So we arrived at the 2nd week of the event Modern Mathematics. First week was very interesting and engaging, hope you enjoyed all the topics presented: Topology Day, Chaos Theory Day, Differential Geometry Day. I am still looking for interesting articles on the modern aspect of Cryptography, so if anyone knows something really good let me know. But today we are going to talk a little about Category Theory.
Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups.
Categories represent abstraction of other mathematical concepts. Many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate and subtle mathematical results in these fields in a much simpler way.
Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies. A basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, and the arrows need not be functions. Any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it.
The “arrows” of category theory are often said to represent a process connecting two objects, or in many cases a “structure-preserving” transformation connecting two objects. There are, however, many applications where much more abstract concepts are represented by objects and morphisms. The most important property of the arrows is that they can be “composed”, in other words, arranged in a sequence to form a new arrow.
It is very important to understand that several terms used in category theory, including the term “morphism”, are used differently from their uses in the rest of mathematics. For example, in category theory, morphisms obey conditions specific to category theory itself.
Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.
“The language of categories is affectionately known as “abstract nonsense,” so named by Norman Steenrod. This term is essentially accurate and not necessarily derogatory: categories refer to “nonsense” in the sense that they are all about the “structure,” and not about the “meaning,” of what they represent.”
― Paolo Aluffi,
In 1942–45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology. Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations. That required defining functors, which required categories.
Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane’s teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure. To achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic formalization of the relation between structures and the processes that preserve them.
Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense).
Category theory has been applied in other fields as well. For example, John Baez has shown a link between Feynman diagrams in Physics and monoidal categories. Another application of category theory, more specifically: topos theory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola.
More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).
Main Source: Wikipedia;
In the end, I want to recommend a couple of good articles if you want to read more on this topic. First on the list is Why Catergory Theory Matters?. The title says everything and I hope it made you curious to find the answer. Next. Wolfram has a good explanation of the definition for category. Totally recommend it. Moreover, here is an interesting article that explain more on the properties of Category Theory and gives an inside into the theory behind it. Also, it uses very funny and easy to remember images that will make you understand better what is happening; just click on the image to read more.
As you are already used for this type of articles, here are some book recommendations on the topic of Category Theory:
There are many other sources for learning about category theory and I advice you to search and give it a try. At least a small glance over it and some of its wonderful concepts will make you see the world with different eyes.
Don’t forget that maths is everywhere! Enjoy! ~LThMath~