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

[Math Amateur]

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

 

[steenis]

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.

 

[steenis]

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

 

[Math Amateur]

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 ...

Appreciate your help ...

Will study the example/lemma more carefully ...

Peter