Need help understanding a monoid as a category

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In summary, a monoid is defined as a category with one object and arrows connecting its elements, which may not necessarily connect the monoid to itself. This contradicts the definition of category arrows, causing confusion about the number and direction of arrows in a monoid category. However, according to the definition, a monoid can be represented as a category with only one object and arrows between that object, which correspond to the elements of the monoid. This may seem ambiguous, but it follows the definition of category arrows.
  • #1
byron.hawkins
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Many textbooks describe a monoid as a category of one object having arrows for its elements. But they also define a category arrow as a binary relation between two (not necessarily distinct) objects of the category. So the elements of a monoid are actually connecting the elements of the monoid set--they do not connect the monoid to itself. Therefore how can we say that the "category arrows" of the "monoid category" connect its elements? It seems like we must either say:

(1) the elements of the monoid set comprise the objects of the category, or
(2) the "monoid category" has only one arrow, namely the category identity (different from the monoid identity).

In the case of (2), we are saying that a monoid is a nested category, with itself in the outer category, and the elements of its set in the inner category. This would make sense to me. But the description given in textbooks... it contradicts the definition of category arrows.

For example, suppose we have a monoid on {0,1,2} with operator "addition modulo 3" and identity element 0. If this monoid is a category such that each arrow joins its numerical elements, then how can this monoid be a one-object category? It has 3 objects, namely 0, 1 and 2. Alternatively, if this monoid is a category with one arrow from the monoid to itself as the "category identity" (not meaning the "monoid identity" 0), then I can see it as a one-object category.

Can someone please help me understand why the textbooks define a monoid as a category this way?
 
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  • #2
A category is determined by a class of objects and also for each to objects A and B a set of arrow Hom(A,B).

If M is a monoid, we can make a category as follows:
There is only one object called *.
Since there is only one object, we only have to specify Hom(*,*). We define Hom(*,*)=M. So the arrows between * are exactly the elements of M.
 
  • #3
Thanks for your reply. So in the case of my example monoid on {0, 1, 2}, would I expect to have 3 arrows, all from * to *? If that is the case, aren't the arrows kind of ambiguous? It seems like the arrows should be between the {0, 1, 2}, not from * to *. Can you help me understand why it is like this?
 

1. What is a monoid?

A monoid is a mathematical structure that consists of a set of elements and an operation that combines any two elements in the set to produce a third element in the set. The operation must also be associative and have an identity element.

2. What is the difference between a monoid and a group?

A monoid is similar to a group, but it does not necessarily have an inverse element for every element in the set, whereas a group does. This means that a monoid is a less restrictive structure than a group.

3. How is a monoid related to a category?

A monoid can be thought of as a category with a single object, where the elements of the monoid are the morphisms in the category. The operation in the monoid corresponds to the composition of morphisms in the category.

4. Can you give an example of a monoid?

One example of a monoid is the set of natural numbers (excluding 0) under addition. The operation of addition is associative and the identity element is 0. Another example is the set of positive integers under multiplication, where the operation is also associative and the identity element is 1.

5. What are some real-world applications of monoids?

Monoids have many applications in computer science, such as in functional programming languages where they are used to model computations. They are also used in algebraic structures like semigroups and rings, and in cryptography for generating and verifying digital signatures.

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