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The question here explains how a cartesian product of sets are specified. It is difficult for me to follow it; for example, I do not know about "symmetric monoidal category".

Can anybody please explain cartesian product construction using simpler concepts? for example, what set of constraint should be checked to see if an object is a cartesian product of another two objects without looking inside the objects?

PS: Question edited to avoid duplication

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qartal
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    Qartal the Q&A you mentioned has 2 parts.The first part proves that the categorical product of 2 objects $Y$, $Z$ (which are sets) in the category $Rel$ is given by the object (which is a set) that you would call "the disjoint union of $Y$ and $Z$".This part has nothing to do with the cartesian product of $Y$ and $Z$.The second part describes/characterizes the object of $Rel$ that you would call "the cartesian product of $X$ and $Y$"... – magma Mar 10 '15 at 09:36
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    ....This description is made using concepts from monoidal categories,so I think that you need to study them a bit if you want to understand this part of the answer given by Zhen Lin. – magma Mar 10 '15 at 09:36
  • That is true, I am interested to understand a characterization of an object in Rel Cat which represent a Cartesian product of two sets. – qartal Mar 10 '15 at 16:21

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Suppose that $P : \mathsf{Rel} \times \mathsf{Rel} \to \mathsf{Rel}$ is a functor equipped with natural isomorphisms $$\hom(P(X,Y),Z) \cong \hom(X,P(Y,Z))$$ and $$P(X,1) \cong X \cong P(X,1).$$

Then $P(-,Y)$ is left adjoint to $P(Y,-)$, so that it preserves colimits, in particular coproducts. This implies $$P(X,Y) \cong P(\coprod_{x \in X} 1,Y) \cong \coprod_{x \in X} P(1,Y) \cong \coprod_{x \in X} Y \cong X \times Y.$$

Martin Brandenburg
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