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I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ...
I am currently focussed on Section I.3 Categories ... ... and am trying to understand Example 3.8 which is introduced as a concrete instance of the coslice categories referred to in Example 3.7 ...
Examples 3.7 and 3.8 read as follows:https://www.physicsforums.com/attachments/5568Since I do not have a basic understanding of the category of Example 3.8 my questions may not be well formulated ... for which I apologise in advance ...
My questions are as follows:Question 1
In the above text by Aluffi we read the following:
" ... ... An object in SET* is then a morphism $$f \ : \ \{ \ast \} \longrightarrow S$$ in Set where $$S$$ is any set. ... ... "My question is as follows: what exactly is $$\ast$$ ... ?
and ... ... is there only one $$\ast$$ for the category ... or one for each set ... if it is just a singleton for each set why not refer to it as a special element $$s \in S$$ ...
Question 2
In the above text by Aluffi we read the following:
" ... ... Thus we may denote object of Set* as pairs $$(S,s)$$ where $$S$$ is any set and $$s \in S$$ is any element of $$S$$ ... ... " My question is as follows: is there only one special element of $$S$$ in the category ... ... or are elements $$(S, s_1)$$ and $$(S, s_2)$$ in the category Set* where $$s_1$$ and $$s_2$$, like $$s$$, belong to the set $$S$$.
Question 3
In the above text by Aluffi we read the following:
" ... ... A morphism between two such objects $$(S,s) \longrightarrow (T,t)$$, corresponds then to a set-function $$\sigma \ : \ S \longrightarrow T$$ such that $$\sigma (s) = t$$. ... ... "
My question is as follows: the prescription $$\sigma \ : \ S \longrightarrow T$$ such that $$\sigma (s) = t$$ does not tell us how the other elements of $$S$$ are mapped ... ... ? ... and there are many alternatives ... and hence presumably, many $$\sigma$$s ... ... ? ... ... can someone clarify this matter ...
Hope someone can help ...
Peter
I am currently focussed on Section I.3 Categories ... ... and am trying to understand Example 3.8 which is introduced as a concrete instance of the coslice categories referred to in Example 3.7 ...
Examples 3.7 and 3.8 read as follows:https://www.physicsforums.com/attachments/5568Since I do not have a basic understanding of the category of Example 3.8 my questions may not be well formulated ... for which I apologise in advance ...
My questions are as follows:Question 1
In the above text by Aluffi we read the following:
" ... ... An object in SET* is then a morphism $$f \ : \ \{ \ast \} \longrightarrow S$$ in Set where $$S$$ is any set. ... ... "My question is as follows: what exactly is $$\ast$$ ... ?
and ... ... is there only one $$\ast$$ for the category ... or one for each set ... if it is just a singleton for each set why not refer to it as a special element $$s \in S$$ ...
Question 2
In the above text by Aluffi we read the following:
" ... ... Thus we may denote object of Set* as pairs $$(S,s)$$ where $$S$$ is any set and $$s \in S$$ is any element of $$S$$ ... ... " My question is as follows: is there only one special element of $$S$$ in the category ... ... or are elements $$(S, s_1)$$ and $$(S, s_2)$$ in the category Set* where $$s_1$$ and $$s_2$$, like $$s$$, belong to the set $$S$$.
Question 3
In the above text by Aluffi we read the following:
" ... ... A morphism between two such objects $$(S,s) \longrightarrow (T,t)$$, corresponds then to a set-function $$\sigma \ : \ S \longrightarrow T$$ such that $$\sigma (s) = t$$. ... ... "
My question is as follows: the prescription $$\sigma \ : \ S \longrightarrow T$$ such that $$\sigma (s) = t$$ does not tell us how the other elements of $$S$$ are mapped ... ... ? ... and there are many alternatives ... and hence presumably, many $$\sigma$$s ... ... ? ... ... can someone clarify this matter ...
Hope someone can help ...
Peter