We now consider how Chu spaces transform. As one would expect, a
function
between Chu spaces sends each word
to
some word
.
But if some Chu spaces are to have nontrivial
structure, not all functions will preserve that structure. Those that
do preserve it we shall call *continuous*. What we shall define
however is not the notion of structure but of continuity. Later we
shall define and defend a suitable notion of structure, and show that,
among all functions between Chu spaces, the continuous ones are exactly
those preserving that structure.

Our basic example of a continuous function will be any *projection*
from
to .
A projection is defined as any dual word
of ,
that is, a function
from
to
for some
.

By way of motivation we give a preliminary definition of continuous function.

*Continuity 1.
The continuous functions are the largest class such that
*

*(i) every continuous function to ** is a projection, and
*

*(ii) the composition of two continuous functions is continuous.*

This is a third order characterization of continuity, being phrased in terms of classes of functions which themselves are second-order entities. We now give a second-order definition of continuity and prove its equivalence to the above.

*Continuity 2. *
* is continuous just when for every
projection *
* of **, ** is a projection of **.*

**Proof:** We first show that continuous-2 implies continuous-1.
For this it suffices to show that the class of continuous-2 functions
meets both 1(i) and 1(ii). Observe first that the identity function on
is a projection of
(the only projection in fact). Now if
is continuous-2 then the composition of the identity on
with *f*, namely *f* itself, must be be a projection, whence 1(i)
is satisfied by continuity-2.

For 1(ii), let
be the composition
of two continuous-2 functions. For any projection
of
,
must be a projection of ,
but then
must be
a projection of .
Hence *gf* is continuous-2 and so continuity-2
satisfies 1(ii). This completes one direction.

We now show that continuous-1 implies continuous-2. For a contradiction,
let
be any function that is continuous-1 but not continuous-2.
By the latter there must exist a projection
such that
is not a projection of .
Hence both *f* and
are
continuous-1, but then their composition
cannot simultaneously
satisfy 1(i) and 1(ii).

We remark in passing that the continuous maps to are exactly the projections. This yields another view of a Chu space, namely as the dual of the space consisting of the continuous functions to . The shorter the words of a space, the fewer the continuous functions to .

Now the operation of composing with *f* defines a function from the dual
words of
to those of
.
When
is extensional
this determines a function
such that for each ,
*g*(*y*) indexes the projection of
that equals
,
i.e.
.
The projection of
that it must equal
is
.
The continuity condition can now be stated
in first order terms (no quantification over functions or predicates)
as the equation

We call this equation the *adjointness condition* on account of its
resemblance to adjoint relationships in linear algebra and categorical
adjunctions. The condition may be understood loosely as saying that *g*
is a form of inverse of *f*, more precisely an adjoint. We call *g*
the *adjoint of* *f*.

Although we obtained the adjointness condition from our original
definition of continuity by assuming that
was extensional, the
condition itself does not make any use of that assumption. In fact pairs
of functions satisfying this condition define the most basic notion of
morphism of Chu spaces. The continuous functions
can then
be defined as those functions
such that there exists
making (*f*,*g*) an adjoint pair. This version of the definition of
continuity makes no assumption about either separability or extensionality
of either
or .

Adjoint pairs
,
where
,
,
and
,
compose via
(*f*',*g*')(*f*,*g*)=(*f*'*f*,*gg*'). That this is itself an adjoint
pair follows from
*t*(*f*'*f*(*a*),*z*) =
*s*(*f*(*a*),*g*'(*z*)) =
*r*(*a*,*gg*'(*z*)).
Hence Chu spaces over
and their adjoint pairs form a category,
denoted
.
We denote by
,
pronounced ``little chu,''
the subcategory of
whose objects are the biextensional Chu
spaces over
and whose morphisms are all adjoint pairs between them
(i.e. a full subcategory of
).

Now the adjointness condition, despite being first-order, is a little bit
magical, and for this reason we started out with higher-order definitions
that did not contain a magic formula and hence were better motivated.
One advantage of the adjointness condition besides its elementary
nature is that it demonstrates the symmetry of continuity with respect
to transposition or duality: the dual (*g*,*f*) of an adjoint pair (*f*,*g*)
from
to
is itself an adjoint pair from
to
,
its adjointness condition being

That is, the dual of a continuous function

We now give a definition of continuity that combines the best features of both the non-magical definitions and the adjoint-pair definition. We exploit the representational aspect of Chu spaces in such a way that duality can be integral to the definition of continuity, yet without pulling any formulas out of a hat.

Lift the representation
of points of
to a representation of
,
simply by forming the composition
.
This represents *f* pointwise in terms of the
representation
of each point in the image of *f*.
But
is the left transpose of a function
,
namely
,
which we can view
as a Chu space
representing *f*,

We define a function to be continuous when the dual of its representation represents a function from to .

That is, for some function ,
the left transpose of
must be
satisfying
.
So our dualization requirement becomes
*s*(*f*(*a*),*y*)=*r*(*a*,*g*(*y*)). But this
is exactly the adjointness condition and hence is equivalent to the
other definitions.

To these four definitions of continuity we may add a fifth in the case
:
a function
is continuous when the inverse
image of each dual word of ,
viewed as a subset of *B*, is a dual
word of .
This is the standard definition of continuity from
point-set topology, where our dual words play the role of open sets.
But this is easily seen to be just a restatement of Continuity-2.

*Concreteness.* Let
denote the underlying set *A* of
,
and for an adjoint pair (*f*,*g*) let *U*(*f*,*g*) denote *f*.
Then *U* is a functor from
to .

Now *f* need not determine *g* uniquely. In particular if both *r*
and *s* are all-zero matrices, every pair (*f*,*g*) of functions between
(*A*,*r*,*X*) and (*B*,*s*,*Y*) is trivially an adjoint pair. Hence *U* is
not a faithful functor, whence
with this choice of forgetful
functor is not concrete.

When
is extensional and
is continuous from
to ,
the adjoint
of *f* making (*f*,*g*) an adjoint pair is uniquely
determined. Hence the restriction of *U* to the extensional Chu spaces
in
is a faithful functor, making that subcategory of
a concrete category.

Now the dual of an extensional Chu space while separable is not extensional. The dual of a biextensional space however is biextensional. We therefore have two self-dual categories, big and little , only the latter of which however is concrete.