Monoid in the category of endofunctors and Monoid as a category with one object

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Quoting from Categories for the Working Mathematician by Saunders Mac Lane:




All told, a monad in X is just a monoid in the category of
endofunctors of X, with product × replaced by composition of
endofunctors and unit set by the identity endofunctor.




At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.






category-theory monoid monoidal-categories monads







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    Quoting from Categories for the Working Mathematician by Saunders Mac Lane:




    All told, a monad in X is just a monoid in the category of
    endofunctors of X, with product × replaced by composition of
    endofunctors and unit set by the identity endofunctor.




    At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.






    category-theory monoid monoidal-categories monads







    share|cite|improve this question























      up vote
      5
      down vote

      favorite
      1









      up vote
      5
      down vote

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      Quoting from Categories for the Working Mathematician by Saunders Mac Lane:




      All told, a monad in X is just a monoid in the category of
      endofunctors of X, with product × replaced by composition of
      endofunctors and unit set by the identity endofunctor.




      At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.






      category-theory monoid monoidal-categories monads







      share|cite|improve this question













      Quoting from Categories for the Working Mathematician by Saunders Mac Lane:




      All told, a monad in X is just a monoid in the category of
      endofunctors of X, with product × replaced by composition of
      endofunctors and unit set by the identity endofunctor.




      At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.






      category-theory monoid monoidal-categories monads




      category-theory monoid monoidal-categories monads






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      Roland

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          The quote you mention does not say that monads are monoids in the usual sense. Rather, they are monoids in the monoidal category of endofunctors of $X$. In general, a monoid in a monoidal category is an oject of that category, so it need not be any kind of set; in particular, it does not have elements. But seeing a monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoids in monoidal categories.



          You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).






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            The quote you mention does not say that monads are monoids in the usual sense. Rather, they are monoids in the monoidal category of endofunctors of $X$. In general, a monoid in a monoidal category is an oject of that category, so it need not be any kind of set; in particular, it does not have elements. But seeing a monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoids in monoidal categories.



            You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).






            share|cite|improve this answer


























              up vote
              6
              down vote













              The quote you mention does not say that monads are monoids in the usual sense. Rather, they are monoids in the monoidal category of endofunctors of $X$. In general, a monoid in a monoidal category is an oject of that category, so it need not be any kind of set; in particular, it does not have elements. But seeing a monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoids in monoidal categories.



              You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).






              share|cite|improve this answer
























                up vote
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                up vote
                6
                down vote









                The quote you mention does not say that monads are monoids in the usual sense. Rather, they are monoids in the monoidal category of endofunctors of $X$. In general, a monoid in a monoidal category is an oject of that category, so it need not be any kind of set; in particular, it does not have elements. But seeing a monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoids in monoidal categories.



                You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).






                share|cite|improve this answer














                The quote you mention does not say that monads are monoids in the usual sense. Rather, they are monoids in the monoidal category of endofunctors of $X$. In general, a monoid in a monoidal category is an oject of that category, so it need not be any kind of set; in particular, it does not have elements. But seeing a monoid as a one-object category means precisely that you identify its elements with the arrows of a category; so you can't do that for monoids in monoidal categories.



                You can however, identify such monoids with one-object enriched categories, exactly in the same way that classical monoids can be identified with classical one-object categories (in fact the classical case is just enrichment over sets with the cartesian product).







                share|cite|improve this answer














                share|cite|improve this answer



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                edited 4 hours ago

























                answered 4 hours ago









                Arnaud D.

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