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Well, my thoughts for now is that relation is a mapping idea defined on $Set$ category (the Set Theory Context). Relations, for instance, they are a more broad idea than functions, because a function is a relation which need always obey the properties of total and functional.

By another hand, on Category Theory world, morphism is even a more abstract ideia for mapping, which is used not only inside of the Set Theory.

However, seems to me that a relation is kind of morphism. But the inverse is not true, right? Or is?

$ relation \Rightarrow morphism $ but $ morphism \not \Rightarrow relation $

What rules a relation need to obey, but morphism don't? What makes they different?

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  • $\begingroup$ Not always, endomorphism implies the reflexive relation. $\endgroup$ Oct 25, 2017 at 0:55
  • $\begingroup$ Can you elaborate it? What are you referring? $\endgroup$ Oct 25, 2017 at 1:14

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In some categories, the morphisms are relations. In particular, there is a category, often referred to as $\mathbf{Rel}$, whose morphisms consist of arbitrary binary relations between sets with relational composition. Conversely, a morphism in some arbitrary category need not even be a set, let alone a relation.

An analogy (that's more than analogy) would be to group theory. A group is a set of elements and a binary operation on that set satisfying some laws. The notion of group does not require the elements to be any particular thing. You can have a group of numbers, or a group of permutation, or a group of points in a manifold.

The situation is the same for categories. A category is a family of hom-sets and a family of binary operations on those hom-sets that satisfy some laws. The elements of those hom-sets, i.e. morphisms, can be anything. You can have a category where the morphisms are numbers, or permutations, or points in a manifold, or, also, functions or relations.

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  • $\begingroup$ Can you give an example where a morphism is a number? $\endgroup$
    – Make42
    Nov 28, 2022 at 16:33
  • $\begingroup$ @Make42 The category with one object and $\Bbb Z$-many arrows from that object to itself, one for each integer, and the composition law is $n+m=(n+m)$ for composing the $n$th arrow with the $m$th arrow $\endgroup$
    – FShrike
    Mar 14, 2023 at 17:37
  • $\begingroup$ @FShrike: This might go into the philosophy of mathematics than mathematics itself, but I do not see how this would make the arrows to be numbers. I would rather say that the arrows are somewhat associated with the numbers and then based on the association we are deriving rules for composition. $\endgroup$
    – Make42
    Mar 17, 2023 at 19:04
  • $\begingroup$ @Make42 You could choose the arrow sets to be your favourite set encoding of the numbers $1,2,3,\cdots$, you could say $C(\ast,\ast)=\Bbb Z$ as a definition. Now the arrows “literally are” integers $\endgroup$
    – FShrike
    Mar 17, 2023 at 19:09
  • $\begingroup$ However there’s the subtlety that we don’t really mean one particular set when we say $\Bbb Z$, we just mean any model of it. Well, this is clearer for $\Bbb R$, where there are Dedekind or Cauchy reals as models, and you need to do work to show they are equivalent, and there are a gazillion other models, … not worth thinking about though $\endgroup$
    – FShrike
    Mar 17, 2023 at 19:10

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