Question

Show that in the category of sets a morphism is an epunorphisin if and only if it onto (sunective)

Short answer

A morphism is an epimorphism in the category of sets if and only if it is onto. This is because the definition of onto suggests that every element in the co-domain is the image of some element in the domain. On the other hand, being an epimorphism means that it works with other morphisms in such a way that the composition does not change the consequent morphisms. Therefore, being onto and being an epimorphism are equivalent in the category of sets.

Step-by-step solution

Define an Epimorphism in Category Theory

In category theory, an epimorphism (also called an epic morphism or simply epi) is a morphism \(f : X \to Y\) such that for all objects \(Z\) and all morphisms \(g_1, g_2 : Y \to Z\), if \(g_1 \circ f = g_2 \circ f\) then \(g_1 = g_2\). That's to say, any two morphisms out of \(Y\) that agree after composing with \(f\) must necessarily be equal.

Define Surjective Function

In the category of sets, a function \(f: X \to Y\) is said to be surjective, or onto, if every element of the codomain \(Y\) is an image of at least one element from the domain \(X\). This means for any \(y\) in \(Y\), there exist an \(x\) in \(X\) such that \(f(x) = y\).

Equate the Two Definitions

Let's start by assuming that \(f\) is an epimorphism, and show that it must be onto. If \(f\) is not onto, then there exists an element \(y\in Y\) that is not in the image of \(f\). Consider the function \(g_1 : Y \to Z\) to be a function that sends \(y\) to some element in the set \(Z\) and \(g_2 : Y \to Z\) to be a function that sends \(y\) to a different element in the set \(Z\). Then, \(g_1\circ f\) and \(g_2\circ f\) are equal, since they agree on the image of \(f\), but \(g_1\) and \(g_2\) are not equal, which contradicts the definition of an epimorphism. Hence, \(f\) must be onto.

Show the Reverse

Now, assume that \(f\) is onto, and show that it must be an epimorphism. If \(g_1\circ f = g_2\circ f\), then for every \(x\in X\), \(g_1(f(x)) = g_2(f(x))\), which means that \(g_1\) and \(g_2\) must agree on all elements in the image of \(f\). But since \(f\) is onto, its image is the entire set \(Y\), so \(g_1 = g_2\). Therefore, \(f\) is an epimorphism.

Key concepts

Category Theory

In the deep and fascinating world of mathematics, category theory serves as a unifying and universal language for describing abstract concepts. It helps mathematicians to talk about different mathematical structures and the relationships between them using very general terms. At its core, category theory consists of objects, morphisms, and arrows. These components can represent various mathematical entities and their interactions.

• Objects: They are any kind of mathematical concept, such as sets, spaces, or groups.
• Morphisms: These are the arrows connecting the objects, often representing functions or transformations.
• Composition: Morphisms can be composed like functions, forming a chain of transformations from one object to another.

Category theory examines these structures as a whole, considering how different categories relate and interact through functors, which are mappings between categories that preserve their structural features.

Surjective Function

A surjective function, often referred to as an onto function, is a type of mapping between two sets. It's a crucial concept in both understanding the structure of functions and connecting them to other branches of mathematics, like category theory. Here’s what makes a function surjective:

• Full Coverage: For a function \( f: X \to Y \) to be surjective, every element in set \(Y\) must be covered by the function.
• Existence of Pre-Image: This means for every \( y \) in set \( Y \), there is at least one \( x \) in set \( X \) such that \( f(x) = y \).

Surjective functions ensure that the output set is completely filled with images from the input set. This understanding is pivotal for identifying certain morphisms in category theory.

Morphisms

Morphisms are the backbone of category theory, describing the relationships or mappings between objects in a category. In the category of sets, morphisms are simply functions between sets. There are several important aspects to understand about morphisms:

• Functions: In mathematics, a morphism could represent different types of functions or transformations, such as linear mappings or continuous functions.
• Identity Morphisms: Each object has an identity morphism, which behaves like a do-nothing function, preserving the original object.
• Composition: Morphisms can often be composed like functions, allowing us to form new morphisms and explore complex relationships.

In essence, morphisms in category theory generalize the idea of functions to a broader context, allowing deeper exploration of mathematical structures.

Sets

Sets are one of the foundational elements in mathematics and serve as the backdrop for many mathematical theories, including category theory. A set is simply a well-defined collection of distinct objects, considered as an object in its own right.

• Elements: The individual objects within a set are known as elements or members.
• Set Notation: Sets are typically denoted by curly braces, such as \( \{a, b, c\} \).
• Types of Sets: There are different kinds of sets, such as finite, infinite, empty, and universal sets.

Sets serve as the starting point for building more complex mathematical frameworks, and they play a pivotal role in category theory when considering the collection of objects and morphisms within a category.