nLab
base change

Contents

Idea

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

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

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

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

The dual concept is cobase change.

Definition

Pullback

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

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

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

(p:KY)(X× YK K p * X Y). (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\mathbf{H} is a topos (or (∞,1)-topos, etc.) f:XYf : X \to Y a morphism in H\mathbf{H}, then base change induces an essential geometric morphism betwen over-toposes/over-(∞,1)-toposes

( ff * f):H/Xf *f *f !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 !f_! is given by postcomposition with ff and f *f^* by pullback along ff.

Proof

That we have adjoint functors/adjoint (∞,1)-functors (f !f *)(f_! \dashv f^*) follows directly from the universal property of the pullback. The fact that f *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 *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 *f^* is a logical functor. Hence (f *f *)(f^* \dashv f_*) is also an atomic geometric morphism.

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

Proof

By prop. 1 f *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\mathbf{H}/X is (p 2:Ω H×XX)(p_2 : \Omega_{\mathbf{H}} \times X \to X). This product is preserved by the pullback by which f *f^* acts, hence f *f^* preserves the subobject classifier.

To show that f *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 * f)(f^* \dashv \prod_f) is called the base change geometric morphism along ff.

The right adjoint f *= ff_* = \prod_f is also called the dependent product relative to ff.

The left adjoint f != ff_! = \sum_f is also called the dependent sum relative to ff.

In the case Y=*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\mathcal{C} is a locally cartesian closed category then for every morphism f:XYf \colon X \to Y in ff the inverse image f *:𝒞 /Y𝒞 /Xf^* \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

See also

  • A. Carboni, G. Kelly, R. Wood, A 2-categorical approach to change of base and geometric morphisms I (numdam)

Revised on December 9, 2013 03:24:11 by Urs Schreiber (82.113.121.146)