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The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.

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enter image description here

Let $A, B$ be sets. A $\textbf{relation from} A$ to $B$, denoted $\alpha:A\to B,$ is a subset $\alpha$ of $A\times B.$ If also $\beta: B\to C$ define $\beta\cdot \alpha: A\to C$ by

$$\beta\cdot \alpha=\{(a,c)\mid \text{ for some }b\in B, (a,b)\in \alpha \text{ and } (b,c)\in \beta \}$$

$(a)$ Show that $\textbf{K},$ whose objects are sets and whose morphisms are relations, is a category with composition as above

$(b)$ Given $\alpha:A\to B$

define

$\alpha^{*}:B\to A$ by $\alpha^{*}=\{(b,a)\mid(a,b)\in \alpha\}.$

Show that a diagram

$$A\xleftarrow{\pi_1}P\xrightarrow{\pi_2}B$$

is a product in $R$ if and only if $$A\xrightarrow{{\pi_1}^{*}}P\xleftarrow{{\pi_2}^{*}}B$$

is a coproduct in $\textbf{K}$

$\color{Red}{Questions:}$

For the exercise above, I am not sure if product and coproduct are defined in the context of $\textbf{Set}.$ If so, I am not sure how to properly define the projection map for the product and injection maps for coproduct. My guess is as follows:

The projection maps are as follows:

$A\times B\xrightarrow{\pi_1}A:(a,b)\mapsto a$ and $A\times B\xrightarrow{\pi_2}B:(a,b)\mapsto b,$ similarly if we replace one of the $A$ or $B$ with $C$. But since the maps $\alpha\in B,$ $\beta\in C,$ $\beta\cdot \alpha \in C$ and $\alpha^{*}\in A.$ Then won't we have:

Let $P=A\times B$

$A\times B\xrightarrow{\pi_1}A:(\alpha^{*}, \alpha)\mapsto \alpha^{*},$

$A\times B\xrightarrow{\pi_2}A:(\alpha^{*}, \alpha)\mapsto \alpha,$

The projection maps for $A\times C$ or $B\times C$ would not make sense since both $\beta\cdot \alpha, \beta$ are in $C$.

For the injection maps,

$A\xrightarrow{{in}_1}A+B:a\mapsto (a,1)$

$B\xrightarrow{{in}_2}A+B:b\mapsto (b,2),$

So, let $P=A+B$, then the injection maps are:

$A\xrightarrow{{in}_1}A+B:\alpha^{*}\mapsto (\alpha^{*},1)$

and

$B\xrightarrow{{in}_2}A+B:\alpha\mapsto (\alpha, 2)$

As in the projection maps case, it would not make sense to have the case $A+C$ or $B+C,$ since both $\beta, \beta\cdot\alpha$ are both in $C.$

In both projections and injection maps, I don't have to write them in terms of elements. What I mean is I don't have to let $\alpha^{*}=(b,a)$ and $\alpha=(a,b)?$

Also, is there ways to involve $\beta\in C$ and $\beta\cdot\alpha\in C$ in the definition of projection and injection maps for product and coproduct respectively.

Thank you in advance

Seth
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    "is a product in $R$" What is $R$? – S.C. Jul 05 '23 at 17:42
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    The question is basically incomprehensible to me. For example, you write "But since the maps $\alpha\in B,$ $\beta\in C,$ $\beta\cdot \alpha \in C$ and $\alpha^{}\in A.$" What? $\alpha$ is an arrow $A\to B$, which in this category is a subset of $A\times B$. It's not an element of $B$. Later you write "don't I have to let $\alpha^ = (b,a)$ and $\alpha = (a,b)$?" Again, $\alpha$ is a subset of $A\times B$, so it's a set of ordered pairs of the form $(a,b)$. It's not itself an ordered pair. Something is very confused here... – Alex Kruckman Jul 05 '23 at 18:53
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    Products and coproducts in $\mathbf{Set}$ are irrelevant to the question. It's all about $\mathbf{K}$. And you're not even being asked to figure out how to describe products and coproducts in $\mathbf{K}$. You're just being asked to show that $(P,\pi_1,\pi_2)$ satisfies the universal property of the product in $\mathbf{K}$ if and only if $(P,\pi_1^,\pi_2^)$ satisfies the universal property of the coproduct in $\mathbf{K}$. – Alex Kruckman Jul 05 '23 at 18:55
  • @AlexKruckman but don't i have to describe what the injections and projections maps are, even in the case of $(P,\pi_1, \pi_2)$ and $(P,{{\pi}_1}^{}, {{\pi}_2}^{})?$ Is it possible to talk about universal properties of product and coproduct without explicit description of respectively their projection and injection maps? – Seth Jul 05 '23 at 19:03
  • @S.C. I attached a screenshot of the original question. – Seth Jul 05 '23 at 19:06
  • You have not defined what "$R$" is, as S.C. has already said. The screenshot does not help with that... Without that we cannot (help you to) answer the question – FShrike Jul 06 '23 at 11:39
  • @FShrike the question did not defined what $R$ is other than it says there is a product in $R$. Am I not reading or misunderstanding the wording of the question? – Seth Jul 06 '23 at 12:15
  • $R$ has to be some kind of category. Alex is suggesting the authors made a typo and that we should have $R=\mathbf{K}$. Do the authors ever define a category $R$, anywhere?? – FShrike Jul 06 '23 at 12:32
  • @FShrike no, not for this question. I am suspecting it is a typo. And assuming it is a typo, how can I talk about the universal properties of product and coproduct without knowing how to explicitly define their projection and injection maps. Also I thought the category $\textbf{K}$, is constructed from the category of $\textbf{Set}.$ Sort of like category of Vector space is constructed from category of Set. – Seth Jul 06 '23 at 13:07
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    You do not need to define any projections or inclusions. You are given a categorical product diagram and want to show a different diagram is a categorical coproduct diagram, and visa versa. You do not need to construct $P$, $P$ is given to you. You do not need to construct $\pi_{1,2}$, they are given to you. You must verify universal properties, nothing more and nothing less – FShrike Jul 06 '23 at 13:22
  • I am pretty sure $R=K$ is intended, on reflection – FShrike Jul 06 '23 at 13:23
  • @FShrike I don't have to worry about the opération $\beta\cdot \alpha$ also? I thought all this time the question implicitly is asking me to try to figure out how to construct them. – Seth Jul 06 '23 at 13:27
  • @Seth You were not asked to construct the products. You were asked to show a given thing is a product – FShrike Jul 06 '23 at 13:38

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