# Is a continuous function a generalization of an adjunction?

In his recent blog post, Bartosz Milewski notes that we can define a continuous function on two topological spaces $$(X, \mathcal{T}_X), (Y, \mathcal{T}_Y)$$ as a pair of functions $$(f,g)$$ with $$f:X\to Y$$ and $$g:\mathcal{T}_Y\to \mathcal{T}_X$$ such that for all $$x\in X$$ and $$O\in \mathcal{T}_Y$$, $$f(x)\in O \iff x\in g(O).$$

This notation is highly suggestive of an adjunction, and he ends with the cryptic remark: "It’s an example of a more general notion of Chu constructions."

Of course, $$f$$ and $$g$$ isn't an adjunction because they don't map between the same collection of objects. We could try to turn it into an adjunction by upgrading $$f$$ to a map $$f: \mathcal{T}_X\to \mathcal{T}_Y$$. Then we could say for all $$U\in \mathcal{T}_X$$, $$V\in \mathcal{T}_Y$$, $$f(U)\subseteq V\iff U\subseteq g(V).$$ This would make $$f$$ and $$g$$ into an adjunction on the posets of open sets, but this would be asserting that $$f$$ is continuous and open (and wouldn't even guarantee that $$f$$ specifies a unique function $$X\to Y$$).

Is there a reasonable way to reinterpret the above continuity condition as a "generalized" adjunction of sorts?

## 1 Answer

The author is alluding to an extremely general construction called the Chu construction which generalizes adjunctions of categories as well as continuous functions, and a bunch of other things. I will discuss only the example of Chu spaces $$\text{Chu}(\text{Set}, 2)$$ over $$2$$. This is the category whose

• objects are triples $$(X, O_X, \in_X)$$ consisting of two sets $$X, O_X$$ and a relation $$\in_X : X \times O_X \to 2$$ between them, and whose
• morphisms $$(X, O_X, \in_X) \to (Y, O_Y, \in_Y)$$ are pairs of functions $$f : X \to Y, f^{\ast} : O_Y \to O_X$$ satisfying the adjunction condition that $$f(x) \in_Y S \Leftrightarrow x \in_X f^{\ast}(Y)$$, where $$x \in X, S \in O_Y$$.

I chose intentionally asymmetric notation that makes this seem as much like the category of topological spaces as possible, but one of the fundamental points of this construction is that it is symmetric with respect to the two sets $$X, O_X$$; that is, the category of Chu spaces has a duality which exchanges these two and also exchanges the functions $$f, f^{\ast}$$ defining its morphisms.

$$\text{Chu}(\text{Set}, 2)$$ is sort of infamous for having the property that surprisingly many other categories of interest embed fully faithfully into it. We can try to construct such categories by picking a concrete category $$(C, U : C \to \text{Set})$$ and an object $$2 \in C$$ whose underlying set is $$2 \in \text{Set}$$, and sending an object $$c \in C$$ to the triple $$(U(c), \text{Hom}(c, 2), U(\text{eval}))$$ consisting of the underlying set of $$c$$, the set of morphisms $$c \to 2$$, and the evaluation map between these. For example:

• With $$C = \text{Top}$$ we can pick $$2$$ to be Sierpinski space, which identifies $$O_X = \text{Hom}(X, 2)$$ with the lattice of open subsets of $$X$$. If $$X, Y$$ are two topological spaces embedded into $$\text{Chu}(\text{Set}, 2)$$ and $$(f, f^{\ast})$$ is a morphism between them, then the adjunction property forces $$f^{\ast} : O_Y \to O_X$$ to be the preimage with respect to $$f : X \to Y$$, and we get that $$f$$ is continuous.
• With $$C = \text{Pos}$$ we can pick $$2$$ to be the poset $$\{ 0 \le 1 \}$$ of truth values, which identifies $$O_X = \text{Hom}(X, 2)$$ with the lattice of upper sets in $$X$$. If $$X, Y$$ are two posets embedded into $$\text{Chu}(\text{Set}, 2)$$ and $$(f, f^{\ast})$$ is a morphism between them, then the adjunction property again forces $$f^{\ast}$$ to be the preimage with respect to $$f$$, and we get that $$f$$ is monotone.

Other interesting examples include the category $$\text{Bool}$$ of Boolean algebras / Boolean rings (where we take $$2$$ to be the $$2$$-element Boolean algebra / ring; this example is closely related to Stone duality) and the category of vector spaces over $$\mathbb{F}_2$$ (where we take $$2$$ to be $$\mathbb{F}_2$$).

The sense in which this deserves to be called a generalization of adjunctions is that we can get adjunctions on the nose in $$\text{Chu}(\text{Cat}, \text{Set})$$, where we send a category $$C$$ to the triple $$(C, C^{op}, \text{Hom})$$. Similarly we can get adjunctions between posets using a different embedding into $$\text{Chu}(\text{Set}, 2)$$ where we send a poset $$P$$ to the triple $$(P, P^{op}, \text{Hom})$$, thinking of posets as enriched categories.