the concept of "morphism, or arrows" is needed. And the state shot can not be indiscriminate, it must be shot in a regular, this law is called the binding law. As we know in elementary school, addition and multiplication all have a binding law. With the intrinsic rules of operations (the same category), there is no need for external operational rules (between different categories). The so-called external rules of operation, refers to two different categories, there is some way to connect them. This is the origin of the concept of "functor" (functor). External operations, of course, can not be arbitrarily defined, must maintain the scope of the internal operational structure (that is, the direction of the arrow) unchanged.

Regardless of scope, internal and external operations, everything is too abstract, you must think of a way to visualize them. The visualization tool for mathematicians is called Exchange graph (commutative diagram). The following is a simple exchange diagram:

Of these, two A are two objects of Category 1 (also for a name), two B is the two object of category 2, two F is the "intrinsic operation" of two categories, and A and B are "external operations" (functor) of Category 2 to Category 1. This exchange diagram means that the functor should not only maintain the invariance of the object, but also maintain the invariance of the state ejection. The simplest example of a state shot is addition. 2. Mathematical definition of category

We can even further crystallize these concepts, for example, whenever we refer to a category, we can refer to two examples: group category or linear space category: homomorphism:
The state-firing is the linear transformation between the homogeneous linear spaces between groups, and the function sub:
The functor between group category and linear Space category is the group representation.

The exchange diagram between the functor and the state-shot in the previous section tells us that the group representation is actually the process of converting the group into a phase algebra. From here you can see the power of category theory in mathematics--a lot of mathematical concepts are expressed in the simplest language.