The Chu construction is a general method for constructing a star-autonomous category from a closed symmetric monoidal category. It is named after Po-Hsiang Chu, a student of Michael Barr, who gave the construction in his master’s thesis at McGill University. It has been extensively developed by Vaughan Pratt for its potential applications in Theoretical Computer Science.
In outline, given a closed symmetric monoidal category $C$ with pullbacks and an object $d$ of $C$, there is a star-autonomous category $Chu(C, d)$ and a strong symmetric monoidal functor
which realizes $C$ as a coreflective subcategory of $Chu(C, d)$. Being star-autonomous, $Chu(C, d)$ is self-dual, hence $C^{op}$ also embeds as a full subcategory of $Chu(C, d)$, this time reflectively.
Many concrete dualities in mathematics can be seen as embedded in a larger ambient self-duality on a Chu construction. This applies in particular to the category of Chu spaces, $Chu(Set, 2)$ (see below).
The objects of $Chu(C, d)$ are triples $(a, b; r: a \otimes b \to d)$ (called $d$-valued pairings between $a$ and $b$), where $a$ and $b$ are objects of $C$ and $r$ is a morphism of $C$. The special triple $(d, I; \rho: d \otimes I \cong d)$, where $\rho$ is an instance of the canonical isomorphism (the right unitor) for the monoidal unit $I$, will play the role of dualizing object in $Chu(C, d)$.
The morphisms of $Chu(C, d)$,
are pairs of morphisms $f: a \to x$, $g: y \to b$ which are adjoint with respect to the pairings, that is, making the diagram
commute. There is an evident self-duality
which takes an object $(a, b; r: a \otimes b \to d)$ to
where $\sigma$ is an instance of the symmetry isomorphism, so that $r^\dagger$ is the evident transpose. On morphisms, it takes a pair $(f, g)$ to $(g, f)$; note well that the directions of the arrows make the functor contravariant on $Chu(C, d)$.
Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of Boolean algebras, or perhaps better Boolean rigs), the star-autonomous structure on $Chu(C, D)$ can pretty much be deduced (or strongly guessed) by the diligent reader, and this is actually a very good exercise. One could sketch this as follows:
The monoidal unit of $Chu(C, d)$ should be the dual of the dualizer, and so is $(I, d; \lambda: I \otimes d \cong d)$ where $\lambda$, the transpose of $\rho$, is a canonical isomorphism for the unit $I$.
The internal hom $A \multimap X$ in $Chu(C, d)$ should internalize the external hom, i.e., the set of maps from the monoidal unit to $A \multimap X$ should be in natural bijection with the set of maps $A \to X$ in $Chu(C, d)$.
This suggests the first component $(A \multimap X)_0$ of the triple
should be the object of adjoint pairs of maps: given
define the first component as pullback:
where exponentials are used to denote internal homs in $C$, $\tilde{r}$ is the result of currying $r$ to $b \to d^a$ and exponentiating, and similarly for $\tilde{s}$.
The pullback is paired with $a \otimes y$, i.e., there is a map
obtained by decurrying either leg of the pullback, so one defines (with fingers crossed) $(A \multimap X)_1$ to be $a \otimes y$, and the pairing to be this map into $d$.
With notation as above, this works out to
where the second component is a pullback, and the pairing omitted but obvious. The main thing to check is the presence of a canonical isomorphism
but this is left as an exercise.
Also one should check that if $D$ is the dualizing object, that $A^* \cong A \multimap D$, but this is straightforward.
There is a strong symmetric monoidal functor
taking $c$ to $(c, d^c; eval: c \otimes d^c \to d)$. (This does not take $d$ to the dualizing object in $Chu(C, d)$, unless of course the canonical map $I \to d^d$ is an isomorphism.) This embedding admits a right adjoint
given by the obvious projection, that is also strong symmetric monoidal. The unit of the adjunction is an isomorphism, hence $C$ is a coreflective (full) subcategory of $Chu(C, d)$.
If $C$ is complete and cocomplete, then so is $Chu(C, d)$. The formula for colimits is the obvious expected one:
where $r$ is the decurrying of
and the formula for limits is obtained by dualizing the formula for colimits in $Chu(C, d)$.
While the Chu construction is worthy of exploration for many types of symmetric monoidal categories $C$, a great deal of attention has been focused just on the particular case $Chu(Set, 2)$ (or $Chu(Set,TV)$, where $TV$ is the set of truth values, to be constructive), called the category of Chu spaces, and on relatives like $Chu(E, \Omega)$ where $E$ is a topos and $\Omega$ its subobject classifier. The reason is that a great many concrete categories of interest are fully embedded in Chu spaces. Moreover, the 2-element set $TV$ carries a panoply of ambimorphic (formerly, schizophrenic) object structures which induce concrete dualities between these categories, and all of these dualities are embedded in (i.e., are restrictions of) the one overarching duality that obtains on Chu spaces.
The way this works is invariably the same: if $U: C \to Set$ is a concrete category and $\mathbf{2}$ is an object of $C$ over the 2-element set $TV$, then there is a functor
which is faithful by the notion of concrete category.
What is striking is that this functor $i$ is also full in many cases of interest. This is because the adjointness condition for a pair $(f, g): i c \to i d$ to be a Chu space morphism, together with faithfulness of $U$, forces
to be a restriction of the preimage function $f^*$ – and then the mere additional fact that $f^*(\phi) = \phi \circ f \in hom(c, \mathbf{2})$ whenever $\phi \in hom(d, \mathbf{2})$ is often enough to force $f$ to be (the underlying function of) a morphism of $C$. All that is required is that there be sufficiently many morphisms $d \to \mathbf{2}$ to detect $C$-structure on $d$. Some examples follow:
As explained above, $Set$ fully embeds in $Chu(Set, 2)$ by $X \mapsto (X, 2^X; eval: X \times 2^X \to 2)$.
For $C = Top$, taking $\mathbf{2}$ to be Sierpinski space, we have for each topological space $c$ an identification $Open(c) = hom(c, \mathbf{2})$. Then the adjointness condition on a morphism $(f, g): i c \to i d$ between the corresponding Chu spaces expresses precisely the continuity condition that the preimage $f^*$ takes opens of $d$ to opens of $c$. Hence the functor $Top \to Chu(Set, 2)$ is full.
For $C = Pos$, the category of posets, take $\mathbf{2}$ to be the partially ordered set of truth values. Here we have that for a partial order $c$, $hom(c, \mathbf{2})$ is the set of upper sets (upward-closed subsets) of $c$. Given a function $f: U c \to U d$ between posets, the condition that the preimage $f^*(v)$ of an upper set of $d$ is an upper set of $c$ is enough to force $f$ to be a poset map (consider $v = \{q \in d: f p \leq q\}$). It follows that the functor $Pos \to Chu(Set, 2)$ is full.
For $C = Sup$, the category of sup-lattices (whose morphisms are those functors between the underlying posets that are left adjoints), take $\mathbf{2}$ to be the partially ordered set of truth values, but this time as the opposite of the poset $TV$. For a sup-lattice $c$, $hom(c, \mathbf{2})$ may be identified with the set of representable functors $c(-, x): c^{op} \to TV$. The Chu condition then is that $d(-, y) \circ f = d(f-, y)$ is representable for every representable $d(-, y)$. But this condition is equivalent $f$‘s being a left adjoint. Therefore the functor $Sup \to Chu(Set, 2)$ is full.
For other examples of concrete categories, the presence of enough elements in $\hom_C(c, \mathbf{2})$ to detect the $C$-structure of $c$ often requires some form of choice principle, such as the axiom of choice or ultrafilter theorem:
It follows that the functor $Vect_{\mathbb{F}_2} \to Chu(Set, 2)$ is full.
Similar considerations apply to Boolean algebras, Stone Boolean algebras, algebraic lattices, and so on.
In all of these cases, the fullness of these embeddings entitles one to identify a topological space, a Boolean algebra, a vector space over $\mathbb{F}_2$, etc. with its corresponding Chu space, and the same consideration applies to the duals (opposites) of these categories.
Now many of these formal categorical duals are themselves concrete categories, as in the famous example of classical Stone duality between Boolean algebras and Stone spaces, i.e., compact Hausdorff totally disconnected topological spaces. In many such Stone duality situations, and certainly wherever Stone duality applies to the categories listed above, a contravariant equivalence between a concrete category $C$ and an algebraic category $D$ (i.e., where $U: D \to C$ is monadic),
is effected by lifting the object $\mathbf{2}$ of $C$ to a $D$-algebra structure in $C$ (making $\mathbf{2}$ an ambimorphic object, carrying $C$- and $D$-structures compatible with one another); equivalently, seeing $hom(-, \mathbf{2}): C^{op} \to Set$ as an algebra over the monad $U F$ for which $U: D \to Set$ is monadic. For example, classical Stone duality is the case where $C$ is the category of Stone spaces, $D$ is the category of Boolean algebras, and the Boolean operations on $\mathbf{2}$ are continuous with respect to its Stone space structure, making $\mathbf{2}$ a Boolean algebra object in the category of Stone spaces. (For much more on this, see Johnstone’s classic treatise Stone Spaces, especially the chapter on general concrete dualities.)
The point is that in each of these situations, a Stone duality is a restriction of the more global duality on Chu spaces, in that the diagram
(where the vertical arrows are full embeddings as described above) commutes up to canonical isomorphism.
The same principle extends to other situations. For example, Pontryagin duality is fully embedded in the larger duality which obtains on $Chu(Top, S^1)$, where $Top$ is a nice category of spaces.
Hi Toby; could I get you to explain the aside about Boolean rigs above? I’m thinking Boolean algebras is appropriate, as we have $I \to x^* \wp x$, $x \otimes x^* \to D$ [where denotes Girards par and denotes the dualizer], together with appropriate triangular equations, categorifying the inequalities $1 \leq (\neg x) \vee v$ and $x \wedge (\neg x) \leq 0$ in a Boolean algebra. —Todd
Now that I go to write Boolean rig, I'm not so sure. I just know that $Chu(P X,\empty)$ at measurable space is not (even classically) a Boolean algebra. I'll get back to you in a day or less. —Toby
Right, I agree. The Chu construction applied to a complete Heyting algebra is merely a $*$-autonomous quantale, not a $*$-autonomous frame (which would be a complete Boolean algebra), as you noted at measurable space. —Todd
One of the simplest occurrences of Chu space constructions, and the one explored in Pratt’s notes (see below), leads to examples that although extremely simple have a well developed theory with connections to areas of logic and to formal concept analysis. This will be explored in a separate entry, Chu spaces, simple examples.