For $f : X \to Y$ a morphism in a category $C$ with pullbacks, there is an induced functor
of over-categories. This is the base change morphism. If $C$ is a topos, then this refines to an essential geometric morphism
The dual concept is cobase change.
For $f : X \to Y$ a morphism in a category $C$ with pullbacks, there is an induced functor
of over-categories. It is on objects given by pullback/fiber product along $f$
The concept of base change generalises from this case to other fibred categories.
For $\mathbf{H}$ a topos (or (∞,1)-topos, etc.) $f : X \to Y$ a morphism in $\mathbf{H}$, then base change induces an essential geometric morphism between over-toposes/over-(∞,1)-toposes
where $f_!$ is given by postcomposition with $f$ and $f^*$ by pullback along $f$.
That we have adjoint functors/adjoint (∞,1)-functors $(f_! \dashv f^*)$ follows directly from the universal property of the pullback. The fact that $f^*$ has a further right adjoint is due to the fact that it preserves all small colimits/(∞,1)-colimits by the fact that in a topos we have universal colimits and then by the adjoint functor theorem/adjoint (∞,1)-functor theorem.
The (co-)monads induced by the adjoint triple in prop. 1 have special names in some contexts:
$f_\ast f^\ast$ is also called the function monad (or “reader monad”, see at monad (in computer science)).
$f_! f^\ast$ is also called the “writer comonad” (in computer science)
in modal type theory $f^\ast f_\ast$ is necessity while $f^\ast f_!$ is possibility.
Here $f^\ast$ is a cartesian closed functor, hence base change of toposes constitutes a cartesian Wirthmüller context.
See at cartesian closed functor for the proof.
$f^*$ is a logical functor. Hence $(f^* \dashv f_*)$ is also an atomic geometric morphism.
This appears for instance as (MacLaneMoerdijk, theorem IV.7.2).
By prop. 1 $f^*$ is a right adjoint and hence preserves all limits, in particular finite limits.
Notice that the subobject classifier of an over topos $\mathbf{H}/X$ is $(p_2 : \Omega_{\mathbf{H}} \times X \to X)$. This product is preserved by the pullback by which $f^*$ acts, hence $f^*$ preserves the subobject classifier.
To show that $f^*$ is logical it therefore remains to show that it also preserves exponential objects. (…)
A (necessarily essential and atomic) geometric morphism of the form $(f^* \dashv \prod_f)$ is called the base change geometric morphism along $f$.
The right adjoint $f_* = \prod_f$ is also called the dependent product relative to $f$.
The left adjoint $f_! = \sum_f$ is also called the dependent sum relative to $f$.
In the case $Y = *$ is the terminal object, the base change geometric morphism is also called an etale geometric morphism. See there for more details
If $\mathcal{C}$ is a locally cartesian closed category then for every morphism $f \colon X \to Y$ in $f$ the inverse image $f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ of the base change is a cartesian closed functor.
See at cartesian closed functor – Examples for a proof.
For $\mathbf{H}$ an (∞,1)-topos and $G$ an group object in $\mathbf{H}$ (an ∞-group), then the slice (∞,1)-topos over its delooping may be identified with the (∞,1)-category of $G$-∞-actions (see there for more):
Under this identification, then left and right base change long a morphism of the form $\mathbf{B}H \to \mathbf{B}G$ (corresponding to an ∞-group homomorphism $H \to G$) corresponds to forming induced representations and coinduced representations, respectively.
As the special case of the above for $H = 1$ the trivial group we obtain the following:
Let $\mathbf{H}$ be any (∞,1)-topos and let $G$ be a group object in $\mathbf{H}$ (an ∞-group). Then the base change along the canonical point inclusion
into the delooping of $G$ takes the following form:
There is a pair of adjoint ∞-functors of the form
where
$hofib$ denotes the operation of taking the homotopy fiber of a map to $\mathbf{B}G$ over the canonical basepoint;
$[G,-]$ denotes the internal hom in $\mathbf{H}$;
$[G,-]/G$ denotes the homotopy quotient by the conjugation ∞-action for $G$ equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial $G$-$\infty$-action
(for $G = S^1$ then this is the cyclic loop space construction).
Hence for
$\hat X \to X$ a $G$-principal ∞-bundle
$A$ a coefficient object, such as for some differential generalized cohomology theory
then there is a natural equivalence
given by
The statement that $i^\ast \simeq hofib$ follows immediately by the definitions. What we need to see is that the dependent product along $i$ is given as claimed.
To that end, first observe that the conjugation action on $[G,X]$ is the internal hom in the (∞,1)-category of $G$-∞-actions $Act_G(\mathbf{H})$. Under the equivalence of (∞,1)-categories
(from NSS 12) then $G$ with its canonical ∞-action is $(\ast \to \mathbf{B}G)$ and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.
Hence
So far this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place.
But now since the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}G}$ is itself cartesian closed, via
it is immediate that there is the following sequence of natural equivalences
Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the base change along it.
See also at double dimensional reduction for more on this.
More generally:
Let $\mathbf{H}$ be an (∞,1)-topos and $G \in Grp(\mathbf{H})$ an ∞-group.
Let moreover $V \in \mathbf{H}$ be an object equipped with a $G$-∞-action $\rho$, equivalently (by the discussion there) a homotopy fiber sequence of the form
Then
pullback along $p_\rho$ is the operation that assigns to a morphism $c \colon X \to \mathbf{B}G$ the $V$-fiber ∞-bundle which is associated via $\rho$ to the $G$-principal ∞-bundle $P_c$ classified by $c$:
the right base change along $p_\rho$ is given on objects of the form $X \times (V/G)$ by
The first statement is NSS 12, prop. 4.6.
The second statement follows as in the proof of prop. 5: Let
be any object, then there is the following sequence of natural equivalences
where again
(symmetric powers)
Let
be the symmetric group on $n$ elements, and
the $n$-element set (h-set) equipped with the canonical $\Sigma(n)$-action. Then prop. 6 says that right base change of any $p_\rho^\ast p^\ast X$ along
is equivalently the $n$th symmetric power of $X$
base change
Base change geometric morphisms may be interpreted in terms of fiber integration. See integral transforms on sheaves for more on this.
A general discussion that applies (also) to enriched categories and internal categories is in
Discussion in the context of topos theory is around example A.4.1.2 of
and around theorem IV.7.2 in
Discussion in the context of (infinity,1)-topos theory is in section 6.3.5 of
See also