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Groupoids and skeletons of monoids and preorders

A monoid (M,e,m)(M,e,m), where MM is a set, ee an element of MM, and mm an associative binary operation on MM, can be viewed as a single object category. Take M0=MM_0 = \\{M\\} or M0=M_0 = \emptyset and M1M_1 to be the elements of MM, that is, morphisms in this category are elements of the monoid (where M0M_0 is the class of objects of the category and M1M_1 is the class of morphisms).

A groupoid is a category where every morphism is an isomorphism. Monoids are groupoids when every element of the monoid has an inverse. Since the morphisms of this category are elements of the monoid, if every element has an inverse, then every morphism is an isomorphism. So, monoids which are groupoids, are those monoids that are groups.

A preorder (P,R)(P,R), is a set PP together with a reflexive, transitive relation RR. A preorder can be viewed as a category by letting P0=PP_0 = P and P1=RP_1=R.

A preorder is a groupoid when the relation RR is symmetric. That is, an equivalence relation on a set is a groupoid. Since, for x,yPx,y\in P, if (x,y)R(x,y)\in R then there is a morphism xyx\to y, and if (y,x)R(y,x)\in R then there is a morphism yxy\to x, inverse to the morphism xyx\to y, so it is an isomorphism. If (x,y)R    (y,x)R(x,y)\in R \implies (y,x)\in R then every morphism xyx\to y has an inverse morphism yxy\to x and so every morphism is an isomorphism.

A skeleton is a category in which, for every object A,BA,B, AB    A=BA\approx B\implies A=B. That is, if two objects are isomorphic then they are equal.

A preorder is a skeleton when the relation RR is antisymmetric. That is, an order is a skeleton. An antisymmetric relation is a relation in which ((x,y)R(y,x)R)    x=y((x,y)\in R \land (y,x)\in R) \implies x=y. Isomorphisms in preorders are those elements of the relation where (x,y)R    (y,x)R(x,y)\in R \implies (y,x)\in R. So in an order, if (x,y)R(x,y)\in R and (y,x)R(y,x) \in R, the morphism xyx\to y is an isomorphism and the objects xx and yy are equal as a result of the antisymmetry of RR.