The notion of category may be defined in a form which only
involves morphisms and does not mention objects. This
definition shows that categories are a generalization of
semigroups in which the closure axiom has been weakened;
rather than requiring that the product of two arbitrary
elements of the system be defined as an element of the
system, we only require the product to be defined in
certain cases.
We define a category to be a set \footnote{ For
simplicity, we will only consider small categories here,
avoiding logical complications related to proper classes.}
(whose elements we shall term morphisms ) and a
function (which we shall term composition)
from a subset of to which satisfies the
following properties:
\item{\mathbf 1.} If are elements of such that and and , then . \item{\mathbf 2} If are elements of such that and , then and and \item{\mathbf 3a} For every , there exists an element such that #
- and #
- and #
- For all such that , we have . \item{\mathbf 3a} For every , there exists an element such that #
- and #
- and #
- For all such that , we have .
This definition may also be stated in terms of predicate calculus.
Defining the three place predicate by if and only if
and , our axioms look
as follows:
\item{\mathbf 0.} . \item{\mathbf 1.} \item{\mathbf 2.} \item{\mathbf 3a.} \item{\mathbf 3b.}
That a category defined in the usual way satisfies these properties
is easily enough established. Given two morphisms and ,
the composition is only defined if
and for suitable objects , i.e if the
final object of equals the initial object of . The three
hypotheses of axiom 1 state that the initial object of equals the
final objects of and and that the initial object of also
equals the final object of ; hence the initial object of equals
the final object of so we may compose with .
Axiom 2 states associativity of composition whilst axioms 3a and 3b
follow from existence of identity elements.
To show that the new definition implies the old one is not so easy
because we must first recover the objects of the category somehow.
The observation which makes this possible is that to each object
we may associate two sets: the set of morphisms which have
as initial object, ,
and the set of morphisms which have as final object,
. Moreover, this
pair of sets determines uniquely. In order
for this observation to be useful for our purposes, we must somehow
characterize these pairs of sets without reference to objects, which
may be done by the further observation that, if we have two sets
and of morphisms such that if and only if
is defined for all and if
and only if is defined for all , then there
exists an object which gives rise to and as above.
This fact may be demonstrated easily enough from the usual definition of
category. We will now reverse the procedure, using our axioms to show
that such pairs behave as objects should, justifying defining objects
as such pairs.
Returning to our new definition, let us now define ,
, , and
as follows:
We now show that, if then either
or . Suppose that and .
Then there exists a morphism such that and . By the
definition of , there exist morphisms and such that
and . By definition of , we have (a,c) \in Dd \in U(d,b) \in D</math> so, by axiom 1,
, i.e. . Likewise, switching the roles of
and we conclude that, if , then . Hence .
Making an argument similar to that of last paragraph, but with instead
of and instead of , we also conclude that, if
then either
or . Because of axiom 3a, we know that, for
every , there exists such that and, by
axiom 3b, there exists such that . Hence, the sets
and are each partitions of .
Next, we show that, if and , then .
By definition, there exists a morphism such that , so
and . Now suppose that . This means
that . By axiom 1, we conclude that , so ab</math> in the foregoing argument,
we conclude that, if , then . Thus, .
By a similar argument to that of the last paragraph, we may also show that,
if and , then . Taken together,
these results tell us that there is a one-to-one correspondence between
of and --- to each , there exists exactly
one such that and vice-versa. In light
of this fact, we shall define and object of our category to be a pair
of subsets of such that if and only if for all
and if and only if for all . Given
two objects and , we define .
We now will verify that, with these definitions, our axioms reproduce the
defining properties of the standard definition of category.
Suppose that and and are objects according
to the above definition and that and f \in Sg \in R(g,f)
\in Dg \circ fhQ</math>. Since
, it follows that . Since as well, it follows
from axiom 2 that , so . Let be any
element of . Since , it follows that . Since (k,g \circ f) \in Dg \circ f
\in Vg \circ f \in P \cap V = {\rm Hom} (A,C)\circ</math> is
defined as a function from .
Next, suppose that and are distinct objects. By the
properties described earlier, and .
Let and be two objects. Since and
, it follows that {\rm Hom} (E,A) \subset Q{\rm Hom} (F,B)
\subset S{\rm Hom} (E.A) \cap {\rm Hom} (F,B) = \emptyset</math>.
Hence, it follows that, given four objects , we have
unless and .
[more to come]