Sub-Category of Short Exact Sequences - Bland pages 85-86

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Section 3.2 on exact sequences in \(\displaystyle \text{Mod}_R\) and need help with the notion of the class \(\displaystyle \mathscr{S} \mathscr{E}\) of short exact sequences in the category \(\displaystyle \text{Mod}_R \).

The subsection of Bland on the topic of short exact sequences in the category \(\displaystyle \text{Mod}_R \) reads as follows:View attachment 3622
View attachment 3623In the above text, Bland writes:

The category \(\displaystyle \mathscr{C}\) can be formed by letting the class \(\displaystyle \mathscr{S} \mathscr{E}\) of short exact sequences in \(\displaystyle \text{Mod}_R \) be the objects of \(\displaystyle \mathscr{C}\).

I do not follow this section of text, since the objects of \(\displaystyle \text{Mod}_R \) are modules not short exact sequences ... so the how can Bland talk about the class \(\displaystyle \mathscr{S} \mathscr{E}\) of short exact sequences in \(\displaystyle \text{Mod}_R \) ... ...

Can someone please clarify this situation?

Peter***EDIT***

I know think that my problem results from a simple misunderstanding of Bland's sentence:

"The category \(\displaystyle \mathscr{C}\) can be formed by letting the class \(\displaystyle \mathscr{S} \mathscr{E}\) of short exact sequences in \(\displaystyle \text{Mod}_R \) be the objects of \(\displaystyle \mathscr{C}\)."Bland is forming a new category of short exact sequences - where the elements of the short exact sequence are modules (and hence from the category \(\displaystyle \text{Mod}_R \) - BUT the objects are short exact sequences ...

Simple misunderstanding ... solved ...

Peter

 

[jvicens]

Dear Peter,

Thank you for your question. It seems like your confusion stems from a misunderstanding of Bland's statement about forming a category based on short exact sequences in \text{Mod}_R. Allow me to clarify this for you.

Firstly, in the category \text{Mod}_R, the objects are modules and the morphisms are module homomorphisms. However, as Bland points out, we can also form a new category, denoted by \mathscr{C}, where the objects are short exact sequences and the morphisms are morphisms of short exact sequences. This means that the objects in this new category are not modules themselves, but rather sequences of modules.

Now, the class \mathscr{S} \mathscr{E} refers to the set of all short exact sequences in \text{Mod}_R. In other words, it is a collection of objects in the category \text{Mod}_R. By letting this class be the objects of the new category \mathscr{C}, we are essentially saying that the objects of \mathscr{C} are short exact sequences, with the understanding that each element of the sequence is a module from the category \text{Mod}_R.

I hope this clarifies things for you. If you have any further questions, please don't hesitate to ask.