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Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that an object is what arrows with the same domain (another derived-only concept) have in common. This helps thinking abstractly from the very beginning.

Question #1: Why isn't Freyd's axiomatization standard?

I guess it's mainly a matter of convenience, but this seems to be rarely mentioned explicitly.

The same goes for graph theory: the usual and beginner's way of thinking is that arcs (seen as pairs of vertices) are 'ontologically' secondary to vertices, just as arrows (seen as functions between sets) would be secondary to objects (seen as sets). But in the most general setting for graph theory - multidigraphs or quivers - it becomes obvious that it is the other way around: arcs can be the only primary objects and vertices can be thought of equivalence classes of arcs according to the two equivalence relations "has the same source (resp. target) as".

Question #2: Is there an introductory textbook to graph theory that emphasizes that general graph theory is "nothing but a theory of two arbitrary equivalence relations".

[Addendum:] The structure of a graph comes in by identifying some equivalence classes of the first equivalence relation with some of the second.

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I removed the mathematics-education tag, which I believe is meant for questions about students/teaching and so on. See meta discussion at… – Joel David Hamkins Jan 26 '10 at 13:59
@JDH: Thanks. I'm glad to know I'm not the only one who is paying any attention to this. – Pete L. Clark Jan 26 '10 at 14:54
I will do, too, in the future. – Hans Stricker Jan 26 '10 at 15:42
up vote 14 down vote accepted

Qu 1: At about the time of Freyd's book there were two approaches to defining categories. One came from algebraic topology and homological algebra, thus from Eilenberg and MacLane and used the objects and arrows definition, the other was motivated by differential geometry and used the arrows only formalism. This second one is perhaps better suited to those areas where the arrows are what is seen first. For instance, when introducing the concept of the fundamental groupoid of a space then you can think of the idea as a set with a partially defined composition satisfying certain rules and that has distinct advantages for that setting. Ehresmann wrote his book on categories from this viewpoint, but some of the basic ideas do end up being less clear from that viewpoint, others are perhaps clearer. That approach is also linked to more algebraic ideas such as inverse semigroups.

The first 'objects plus arrows' approach is thought to be more accessible to researchers with a `standard' mathematical background e.g. from algebra or algebraic topology. There the objects are usually thought of as being what is being studied and the morphisms are a tool for that study.

The recent use of categorification, quasicategories, internal categories, enriched categories and other similar ideas, tends to show that both viewpoints are best kept in balance, meeting in that higher categorical area.

Put simply I thing the answer to your question is: historically the comparison of objects using arrows was the more pressing application to start with. It came to dominate. More researchers came to use it.

Now the question arises `which to use with a given audience? and that is a hard one to answer.

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Thanks for this profound answer, I guess I am going to "accept" it. It gives some valuable hints on how to look at category theory from a meta-point of view. – Hans Stricker Jan 26 '10 at 10:47

The first obvious answer is that the intuition behind categories involves objects, so there is no need to remove them from the definition, even if the resulting formulation turns out to be more economic. Try teaching a second or third-year student categories without mentioning objects...

A second possible reason is that it may be involved to ask that morphism between two objects form a set, but objects are allowed to be a proper class. Without mentioning objects, one should say that arrows can form a proper class, but every way I can think of to ask for Hom(A, B) to be a set really boils down to putting objects back in the picture.

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Andrea, I did introduce students to categories starting with the Moore category of paths, and you then get the nice effect that there is a feeling that when you introduce the object-arrow examples, the graph theoretic and algebraic aspects interact beautifully. This is a good spin-off. The partial algebraic method without objects initially has other advantages, but, of course, it is still very useful to get the object/arrow aspect understood. Ehresmann would have disagreed with your initial statement about the intuition behind categories. It is not a question of elegance, rather geometry! – Tim Porter Jan 26 '10 at 18:43

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