Some examples of monoids are: * finite strings of an alphabet `A`, with `*` being concatenation and `z` being the empty string * all functions `X->X` over a set `X`, with `*` being composition and `z` being the identity function over `X` * the natural numbers with `*` being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.) * if we let `*` be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item. 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`. To have a category, the elements and morphisms have to satisfy some constraints: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. 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.) Some examples of categories are: * Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., sin and cos) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set. * any monoid `(S,*,z)` generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where `s3=s1*s2`. The identity morphism for the (single) category element `x` is the monoid's identity `z`. * a **preorder** is a structure `(S, <=)` consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither `x<=y` nor `y<=x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1<=s2` and `s2<=s1` but `s1` and `s2` are not identical). Some examples: * sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a pre-order without anti-symmetry) * sets ordered by size (this illustrates it too) Any pre-order `(S,<=)` generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff `s1<=s2`. 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: (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(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}