nLab base change

This entry is about base change of slice categories. For base change in enriched category theory see at change of enriching category.

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Idea

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

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

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

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

More generally, such a triple adjunction holds whenever $C$ is locally cartesian closed, and indeed this characterises locally cartesian closed categories. 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

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

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

$$(p : K \to Y) \mapsto \left( \array{ X \times_Y K &\to & K \\ {}^{\mathllap{p^*}}\downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y } \right) \,.$$

On morphisms, it follows from the universal property of pullback

$$\left\lbrace \array{ K &&\stackrel{g}{\to}&& K' \\ & {}_p \searrow && \swarrow_{p'} \\ && Y } \right\rbrace \mapsto \left\lbrace \array{ X \times_Y K &&\stackrel{g^*}{\to}&& X \times_Y K' \\ & {}_{p^*} \searrow && \swarrow_{p'^*} \\ && X } \right\rbrace$$

by observing that this square commutes

$$\array{ &&&& X \times_Y K \\& && {}^{p^*}\swarrow && \searrow^{g \circ p_K} \\ && X &&&& K' \\ & && {}_f\searrow & & \swarrow_{p'} && \\ &&&& Y &&&& } \,.$$

In a fibered category

The concept of base change generalises from this case to other fibred categories.

Base change geometric morphisms

Proposition

For $\mathbf{H}$ a topos (or (∞,1)-topos, etc.) $f : X \to Y$ a morphism in $\mathbf{H}$, then base change induces an essential geometric morphism between over-toposes/over-(∞,1)-toposes

$$(\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{H}/Y$$

where $f_!$ is given by postcomposition with $f$ and $f^*$ by pullback along $f$.

Proof

That we have adjoint functors/adjoint (∞,1)-functors $(f_! \dashv f^*)$ 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.

Remark

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

• $f_\ast f^\ast$ is also called the function monad (or “reader monad”, see at monad (in computer science)).

• $f_! f^\ast$ is also called the “writer comonad” (in computer science)

• in modal type theory $f^\ast f_\ast$ is necessity while $f^\ast f_!$ is possibility.

Proposition

Here $f^\ast$ is a cartesian closed functor, hence base change of toposes constitutes a cartesian Wirthmüller context.

See at cartesian closed functor for the proof.

Proposition

$f^*$ is a logical functor. Hence $(f^* \dashv f_*)$ is also an atomic geometric morphism.

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

Proof

By prop. $f^*$ is a right adjoint and hence preserves all limits, in particular finite limits.

Notice that the subobject classifier of an over topos $\mathbf{H}/X$ is $(p_2 : \Omega_{\mathbf{H}} \times X \to X)$. 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 $(f^* \dashv \prod_f)$ 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 $\mathcal{C}$ is a locally cartesian closed category then for every morphism $f \colon X \to Y$ in $\mathcal{C}$ the inverse image $f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ of the base change is a cartesian closed functor.

See at cartesian closed functor – Examples for a proof.

Examples

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

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

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

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

Along $\ast \to \mathbf{B}G$

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

Proposition

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

$$i \;\colon\; \ast \to \mathbf{B}G$$

into the delooping of $G$ takes the following form:

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

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

where

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

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

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

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

Hence for

• $\hat X \to X$ a $G$-principal ∞-bundle

• $A$ a coefficient object, such as for some differential generalized cohomology theory

then there is a natural equivalence

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

given by

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

Proof

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

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

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

(from NSS 12) then $G$ with its canonical ∞-action is $(\ast \to \mathbf{B}G)$ and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.

Hence

$$[G,X]/G \simeq [\ast, X \times \mathbf{B}G]_{\mathbf{B}G} \;\;\;\;\; \in \mathbf{H}_{/\mathbf{B}G} \,.$$

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

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

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

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

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

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

See also at double dimensional reduction for more on this.

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

More generally:

Proposition

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

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

$$\array{ V \\ \downarrow \\ V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G }$$

Then

1. pullback along $p_\rho$ is the operation that assigns to a morphism $c \colon X \to \mathbf{B}G$ the $V$-fiber ∞-bundle which is associated via $\rho$ to the $G$-principal ∞-bundle $P_c$ classified by $c$:

   $$(p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V$$

2. the right base change along $p_\rho$ is given on objects of the form $X \times (V/G)$ by

   $$(p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\; [V,X]/G$$

Proof

The first statement is NSS 12, prop. 4.6.

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

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

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

$$\begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} (V/G), \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}_{/\mathbf{B}G} ( (p_\rho)_!( P_c \times_G (V/G) ), p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)} ( P_c \times_G V, (p_\rho)^\ast p^\ast X ) \\ & \simeq \mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G)) \end{aligned}$$

where again

$$p \colon \mathbf{B}G \to \ast \,.$$

Example

(symmetric powers)

Let

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

be the symmetric group on $n$ elements, and

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

the $n$-element set (h-set) equipped with the canonical $\Sigma(n)$-action. Then prop. says that right base change of any $p_\rho^\ast p^\ast X$ along

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

is equivalently the $n$th symmetric power of $X$

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

Related concepts

• base change

  • dependent sum, dependent product

  • dependent sum type, dependent product type

  • necessity, possibility, reader monad, writer comonad

• proper base change theorem

• Base change geometric morphisms may be interpreted in terms of fiber integration. See integral transforms on sheaves for more on this.

• change of enriching category

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

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

• Peter Johnstone, Sketches of an Elephant

and around theorem IV.7.2 in

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

Discussion in the context of (infinity,1)-topos theory is in section 6.3.5 of

• Jacob Lurie, Higher Topos Theory

See also

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