Question about the definition of a category

I am confused about the definition of a category given in the Wikipedia article on Category theory:

It seems to me that the structure being described (the “arrows” between objects in some class) is just a binary relation that is both reflexive and transitive. If so, what is meant by the set of all morphisms (arrows) from one object to another? The definition says that every morphism (arrow) has a unique source object and target object. Is there not then at most one arrow from one object to another?

EDIT 1

I am looking for formal axioms of category theory expressed only in the notation of first-order logic and set theory — no words, just variables, functions, logical connectors, quantifiers, predicates, ‘=’ signs, ‘$\in$’, etc.

EDIT 2

How about: class $ob$ is a category iff…

1. $\forall a,b \in ob\exists hom \forall f(f\in hom \leftrightarrow \forall d\in a(f(d)\in b))$

2. $\forall a\in ob\exists i \forall b\in a(i(b)=b)$

Note that the required properties of composition are a direct result of functionality of each morphism.

EDIT 3

I must be trying the patience of the moderators here. Sorry guys! This has turned into a rather open-ended discussion. To be continued at the sci.logic and sci.math newsgroups:

EDIT 4

I’m not sure this is right either, but you might consider the following.

I define the Category and Arrow predicates as follows:

$\forall x (Category(x) \leftrightarrow \forall y\in x \exists f \forall z\in y (f(z)=z))$

This would be redundant if the elements of x were all sets because you can prove the existence of an identity function on every set.

$\forall x\forall a,b\in x (Arrow(x,a,b) \leftrightarrow a\in x \land b\in x \land \exists f \forall c\in a (f(c)\in b) \land \forall c\in b \rightarrow \exists d\in a (f(d)=c)$

It is then easy to prove that the Arrow relation is reflexive:

$\forall x (Category(x) \rightarrow \forall a\in x Arrow(x,a,a))$

And that the Arrow relation is transitive:

$\forall x (Category(x)\rightarrow \forall a,b,c\in x (Arrow(x,a,b) \land Arrow(x,b,c) \rightarrow Arrow(x,a,c)))$

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You’re misinterpreting the meaning of the word “unique” (which was poor word choice on the part of whoever wrote that, so I am removing it). It just means that an arrow doesn’t have more than one source or target.

Here is a formal definition of a small category (this will allow me to ignore size issues which I think are irrelevant when first trying to understand category theory). I’m afraid I’m too attached to words to follow the “no words” edict, but I hope this will be formal enough. A category consists of the following data:

• A set $\text{Ob}$ (objects),
• For every $a, b \in \text{Ob}$, a set $\text{Hom}(a, b)$ (morphisms from $a$ to $b$),
• For every $a \in \text{Ob}$, an element $\text{id}_a \in \text{Hom}(a, a)$ (identity),
• For every $a, b, c \in \text{Ob}$, a function $\circ : \text{Hom}(a, b) \times \text{Hom}(b, c) \to \text{Hom}(a, c)$ (composition).

(I am writing function composition in the opposite of the usual order. You should think of a morphism $f \in \text{Hom}(a, b)$ as an arrow $f : a \to b$ pointing from $a$ on the left to $b$ on the right.)

This data is subject to the following axioms:

• Identity: for every $f \in \text{Hom}(a, b)$, we have $\text{id}_a \circ f = f$, and for every $g \in \text{Hom}(b, a)$, we have $g \circ \text{id}_a = g$.
• Associativity: for every $f \in \text{Hom}(a, b), g \in \text{Hom}(b, c), h \in \text{Hom}(c, d)$, we have $f \circ (g \circ h) = (f \circ g) \circ h$.

Some concrete classes of examples to keep in mind (“categories-as-mathematical-objects” rather than “categories-as-settings-to-study-mathematical-objects”) are the following:

• A monoid is a category with one object; that is, $\text{Ob}$ is a one-element set. The elements of the monoid are the morphisms from the unique object to itself.
• A poset is a category in which $\text{Hom}(a, b)$ has either $1$ or $0$ elements (corresponding to whether $a \le b$ or not); moreover, if $\text{Hom}(a, b)$ and $\text{Hom}(b, a)$ both have one element, then $a = b$. The existence of identities expresses reflexivity, the composition law expresses transitivity, and associativity is automatic.
• A groupoid is a category in which every morphism $f : a \to b$ has an inverse $g : b \to a$, which is a morphism satisfying $f \circ g = \text{id}_a, g \circ f = \text{id}_b$. Groupoids are a simultaneous generalization of groups, equivalence relations, and group actions. An important example is the fundamental groupoid $\Pi_1(X)$ of a topological space $X$, which is the groupoid whose objects are the points of $X$ and whose morphisms are the homotopy classes of paths between points in $X$; composition is given by concatenating paths.

One example of “categories-as-settings-to-study-mathematical-objects”:

• The “matrix category” $\text{Mat}$ is the category whose objects are the non-negative natural numbers $\mathbb{Z}_{\ge 0}$ and whose morphisms $\text{Hom}(n, m)$ are the $n \times m$ matrices, say over some ring (or $m \times n$; whichever convention makes composition correspond to matrix multiplication).

There can be many distinct arrows between two given objects. For example, consider the category of sets: The objects are all sets (or all sets in a given Grothendieck universe, to avoid foundational issues), and the arrows are functions between sets. If $X, Y$ are sets, there are usually many different functions $f: X \to Y$, and each function is an arrow from $X$ to $Y$. The situation is similar in many other commonly used categories.

Category theory is best learned by example; it’s difficult to get intuition for it in the abstract. If you’re having trouble understanding some categorical concept, see how it applies in a few familiar categories, like sets, groups, or modules.

Paraphrased from Categories, Allegories by Freyd and Scedrov:

The theory of CATEGORIES is given by two unary operations and a binary partial operation. The elements are called “morphisms” or “maps”. The operations are pronounced as

• $\square x$ : the source of $x$
• $x \square$ : the target of $x$
• $xy$ : the composition of $x$ and $y$

the axioms are

• $xy$ is defined iff $x \square = \square y$
• $(\square x)\square = \square x$
• $\square(x\square) = x \square$
• $(\square x)x = x$
• $x(x\square) = x$
• $\square(xy) = \square(x(\square y))$
• $(xy)\square = ((x\square)y)\square$
• $x(yz) = (xy)z$

(in the above, in any equation, one side is defined iff the other side is defined)

Note that this set of axioms is modeled after syntax where functions act on the right, rather than on the left as usual. Note that if the target of $x$ is the source of $y$, then $xy$ is the product guaranteed to be defined, rather than $yx$.

In this definition, the notion of “object” is defined to be any morphism of the form $\square x$.

You can find find a fully formal theory of categories in nlab:

http://ncatlab.org/nlab/show/fully+formal+ETCS

This is an “arrows only” axiomatization.

Concrete Categories

Let $ob$ be a class such that

$\forall a (a\in ob \leftrightarrow P(a))$

for some property $P$.

Then $ob$ is a concrete category iff…

(1) $\forall a,b \in ob(\exists hom (\forall f(f\in hom \leftrightarrow (\forall d\in a(f(d)\in b) \land \exists d\in ob \forall e\in d\exists g\in a (f(g)=e)))))$

where the $f$’s are morphisms (functions) that map set $a$ to set $b$ preserving property P, and $hom$ is the set of all such morphisms.

(2) $\forall a\in ob(\exists i \forall b\in a(i(b)=b))$

where the $i$’s correspond to identity morphisms.