Category Theory

The field of study which is more concerned with the way of studying the properties of a particular mathematical concept is referred to as the collection of the arrows and the objects in the category theory. These collections of the arrows and the objects do satisfy the specific basic conditions. There are various significant areas in mathematics which can be formed into various categories and category theory do permits subtle and intricate mathematical results to work in these fields and thereby it can be proved in a simpler method than without using the different categories.

Objects and Arrows:

The category of sets is the most available example of a category. In the category of sets the objects are considered to the sets and the arrows are considered to the functions from one set to another set. Anyhow it is quite important to note that there is no compulsory need for the objects of a category to be sets and similarly for the arrows to be the functions. But, it would be a valid category if the mathematical concept meets the basic conditions of the mathematics concerned with the behavior of the objects and the arrows.

Groupoid:

Groupoid can be considered as the simplest example of a category theory. Groupoid can be defined as a category which has got the arrows or the morphisms which are invertible. In the field of topology, the concept of groupoid is given due consideration. Almost all fields of mathematics and certain areas in the theoretical computer science which is more concerned with the types and some areas of the mathematical physics which is specifically used to describe the vector spaces have started using the concept of categories.

Samuel Eilenburg and Saunders Mac Lane are the first people to introduce the concept of categories. This concept was introduced in the year 1942-45 in relevance to the algebraic topology.

Faces of theory:

There are several faces of the category theory which are not just known to the specialists in the category theory but also to several other mathematicians. “General abstract nonsense” is the term which dates back to the 1940s referring to the high level of abstraction when compared with more number of the classical branches of the mathematics. The category theory which organizes and suggests manipulation in abstract algebra is referred to as homological algebra. On the other hand the visual method that is used to argue with the abstract “arrows” that are joined in diagrams is referred to as Diagram chasing.

Questions:

• What is category theory?
• What is homological algebra?