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base change

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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} 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 between 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.

Remark

The (co-)monads induced by the adjoint triple in prop. 1 have special names in some contexts:

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.

Examples

Along BHBG\mathbf{B}H \to \mathbf{B}G

For H\mathbf{H} an (∞,1)-topos and GG an group object in H\mathbf{H} (an ∞-group), then the slice (∞,1)-topos over its delooping may be identified with the (∞,1)-category of GG-∞-actions (see there for more):

Act G(H)H /BG. Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} \,.

Under this identification, then left and right base change long a morphism of the form BHBG\mathbf{B}H \to \mathbf{B}G (corresponding to an ∞-group homomorphism HGH \to G) corresponds to forming induced representations and coinduced representations, respectively.

Along *BG\ast \to \mathbf{B}G

As the special case of the above for H=1H = 1 the trivial group we obtain the following:

Proposition

Let H\mathbf{H} be any (∞,1)-topos and let GG be a group object in H\mathbf{H} (an ∞-group). Then the base change along the canonical point inclusion

i:*BG i \;\colon\; \ast \to \mathbf{B}G

into the delooping of GG takes the following form:

There is a pair of adjoint ∞-functors of the form

Hi *[G,]/Gi *hofibH /BG, \mathbf{H} \underoverset { \underset{i_\ast \simeq [G,-]/G}{\longrightarrow}} { \overset{i^\ast \simeq hofib}{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,,

where

  • hofibhofib denotes the operation of taking the homotopy fiber of a map to BG\mathbf{B}G over the canonical basepoint;

  • [G,][G,-] denotes the internal hom in H\mathbf{H};

  • [G,]/G[G,-]/G denotes the homotopy quotient by the conjugation ∞-action for GG equipped with its canonical ∞-action by left multiplication and the argument regarded as equipped with its trivial GG-\infty-action

    (for G=S 1G = S^1 then this is the cyclic loop space construction).

Hence for

then there is a natural equivalence

H(X^,A)originalfluxesoxidationreductionH(X,[G,A]/G)doublydimensionally reducedfluxes \underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}(X \;,\; [G,A]/G) } }

given by

(X^A)(X [G,A]/G BG) \left( \hat X \longrightarrow A \right) \;\;\; \leftrightarrow \;\;\; \left( \array{ X && \longrightarrow && [G,A]/G \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right)
Proof

The statement that i *hofibi^\ast \simeq hofib follows immediately by the definitions. What we need to see is that the dependent product along ii is given as claimed.

To that end, first observe that the conjugation action on [G,X][G,X] is the internal hom in the (∞,1)-category of GG-∞-actions Act G(H)Act_G(\mathbf{H}). Under the equivalence of (∞,1)-categories

Act G(H)H /BG Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}

(from NSS 12) then GG with its canonical ∞-action is (*BG)(\ast \to \mathbf{B}G) and XX with the trivial action is (X×BGBG)(X \times \mathbf{B}G \to \mathbf{B}G).

Hence

[G,X]/G[*,X×BG] BGH /BG. [G,X]/G \simeq [\ast, X \times \mathbf{B}G]_{\mathbf{B}G} \;\;\;\;\; \in \mathbf{H}_{/\mathbf{B}G} \,.

So far this is the very definition of what [G,X]/GH /BG[G,X]/G \in \mathbf{H}_{/\mathbf{B}G} is to mean in the first place.

But now since the slice (∞,1)-topos H /BG\mathbf{H}_{/\mathbf{B}G} is itself cartesian closed, via

E× BG()[E,] BG E \times_{\mathbf{B}G}(-) \;\;\; \dashv \;\;\; [E,-]_{\mathbf{B}G}

it is immediate that there is the following sequence of natural equivalences

H /BG(Y,[G,X]/G) H /BG(Y,[*,X×BG] BG) H /BG(Y× BG*,X×BGp *X) H(p !(Y× BG*)hofib(Y),X) H(hofib(Y),X) \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} \ast, \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}( \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}}, X ) \\ & \simeq \mathbf{H}(hofib(Y),X) \end{aligned}

Here p:BG*p \colon \mathbf{B}G \to \ast denotes the terminal morphism and p !p *p_! \dashv p^\ast denotes the base change along it.

See also at double dimensional reduction for more on this.

Along V/GBGV/G \to \mathbf{B}G

More generally:

Proposition

Let H\mathbf{H} be an (∞,1)-topos and GGrp(H)G \in Grp(\mathbf{H}) an ∞-group.

Let moreover VHV \in \mathbf{H} be an object equipped with a GG-∞-action ρ\rho, equivalently (by the discussion there) a homotopy fiber sequence of the form

V V/G p ρ BG \array{ V \\ \downarrow \\ V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G }

Then

  1. pullback along p ρp_\rho is the operation that assigns to a morphism c:XBGc \colon X \to \mathbf{B}G the VV-fiber ∞-bundle which is associated via ρ\rho to the GG-principal ∞-bundle P cP_c classified by cc:

    (p ρ) *:cP c× GV (p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V
  2. the right base change along p ρp_\rho is given on objects of the form X×(V/G)X \times (V/G) by

    (p ρ) *:X×(V/G)[V,X]/G (p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\; [V,X]/G
Proof

The first statement is NSS 12, prop. 4.6.

The second statement follows as in the proof of prop. 5: Let

(Y c BG)H /BG \left( \array{ Y \\ \downarrow^{\mathrlap{c}} \\ \mathbf{B}G } \right) \;\in\; \mathbf{H}_{/\mathbf{B}G}

be any object, then there is the following sequence of natural equivalences

H /BG(Y,[V,X]/G) H /BG(Y,[V/G,X×BG] BG) H /BG(Y× BG(V/G),X×BGp *X) H /BG((p ρ) !(P c× G(V/G)),p *X) H /(V/G)(P c× GV,(p ρ) *p *X) H /(V/G)(P c× GV,X×(V/G)) \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} (V/G), \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}_{/\mathbf{B}G} ( (p_\rho)_!( P_c \times_G (V/G) ), p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)} ( P_c \times_G V, (p_\rho)^\ast p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G)) \end{aligned}

where again

p:BG*. p \colon \mathbf{B}G \to \ast \,.
Example

(symmetric powers)

Let

G=Σ(n)Grp(Set)Grp(Grpd)LConstH G = \Sigma(n) \in Grp(Set) \hookrightarrow Grp(\infty Grpd) \overset{LConst}{\longrightarrow} \mathbf{H}

be the symmetric group on nn elements, and

V={1,,n}SetGrpdLConstH V = \{1, \cdots, n\} \in Set \hookrightarrow \infty Grpd \overset{LConst}{\longrightarrow} \mathbf{H}

the nn-element set (h-set) equipped with the canonical Σ(n)\Sigma(n)-action. Then prop. 6 says that right base change of any p ρ *p *Xp_\rho^\ast p^\ast X along

{1,,n}/Σ(n)BΣ(n) \{1, \cdots, n\}/\Sigma(n) \longrightarrow \mathbf{B}\Sigma(n)

is equivalently the nnth symmetric power of XX

[{1,,n},X]/Σ(n)(X n)/Σ(n). [\{1,\cdots, n\},X]/\Sigma(n) \simeq (X^n)/\Sigma(n) \,.

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 September 30, 2017 02:41:46 by Tim Porter (2a01:cb04:5b1:2f00:b190:87b4:5767:5a5d)