nLab
base change

Context

Limits and colimits

Topos theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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)