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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:

Since 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:

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## ...

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##.

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:

Since 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