Arrow Worlds

An arrow world, W, is an arrow land of which no arrow has empty Prior or Post: equivalently, (|W)=(W|). To put this yet another way, every arrow appears in at least one homset.

Any Cartesian product of arrow worlds forms an arrow world and the dual of any arrow world is an arrow world.

In an arrow world, W, we say of an arrow h that it is

initial

iff, for each arrow f of W and each a in Prior(h), Hom(a, f) has exactly one member (which is, then, initial);

terminal

iff, for each arrow f of M and each w in Post(h), Hom(f, w) has exactly one member (which is, then, terminal);

null

iff it is both initial and terminal;

self-composable

iff (h, h) is composable; and

auto

(or an automorphism) iff there is a self-composable arrow in Hom(h,h).

Further Reading

Arrow worlds only get seriously interesting when we come to turn their composability into an actual composition. They do, however, suffice to describe transformations, from which functors spring.

Examples

I'll start with one which I happen to like: maybe I'll add more later. Also, any example of a category is an example of an arrow world and many examples of arrow lands will be arrow worlds (at least sometimes). For example, a partial order is an arrow world whenever there's an identity composable after any maximal arrow and before any minimal one: all other arrows are then `between' some others, and so appear in a homset; the given identities are just enough to fit the extremal arrows into homsets.

Functions

[I used to define functions here: so if you're joining the discourse here in search of the definitions of (partial) functions and the categories Partial and Set, you've discovered a dangling link - Sorry, dash onwards !]

In the familiar world of sets and functions, we can define a composition on arbitrary functions: this allows us to define an arrow world in which every function is composable after any other. I call the resulting arrow world Partial. It lacks, among other things, any identities: so it is not a category.