Unstructured Sets and Monoid Morphisms in Category Theory

[Math Amateur]

I am reading the book: "An Introduction to Category Theory" by Harold Simmonds and am currently focused on Section 1.2: Categories of Unstructured Sets ...

I need some help in order to fully understand Example 1.2.1 on page 9 ... ...

Example 1.2.1 reads as follows:
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In the above example we read the following: " ... ... A monoid morphism

\(\displaystyle R \stackrel{ \phi }{ \longrightarrow } S\)

between two monoids is a function that respects the furnishings, that is

\(\displaystyle \phi ( r \star s ) = \phi (r) \star \phi (s)\) and \(\displaystyle \phi (1) = 1\)

for all \(\displaystyle r,s \in R\). (Notice that we have overloaded the operation symbol and the unit symbol ... ... )

... ... "
My question is as follows:Where in the definition of a category does it follow that we must have \(\displaystyle \phi ( r \star s ) = \phi (r) \star \phi (s)\) and \(\displaystyle \phi (1) = 1\) for all \(\displaystyle r,s \in R\) ...

and further ... ...

what does Simmons mean when he writes "Notice that we have overloaded the operation symbol and the unit symbol ... ... " ? Help will be appreciated ...

Peter***EDIT*** Oh! Reading the section again, maybe Simmons is not indicating why monoids are a category ... but simply describing a monoid morphism ... s that correct?====================================================================================The above post refers to the definition of a category ... so I think MHB readers would benefit from having access to Simmons' definition of a category ... so I am providing access to the same ... as follows:
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https://www.physicsforums.com/attachments/8339
https://www.physicsforums.com/attachments/8340Hope that helps ...

Peter

 

[Evgeny.Makarov]

Reading the section again, maybe Simmons is not indicating why monoids are a category ... but simply describing a monoid morphism

Exactly. Definitions of categories of algebraic structures usually don't mention elements of those structures: individual monoid elements, vectors and so on. Instead categories are defined in terms of appropriate morphisms, i.e., maps that commute with the operations of the structure at hand.

what does Simmons mean when he writes "Notice that we have overloaded the operation symbol and the unit symbol ... ... " ?

In the definition of morphism we have two monoids: $R$ and $S$. Yet operations in both of them are denoted by $\star$, and units in both of them are denoted by 1. Overloading (at least in programming) is precisely this: when the same identifier denotes different things in different contexts. Formally one has to write $\phi(r\star_R s)=\phi(r)\star_S\phi(s)$ and $\phi(1_R)=1_S$ where $\star_R$ and $1_R$ are the operation and unit of $R$ and $\star_S$ and $1_S$ are the operation and unit of $S$.

Spoiler

It's an interesting technical problem in automated reasoning to implement overloading in this context. Fortunately, with the technique called "type classes", which originated in Haskell programming language, this is possible. This allows writing $\phi(r\star s)=\phi(r)\star\phi(s)$ without additional annotations, and the computer is able to figure out that the first $\star$ means $\star_R$ while the second one means $\star_S$.

Do we have an off-topic tag on this forum?

 

[steenis]

In addition to the last post

Simmons is indeed describing the properties of a monoid here

But the highlight of this section is the definition of the category Mon

Can you now, using what you have learned about monoids, decribe the objects, arrows, identity arrows, and the composition rule of Mon ?

Later, in exanple 1.5.1, you will learn that one monoid M, can be viewed as a category too

 

[Math Amateur]

In addition to the last post

Simmons is indeed describing the properties of a monoid here

But the highlight of this section is the definition of the category Mon

Can you now, using what you have learned about monoids, decribe the objects, arrows, identity arrows, and the composition rule of Mon ?

Later, in exanple 1.5.1, you will learn that one monoid M, can be viewed as a category too

Thanks to Evgeny and Steenis for their help ...

Note that I found sections on monoids in the following two books:

(1) Category Theory (Second Edition) by Steve Awodey ... ... Section 1.4, Example 13

(2) Conceptual Mathematics: A First Introduction to Categories (Second Edition) ... ... Session 13: Monoids I am studying the material on monoids in the above two publications ...

Thanks again,

Peter