Category Theory Homework Help - K-12 Grade Level, College Level Mathematics

Introduction to Category Theory

The category theory is a branch of study in mathematics which examines in an abstract manner, the properties of particular mathematical concepts by formalizing them as collections of objects and arrows (also known as morphisms, however this term also has a precise, non category-theoretical meaning) where these collections satisfy some fundamental conditions. Many significant regions of mathematics can be formalized as categories and the use of category theory permits many intricate and subtle mathematical outcomes in such fields to be stated and proved in a much easier way than without the use of categories.

The most familiar illustration of a category is the category of sets where the arrows are functions and the objects are sets from one set to the other. Though it is necessary to note that the objects of a category is not require to be sets nor the arrows functions; any method of formalizing a mathematical concept such that it meets the basic conditions on the behavior of objects and arrows is a valid category and all the outcomes of category theory will apply to it.

One of the easiest instances of a category is that of groupoid, explained as a category whose arrows or morphisms are all invertible. The groupoid concept is necessary in topology. The categories now look in most of the branches of mathematics, some regions of theoretical / hypothetical computer science where they correspond to the kinds and mathematical physics where they can be used to explain vector spaces. The categories were first introduced by Saunders Mac Lane and Samuel Eilenberg in the year 1942-45, in association with algebraic topology.

The category theory has different faces known not merely to specialists but to other mathematicians. The term dating from the year 1940s   "general abstract nonsense" defines to its high level of abstraction; which is compared to more classical branches of mathematics. The Homological algebra is a category theory in its aspect of organizing and suggesting manipulations in an abstract algebra. The visual method of arguing is diagram chasing with abstract "arrows" connected in diagrams. Note that the arrow between categories is known as functors which are subject to particular defining Commutativity conditions.

 

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Furthermore, categorical diagrams and series can be defined as functors (viz. Mitchell, 1965). An arrow between the two functors is a natural transformation whenever it is subject to certain naturality or commutativity conditions. The functors and natural transformations are the key concepts in category theory. Topos theory is a kind of abstract sheaf theory with the geometric origins and leads to ideas such as pointless topology. The topos can also be considered as a specific kind of category with two additional topos axioms.

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