nLab
base change

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Limits and colimits

limits and colimits

1-Categorical

2-Categorical

(∞,1)-Categorical

Model-categorical

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:X→Y a morphism in a category C with pullbacks, there is an induced functor

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

The dual concept is cobase change.

Definition

Pullback

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

f *:C/Y→C/Xf^* : C/Y \to C/X

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

(p:K→Y)↦(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 is a topos (or (∞,1)-topos, etc.) f:X→Y a morphism in H, then base change induces an essential geometric morphism betwen over-toposes/over-(∞,1)-toposes

(∑ f⊣f *⊣∏ f):H/X→f *←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 ! is given by postcomposition with f and f * by pullback along f.

Proof

That we have adjoint functors/adjoint (∞,1)-functors (f !⊣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

f * is a logical functor. Hence (f *⊣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 H/X is (p 2:Ω H×X→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 *⊣∏ f) is called the base change geometric morphism along f.

The right adjoint f *=∏ f is also called the dependent product relative to f.

The left adjoint f !=∑ 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→Y in f the inverse image f *:𝒞 /Y→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 November 14, 2012 01:27:19 by Urs Schreiber (82.169.65.155)
Edit | Back in time (21 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: classifying topos, sheaf and topos theory, topos, sheaf, presheaf, site, ETCS, Grothendieck topos, end, Kan extension, subobject classifier, terminal object, finitely complete category, Heyting algebra, limit, cone, pullback, global element, cover, sieve, Grothendieck topology, Lawvere-Tierney topology, initial object, locally cartesian closed category, copower, power, image, geometric morphism, locale, product, coherent category, Heyting category, kernel pair, pretopos, Elephant, disjoint coproduct, coproduct, (infinity,n)-category, internal logic, complete category, colimit, pushout, union, intersection, coverage, homotopy limit, 2-limit, strict 2-limit, Grothendieck pretopology, subcanonical coverage, join, meet, comma category, large site, comma object, coequalizer, homotopy pullback, weighted limit, limit in a quasi-category, internal hom, disjoint union, quotient object, Isbell envelope, wedge sum, fundamental group of a topos, frame, sketch, Grothendieck's Galois theory, projective limit, wide pullback, equalizer, universe in a topos, Yoneda extension, exact functor, directed limit, cocone, abelian sheaf, direct sum, direct product, mapping cone, local isomorphism, Quillen exact category, base change, cobase change, category of sheaves, sheafification, category of open subsets, dense monomorphism, Sheaves in Geometry and Logic, 2009 March changes, ringed site, direct image, inverse image, natural numbers object, final functor, ind-object in an (infinity,1)-category, co-Yoneda lemma, induced representation, directed colimit, D-module, limits and colimits by example, subquotient, dependent product, quasicoherent sheaf, ringed space, exponential object, Verdier site, geometric embedding, synthetic differential geometry, coimage, commutativity of limits and colimits, algebraic geometry, algebraic stack, separated presheaf, quasitopos, stalk, point of a topos, spectral sequence, fiber sequence, numerable open cover, concrete sheaf, effective topos, big and little toposes, Models for Smooth Infinitesimal Analysis, stability under pullback, fiber, flabby sheaf, soft sheaf, fine sheaf, ground ring, higher topos theory, localic topos, (0,1)-topos, locally ringed space, structure sheaf, inverse limit, direct limit, inductive limit, topos theory - contents, Q-category, 1-topos, proper map, ringed topos, smooth topos, Kock-Lawvere axiom, Specht module, ionad, cocylinder, etale morphism of schemes, filtered limit, infinity-limits - contents, small presheaf, geometric theory, mapping cocone, category over a category, Beck-Chevalley condition, reflexive coequalizer, canonical topology, Johnstone's topological topos, cokernel pair, fully formal ETCS, open cover, little site, inhabited object, logical functor, concrete site, bounded geometric morphism, pro-object in an (infinity,1)-category, locally connected geometric morphism, etale geometric morphism, locally constant sheaf, fpqc site, preserved limit, DistLat, open map, Loc, global section, essential geometric morphism, category of presheaves, sequential limit, subtopos, final (infinity,1)-functor, homotopy Kan extension, (infinity,1)-Kan extension, connected topos, forcing, reflected limit, created limit, connected limit, locally connected topos, good open cover, finite limit, absolute colimit, big site, topological locale, frame of opens, sifted colimit, double negation, posite, Diaconescu's theorem, base topos, homotopy equivalence of toposes, localic geometric morphism, hyperconnected geometric morphism, indecomposable object, coherent coverage, poset of subobjects, line object, inverter, Topos, local geometric morphism, strongly connected site, cohesive topos, totally connected geometric morphism, locally connected site, compactness and stable closure, conical limit, Barr's theorem, Galois topos, local site, constructive Gelfand duality theorem, colimits in categories of algebras, van Kampen theorem for toposes, surjective geometric morphism, product of simplices, cosheaf, equifier, inserter, PIE-limit, integral transforms on sheaves, dg-geometry, Lawvere interval, etale site, strongly connected topos, over-topos, topos of algebras over a monad, isoinserter, Tannaka duality for geometric stacks, locally ringed topos, little etale topos, standard site, fppf site, projection, connected site, (geometric surjection, embedding) factorization system, variable, crystalline site, regular coverage, Handbook of Categorical Algebra, dense sub-site, small-generated site, complete small category, syntactic site, cartesian site, categorical semantics, Bohr topos, atomic geometric morphism, syntactic category, coherent topos, regular topos, internal site, indexed topos, geometric transformation, n-topos, finite (infinity,1)-limit, product topological space, internal sheaf, real numbers object, contents of contents, Picard scheme, homotopy coend, M-complete category, flexible limit, saturated class of limits, stable factorization system, existential quantifier, HoTT methods for homotopy theorists, proper geometric morphism, small limit, open geometric morphism, dependent type theory, (hyperconnected, localic) factorization system, contractible type, cartesian closed functor, Artin gluing, topos of types, Mayer-Vietoris sequence, sifted (infinity,1)-colimit, category object in an (infinity,1)-category, Giraud's theorem, closed cover, separated geometric morphism, unbounded topos, exactness property, plus construction on presheaves, sheaf of meromorphic functions, morphism of sites, sheaf of abelian groups, realizability topos, geometry of physics, base change spectral sequence, cofiber, predicative topos, Galois cohomology, dependent sum, coherence theorem for bicategories with finite limits, strict initial object, cofiber sequence, van Kampen colimit