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

[Math Amateur]

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

 

[GJA]

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?

 

[Math Amateur]

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