Functor From Any Category to Preorder as a Category .... Awodey Exercise 8, Chapter 1 ....

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 1: Categories

I need some help in order to make a meaningful start on Awodey Exercise 8, Chapter 1 Awodey Exercise 8, Chapter 1 reads as follows:View attachment 8388

I am unable to make a meaningful start on this exercise ... can someone please help me to formulate a solution to the above exercise ...I would like to utilise the exact form of Awodey's definition of functor (see below) in the solution to the exercise ...
Help will be much appreciated ...

Peter=======================================================================================
It may well help readers of the above post to have access to Awodey's definition of a preorder and its consideration as a category ... so I am providing the same ... as follows:View attachment 8389It may also help readers of the above post to have access to Awodey's definition of a functor ... so I am providing the same ... as follows:View attachment 8390
View attachment 8391Hope that helps ...

Peter

 

[steenis]

Given is a category $C$

Define the category $P(C)$ as follows:

objects: the objects of $P(C)$ are the objects of $C$
arrows: $f:A \leq B$ is an arrow in $P(C)$ if and only if $f:A \rightarrow B$ is an arrow in $C$

Prove that $P(C)$ is a category

This makes $P$ a functor $P:C \rightarrow P(C)$ by defining $P$ as follows
for objects: if $A \in C$ is an object, then $PA=A$

for arrows: if $f:A \rightarrow B$ is an arrow in C, then $Pf=f:A \leq B$

Show that this is a functor

Using this, we have to define a functor $\mathscr{P}: "categories" \rightarrow "preorders"$, but I have to think about that for a while.

 

[Math Amateur]

Given is a category $C$

Define the category $P(C)$ as follows:

objects: the objects of $P(C)$ are the objects of $C$
arrows: $f:A \leq B$ is an arrow in $P(C)$ if and only if $f:A \rightarrow B$ is an arrow in $C$

Prove that $P(C)$ is a category

This makes $P$ a functor $P:C \rightarrow P(C)$ by defining $P$ as follows
for objects: if $A \in C$ is an object, then $PA=A$

for arrows: if $f:A \rightarrow B$ is an arrow in C, then $Pf=f:A \leq B$

Show that this is a functor

Using this, we have to define a functor $\mathscr{P}: "categories" \rightarrow "preorders"$, but I have to think about that for a while.

Hi Hugo ... ...

Thanks for a very helpful post above ... much appreciated ...

I will try to fill in the details following what you have said ...
We are given a category \(\displaystyle \mathscr{C}\) with objects \(\displaystyle A, B, C, D,\) ... ...

and arrows \(\displaystyle f: A \to B, \ g : B \to C,\) ... ...Now ... \(\displaystyle \mathscr{C}\) determines a preorder \(\displaystyle P( \mathscr{C} )\) when we define a binary relation \(\displaystyle \leq\) on the objects of \(\displaystyle \mathscr{C}\) by

\(\displaystyle A \leq B\) if and only if there is an arrow \(\displaystyle A \to B\) ... ...Now we need to show that \(\displaystyle P( \mathscr{C} )\) is a category before we can assert that there is a functor between \(\displaystyle \mathscr{C}\) and \(\displaystyle P( \mathscr{C} )\) ...

So ... the objects of \(\displaystyle P( \mathscr{C} )\) are the objects \(\displaystyle A, B, C, D,\) ... ... of category \(\displaystyle \mathscr{C}\) ... ...

and the arrows of \(\displaystyle P( \mathscr{C} )\) are of the form \(\displaystyle f_{ \Large_P } : A \to B\) where \(\displaystyle f_{ \Large_P } \Longrightarrow A \leq B\)

Now, consider arrows \(\displaystyle f_{ \Large_P }: A \to B, g_{ \Large_P } : B \to C\) implying respectively that \(\displaystyle A \leq B\) and \(\displaystyle B \leq C\) ...

We define composition as "and" in the following sense ...

\(\displaystyle g_{ \Large_P } \circ f_{ \Large_P } : A \to D\) is defined as \(\displaystyle A \leq B\) and \(\displaystyle B \leq C \Longrightarrow A \leq C\) (a preorder is transitive!)

We also have \(\displaystyle 1_{ A_P } : A \to A\) is \(\displaystyle A \leq A\) (a preorder is reflexive!)
Now ... consider the functor \(\displaystyle F : \mathscr{C} \to P( \mathscr{C} )\) by defining \(\displaystyle F\) as follows ...

For an object \(\displaystyle A \in \mathscr{C}\) we have \(\displaystyle F(A) = A\) (where \(\displaystyle F(A) \in P( \mathscr{C} )\) ... )

For arrows \(\displaystyle f : A \to B\) in \(\displaystyle \mathscr{C}\) we have \(\displaystyle F(f) = f_{ \Large_P }\) in \(\displaystyle P( \mathscr{C} )\) where \(\displaystyle f_{ \Large_P } \Longrightarrow A \leq B \) ...Now ... \(\displaystyle F\) is a functor since ( checking against Awodey's conditions (a), (b) and (c) ... pages 8 - 9 ... )(a) \(\displaystyle F( f : A \to B ) = f_{ \Large_P } : A \to B = F(f) : F(A) \to F(B)\)

... so \(\displaystyle F\) preserves arrows between given objects ...
(b) \(\displaystyle F( 1_A ) = 1_{ A_P }\) since a preorder is reflexive ...
(c) \(\displaystyle F( g \circ f ) = g_{ \Large_P } \circ f_{ \Large_P } = F(g) \circ F(f)\) since composition is "and" preorder is transitive ...Is that correct?I must say that I am perplexed and mystified as to the meaning of the rest of the question ... hope that you can help ..

Peter

 

[steenis]

Recall the definition of $PC$ for a category $C$, with a slightly different notation:
For objects $a \in C$: $Pa=a$ in $PC$
For arrows $f:a \rightarrow b$ in $C$: $f:a \leq b$ in $PC$
($f:a \rightarrow b$ is an arrow in C if and only if $f:a \leq b$ is an arrow in PC)

Notations:
$Cat$ = “categories”, the category of “all ?” categories
$Pre$ = “preorders”, the category of all categories equipped with a preorder, see item 7 on page 9.

A category with a preorder is a category. One can define an “inclusion” of $Pre$ in $Cat$ by this functor $I:Pre \rightarrow Cat$ defined by:
For objects $Q \in Pre$: $IQ=Q$
For arrows $F$ in $Pre$: $IF=F$, notice that an arrow in $Pre$ is a functor $F:Q_1 \rightarrow Q_2$ between categories in $Pre$, so $F:Q_1 \rightarrow Q_2$ is also a functor, that is, an arrow in $Cat$ between categories in $Cat$

Definition of $\mathscr{P}:Cat \rightarrow Pre$
For objects $C \in Cat$: $\mathscr{P}C = PC$, with the same P as above
For arrows $F:C \rightarrow D$ in $Cat$, (which are functors between categories), we define the functor $\mathscr{P}F:PC \rightarrow PD$ in $Pre$ as follows:

For objects $a \in PC$, we define $\mathscr{P}F(a) = Fa$ (remember, if $a$ is a member of the category PC, then $a$ is also a member of the category $C$, that is, $C$ and $PC$ have the same objects)
For arrows $f:a \leq b$ in $PC$, we define $\mathscr{P}F(f) = F(f): Fa \leq Fb$
Because if $f:a \leq b$ is an arrow in $PC$, then $f:a \rightarrow b$ is an arrow in C, then $Ff:Fa \rightarrow Fb$ is an arrow in $D$ and $F(f): Fa \leq Fb$ is an arrow in PD. Therefore we define $\mathscr{P}F(f) = F(f): Fa \leq Fb$

If $1_a$ is the identity arrow of the object $a \in PC$, that is, $1_a:a \leq a$, then $F1_a:Fa \leq Fa$, thus $F1_a = 1_{Fa}$

If $f:a \leq b$ and $g:b \leq c$ in PC, then $g \circ f:a \leq c$ in $PC$.
Then, $Ff:Fa \leq Fb$ and $Fg:Fb \leq Fc$ and $F(g \circ f)=Fg \circ Ff:a \leq c$ in $PD$

Pretty complex he? But in fact, the only thing we do is replacing $f:a \rightarrow b$ with $f:a \leq b$, vice versa.

 

[Math Amateur]

Recall the definition of $PC$ for a category $C$, with a slightly different notation:
For objects $a \in C$: $Pa=a$ in $PC$
For arrows $f:a \rightarrow b$ in $C$: $f:a \leq b$ in $PC$
($f:a \rightarrow b$ is an arrow in C if and only if $f:a \leq b$ is an arrow in PC)

Notations:
$Cat$ = “categories”, the category of “all ?” categories
$Pre$ = “preorders”, the category of all categories equipped with a preorder, see item 7 on page 9.

A category with a preorder is a category. One can define an “inclusion” of $Pre$ in $Cat$ by this functor $I:Pre \rightarrow Cat$ defined by:
For objects $Q \in Pre$: $IQ=Q$
For arrows $F$ in $Pre$: $IF=F$, notice that an arrow in $Pre$ is a functor $F:Q_1 \rightarrow Q_2$ between categories in $Pre$, so $F:Q_1 \rightarrow Q_2$ is also a functor, that is, an arrow in $Cat$ between categories in $Cat$

Definition of $\mathscr{P}:Cat \rightarrow Pre$
For objects $C \in Cat$: $\mathscr{P}C = PC$, with the same P as above
For arrows $F:C \rightarrow D$ in $Cat$, (which are functors between categories), we define the functor $\mathscr{P}F:PC \rightarrow PD$ in $Pre$ as follows:

For objects $a \in PC$, we define $\mathscr{P}F(a) = Fa$ (remember, if $a$ is a member of the category PC, then $a$ is also a member of the category $C$, that is, $C$ and $PC$ have the same objects)
For arrows $f:a \leq b$ in $PC$, we define $\mathscr{P}F(f) = F(f): Fa \leq Fb$
Because if $f:a \leq b$ is an arrow in $PC$, then $f:a \rightarrow b$ is an arrow in C, then $Ff:Fa \rightarrow Fb$ is an arrow in $D$ and $F(f): Fa \leq Fb$ is an arrow in PD. Therefore we define $\mathscr{P}F(f) = F(f): Fa \leq Fb$

If $1_a$ is the identity arrow of the object $a \in PC$, that is, $1_a:a \leq a$, then $F1_a:Fa \leq Fa$, thus $F1_a = 1_{Fa}$

If $f:a \leq b$ and $g:b \leq c$ in PC, then $g \circ f:a \leq c$ in $PC$.
Then, $Ff:Fa \leq Fb$ and $Fg:Fb \leq Fc$ and $F(g \circ f)=Fg \circ Ff:a \leq c$ in $PD$

Pretty complex he? But in fact, the only thing we do is replacing $f:a \rightarrow b$ with $f:a \leq b$, vice versa.

Thanks Hugo ...Yes indeed ... pretty complex ... especially considering what is in fact essentially underlying the category theory ... no wonder some people refer to category theory as "abstract nonsense" ...

Still reflecting on what you have written ...

Thanks again...

Peter