```-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`
-```
+ +
```+	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
+```
Some examples of monoids are: @@ -39,15 +40,15 @@ Categories ---------- A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension." -When a morphism `f` in category **C** has source `C1` and target `C2`, we'll write `f:C1->C2`. +When a morphism `f` in category C has source `C1` and target `C2`, we'll write `f:C1->C2`. 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 1E, which is such that for every morphism `f:C1->C2`: 1C2 o f = f = f o 1C1
-```
+
```+	(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 1E, which is such that for every morphism f:C1->C2: 1C2 o f = f = f o 1C1
+```
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. @@ -70,16 +71,18 @@ Some examples of categories are: Functors -------- -A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category **C** to category **D** must: +A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category C to category D must: - (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(1C1) = 1F(C1). - (iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o F(f) +
```+	(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(1C1) = 1F(C1).
+	(iv) "distribute over composition", that is for any morphisms f and g in C: F(g o f) = F(g) o F(f)
+```