nLab base change

Context

Limits and colimits

limits and colimits

(∞,1)-Categorical

Model-categorical

Topos theory

Could not include topos theory - contents

Contents

Idea

For $f : X \to Y$ a morphism in a category $C$ with pullbacks, there is an induced functor

$f^* : C/Y \to C/X$

of over-categories. This is the base change morphism. If $C$ is a topos, then this refines to an essential geometric morphism

$(f_! \dashv f^* \dashv f_*) : C/X \to C/Y \,.$

The dual concept is cobase change.

Definition

Pullback

For $f : X \to Y$ a morphism in a category $C$ with pullbacks, there is an induced functor

$f^* : C/Y \to C/X$

of over-categories. It is on objects given by pullback/fiber product along $f$

$(p : K \to Y) \mapsto \left( \array{ X \times_Y K &\to & K \\ {}^{\mathllap{p^*}}\downarrow && \downarrow \\ X &\to& Y } \right) \,.$

In a fibered category

The concept of base change generalises from this case to other fibred categories.

Base change geometric morphisms

Proposition

For $\mathbf{H}$ is a topos (or (∞,1)-topos, etc.) $f : X \to Y$ a morphism in $\mathbf{H}$, then base change induces an essential geometric morphism betwen over-toposes/over-(∞,1)-toposes

$(\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{H}/Y$

where $f_!$ is given by postcomposition with $f$ and $f^*$ by pullback along $f$.

Proof

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.

Proposition

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.

Proposition

$f^*$ is a logical functor. Hence $(f^* \dashv f_*)$ is also an atomic geometric morphism.

This appears for instance as (MacLaneMoerdijk, theorem IV.7.2).

Proof

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. (…)

Definition

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

Properties

Proposition

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.

Applications

Base change geometric morphisms may be interpreted in terms of fiber integration. See integral transforms on sheaves for more on this.

References

A general discussion that applies (also) to enriched categories and internal categories is in

• Dominic Verity, Enriched categories, internal categories and change of base Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 (TAC)

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