(see also Chern-Weil theory, parameterized homotopy theory)
For $\mathcal{T}$ a topos and $X \in \mathcal{T}$ any object the over category $\mathcal{T}/X$ – the slice topos or over-topos – is itself a topos: the “big little topos” incarnation of $X$. This fact is sometimes called the “Fundamental Theorem of Topos Theory”.
More generally, given a fibred product-preserving functor $u : E \to F$ between toposes, the comma category $(id_F/u)$ is again a topos, called the Artin gluing.
For $\mathcal{T}$ a topos and $X \in \mathcal{T}$ any object, the slice category $\mathcal{T}{/X}$ is itself again a topos.
A proof is spelled out for instance in MacLane-Moerdijk, IV.7 theorem 1. In particular we have
If $\Omega \in \mathcal{T}$ is the subobject classifier in $\mathcal{T}$, then the projection $\Omega \times X \to X$ regarded as an object in the slice over $X$ is the subobject classifier of $\mathcal{T}{/X}$.
The power object of a map $f: A \to X$ is given by the equalizer of the maps $p, t$:
where $p$ is the projection map and $t$ is the composition $\wedge \circ (1 \times P f) \circ (1 \times \{\cdot\}_X)$. In the internal language, this says
The map to $X$ is given by projection onto the second factor.
The fact that the slice $\mathcal{T}/X$ is a topos, and particularly the construction of power objects above, can be deduced from a more general result: that the category of coalgebras of a pullback-preserving comonad $G: \mathcal{T} \to \mathcal{T}$ is a topos. See at topos of coalgebras over a comonad. In the case of a slice topos, the comonad would be $X \times -: \mathcal{T} \to \mathcal{T}$ (with comultiplication induced by the diagonal $X \to X \times X$, and counit induced by the projection $!: X \to 1$). This result also subsumes the weaker result where $G$ is assumed to preserve finite limits. See the Elephant, Section A, Remark 4.2.3. A proof of a still more general result may be found here.
For $\mathcal{T}$ a Grothendieck topos and $X \in \mathcal{T}$ any object, the canonical projection functor $X_! : \mathcal{T}/X \to \mathcal{T}$ is part of an essential geometric morphism
The functor $X^*$ is given by taking the product with $X$:
since commuting diagrams
are evidently uniquely specified by their components $A \to K$.
Moreover, since in the Grothendieck topos $\mathcal{T}$ we have universal colimits, it follows that $(-) \times X$ preserves all colimits. Therefore by the adjoint functor theorem a further right adjoint $X_*$ exists.
One also says that $X_!$ is the dependent sum operation and $X_*$ the dependent product operation. As discussed there, this can be seen to compute spaces of sections of bundles over $X$.
Moreover, in terms of the internal logic of $\mathcal{T}$ the functor $X_!$ is the existential quantifier $\exists$ and $X_*$ is the universal quantifier $\forall$.
A geometric morphism $\mathcal{E} \to \mathcal{T}$ equivalent to one of the form $(X_! \dashv X^* \dashv X_*)$ is called an etale geometric morphism.
More generally:
For $\mathcal{E}$ a Grothendieck topos and $f : X \to Y$ a morphism in $\mathcal{E}$, there is an induced essential geometric morphism
where $f_!$ is given by postcomposition with $f$ and $f^*$ by pullback along $X$.
By universal colimits in $\mathcal{E}$ the pullback functor $f^*$ preserves both limits and colimits. By the adjoint functor theorem and using that the over-toposes are locally presentable categories, this already implies that it has a left adjoint and a right adjoint. That the left adjoint is given by postcomposition with $f$ follows from the universality of the pullback: for $(a : A \to X)$ in $\mathcal{E}/X$ and $(b : B \to Y)$ in $\mathcal{E}/Y$ we have unique factorizations
in $\mathcal{E}$, hence an isomorphism
We discuss special properties of over-presheaf toposes.
Let $C$ be a small category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$.
Write
$PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$
and write $PSh(C)/Y(c)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the Yoneda embedding.
There is an equivalence of categories
The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map
A weak inverse of $e$ is given by the functor
which sends $\eta : F' \to Y(c)$ to $F \in PSh(C/c)$ given by
where $F'(d)|_c$ is the pullback
Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $c$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:
Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the closed monoidal structure on presheaves.
Consider $\int_C Y(c)$ , the category of elements of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.
Hence, the above proposition 6 can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:
Let $P:C^{op}\to Set$ be a presheaf. Then there is an equivalence of categories
On objects this takes $F : (\int_C P)^{op} \to Set$ to
with obvious projection to $P$. The inverse takes $f : Q \to P$ to
For a proof see Kashiwara-Schapira (2006, p.26). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).
In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.
For $(f^* \dashv f_*) : \mathcal{T} \to \mathcal{E}$ a geometric morphism of toposes and $X \in \mathcal{E}$ any object, there is an induced geometric morphism between the slice-toposes
where the inverse image $f^*/X$ is the evident application of $f^*$ to diagrams in $\mathcal{E}$.
The slice adjunction $(f^*/X \dashv f_*/X)$ is discussed here: the left adjoint $f^*/X$ is the evident induced functor. Since limits in an over-category $\mathcal{E}/X$ are computed as limits in $\mathcal{E}$ of diagrams with a single bottom element $X$ adjoined, $f^*/X$ preserves finite limits, since $f^*$ does, so that $(f^*/X \dashv f_*/X)$ is indeed a geometric morphism.
We discuss topos points of over-toposes.
Let $\mathcal{E}$ be a topos, $X \in \mathcal{E}$ an object and
a point of $\mathcal{E}$. Then for every element $x \in e^*(X)$ there is a point of the slice topos $\mathcal{E}/X$ given by the composite
Here $(e^*/X \dashv e_*/X)$ is the slice geometric morphism of $e$ over $X$ discussed above and $(x^* \dashv x_*)$ is the étale geometric morphism discussed above induced from the morphism $* \stackrel{x}{\to} e^*(X)$.
Hence the inverse image of $(e,x)$ sends $A \to X$ to the fiber of $e^*(A) \to e^*(X)$ over $x$.
If $\mathcal{E}$ has enough points then so does the slice topos $\mathcal{E}/X$ for every $X \in \mathcal{E}$.
That $\mathcal{E}$ has enough points means that a morphism $f : A \to B$ in $\mathcal{E}$ is an isomorphism precisely if for every point $e : Set \to \mathcal{E}$ the function $e^*(f) : e^*(A) \to e^*(B)$ is an isomorphism.
A morphism in the slice topos, given by a diagram
in $\mathcal{E}$ is an isomorphism precisely if $f$ is. By the above observation we have that under the inverse images of the slice topos points $(e,x \in e^*(X))$ this maps to the fibers of
over all points $* \stackrel{x}{\to} e^*(X)$. Since in Set every object $S$ is a coproduct of the point indexed over $S$, $S \simeq \coprod_S *$ and using universal colimits in $S$, we have that if $x^* e^*(f)$ is an isomorphism for all $x \in e^*(X)$ then $e^*(f)$ was already an isomorphism.
The claim then follows with the assumption that $\mathcal{E}$ has enough points.
over-topos
Masaki Kashiwara, Pierre Schapira, Categories and Sheaves , Springer Heidelberg 2006.
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (Especially section IV.7)