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I'm struggling with comprehending what monoids are in terms of category theory.

In examples they view integer numbers as a monoid. I think I get the set theoretic definition. We have a set and a associative binary operator (addition) and the neutral element (zero).

Then they are saying something like that - view the whole set as a single object and binary operator as bunch of morphisms for every element of the set.

Like add0 is an identity morphism. Which would really give us the same object i.e. the same set of all integer numbers. I think I understand this.

But let's view the morphism add1. After applying it to the our single object (the set of all integers) we would have a set {1,2,3…} not the {0,1,2,3…}. Aren't domain and codomain different in that case?

That's what's bothering me. Can someone clarify that to me?

Here is the text that gives me problems.

The text

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3 Answers 3

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No, that's not the way to view it. In order to view a monoid as a category, you have a single object $\mathsf{Andreas}$, and each element of the monoid is one morphism $\mathsf{Andreas}\to\mathsf{Andreas}$ in the category. The monoid operation is the composition in the category.

So for the integers, you don't have a morphism "add 1", but a morphism that is simply called $1$. And composition in the category works such that $1$ composed with $1$ is the morphism called $2$.

This is an example of a category where the morphisms are not functions.

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    $\begingroup$ @user1685095: The domain of each morphism is $\bullet$ (that is, the object of the category), and so is the codomain. The very point here is that a morphism is not necessarily a function. $\endgroup$ Jun 20, 2015 at 15:13
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    $\begingroup$ @user1685095: $\bullet$ is the name of the object in the category. Its precise identity is not important. $\endgroup$ Jun 20, 2015 at 15:26
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    $\begingroup$ @user1685095 add$1$ can be looked at as a function having $\mathbb N$ as domain and as codomain. It is prescribed by $n\mapsto n+1$. There is a category with objectset $\{\mathbb N\}$ and morphismset $\{\text{add}n\mid n\in\mathbb N\}$. So there is only one object and it can equally well be denoted as $\bullet$. Monoids can be identified as categories that have exactly one object. Composition of morphisms corresponds with addition: $\text{add}k\circ\text{add}m=\text{add}(k+m)$. $\endgroup$
    – drhab
    Jun 20, 2015 at 15:47
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    $\begingroup$ Since the unique but unspecified object seems to cause difficulties for the OP, I hereby volunteer to make things specific by being that object. That is, the category has one object, namely me, and its morphisms are the integers. Composition of morphisms is addition of integers. $\endgroup$ Jun 20, 2015 at 16:00
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    $\begingroup$ @Phạm Văn Thông, unfortunately not. First, as hmakholm left over Monica said, the morphisms 0, 1, 2, 3 etc. are (or correspond to) elements of the monoid. Composition is the monoid operation. So if the monoid operation is +, 1 composed with 2 is 3, showing that neither are identity morphisms. Second & perhaps more deeply, assume morphisms i and j are both identities. Then what is j composed with i, j ∘ i? Focusing on j as identity, it has to be i. But focusing on i as identity, it has to be j. So j=i, and we have only one identity morphism. So only one of 0, 1, 2, 3 ... could be one anyway. $\endgroup$ Jun 13, 2021 at 17:44
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You already know that a monoid $M$ is a set with a unit $e$ and a binary operation. More precisely, if $a,b,c\in M$ then $$a\circ b\in M$$ $$(a\circ b)\circ c=a\circ (b\circ c)$$ $$e\circ a=a\circ e=a$$

Now, take any category $C$ with one object, $c$. Since $C$ is a category, we need to say what the arrows are. That is, what the morphisms $c\rightarrow c$ are. There must be a unit $1_{c}:c\rightarrow c$, the arrows must be composeable and the composition must be associative. More precisely, if $f,g$ and $h$ are arrows, then $$f\circ g\in Morph(C)$$ $$(f\circ g)\circ h= f\circ (g\circ h)$$ $$1_{c}\circ f=f\circ 1_{c}=f$$Notice we are $\textit not$ talking about sets here. Just objects and arrows, in the abstract.

But now if we just observe that the operations on $M$ are $\textit exactly$ the same as the operations on $Morph(C)$, we may regard the category $C$ as the monoid $M$. This correspondence is reversible: given category $C=\left \{ c \right \}$ we obtain a monoid $M$ whose elements are the arrows of $C$.

Thus the two descriptions are equivalent.

All this works because the binary operations are the same for both structures.

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  • $\begingroup$ What's special about the case when $Ob(C)=\{c\}$? Why can't we interpret a monoid as a category with more than one object? The same axioms will hold for $Morph(C)$. $\endgroup$
    – user557
    Jan 27, 2019 at 16:15
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    $\begingroup$ You want to identify Morph$C$ with $M$? But in an arbitrary category, not all morphisms are composable, right? If you have $f,g:A\to B$ then $f\circ g$ doesn't even exist. What's special about the above construction is that there is an $\textit {exact}$ i.e. bijective correspondence between the operations. $\endgroup$ Jan 27, 2019 at 16:37
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    $\begingroup$ Then we can consider a category with more than one object and the set of all morphisms from an object $A$ to $A$. This set with the operation of composition should be a monoid. Can we then say that conversely, a monoid can be interpreted as the set of arrows from a fixed object to itself in an arbitrary category? This is also a bijective correspondence between the arrows from that object to itself and the monoid operation. $\endgroup$
    – user557
    Jan 27, 2019 at 17:00
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    $\begingroup$ "...a monoid can be interpreted as the set of arrows from a fixed object to itself in an arbitrary category"...yes, and this is the essence of my answer. If you have a category $C$ with small hom-sets, then for each $c\in C$, then each Hom$[c,c]$ will give a monoid, distinct if the cardinalities are different. $\endgroup$ Jan 27, 2019 at 17:06
  • $\begingroup$ Then I guess the only reason to mention categories with one object is to be able to make statements like "a monoid is a category with one object" as opposed to "a monoid is a set of morphisms from a fixed object to itself in a category". So a monoid and a category with one object are equivalent concepts (like groups and $\mathbb Z$-modules). Given a category with one object, one obtains a monoid as in your answer. Conversely, given a monoid one constructs the corresponding category by taking the only object to be $\{\ast\}$ and the only morphism $\ast\mapsto \ast$. Is that correct? $\endgroup$
    – user557
    Jan 27, 2019 at 17:40
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In addition to Henning Makholm's crisp and clear answer, you might find the opening six pages of my Notes on Category Theory helpful. They too give the example of a monoid as a category, but also give some other examples of categories where the arrows are not functions in any ordinary sense. Another important illustration is the case of a posets treated as a category.

In fact these examples suggest why we might well prefer to talk of 'arrows' rather than 'morphisms' (because the very term 'morphism' comes with baggage, and almost inevitably makes us think of a function -- but to repeat, arrows need not be functions).

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  • $\begingroup$ A footnote to another solution should, well, be a footnote on that solution (or rather a comment or two.) $\endgroup$
    – rschwieb
    Jun 20, 2015 at 16:03
  • $\begingroup$ Nice notes Peter. And nice blog too. I will contact you there. $\endgroup$
    – magma
    Jun 24, 2015 at 2:03
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    $\begingroup$ the link is broken $\endgroup$
    – glS
    May 14, 2021 at 9:10

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