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Need help understanding a monoid as a category

  1. Apr 12, 2012 #1
    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?
  2. jcsd
  3. Apr 12, 2012 #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.
  4. Apr 12, 2012 #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?
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