Categories of Pointed Sets - Aluffi, Example 3.8

[Math Amateur]

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 \(\displaystyle f \ : \ \{ \ast \} \longrightarrow S\) in Set where \(\displaystyle S\) is any set. ... ... "My question is as follows: what exactly is \(\displaystyle \ast\) ... ?

and ... ... is there only one \(\displaystyle \ast\) 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 \(\displaystyle s \in S\) ...
Question 2

In the above text by Aluffi we read the following:

" ... ... Thus we may denote object of Set* as pairs \(\displaystyle (S,s)\) where \(\displaystyle S\) is any set and \(\displaystyle s \in S\) is any element of \(\displaystyle S\) ... ... " My question is as follows: is there only one special element of \(\displaystyle S\) in the category ... ... or are elements \(\displaystyle (S, s_1)\) and \(\displaystyle (S, s_2)\) in the category Set* where \(\displaystyle s_1\) and \(\displaystyle s_2\), like \(\displaystyle s\), belong to the set \(\displaystyle S\).

Question 3

In the above text by Aluffi we read the following:

" ... ... A morphism between two such objects \(\displaystyle (S,s) \longrightarrow (T,t)\), corresponds then to a set-function \(\displaystyle \sigma \ : \ S \longrightarrow T\) such that \(\displaystyle \sigma (s) = t\). ... ... "


My question is as follows: the prescription \(\displaystyle \sigma \ : \ S \longrightarrow T\) such that \(\displaystyle \sigma (s) = t\) does not tell us how the other elements of \(\displaystyle S\) are mapped ... ... ? ... and there are many alternatives ... and hence presumably, many \(\displaystyle \sigma\)s ... ... ? ... ... can someone clarify this matter ...
Hope someone can help ...

Peter

 

[Evgeny.Makarov]

In the above text by Aluffi we read the following:

" ... ... An object in SET* is then a morphism \(\displaystyle f \ : \ \{ \ast \} \longrightarrow S\) in Set where \(\displaystyle S\) is any set. ... ... "

My question is as follows: what exactly is \(\displaystyle \ast\) ... ?

Doesn't the text say that $A=\{*\}$ is a fixed singleton? We fix an arbitrary one-element set $A$ and call its element $*$.

is there only one \(\displaystyle \ast\) for the category ... or one for each set

Let's read the text again: Let $\mathsf{C}=\mathsf{Set}$ and $A=$ a fixed singleton $\{*\}$. That is, we consider a class of all sets and a particular one-element set. Where does the idea that there is one $*$ for each set come from? However, it is true that we could consider different one-element sets and get, strictly speaking, different categories $\mathsf{Set}^*$, but they will be isomorphic.

is there only one special element of \(\displaystyle S\) in the category ... ... or are elements \(\displaystyle (S, s_1)\) and \(\displaystyle (S, s_2)\) in the category Set* where \(\displaystyle s_1\) and \(\displaystyle s_2\), like \(\displaystyle s\), belong to the set \(\displaystyle S\).

I assume that by "elements \(\displaystyle (S, s_1)\) and \(\displaystyle (S, s_2)\) in the category Set*" you mean objects of the category Set*. Yes, for each nonempty set $S$ and each $s\in S$ there is an object $(S,s)$ in Set*.

My question is as follows: the prescription \(\displaystyle \sigma \ : \ S \longrightarrow T\) such that \(\displaystyle \sigma (s) = t\) does not tell us how the other elements of \(\displaystyle S\) are mapped ... ... ? ... and there are many alternatives ... and hence presumably, many \(\displaystyle \sigma\)s ... ... ?

Yes. Just like there are many morphisms between given objects of the category Set, there are in general many morphisms between any two objects in Set*.

 

[Deveno]

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:Question 1

In the above text by Aluffi we read the following:

" ... ... An object in SET* is then a morphism \(\displaystyle f \ : \ \{ \ast \} \longrightarrow S\) in Set where \(\displaystyle S\) is any set. ... ... "My question is as follows: what exactly is \(\displaystyle \ast\) ... ?

and ... ... is there only one \(\displaystyle \ast\) 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 \(\displaystyle s \in S\) ...

It is a "generic" singleton set. If you prefer, you may think of it as the set $\{1\}$, although that notation implies we have some sort of numeric structure, which is not the case. In the category $\mathbf{Set}$, singletons are unique-up to set-isomorphism (bijection).

Question 2

In the above text by Aluffi we read the following:

" ... ... Thus we may denote object of Set* as pairs \(\displaystyle (S,s)\) where \(\displaystyle S\) is any set and \(\displaystyle s \in S\) is any element of \(\displaystyle S\) ... ... " My question is as follows: is there only one special element of \(\displaystyle S\) in the category ... ... or are elements \(\displaystyle (S, s_1)\) and \(\displaystyle (S, s_2)\) in the category Set* where \(\displaystyle s_1\) and \(\displaystyle s_2\), like \(\displaystyle s\), belong to the set \(\displaystyle S\).

It is possible to have $(S,s_1)$ and $(S,s_2)$ where $S = S$ but $s_1 \neq s_2$-but the idea is we are thinking of each $S$ as "coming with" a distinguished point", or "base point" that "roots it." I admit it may not immediately be clear *why* we are doing this, but perhaps this might make it clear.

In the category $\mathbf{Set}$, there is an initial object $\emptyset$: that is, there is a unique function $\emptyset \to A$ for any $A$, the empty function. We also have terminal objects, there is a unique function $A \to \{\ast\}$ (these are called "constant functions"). Note that the initial object and the terminal object are "different". Using pointed sets fixes this asymmetry.

Question 3

In the above text by Aluffi we read the following:

" ... ... A morphism between two such objects \(\displaystyle (S,s) \longrightarrow (T,t)\), corresponds then to a set-function \(\displaystyle \sigma \ : \ S \longrightarrow T\) such that \(\displaystyle \sigma (s) = t\). ... ... "


My question is as follows: the prescription \(\displaystyle \sigma \ : \ S \longrightarrow T\) such that \(\displaystyle \sigma (s) = t\) does not tell us how the other elements of \(\displaystyle S\) are mapped ... ... ? ... and there are many alternatives ... and hence presumably, many \(\displaystyle \sigma\)s ... ... ? ... ... can someone clarify this matter ...
Hope someone can help ...

Peter

Morphisms in a category are typically NOT unique, just as morphisms in $\mathbf{Set}$ are non-unique (we typically have MANY functions $f:A \to B$, for any pair $(A,B) \in [\mathcal{Obj}(\mathbf{Set})]^2$). Just so, a morphism in $\mathbf{Set}_{\ast}$ is a function $f: (A,a_0) \to (B,b_0)$ such that $f:A \to B$ is a function with $f(a_0) = b_0$. Such maps are called "base-point preserving".

 

[Math Amateur]

It is a "generic" singleton set. If you prefer, you may think of it as the set $\{1\}$, although that notation implies we have some sort of numeric structure, which is not the case. In the category $\mathbf{Set}$, singletons are unique-up to set-isomorphism (bijection). It is possible to have $(S,s_1)$ and $(S,s_2)$ where $S = S$ but $s_1 \neq s_2$-but the idea is we are thinking of each $S$ as "coming with" a distinguished point", or "base point" that "roots it." I admit it may not immediately be clear *why* we are doing this, but perhaps this might make it clear.

In the category $\mathbf{Set}$, there is an initial object $\emptyset$: that is, there is a unique function $\emptyset \to A$ for any $A$, the empty function. We also have terminal objects, there is a unique function $A \to \{\ast\}$ (these are called "constant functions"). Note that the initial object and the terminal object are "different". Using pointed sets fixes this asymmetry.
Morphisms in a category are typically NOT unique, just as morphisms in $\mathbf{Set}$ are non-unique (we typically have MANY functions $f:A \to B$, for any pair $(A,B) \in [\mathcal{Obj}(\mathbf{Set})]^2$). Just so, a morphism in $\mathbf{Set}_{\ast}$ is a function $f: (A,a_0) \to (B,b_0)$ such that $f:A \to B$ is a function with $f(a_0) = b_0$. Such maps are called "base-point preserving".

Thanks Evgeny, Deveno ... appreciate your help ...

Reflecting on your posts now ... still a bit puzzled though ... especially about {*} ...

Peter

 

[Deveno]

Thanks Evgeny, Deveno ... appreciate your help ...

Reflecting on your posts now ... still a bit puzzled though ... especially about {*} ...

Peter

In category theory, one is concerned with how things BEHAVE, not what they "are". For example, the groups:

$\{1,i,-1,-i\}$ under complex multiplication and $\{[0]_4,[1]_4,[2]_4,[3]_4\}$ under addition modulo $4$ are regarded as "the same", since there exists a group isomorphism between them (what kind of things "isomorphisms" are will vary from category to category).

With sets, an "isomorphism" is a bijective function. It is pretty clear that if $\{a\}$ and $\{b\}$ are two one-element sets, then:

$f:\{a\} \to \{b\}$ given by $f(a) = b$ is a bijection between them. So insofar as their "set-behavior" is concerned, there really isn't any significant difference between them except for the SYMBOL we attach to their sole element. Put another way, the most significant ALGEBRAIC property of a "set without algebraic structure" is its SIZE (its cardinality).

For example, the sets $\{1,2,3\}$ and $\{\text{Alice},\text{Bob},\text{Carol}\}$ are "essentially the same set" (up to a re-naming). We might represent both/either by $\{a,b,c\}$.

In algebraic topology, it is common for one to make a structure out of "loops in a space", $X$. A loop is just a continuous function:

$f:\Bbb I = [0,1] \to X$, where $f(0) = f(1)$.

To compose loops, its typical to run one after another, but this only makes sense if both loops start and end at "the same place". So one "picks a basepoint". It often turns out it doesn't matter "which point in $X$" one chooses, as long as you "stick with it". So we pick a generic point, and call it $\{\ast\}$ (or we could call it $x_0$, no real difference).

Another facet of category theory is the emphasis on arrows rather than objects. With sets, we don't "need elements", we can instead speak of a mapping $\{\ast\} \to A$, which picks out the element of $A$ that is the image of this mapping. Such a mapping is necessarily injective, and thus can be considered a kind of "inclusion" function.

 

[Math Amateur]

In category theory, one is concerned with how things BEHAVE, not what they "are". For example, the groups:

$\{1,i,-1,-i\}$ under complex multiplication and $\{[0]_4,[1]_4,[2]_4,[3]_4\}$ under addition modulo $4$ are regarded as "the same", since there exists a group isomorphism between them (what kind of things "isomorphisms" are will vary from category to category).

With sets, an "isomorphism" is a bijective function. It is pretty clear that if $\{a\}$ and $\{b\}$ are two one-element sets, then:

$f:\{a\} \to \{b\}$ given by $f(a) = b$ is a bijection between them. So insofar as their "set-behavior" is concerned, there really isn't any significant difference between them except for the SYMBOL we attach to their sole element. Put another way, the most significant ALGEBRAIC property of a "set without algebraic structure" is its SIZE (its cardinality).

For example, the sets $\{1,2,3\}$ and $\{\text{Alice},\text{Bob},\text{Carol}\}$ are "essentially the same set" (up to a re-naming). We might represent both/either by $\{a,b,c\}$.

In algebraic topology, it is common for one to make a structure out of "loops in a space", $X$. A loop is just a continuous function:

$f:\Bbb I = [0,1] \to X$, where $f(0) = f(1)$.

To compose loops, its typical to run one after another, but this only makes sense if both loops start and end at "the same place". So one "picks a basepoint". It often turns out it doesn't matter "which point in $X$" one chooses, as long as you "stick with it". So we pick a generic point, and call it $\{\ast\}$ (or we could call it $x_0$, no real difference).

Another facet of category theory is the emphasis on arrows rather than objects. With sets, we don't "need elements", we can instead speak of a mapping $\{\ast\} \to A$, which picks out the element of $A$ that is the image of this mapping. Such a mapping is necessarily injective, and thus can be considered a kind of "inclusion" function.

Thanks for a really helpful and clarifying post, Deveno ...

Great to know about the link with algebraic topology ...

Sorry to be a bit slow in responding ... had to travel out of Tasmania for a whole ...

Peter