Continuous Maps as Lax Monad Morphisms

 

Topology as a Monad on the Power Set

We can view topological spaces and continuous functions through the lens of closure operators and monads.

From topology to a closure monad

Recall that a topology on a set $X$$X$ corresponds bijectively to a family of closed sets, which in turn gives a closure operator.
Consider the power set $\hat{X}=\mathcal{P}(X)$$\widehat{X} = \mathcal{P}(X)$ as a poset category (ordered by inclusion).
The closure operator defines a monad $(c,\eta ,\mu )$$(c, \eta, \mu)$ on $\hat{X}$$\widehat{X}$:

• $c:\hat{X}\to \hat{X}$$c : \widehat{X} \to \widehat{X}$ sends each subset to its closure.

• The unit ${\eta }_U:U\hookrightarrow c(U)$$\eta_U : U \hookrightarrow c(U)$ witnesses $U\subseteq \bar{U}$$U \subseteq \overline{U}$.

• The multiplication ${\mu }_U:c^2(U)\to c(U)$$\mu_U : c^2(U) \to c(U)$ is the identity $c(c(U))=c(U)$$c(c(U)) = c(U)$, i.e. $\overline{\stackrel{―}{U}}=\bar{U}$$\overline{\overline{U}} = \overline{U}$ (idempotence).

Direct image functor

A function $f:X\to Y$$f : X \to Y$ between two sets induces a functor ${\mathrm{\exists }}_f:\hat{X}\to \hat{Y}$$\exists_f : \widehat{X} \to \widehat{Y}$ sending each subset $A\subseteq X$$A \subseteq X$ to its direct image $f(A)\subseteq Y$$f(A) \subseteq Y$. If $X$$X$ and $Y$$Y$ carry topologies, we have two closure monads: $(\hat{X},c,\eta ,\mu )$$(\widehat{X}, c, \eta, \mu)$ and $(\hat{Y},c^{\prime },{\eta }^{\prime },{\mu }^{\prime })$$(\widehat{Y}, c', \eta', \mu')$.

Lax monad morphism

Let $(C,T,\eta ,\mu )$$(C, T, \eta, \mu)$ and $(D,T^{\prime },{\eta }^{\prime },{\mu }^{\prime })$$(D, T', \eta', \mu')$ be monads on categories $C$$C$ and $D$$D$.
A functor $F:C\to D$$F : C \to D$ is a lax monad morphism if there exists a natural transformation

such that the following diagrams commute:

• Unit coherence: $\phi \circ F\eta ={\eta }_F^{\prime }$$\varphi \circ F\eta = \eta'_F$

  

• Multiplication coherence: $\phi \circ F\mu ={\mu }_F^{\prime }\circ T^{\prime }\phi \circ {\phi }_T$$\varphi \circ F\mu = \mu'_F \circ T'\varphi \circ \varphi_T$

  

Continuity as a lax monad morphism

We claim:

$f:X\to Y$$f : X \to Y$ is continuous  iff  ${\mathrm{\exists }}_f:\hat{X}\to \hat{Y}$$\exists_f : \widehat{X} \to \widehat{Y}$ is a lax monad morphism between the closure monads.

Indeed, the natural transformation required is an inclusion：

$$\sigma :{\mathrm{\exists }}_f\circ c\subseteq c\circ {\mathrm{\exists }}_f,f(\bar{A})\subseteq \overline{f(A)}$$

This inequality is precisely the usual characterisation of continuity in terms of closures. The remaining coherence conditions (unit and multiplication) automatically hold because we are in a poset category (all diagrams commute trivially) and the closure operator is idempotent.

Thus, the monad framework gives a very compact definition: a continuous map is exactly a functor between power sets that laxly preserves the closure monad structure.