Given that the name of this site is Epic Math, I felt I should write about a topic where the word epic is a technical term! It is hardly surprising that such a term exists, as many otherwise non-mathematical words such as almost, simple, open, connected, regular, normal, field, ring, onto, map, twin, lucky, and even sexy have technical definitions.
The term epic is found in category theory, which is an extremely abstract branch of mathematics that formally deals with many other fields of math. It is so strange that some mathematicians have labeled it “abstract nonsense.”
In category theory the main subject of study is the category, which consists of three parts:
- A class of objects,
- A class of morphisms which go from one object to another object, and
- A binary operator called the composition function of morphisms, satisfying associativity and existence of identity.
Do not worry if you do not know what objects or morphisms are—they are very abstractly defined ideas. Here are several examples to make the theory more concrete:
Example 1: Let us write three objects . We can also write morphisms and . Since categories have a composition function, then there is also a morphism such that .
Example 2: Using the notation from the above example, let all be , the set of real numbers. Let the morphisms and be functions from to , which should be familiar to most readers. Then is just standard composition of functions, so . For instance, if and , then .
Example 3: Let , the -dimensional vector space over the field . Let and . Then let the morphisms and be linear transformations. Since they are linear transformations over finite vector spaces, the morphisms can be represented as matrices, i.e. let denote the matrix of , and let be the matrix of . Then the composition is given by the matrix with dimension obtained from matrix multiplication. This is the correspondence between the composition of linear transformations and the multiplication of matrices seen in linear algebra.
Example 4: Let all be (the plane), and let be rotations about the origin by some angle. Let the composition denote first applying , then applying . Then there is a rotation so that . For instance, let be rotation by 90 degrees and be rotation by 180 degrees. Then is rotation by 270 degrees.
Example 5 (Abstract): Let all be the set of functions from to , denote this set . Then let be functions from to itself. Let composition be the standard composition of functions. Then given there is a function from to itself that is equal to . For instance, let map all functions to , i.e. increases every value of a function by 1. And let map all functions to , i.e. doubles every value of a function. Then maps all functions to . For example, if , then and .
An epimorphism (also called an epic morphism) is a morphism such that for all morphisms , we have .
Here are a few basic useful properties of epimorphisms:
- Every isomorphism is an epimorphism.
- If and are epimorphisms, then is an epimorphism.
- If is an epimorphism, then is an epimorphism.
- The epic property is preserved under equivalence of categories.
Example 6: Let be , so our morphisms are functions. Then the function is an epimorphism. Since are equal, then for all , , i.e. that . Substitute , which is a surjective function. Then for all , so and thus is an epimorphism.
Example 7: Let be as before, so our morphisms are functions. Then the function is not an epimorphism. For example, let , and let be the function that has value 0 in the interval and 1 everywhere else. Then clearly as both are equal to 0 for all . However , so is not an epimorphism.
In fact, we can extend the process in Example 5 to show that every surjective function from a set to a set is an epimorphism, and every non-surjective function from a set to a set is not an epimorphism. Thus in the category of sets with functions, epimorphisms correspond to surjective functions. In general categories, however, epimorphisms do not always correspond to surjections. For example, in the category of rings with ring homomorphisms, the inclusion map is an epimorphism but not a surjection.
For more information about Epimorphisms (or epic morphisms), check out the following:
For more information about the background and/or theorems used, see: