Categories - Bland Chapter 3 - Problem 1 - Problem Set 3.1

[Math Amateur]

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

I am trying to understand Section 3.1 on Categories.

At present I am working on Problem 1 in Problem Set 3.1 and I need some help in understanding the problem and its solution.

Problem 1 (Problem Set 3.1) reads as follows:
View attachment 3599
Potential helpers on this problem will need to know Bland's definition of a category and his notation ... the definition reads as follows:View attachment 3600
View attachment 3601Thus for \(\displaystyle \mathscr{C}\) we have that

\(\displaystyle \mathscr{O}\) is the class of positive integers greater than or equal to \(\displaystyle 2\)

and

the set of morphisms \(\displaystyle \text{Mor} (m,n)\) is the set of \(\displaystyle m \times n\) matrices over a commutative ring \(\displaystyle R\)
Now my problem concerns verifying and interpreting \(\displaystyle C1\).
\(\displaystyle C1\) requires that for all \(\displaystyle m, n \in \mathscr{O}\) there is a (possibly empty) set \(\displaystyle \text{Mor} (m,n)\) called the set of morphisms \(\displaystyle f \ : \ m \rightarrow n\) from \(\displaystyle m\) to \(\displaystyle n\) such that:

\(\displaystyle \text{Mor} (m,n) \cap \text{Mor} (s,t) = \emptyset \ \text{ if } \ (m,n) \ne (s,t)\)This of course is true since there is no common element between the set of \(\displaystyle m \times n\) matrices and the set of \(\displaystyle s \times t\) matrices when \(\displaystyle (m,n) \ne (s,t)\)... ... BUT ... what do we make of \(\displaystyle f \ : \ m \rightarrow n\)? ... ... what does it represent? ... ... do we just ignore this as something that just does not fit the present case, or perhaps some formalism that does not matter?But then ... why does it not matter that there is no interpretation for \(\displaystyle f \ : \ m \rightarrow n\), that is no interpretation for \(\displaystyle f\) ... ...I hope someone can clarify this (possibly simple) point.

Peter

 

[Fallen Angel]

Hi Peter,

I think the natural way of interpreting this is seeing $f$ as a morphism from $R^{m}$ to $R^{n}$ using matrix multiplication.

$\begin{array}{cccc}f: & R^{m} & \longrightarrow & R^{n}\\ & x & \mapsto & Ax \end{array}$

where $A\in Mor(m,n)$

 

[Math Amateur]

Hi Peter,

I think the natural way of interpreting this is seeing $f$ as a morphism from $R^{m}$ to $R^{n}$ using matrix multiplication.

$\begin{array}{cccc}f: & R^{m} & \longrightarrow & R^{n}\\ & x & \mapsto & Ax \end{array}$

where $A\in Mor(m,n)$

Well! ... ... that was really helpful Fallen Angel!

Thanks for that insight!

Appreciate your time and support!

Peter