For the closure of a subset of a topological space, see at closed subset.
A closure operator is a monad on a poset, typically a poset of subobjects (of some object) in some category. In logic, this is often referred to as a (monadic) modal operator. The elements of the poset that are fixed by the closure operator are called closed (or perhaps modal).
Dually, a comonad on a poset is called a co-closure operator and the elements fixed by it are called co-closed.
More generally, in type theory/category theory, we may think of any idempotent monad on a category as being a closure operator, and of any idempotent comonad as a co-closure operator.
For $\mathcal{C}$ a category, a closure operator $\diamond$ on $\mathcal{C}$ is an idempotent monad on $\mathcal{C}$, hence an endofunctor $\diamond \colon \mathcal{C} \to \mathcal{C}$ equipped with unit and product natural transformations
$\eta_{\diamond} \;\colon\; id_{\mathcal{C}} \to \diamond$
$\mu_{\diamond} \;\colon\; \diamond \circ \diamond \to \diamond$
such that the monad-axioms hold and such that the following equivalent conditions hold (idempotency)
the product map is an isomorphism;
$\eta_{\diamond(-)}$ is an isomorphism.
For $\diamond \colon X \to X$ a closure operator, def. 1, and for $X \in \mathcal{C}$ an object, we say that
$\diamond X \in \mathcal{C}$ is the $\diamond$-closure of $X$;
$\eta_{X} \colon X \to \diamond X$ is the closing map or map to the closure;
$X$ is $\diamond$-closed precisely if $\eta_X$ is an isomorphism.
We write
for the full subcategory on the $\diamond$-closed objects.
A closure operator on a power set is also called a Moore closure. See there for more.
One well-known example of a Moore closure is topological closure, which is precisely a Moore closure that preserves finite joins (unions). Similarly, one can view topological interior as a co-closure operator on a power set that preserves finite meets, or dually as a closure operator on the opposite of a power set that preserves finite joins (given by intersections of subsets).
A Lawvere-Tierney topology is a closure operator on the subobject classifier $\Omega$ of a topos $E$, viewed as an internal meet-semilattice. More precisely, it is a just such a closure operator that preserves internal finite meets. Externally, $\hom(-, \Omega) \colon E^{op} \to Set$ provides an example of a universal closure operator.
We discuss here how a closure operator on a topos may induce closure operators on each of its slice categories.
Throughout, our topos is denoted $\mathcal{C}$.
Given a monad $\diamond \colon \mathcal{C} \to \mathcal{C}$ we write $\eta_\diamond \colon id_{\mathcal{C}} \to \diamond$ for its unit, and write $\mu_\diamond \colon \diamond \circ \diamond \to \diamond$ for its multiplication. As we proceed, we add assumptions on $\diamond$, such as that it preserves certain limits and/or that it be idempotent.
For $X \in \mathcal{C}$ any object, we write $\mathcal{C}_{/X}$ for the slice topos over it. The corresponding base change geometric morphism (dependent sum $\dashv$ context extension $\dashv$ dependent product) we write
We denote an object $p \in \mathcal{C}_{/X}$ in the slice also by the corresponding morphism
in $\mathcal{C}$, which is the image under dependent sum of the unique morphism from $p$ to the terminal object in $\mathcal{C}_{/X}$. Accordingly, a morphism $\phi \colon p_1 \to p_2$ in the slice we also denote by the corresponding triangular commuting diagram
in $\mathcal{C}$.
Here we study the following endofunctors on slices induced from a monad on the total topos.
For $\diamond \colon \mathcal{C} \to \mathcal{C}$ an monad on a topos $\mathcal{C}$, and for $X \in \mathcal{C}$ any object, the induced operator
on the slice topos $\mathcal{C}_{/X}$ is the functor which sends an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ to $(X \underset{\diamond X}{\times} \diamond E \stackrel{\eta_\diamond(X)^\ast \diamond p}{\to} X)$, hence to the left vertical morphism in the pullback diagram
regarded as an object in $\mathcal{C}_{/X}$, and which sends morphisms to the corresponding universal maps between these pullbacks:
We now want to identify conditions under which $\diamond_{/X}$ is itself a monad. First observe that the unit-like map is canonically present.
In the situation of def. 3, there is a natural transformation
from the identity on the slice to the induced operator on the slice, whose component over an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ is the universal morphism into the defining pullback in def. 3 induced from the naturality of the $\diamond$-unit $\eta_{\diamond}$:
We have to show that for all morphisms
in $\mathcal{C}_{/X}$ the induced diagram
in $\mathcal{C}$ commutes. Inspection of the defining pullback diagram shows that both composites in this diagram constitute cones over the pullback diagram that defines the bottom right object. Therefore by the universal property of the pullback they have to coincide.
Next, to have also a product operation on the induced operator $\diamond_{/X}$ we need that $\diamond$ preserves some pullbacks:
Assume that the monad $\diamond \colon \mathcal{C} \to \mathcal{C}$ preserves pullbacks over objects in its image. Then for each $X \in \mathcal{C}$ the induced endofunctor $\diamond_{/X}$ of def. 3 comes with a natural transformation
whose component on an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ is the pullback of the component $\mu_{\diamond} E$ of the product of $\diamond$ itself over the component $\mu_\diamond X$ along the unit components $\eta_{\diamond} X$.
First we produce the component map as claimed, then we show that it is indeed the component of a natural transformation.
So for $p \in \mathcal{C}_{/X}$ an object in the slice, consider the defining pullback diagram of $\diamond_{/X} p$ from def. 3
By the assumption that $\diamond$ preserves pullback diagrams of this form, application of $\diamond$ yields the pullback diagram
Pasting to this the pullback of its left vertical morphism along $\eta_\diamond(X)$ yields
where the total rectangle is also a pullback, by the pasting law.
We now build a morphism of diagrams from the underlying cospan of this diagram to another cospan, such that the induced map on pullbacks is the component of the natural transformation that we are looking for,
To this end, first paste to the above diagram the naturality square of the monad multiplication map $\mu_\diamond \colon \diamond \circ \diamond \to \diamond$ to obtain
Then fill in the commuting diagram that exhibits the unitality axiom of $\diamond$ to obtain
Finally paste in an identity square, just as to manifestly exhibit a morphism of diagrams
Now observe that the total front cospan of morphisms is such that the limit cone over it is the pullback that defines $X \underset{\diamond X}{\times} \diamond E$. By functoriality of pullbacks (by their universal property), this induces a component morphism
as claimed.
Since this is built just from universal constructions, the fact that this morphism is indeed natural follows as in prop. 1.
So far we have constructed from a monad that preserves pullbacks over objects in its image an operator on slices which is equipped with a unit-like and a multiplication-like transformation. We now claim that this yields indeed a monad on the slice.
For $\diamond \colon \mathcal{C} \to \mathcal{C}$ a monad which preserves pullbacks over objects in its image, and for $X \in \mathcal{C}$ any object, the natural transformations
$\eta_{\diamond_{/X}} \colon id_{\mathcal{C}_{/X}} \to \diamond_{/X}$ of prop. 1
$\mu_{\diamond_{/X}} \colon \diamond_{/X} \circ \diamond_{/X} \to \diamond_{/X}$ from prop. 2
constitute the unit and product of a monad structure $(\diamond_{/X}, \mu_{\diamond_{/X}}, \eta_{\diamond_{/X}})$ on the slice operator $\diamond_{/X}$ of def. 3.
If moreover $\diamond$ is idemponent, then so is $\diamond_{/X}$.
By forming cospan morphisms and inducing maps between the corresponding pullbacks, this follows from the monad structure $(\diamond, \mu_{\diamond}, \eta_{\diamond})$ by the same arguments as in the proof of prop. 2.
If $\diamond$ is an idempotent monad, hence a closure operator, then, by the discussion there, the monad unit exhibits an equivalence of categories between the objects in the image of $\diamond$ and the $\diamond$-closed objects.
Therefore in this case the condition that $\diamond$ preserves pullbacks over objects in its image is equivalently that it preserves pullbacks over $\diamond$-closed objects. In this form we will mostly state this condition in the following.
Let $\diamond \colon \mathcal{C} \to \mathcal{C}$ be an idempotent monad that preserves pullbacks over $\diamond$-closed objects. Then the closed objects $p \in \mathcal{C}_{/X}$, def. 2, of the induced idempotent monad $\diamond_{/X}$ on the slice over any $X \in \mathcal{C}$ are precisely those objects $(E \stackrel{p}{\to} X)$ for which the naturality square of the $\diamond$-unit is a pullback square in $\mathcal{C}$.
By def. 2 we need to show that for $p \in \mathcal{C}_{/X}$ the corresponding component of the $\diamond_{/X}$-unit $\eta_{\diamond_{/X}}(p)$ is an isomorphism precisely if
is a pullback diagram in $\mathcal{C}$. By prop. 3 and prop. 1, the universal map from this diagram, regarded as a cone over the underlying cospan, to the limiting cone is precisely $\eta_{\diamond_{/X}}(p)$. Hence the claim follows by the universal property of the pullback.
As a special case of def. 2 we are therefore now interested in the following.
For $\diamond \colon \mathcal{C} \to \mathcal{C}$ an idempotent monad which preserves pullbacks over $\diamond$-closed objects, write
of the full subcategory of the slice topos on the $\diamond_{/X}$-closed objects.