Yoga of Synthesis: At the gateway of new Civilisation, Culture and Ideals of the Aquarian Age

 We continue describing some examples of categories from the last post. This will be a very small list of examples and many books on the subject list long tables of categories.  I am attaching two slides to quote examples of some common categories in mathematics and physics. Unless you already have a mathematics background it will be OK to not understand what these categories are.   But if you have at least some mathematics background then you might understand that the first slide describes what structure we are considering on objects. All the objects in these categories are constructed from plain sets along with additional well-defined axioms on their elements.   Using above objects we can form corresponding categories as shown below. Don't worry if you are not able to understand these examples since they need some mathematical background. However I will try to give some visual intuition regarding what these structures are and how an arrow preserves these ...

If your perspective on nature has evolved up-to the point where you observe that life cannot be just a collection of random events but is structured and therefore naturally intelligent then category theory is a very interesting mathematical language that takes interest in structures and their inter-relationships. We shall look at some basic examples of categories today.  Before trying to understand the examples, a word on the notion of object and arrow is due. An object carries a structure while an arrow or morphism refers to a correspondence between two objects carrying that common structure. Through the notion of correspondence we intuit that given an unknown object we can study it by setting up various possible correspondences with a known object. This is one of the most fundamental philosophy underlying the notion of arrows and their composition and therefore category theory itself. Here the mathematicians treat arrows and their composition more fundamentally than objects thems...

 In this post we will complete the definition of a category partially done in the last post.  The third axiom says that for every object in a category there exists an identity arrow of the object. This intuitively says that for every object we can always do nothing and leave the object as it is or in other words transform the object in such a way that it leaves the object exactly in the same state from which we originally started. This translates to no rotation of the propeller (which is same as rotation by 0 degrees or rotation by 360 degrees). This rotation (or effectively no rotation) is the identity arrow of the propeller. Similarly in the case of second object, no permutation (which is same as the permutation in which we send every element to itself) is the identity arrow. Take a visual look to make sure you understand that identity arrow will always exists given an object because we have nothing in effect to do to the given object.  Now if we ponder on the clockwise...

In this post, the goal is to intuitively understand the definition of a category through two very simple practical and concrete examples.   1.The first example is a four blade propeller, whose blades we shall label as 1,2,3 and 4.   2. Another example is a four element set A = {1,2,3,4}.  Let us form two categories containing these objects. The first example of a propeller will give rise to a category we shall call PROP (the label reflects that its object is a propeller). The second example of a finite set will lead to a second category which we will call SET. Notation wise the categories are denoted using capital bold letters.  The above figure is a visual depiction of the first axiom of a category. The first axiom says that every category consists of a collection of objects. In both categories that we have formed there is just a single object. But in general a category can have finitely or even infinitely many (collection of) objects. In fact mathematicia...

On the first blog of the post, I wrote about the fact of around four theosophists in Grothendieck's list of mutants. Grothendieck is considered to be the greatest mathematicians of the last century informally by some of the eminent mathematicians. Although Samuel Eilenberg and Saunders Maclane gave the first definition of category, functor and natural transformations in their 1945 paper, the recognition and development of a separate mathematical discipline of category theory has been generally attributed to Grothendieck's influence by the mathematical community. The following parallels between theosophy (as an occult philosophy) and category theory in contrast to current set-theoretic approach to science might give us a clue to the deeper thought patterns of this mathematician:  1. The study of theosophy like that of geometry proceeds from the universal to particulars. This is also the approach of category theory which can be thought of universal mathematics. It is in distinct...