Transformation

We now consider how Chu spaces transform. As one would expect, a function $f:{\cal A}\to{\cal B}$ between Chu spaces sends each word $a\in{\cal A}$ to some word $f(a)\in{\cal B}$. 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 ${\cal A}=(A,r,X)$ to $\bot$. A projection is defined as any dual word of ${\cal A}$, that is, a function $\kern.25em\lower.2em\hbox{$\mathaccent''7B14{\vphantom~}$ }\kern-.55emr(x)$ from ${\cal A}$ to $\Sigma$ for some $x\in X$.

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 $\bot$ 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. $f:{\cal A}\to{\cal B}$ is continuous just when for every projection $\pi:{\cal B}\to\bot$ of ${\cal B}$, $\pi f$ is a projection of ${\cal A}$.

Proposition 1   Continuity-1 and continuity-2 are equivalent.

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 $\bot$ is a projection of $\bot$ (the only projection in fact). Now if $f:{\cal A}\to\bot$ is continuous-2 then the composition of the identity on $\bot$ with f, namely f itself, must be be a projection, whence 1(i) is satisfied by continuity-2.

For 1(ii), let ${\cal A}\stackrel f\to{\cal B}\stackrel g\to{\cal C}$ be the composition of two continuous-2 functions. For any projection $\pi:{\cal C}\to\bot$ of ${\cal C}$, $\pi g$ must be a projection of ${\cal B}$, but then $\pi gf$ must be a projection of ${\cal A}$. 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 $f:{\cal A}\to{\cal B}$ be any function that is continuous-1 but not continuous-2. By the latter there must exist a projection $\pi:{\cal B}\to\bot$ such that $\pi f$ is not a projection of ${\cal A}$. Hence both f and $\pi$ are continuous-1, but then their composition $\pi f$ cannot simultaneously satisfy 1(i) and 1(ii). $\rule{2mm}{3mm}$

We remark in passing that the continuous maps to $\bot$ 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 $\bot$. The shorter the words of a space, the fewer the continuous functions to $\bot$.

Now the operation of composing with f defines a function from the dual words of ${\cal B}=(B,s,Y)$ to those of ${\cal A}=(A,r,X)$. When ${\cal A}$ is extensional this determines a function $g:Y\to X$ such that for each $y\in Y$, g(y) indexes the projection of ${\cal A}$ that equals $\kern.25em\lower.2em\hbox{$\mathaccent''7B14{\vphantom~}$ }\kern-.55ems(y) f$, i.e. $\lambda a.s(f(a),y)$. The projection of ${\cal A}$ that it must equal is $\lambda a.r(a,g(y))$. The continuity condition can now be stated in first order terms (no quantification over functions or predicates) as the equation

s(f(a),y)=r(a,g(y)).

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 ${\cal A}$ 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 $f:{\cal A}\to{\cal B}$ can then be defined as those functions $f:A\to B$ such that there exists $g:Y\to X$ making (f,g) an adjoint pair. This version of the definition of continuity makes no assumption about either separability or extensionality of either ${\cal A}$ or ${\cal B}$.

Adjoint pairs ${\cal A}\stackrel{(f,g)}\longrightarrow{\cal B}\stackrel{(f',g')} \longrightarrow{\cal C}$, where ${\cal A}=(A,r,X)$, ${\cal B}=(B,s,Y)$, and ${\cal C}=(C,t,Z)$, 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 $\Sigma$ and their adjoint pairs form a category, denoted ${\bf Chu}_\Sigma$. We denote by ${\bf chu}_\Sigma$, pronounced ``little chu,'' the subcategory of ${\bf Chu}_\Sigma$ whose objects are the biextensional Chu spaces over $\Sigma$ and whose morphisms are all adjoint pairs between them (i.e. a full subcategory of ${\bf Chu}_\Sigma$).

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 ${\cal A}$ to ${\cal B}$ is itself an adjoint pair from ${\cal B}^\bot$ to ${\cal A}^\bot$, its adjointness condition being

$$r\breve{~}(g(y),a)=s\breve{~}(y,f(a)).$$

That is, the dual of a continuous function f is the adjoint of f whose existence the continuity of f requires. This dualizability is not at all apparent from either of our first two definitions of continuity.

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 $\kern.25em\lower.2em\hbox{$\mathaccent''7B5E{\vphantom~}$ }\kern-.55ems:B\to\Sigma^Y$ of points of ${\cal B}=(B,s,Y)$ to a representation of $f:{\cal A}\to{\cal B}$, simply by forming the composition $\kern.25em\lower.2em\hbox{$\mathaccent''7B5E{\vphantom~}$ }\kern-.55ems f:A\to\Sigma^Y$. This represents f pointwise in terms of the representation $\kern.25em\lower.2em\hbox{$\mathaccent''7B5E{\vphantom~}$ }\kern-.55ems(f(a))$ of each point in the image of f. But $\kern.25em\lower.2em\hbox{$\mathaccent''7B5E{\vphantom~}$ }\kern-.55ems f$ is the left transpose of a function $\varphi:A\times Y\to\Sigma$, namely $\varphi(a,y)=\kern.25em\lower.2em\hbox{$\mathaccent''7B5E{\vphantom~}$ }\kern-.55ems(f(a))=s(f(a),y)$, which we can view as a Chu space ${\cal F}=(A,\varphi,Y)$ representing f,

We define a function $f:{\cal A}\to{\cal B}$ to be continuous when the dual ${\cal F}^\bot$ of its representation ${\cal F}$ represents a function from ${\cal B}^\bot$ to ${\cal A}^\bot$.

That is, for some function $g:Y\to X$, the left transpose of $\varphi\breve{~}$ must be $\kern.25em\lower.2em\hbox{$\mathaccent''7B14{\vphantom~}$ }\kern-.55emr g:Y\to\Sigma^A$ satisfying $(\kern.25em\lower.2em\hbox{$\mathaccent''7B14{\vphantom~}$ }\kern-.55emr g)(y)(a)=r(a,g(y))$. 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 $\Sigma=2=\{0,1\}$: a function $f:{\cal A}\to{\cal B}$ is continuous when the inverse image of each dual word of ${\cal B}$, viewed as a subset of B, is a dual word of ${\cal A}$. 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 $U({\cal A})$ denote the underlying set A of ${\cal A}=(A,r,X)$, and for an adjoint pair (f,g) let U(f,g) denote f. Then U is a functor from ${\bf Chu}_\Sigma$ to ${\bf Set}$.

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 ${\bf Chu}_\Sigma$ with this choice of forgetful functor is not concrete.

When ${\cal A}$ is extensional and $f:A\to B$ is continuous from ${\cal A}$ to ${\cal B}$, the adjoint $g:Y\to X$ of f making (f,g) an adjoint pair is uniquely determined. Hence the restriction of U to the extensional Chu spaces in ${\bf Chu}_\Sigma$ is a faithful functor, making that subcategory of ${\bf Chu}_\Sigma$ 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 ${\bf Chu}$ and little ${\bf chu}$, only the latter of which however is concrete.