Some examples of monoids are: @@ -41,11 +43,13 @@ When a morphism `f` in category **C** has source `C1` and target `C2`, we'll wri To have a category, the elements and morphisms have to satisfy some constraints: - (i) the class of morphisms has to be closed under composition: where f:C1->C2 and g:C2->C3, g o f is also a morphism of the category, which maps C1->C3. - (ii) composition of morphisms has to be associative - (iii) every element e of the category has to have an identity morphism id[e], which is such that for every morphism f:C1->C2: id[C2] o f = f = f o id[C1] ++for all `s1`, `s2`, `s3` in `S`: +(i) `s1*s2` etc are also in `S` +(ii) `(s1*s2)*s3` = `s1*(s2*s3)` +(iii) `z*s1` = `s1` = `s1*z` +

-These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `e` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism. +These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism. A good intuitive picture of a category is as a generalized directed graph, where the category elements are the graph's nodes, and there can be multiple directed edges between a given pair of nodes, and nodes can also have multiple directed edges to themselves. (Every node must have at least one such, which is that node's identity morphism.) @@ -70,7 +74,7 @@ A **functor** is a "homomorphism", that is, a structure-preserving mapping, betw (i) associate with every element C1 of **C** an element F(C1) of **D** (ii) associate with every morphism f:C1->C2 of **C** a morphism F(f):F(C1)->F(C2) of **D** - (iii) "preserve identity", that is, for every element C1 of **C**: F of C1's identity morphism in **C** must be the identity morphism of F(C1) in **D**: F(id[C1]) = id[F(C1)]. + (iii) "preserve identity", that is, for every element C1 of **C**: F of C1's identity morphism in **C** must be the identity morphism of F(C1) in **D**: F(1+(i) the class of morphisms has to be closed under composition: where `f:C1->C2` and `g:C2->C3`, `g o f` is also a morphism of the category, which maps `C1->C3`. +(ii) composition of morphisms has to be associative +(iii) every element `E` of the category has to have an identity morphism 1_{E}, which is such that for every morphism `f:C1->C2`: 1_{C2}o f = f = f o 1_{C1}+