# Universal Propeties in Category Theory .... Leinster Lemma 0.7 ....

• MHB
• Math Amateur
In summary, Leinster explains the idea of universal properties in a clear and concise manner, and provides a useful example to illustrate the concept.
Math Amateur
Gold Member
MHB
I am reading Tom Leinster's book: "Basic Category Theory" and am focused on Chapter 1: Introduction where Leinster explains the basic idea of universal properties ...

I need help in order to fully understand the proof of Lemma 0.7 ...

Lemma 0.7 and its proof read as follows:View attachment 8333In the above proof by Leinster we read the following:

" ... ... So by the uniqueness part of the universal property of b ... ... "My question is ... isn't this assuming what we are trying to prove ... that is that b is essentially unique ...Help will be appreciated ...

Peter

No, it is given that b and b' have a universal property that means that they have some kind of uniqueness property, using that, you can prove that T and T' are isomorphic.

In this example, you could say that $(T,b)$ has the following universal property:

For every vector space $W$ and every bilinear map $f:U \times V \Longrightarrow T$ there is a unique $\bar{f}:T \Longrightarrow W$ such that $\bar{f} \circ b=f$

So in Lemma 0.7: $(T,b)$ has this universal property
For the vector space $T'$ and the bilinear map $b':U \times V \Longrightarrow T'$, there is a unique map ...

Conversely, $(T',b')$ has also this universal property
For the vector space $T$ and the bilinear map $b:U \times V \Longrightarrow T$, there is a unique map ...

Make a lot of diagrams, see the book.

In the end, you will see that $T \cong T'$

Another Edit
In the book of Bland, in chapter 2.1, you saw the universal property of direct products and direct sums. You can read it and see if you can recognize the universal properties. In Simmons you will see universal properties in chapter 2.5 and others. It is very important

Last edited:
steenis said:
In this example, you could say that $(T,b)$ has the following universal property:

For every vector space $W$ and every bilinear map $f:U \times V \Longrightarrow T$ there is a unique $\bar{f}:T \Longrightarrow W$ such that $\bar{f} \circ b=f$

So in Lemma 0.7: $(T,b)$ has this universal property
For the vector space $T'$ and the bilinear map $b':U \times V \Longrightarrow T'$, there is a unique map ...

Conversely, $(T',b')$ has also this universal property
For the vector space $T$ and the bilinear map $b:U \times V \Longrightarrow T$, there is a unique map ...

Make a lot of diagrams, see the book.

In the end, you will see that $T \cong T'$

Another Edit
In the book of Bland, in chapter 2.1, you saw the universal property of direct products and direct sums. You can read it and see if you can recognize the universal properties. In Simmons you will see universal properties in chapter 2.5 and others. It is very important

Thanks for a most helpful post Steenis ...

Will study the example/lemma more carefully ...

Peter

## 1. What is the significance of universal properties in category theory?

Universal properties are an important concept in category theory that allows us to understand and compare different objects in a category. They provide a way to describe and characterize objects based on their relationships with other objects, rather than their specific properties. This makes them powerful tools for generalizing mathematical concepts and proving theorems.

## 2. How does Leinster Lemma 0.7 relate to universal properties?

Leinster Lemma 0.7 is a specific lemma that applies to universal properties in category theory. It states that if an object in a category has a certain universal property, then it must also have another universal property. This lemma is useful in proving theorems and understanding the relationships between different objects in a category.

## 3. Can you give an example of a universal property in category theory?

One example of a universal property is the universal mapping property for products. This property states that for any two objects A and B in a category, there exists a unique object A x B (the product of A and B) and two unique morphisms from A x B to A and B, respectively. This property characterizes the product of two objects in a category and allows us to define and compare products in different categories.

## 4. How are universal properties used in practical applications?

Universal properties have many practical applications in mathematics and other fields. In algebraic geometry, they are used to define and study varieties and sheaves. In computer science, they are used in functional programming and type theory. They are also used in physics, particularly in the study of quantum field theory and topological field theory.

## 5. Are there any limitations or criticisms of universal properties in category theory?

While universal properties are a useful tool in category theory, they do have some limitations. One criticism is that they can be difficult to understand and apply in certain situations. Additionally, some argue that universal properties are not always necessary and can sometimes lead to unnecessarily complicated proofs. However, overall, universal properties are a valuable and widely used concept in category theory.

Replies
2
Views
1K
Replies
3
Views
1K
Replies
3
Views
1K
Replies
2
Views
1K
Replies
3
Views
1K
Replies
2
Views
1K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
4
Views
1K
Replies
5
Views
1K