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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:
View attachment 8336
In the above example we read the following: " ... ... A monoid morphism
$$R \stackrel{ \phi }{ \longrightarrow } S$$
between two monoids is a function that respects the furnishings, that is
$$\phi ( r \star s ) = \phi (r) \star \phi (s)$$ and $$\phi (1) = 1$$
for all $$r,s \in R$$. (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 $$\phi ( r \star s ) = \phi (r) \star \phi (s)$$ and $$\phi (1) = 1$$ for all $$r,s \in R$$ ...
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:
View attachment 8337
View attachment 8338
https://www.physicsforums.com/attachments/8339
https://www.physicsforums.com/attachments/8340Hope that helps ...
Peter
I need some help in order to fully understand Example 1.2.1 on page 9 ... ...
Example 1.2.1 reads as follows:
View attachment 8336
In the above example we read the following: " ... ... A monoid morphism
$$R \stackrel{ \phi }{ \longrightarrow } S$$
between two monoids is a function that respects the furnishings, that is
$$\phi ( r \star s ) = \phi (r) \star \phi (s)$$ and $$\phi (1) = 1$$
for all $$r,s \in R$$. (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 $$\phi ( r \star s ) = \phi (r) \star \phi (s)$$ and $$\phi (1) = 1$$ for all $$r,s \in R$$ ...
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:
View attachment 8337
View attachment 8338
https://www.physicsforums.com/attachments/8339
https://www.physicsforums.com/attachments/8340Hope that helps ...
Peter
Last edited: