Diagram Chasing .... Simmons Example 2.1.2 .... .... very basic question ....

In summary, Peter is struggling to understand Example 2.2.1 (b) in Section 2.1 of "An Introduction to Category Theory" by Harold Simmonds. He asks for help in understanding how letting the function l collapse multiple elements to the same element can cause the left hand triangle to not commute. GJA suggests using the set $\{0,1\}$ with certain mappings as an example, but Peter still cannot understand. He asks for further clarification.
  • #1
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I am reading the book: "An Introduction to Category Theory" by Harold Simmonds and am currently focused on Section 2.1: Diagram Chasing ...

I need some help in order to fully understand Example 2.2.1 (b) on page 36 ... ...

Example 2.1.2 reads as follows:

https://www.physicsforums.com/attachments/8385

https://www.physicsforums.com/attachments/8384In Example (b) above we read the following:

" ... ... It is not hard to find an appropriate example in Set. Simply let l collapse a lot of elements to the same element. ... ... "Can someone explain (perhaps including a simple example ...) how letting l be such that it collapses a lot of elements to the same element, can lead to the left hand triangle not commuting ... Hope someone can help ...

Peter
 
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  • #2
Hi Peter,

How about taking each node to be the set $\{0,1\}$, where $f, g, $ and $l$ take both 0 and 1 to 0, and $h$ and $k$ are the identity maps?
 
  • #3
GJA said:
Hi Peter,

How about taking each node to be the set $\{0,1\}$, where $f, g, $ and $l$ take both 0 and 1 to 0, and $h$ and $k$ are the identity maps?
HI GJA ... thanks for the suggestion ...

I've done the calculations... and yes ... the outer cell and the right hand triangle commute ... yet the left hand triangle does not commute ...

BUT ...

I still cannot "see" what is happening and exactly why the function $l$ collapsing 0 and 1 to the element 0 causes this ...

Can you help? Sorry if I'm being slow in this matter ...Peter
 

FAQ: Diagram Chasing .... Simmons Example 2.1.2 .... .... very basic question ....

1. What is diagram chasing in mathematics?

Diagram chasing is a technique commonly used in mathematics to prove theorems or establish mathematical properties. It involves following a path through a diagram by applying certain rules and using the given information to reach a desired conclusion.

2. Can you provide an example of diagram chasing?

One example of diagram chasing is demonstrated in Simmons Example 2.1.2, where the author uses diagram chasing to prove that the product of two injective homomorphisms is also injective.

3. Is diagram chasing a difficult concept to understand?

Diagram chasing can be challenging for those new to mathematics, but with practice and a thorough understanding of the basic principles, it can become easier to grasp. It is important to have a strong foundation in algebra and category theory to fully understand the concept.

4. What are the benefits of using diagram chasing?

Diagram chasing can be a powerful tool in mathematics as it allows for the visualization and manipulation of complex mathematical arguments. It also helps to bridge the gap between abstract mathematical concepts and their concrete representations.

5. How can I improve my diagram chasing skills?

To improve your diagram chasing skills, it is important to practice regularly and to fully understand the basic principles. It can also be helpful to work through examples and exercises with a mentor or study group to gain a better understanding of the technique.

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