Categories presented with Arrows only, no objects: partial monoids - MathOverflow

Categories presented with Arrows only, no objects: partial monoids

2011-09-21T21:15:34Z

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:

http://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?

Answer by Giorgio Mossa for Categories presented with Arrows only, no objects: partial monoids

2011-09-21T21:50:38Z

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.

Answer by kow for Categories presented with Arrows only, no objects: partial monoids

2011-09-22T01:11:08Z

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.