Adjoint definition of continuity

As category theory is basically the most fundamental object of mathematics, I am always looking for the most categorically appropriate way to define things. I have a new idea of a way of defining continuous maps of topological spaces $f: (X,\tau_1) \to (Y,\tau_2)$ which I think is really nice. Start by generalizing functions from taking values in sets to them taking values in topological spaces.

• Topological image: let $f: X \to Y$ be a function and let $\tau_1$ be a topology of $f$ then the topological image of $\tau_1$ is $f(\tau_1) = \{ U \subseteq Y : f^{-1}(U) \in \tau_1 \}$
• Topological inverse image: let $f: X \to Y$ be a function and let $\tau_2$ be a topology on $Y$ then the topological inverse image is $f^{-1}(\tau_2) = \{ f^{-1}(U) : U \in \tau_2 \}$.

Then if you recall the definition of a monotone galois connection, it states that $F: A \to B$ and $G: B \to A$ are monotone galois connections provided that: \[ F(a) \subseteq b \Leftrightarrow a \subseteq F(b) \] Let $f: A \to B$ be a function then its topological image and inverse image functions are monotone maps on the lattices of topological spaces of $A$ and $B$ with the topological image $f : Top(A) \to Top(B)$ and inverse image $f^{-1} : Top(B) \to Top(A)$ forming an adjoint pair. Then the if and only if condition is also the definition of continuity: \[ f(\tau_1) \subseteq \tau_2 \Leftrightarrow \tau_1 \subseteq f^{-1}(\tau_2) \Leftrightarrow \text{f is continuous} \] The key point is that the topological image and inverse image describe the extremal solutions to the continuity problem, and so they form an adjoint pair. We can now describe which functions of topological spaces are continuous and which topological spaces make a function continuous, so for example if we only have a topology on the input set we can get a topological on the output set using the topological image.

We can generalize the topological image and inverse image to families of functions to get the weakest topology that makes the family of functions continuous. So for example, in smooth manifolds and observables we define the topology on the dual space of an $\mathbb{R}$-algebra $F$ as the weakest topology that makes all $\mathbb{R}$-homomorphisms $m: F \to \mathbb{R}$ continuous. So these kinds of definitions where we need to form a minimal or maximal topology to make some set of functions continuous appears all the time, now we can give this concept its appropriate role.

Everything as much as possible should be defined in terms of adjoints like these, because by doing so you not only give a definition of something but also the way to compute its extremal solutions. So adjoints are one of the most fundamental objects of category theory, and I think at a later date I might elucidate how their fundamental importance extends beyond topology to almost every branch of math. But that is a discussion for another time.

References:
Galois connection
Adjoint functor