Skip to main content

Synthesis of Modern Mathematics : Intuitive Category Theory (Definition II)

 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 rotation by 180 degrees it can be discerned that it is basically two successive clockwise rotations by 90 degrees each. This is noting but composition axiom of a category in disguise. In the fist example observe that if we rotate the propeller by 90 degrees and further rotate it again by 90 degrees then those two rotations can be composed to form a new (composite) rotation which is nothing but rotation by 180 degrees. 


Let us look at another example of the propeller in which we rotate the blades first by 90 degrees and then by 180 degrees both clockwise then we have a composite arrow which is simply a rotation clockwise by 270 degrees.

Similarly in the case of object set S as shown the map f can be composed by map g and the effect of mapping the elements 1,2,3,4 by the composite map h is same as first mapping the elements by f then followed by mapping by g. We can verify this visually by using four different colours to track the elements as shown below.


We just intuitively demonstrated the composition axiom of a category. An arrow with a codomain object A can always be composed with another arrow having object A as its domain. Note that in the notation order is in the reverse way. That if f is followed by g we write as "g composed with f".

Rotation by 90 degrees followed by no rotation is same as no rotation first followed by rotation by 90 degrees which is identical to simple rotation by 90 degrees. This is the simple axiom of the unit law in a category.

The last axiom is pertaining to the composition of three arrows. We demonstrate it using the example of propeller. Observe that rotation by 90 degrees followed by rotation by 90 degrees which is again followed by rotation by 90 degrees is identical to rotation by 180 degrees followed by 90 degrees which is identical to rotation by 90 degrees followed by rotation by 180 degrees. This is a single transformation which is nothing but rotation by 270 degrees. It does not matter how we club the pair of transformations as seen in the figure. 

That is all there is to the definition of a category often dreaded as abstract nonsense yet so concrete and visually demonstrated in very basic objects of geometry and algebra.  For those who are well versed with the concept of a group will recognize that these examples are special cases of categories which are also groups (there is a single object propeller or a set and all the arrows have inverses, as an example anti-clockwise rotation by 90 degrees is the inverse of clockwise rotation by 90 degrees). The first category is nothing but a also cyclic group of order 4 while the second example is a permutation group. I have chosen these examples to emphasise the fact that historically category theory was born in trying to generalise group (as invariance and relativity was hot topic of research in those days)  or the concept that the study of invariant transformations characterize the structure of an object. This is thoroughly covered in the book  "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory" by Marquis, Jean-Pierre.

 If you loved following the first example then take a look at Nathan Carter's visual group theory for intuitive visual approach to group theory (which is special case of category theory). The visual approach here is motivated from that book. If you are more comfortable with the second example then refer the book "Conceptual Mathematics: A First Introduction to Categories" by Lawvere a very well-respected category theorist. 

All these axioms are summarized in the slide attached below:

In the next post we shall see several examples of categories pervading throughout mathematics and its applied fields including physics and engineering.

Comments

Post a Comment

Popular posts from this blog

More on Synthesis

 Again we shall contemplate on a few more words on the wisdom of synthesis (navigating the "narrow razor-edged Path" between duality or the "noble middle path" of Buddha) from our beloved DK. However I wish to emphasize that Djwal Khul and Alice Bailey demonstrated practical (not just theoretically) synthesis. For instance Alice Bailey being born in the west was open enough to acknowledge and bring forth teachings from a far wiser eastern/oriental master Djwal Khul. She could both surrender herself (a difficult quality to develop for the western concrete mind with its overemphasis on freedom) and at the same time was careful enough to guard herself against foolish blind devotion (a vice of oriental masses). Another disciple who demonstrated practical synthesis was Blavatsky (synthesis of religions). She established Theosophical society headquarters in the west and soon shifted it to the east although she worked and died for the cause of theosophy in the west. One mo...
Post a Comment
Read more

Third Initiation : Disciple and Mental Body

 We studied initiations until the second in our ealier posts. Today let us look at the third especially from the perspective of a disciple (rather than the ray as well as the external ceremonial aspect conferred during the wesak full moon which we will cover later). The third initiation marks the culmination of complete mastery of the mental (in the same way that the earlier two initiations resulted into the control of physical and astral planes). It is a distinct initiation, which marks a definite transition out of the form consciousness (lower personality of mental, emotional(astral) and physical planes), and the initiate can function at will on the plane of the intuition though soul (ego). Between the second and third initiation work is struggling with and mastering the energies of the mental world. What kind of work ? DK sums that for us (details found in TWM book) as -  "A. Work on the mental plane produces realisation of duality. The disciple seeks to blend and merge the...
Post a Comment
Read more

Aphorisms as seed-thoughts

 Subsequent to our discussion on seed-thoughts, another great example of seed-thouht I think is an aphorism. The complete agni-yoga series of books by Roerichs are aphorisms of Ethical living (and give Ray 1 flavour of fiery will to the yoga of synthesis). Similarly the temple aphorisms of Francia A. La Due give a Ray 5 flavour of occult science to the works of H.P. Blavatsky and William Q. Judge. We quote those aphorims (as seed-thoughts) from temple teachings of Master H. (with his image):  "Days come and days go, but if thou watchest thou shalt see: THE LOAD thou hast laid on the heart of a friend will God transfer to thine own heart; heavy as it presses on the heart of thy friend, heavier will it press on thine own heart in the days to come. THE STONE thou hast cast from the path of the blind will smite the adder lying in wait for thee. THE WEIGHT thou hast clamped on the feet of another will drag thine own feet into Hadean desolation. THE SHELTER thou hast given the wayfa...
Post a Comment
Read more