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Click to add text What is a Category? A Category consists of a collection of objects and a collection of arrows between objects, where the arrows can be combined to make Category Example A: The 'Divides' Category written ab . For example 3 divides 9 evenly so the arrow would be 39 between 3 and 9. Also, since every number divides itself , every number has an arrow to itself . Functor Example 1: The Matrix Functor Functor Example 2: The Forgetful Functor This functor takes a communitive ring and makes it into a monoid * of 2 by 2 matrices. The homomorphisms in CRing become entry wise homomorphisms on matrices. (*like a group without inverses). This functor "forgets" addition in the commutative ring: it takes all elements of the communitive ring under just the multiplication operation, making the set into a monoid * using the multiplication operation. (*like a group without inverses). Creating a Natural Transformation Between Two Functors Category Example B: The CRing Category The objects in this category are communitive rings . If you are familiar with groups, rings can be thought of as groups with two commutative operations (addition and multiplication) that satisfy the distributive property. One standard example of a communitive ring is multiplication and addition in the integers. Another common example is Z n , the numbers {0,1,2,...n}. In Z n numbers are represented by the remainder after dividing by n. The arrows in CRing are homomorphisms between rings. This is just a special word to say a function that respects multiplication and addition. An Introduction to Mathematical Categories Courtney Fleming, Joseph Ruiz, Gabriel Vigil, Advisor: Dr. Mandi Schaeffer Fry Metropolitan State University of Denver, Department of Mathematical and Computer Sciences What is a Functor? A Functor is a map between categories that preserves arrows. What is a Natural Transformation? A natural transformation is a set of maps between the image of objects between two functors. A specific map in the set of maps in a natural transformation is called a component of a natural transformation. A natural transformation also has some important commutative properties (see below for an example). An example of a natural transformation is the determinant map between M 2x2 ( ) and F( ) (see below). The power of a natural transformation is that it can transform one functor into a simpler fucntor. An example of a natural transformation between The Matrix Functor M 2x2 and the Forgetful Functor F is the determinant. The determinant can be defined on any matrix monoid in the same way. The component maps of the natural transformation are given below. These maps take us from The Matrix Functor to The Forgetful Functor in communitive way. This calculation is a demonstration of how the commutative diagram to the left works. We see on this matrix performing the determinant then the map f or the map f then the determinant gives the same values, so the transformation is natural between the two functors. The natural transformation being communitive between functors means if we have map from one object to another in our starting category, then we can take any path in the square below and arrive at the same result.
