Please help me understand 'arrows' (morphism) in Category theory. For a Category A let natural numbers be 'objects' and let's assume that I want to define summation (+) as the composition then what can I define the 'arrow' as? (I am sorry if the question doesn't make much sense, I am still trying to understand basics of Category theory. All I understand about arrows is that it can be anything and may not actually be a mapping from one set to another)

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    $\begingroup$ If you want to define summation as the composition then the natural numbers should be the morphisms, not the objects. You can take there to be one object. $\endgroup$ Jul 29 '13 at 23:30
  • $\begingroup$ Thanks, what you wrote makes more sense, but then what would be my 'object(s)'. Can you please elaborate? $\endgroup$
    – Dev Maha
    Jul 30 '13 at 0:00
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    $\begingroup$ The object would just be a placeholder, it's only really there so that the arrows have a source and target. All of the information about the addition of natural numbers is really in the arrows. $\endgroup$
    – Dan Rust
    Jul 30 '13 at 0:09
  • $\begingroup$ Now it makes sense. This is the exact reason I asked this query because I wanted to know if there could be something which is 'just' a place holder. Thanks for confirming. $\endgroup$
    – Dev Maha
    Jul 30 '13 at 1:05
  • $\begingroup$ @DanielRust, so would it be correct to say that the 'composition' is actually on the arrow and has nothing to do with the object? [I read another example which kind of confirms this. In that, natural numbers are objects, f:N->M, arrow is a NxM matrix and matrix multiplication is the composition] $\endgroup$
    – Dev Maha
    Jul 30 '13 at 1:09

I assume your natural numbers contain $0$. For $a,b$ natural numbers, let $\hom(a,b)$ be empty if $b< a$ and otherwise let it contain a single element/arrow taking $a$ to $b$ so addition by $b-a$.

You then have an identity in $\hom(a,a)$ and composition is associative and corresponds to addition.

I am not completely sure if this is what you have in mind. This is a bit of an unusual choice of an example to understand categories, in my opinion, but since you asked... :-)

In view of the comments: as Qiaochu Yuan said it seems more natural to have a category with one object, the object is the natural numbers. For fixed $a$, the map $f_a: \mathbb{N}\to \mathbb{N}$, $x \mapsto x+a$ is then an element of $\hom(\mathbb{N}, \mathbb{N})$, and let us say these are all the elements of the hom-set.

Composition of $f_a \circ f_b$ is $f_{a+b}$, this is associative, and you also have an identity element $f_0$.

  • $\begingroup$ Thank you for replying and consolidating ideas from the comments (this way I have at least something to mark as 'right answer') $\endgroup$
    – Dev Maha
    Jul 30 '13 at 1:13
  • $\begingroup$ You are welcome. In general I am/was not sure how well you understand categories already. The most natural examples are others. Say, fix a field, the reals for example, and consider all vector spaces over this field as obejects, and the morphism are just linear maps. Or groups as objects, and groups homomorphisms as morphisms, and so on for the different algebraic structures. But perhaps this is already clear to you and you are searching for more "exotic" examples. $\endgroup$
    – quid
    Jul 30 '13 at 1:22
  • $\begingroup$ To follow up on another comment of yours. By definiton of a category you always need to have a composition of the arrows. But also you can have 'operations' or compositions so to say on the objects or in the objects. Consider the example of groups. The arrows are homomorphisms of groups if you have g from G to H and one f from H to K than you can consider their composition f o g from G to K. This is the compoisiton of the morphisms/arrows. But on each group/object you also have an internal law/composition but these are different things. $\endgroup$
    – quid
    Jul 30 '13 at 1:28
  • $\begingroup$ Thanks again, I learnt something new from your reply. $\endgroup$
    – Dev Maha
    Jul 30 '13 at 2:20

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