nLab base change

Context

Limits and colimits

limits and colimits

topos theory

Contents

Idea

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

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

$\left({f}_{!}⊣{f}^{*}⊣{f}_{*}\right):C/X\to C/Y\phantom{\rule{thinmathspace}{0ex}}.$(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

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

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

$\left(p:K\to Y\right)↦\left(\begin{array}{ccc}X{×}_{Y}K& \to & K\\ {}^{{p}^{*}}↓& & ↓\\ X& \to & Y\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $H$ is a topos (or (∞,1)-topos, etc.) $f:X\to Y$ a morphism in $H$, then base change induces an essential geometric morphism betwen over-toposes/over-(∞,1)-toposes

$\left(\sum _{f}⊣{f}^{*}⊣\prod _{f}\right):H/X\stackrel{\stackrel{{f}_{!}}{\to }}{\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}}H/Y$(\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 $\left({f}_{!}⊣{f}^{*}\right)$ 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

${f}^{*}$ is a logical functor. Hence $\left({f}^{*}⊣{f}_{*}\right)$ 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 $H/X$ is $\left({p}_{2}:{\Omega }_{H}×X\to X\right)$. 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 $\left({f}^{*}⊣{\prod }_{f}\right)$ 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 $𝒞$ is a locally cartesian closed category then for every morphism $f:X\to Y$ in $f$ the inverse image ${f}^{*}:{𝒞}_{/Y}\to {𝒞}_{/X}$ 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