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?

  • $\begingroup$ Hi Ben, What do you mean by dcpo? $\endgroup$ – Rachel Jan 17 '13 at 10:00
  • $\begingroup$ @Rachel: dcpo $\endgroup$ – Peter Heinig Aug 7 '17 at 19:51

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.

  • $\begingroup$ Giorgio, Thank you so much for your answer. One question, where is the last statement proved in the literature? $\endgroup$ – Ben Sprott Sep 21 '11 at 23:50
  • $\begingroup$ If we define the ordering relation amongst these partial algebras as the functor defined in Giorgio's post, then by the dcpo axiom "every direct set has a supremum", we get collections of categories which themselves have a form of limit as a universal property. So, a dcpo of categories has something like all finite limits. That's a question. Next I am going to look at the compact elements. $\endgroup$ – Ben Sprott Sep 22 '11 at 0:18
  • $\begingroup$ Can someone help me out? If you have arrows $f_1, f_2, g, h$, it's not so clear to me under these axioms that if $(f_1,g), (f_2,g)\in D_\mathcal{C}$, then $(f_1,h)\in D_\mathcal{C}\Rightarrow (f_2,h)\in D_\mathcal{C}$. Of course, this would necessarily hold if these arrows were the realization of the arrows in some actual category. $\endgroup$ – Kevin Ventullo Sep 22 '11 at 2:02
  • $\begingroup$ @Ben Sprott If I'm not wrong I've found this in the Adamek&others "Abstract and concrete Categories: The joy of cats". $\endgroup$ – Giorgio Mossa Sep 22 '11 at 9:18

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)

  1. Existence of identities (at notional source and target)
  2. Proper domains and codomains of composites (in terms of identities)
  3. Associativity
  4. `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.

  • $\begingroup$ any English copy available? $\endgroup$ – Musa Al-hassy Oct 14 '15 at 1:58

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