-------
A **monoid** is a structure `(S, *, z)` consisting of an associative binary operation `*` over some set `S`, which is closed under `*`, and which contains an identity element `z` for `*`. That is:
- 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
+<blockquote><pre>
+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`
+</pre></blockquote>
Some examples of monoids are:
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]
+<blockquote><pre>
+(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<sub>E</sub>, which is such that for every morphism `f:C1->C2`: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>
+</pre></blockquote>
-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.)
(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<sub>C1</sub>) = 1<sub>F(C1)</sub>.
(iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o F(f)
A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of **C** to itself is denoted `1C`.
In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
(*
- P2. every element C1 of a category **C** has an identity morphism id[C1] such that for every morphism f:C1->C2 in **C**: id[C2] o f = f = f o id[C1].
- P3. functors "preserve identity", that is for every element C1 in F's source category: F(id[C1]) = id[F(C1)].
+ P2. every element C1 of a category **C** has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1->C2 in **C**: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>.
+ P3. functors "preserve identity", that is for every element C1 in F's source category: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
*)
Let's remind ourselves of some principles: