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](https://mathoverflow.net/questions/68775/products-in-a-category-without-reference-to-objects-or-sources-and-targets/68847#68847)


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?



Categories presented with Arrows only, no objects: partial monoids
Of course you can define a (just-arrow) category $\mathcal C$ like a partial algebra which consist of:


a set $\mathcal C$ (namely the set of arrows of your category), a set $D\_\mathcal{C} \subseteq \mathcal C \times \mathcal C$ (the set of pair of composable arrows) and 
a map $\circ \colon D\_\mathcal{C} \to \mathcal C$, which is the composition for this "category".
In this structure we call identities all the elements $f \in \mathcal C$ such that for each $g,h \in \mathcal C$ with $(g,f),(f,h) \in D\_\mathcal{C}$ we have $g\circ f=g$ and $f \circ h=h$.
The composition have to satisfy the following axioms:


\*for each triple $h,g,f \in \mathcal C$ we have that these three statements are equivalent:


$(g,f) \in D\_\mathcal{C}$ and $(h,g\circ f) \in D\_\mathcal{C}$ 


$(h,g) \in D\_\mathcal{C}$ and $(h\circ g, f) \in D\_\mathcal{C}$ 


$(h,g) \in D\_\mathcal{C}$ and $(g,f) \in D\_\mathcal{C}$


and in this case the equality $h\circ(g \circ f)=(h \circ g) \circ f$ holds;


\*for each $f \in \mathcal C$ there are two arrows $g,h \in \mathcal C$ which are identities such that $(f,g), (h,f) \in D$ and $f \circ g=f=h \circ f$.


With these data you have a concept of category just-arrow. 
With this definition of category a functor $F$ from the category $\mathcal C$ to the category $\mathcal D$ is just a function $F \colon \mathcal C \to \mathcal D$ between the sets of the arrows such that:


* for each pair $f,g \in \mathcal C$ if $(g,f) \in D\_\mathcal{C}$ then $(\mathcal F(g),\mathcal F(f)) \in D\_\mathcal{D}$ and $\mathcal F(g \circ f)= \mathcal F(g) \circ \mathcal F(f)$;
* for each identity $f \in \mathcal C$ also $\mathcal F(f)$ is an identity.


The category of just-arrow categories and functors between them is proven to be equivalent to $\mathbf{Cat}$, the category of (ordinary) categories and functors between them.


