# Is there any difference of relation and morphism?

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?

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

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.
• @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 Mar 14, 2023 at 17:37
• @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 Mar 17, 2023 at 19:09
• 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 Mar 17, 2023 at 19:10