TITLE: Categories presented with Arrows only, no objects: partial monoids
QUESTION [9 upvotes]: Hi,
I received an answer to a question a while back.  The question was about how we can present a category as a collection of arrows and a large list of algebraic relations between them.  One of the answers I got was about Freyd's "Categories, Alegories", and here it is:
products in a category without reference to objects or sources and targets
Can anyone (Wouter maybe?), give much more detail about this presentation.  Can anyone give the precise definition of these "kinds" of partial monoids a la Freyd?
As a side note, could someone suggest a good way to define a dcpo of such partial monoids?

REPLY [4 votes]: One reference that might be of interest to you is C. Ehresmann's Catégories et structures (Dunod, 1965).  It starts Chapter 1 with the following definition (my translation)

Let $C$ be a class; a (partially defined) law of composition on $C$ is a
  function $\kappa$ from some subclass $K$ of the product class $C\times C$ 
  into $C$; the class $K$ is called the class of composables; if $(g,f)\in K$,
  we call $\kappa(g,f)$ the composite of $g$ and $f$.  The pair $(C,\kappa)$
  of a class $C$ and a law of composition on $C$ is called a multiplicative class.

By page 5, we've reached the definition of a category, as a multiplicative class satisfying four further axioms, being (paraphrased)

Existence of identities (at notional source and target)
Proper domains and codomains of composites (in terms of identities)
Associativity
`Enough composites', that is, if $\mathrm{dom}\ g = \mathrm{cod}\ f$, then $g$ and $f$ are composable.

The book goes on to cover most of basic category theory, as far as I can tell.