nLab geometric function object

Redirected from "examples for geometric function objects".

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Idea

This entry list details on concrete constructions for examples of geometric function theories, or closely related structures.

Recall the notion of geometric function object from geometric function theory:

Given an (∞,1)-topos H\mathbf{H} of ∞-stacks – in the simplest case just Top or ∞-Grpd – a geometric function theory is some kind of assignment

C:H(,1)Cat C : \mathbf{H} \to (\infty,1)Cat

such that for XHX \in \mathbf{H} the object C(X)C(X) behaves to some useful extent like a collection of “functions on XX”.

More concretely, this will usually be taken to mean that CC satisfies properties of the following kind:

  • existence of pull-push – For every morphism f:ABf : A \to B in H\mathbf{H} there is naturally (functorially) an adjunction f *:C(A)C(B):f *f_* : C(A) \stackrel{\leftarrow}{\to} C(B) : f^* with f *f_* playing the role of push-forward of functions along ff and f *f^* playing the role of pullback of functions along ff;

  • respect for composition of spans – Pull-push through spans should be functorial: if

    Y 1× X 2Y 2 p 1 p 2 Y 1 Y 2 t u v w X 1 X 2 X 4 \array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & {}^t\swarrow && \searrow^u && {}^v\swarrow && \searrow^w \\ X_1 &&&& X_2 &&&& X_4 }

    is a composite of two spans, then the pull-push through both spans seperately should be equivalent to that through the total span

    w *v *u *t *w *p 2 *p 1 *t *. w_* v^* u_* t^* \simeq {w}_* {p_2}_* p_1^* t^* \,.

    Of course this just means that the two ways to pull-push through the pullback diamond

    Y 1× X 2Y 2 p 1 p 2 Y 1 Y 2 u v X 2 \array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& }

    should coincide.

  • respect for fiber products – With respect to some suitable tensor product of geometric functions one has for each (homotopy) fiber product X× ZYX \times_Z Y in H\mathbf{H} that

    C(X× ZY)C(X) C(Z)C(Y). C(X \times_Z Y) \simeq C(X) \otimes_{C(Z)} C(Y) \,.

over-categories and groupoidification

This first example is rather minimalistic and may feel a bit tautological, as compared to more involved constructions as discussed below. It does nevertheless have interesting applications and, due to its structural simplicity, should serve as a good model on which to study the structural aspects of geometric function theory.

So consider here the assignment

C:=H/():H(,1)Cat C := \mathbf{H}/(-) : \mathbf{H} \to (\infty,1)Cat

that sends each object XHX \in \mathbf{H} to its over category H/X\mathbf{H}/X.

Checking that this assignment does satisfy a good deal of the properties of a geometric function object amounts to recalling the properties of over categories.

So an object in C(X)C(X) is a morphism ψ:ΨX\psi : \Psi \to X in H\mathbf{H}. A morphism (ψ,Ψ)(ψ,Ψ)(\psi,\Psi) \to (\psi',\Psi') is a diagram

Ψ Ψ ψ ψ X \array{ \Psi &&\to&& \Psi' \\ & {}_\psi\searrow && \swarrow_{\psi'} \\ && X }

in H\mathbf{H}.

For f:XYf : X \to Y a morphism in H\mathbf{H} the push-forward functor

f *:C(X)C(Y) f_* : C(X) \to C(Y)

is simply given by postcomposition with ff:

f *:(Ψ ψ X)(Ψ ψ X f Y). f_* \;\;:\;\; \left( \array{ \Psi \\ \downarrow^\psi \\ X } \right) \;\; \mapsto \;\; \left( \array{ \Psi \\ \downarrow^\psi \\ X \\ \downarrow^f \\ Y } \right) \,.

The pullback functor

f *:C(Y)C(X) f^* : C(Y) \to C(X)

is literally given by the (homotopy) pullback

f *Ψ Ψ f *ψ ψ X f Y \array{ f^* \Psi &\to& \Psi \\ \downarrow^{f^* \psi} && \downarrow^\psi \\ X &\stackrel{f}{\to}& Y }

of a morphism ψ:ΨY\psi : \Psi \to Y along ff.

A quick way to check that pushforward f *f_* and pullback f *f^* defined this form a pair of adjoint functors is to notice the hom-isomorphism

Hom H/X(Ψ,f *Φ)Hom H/X(f *Ψ,Φ) Hom_{\mathbf{H}/X}(\Psi, f^* \Phi) \simeq Hom_{\mathbf{H}/X}(f_* \Psi, \Phi)

which is established by the essential uniqueness of the universal morphism into the pullback

Ψ !k¯ k f *Φ Φ f *ψ ψ X f Y \array{ \Psi &&\to&& \\ & \searrow^{\exists ! \bar k} & && \downarrow^{k} \\ \downarrow && f^* \Phi &\to& \Phi \\ &\searrow& \downarrow^{f^* \psi} && \downarrow^\psi \\ && X &\stackrel{f}{\to}& Y }

Here the outer diagram exhibits a morphism k:f *ΨΦk : f_* \Psi \to \Phi. The universal property of the pullback says that this essentially uniquely corresponds to the adjunct morphism k¯:Ψf *Φ\bar k : \Psi \to f^* \Phi.

The fact that the pull-push respects composition of spans is a direct consequence of the way pullback diagrams compose under pasting: recall that in a diagram

A B C D E F \array{ A &\to& B &\to& C \\ \downarrow && \downarrow && \downarrow \\ D &\to& E &\to& F }

for which the left square is a pullback, the total rectangle is a pullback precisely if the right square is, too.

Apply this to the pull-push of an object (Ψ ψ Y 1)C(Y 1)\left(\array{ \Psi \\ \downarrow^{\psi} \\ Y_1}\right) \in C(Y_1) through a pullback diamond (see the introduction above)

Y 1× X 2Y 2 p 1 p 2 Y 1 Y 2 u v X 2 . \array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& } \,.

This is described by the diagram

p 1 *Ψ q 1 p 1 *ψ Ψ Y 1× X 2Y 2 f p 1 p 2 Y 1 Y 2 u v X 2 . \array{ && p_1^* \Psi \\ & {}^{q_1}\swarrow && \searrow^{p_1^* \psi} \\ \Psi&&&& Y_1 \times_{X_2} Y_2 \\ &\searrow^f && {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& } \,.

By the above definitions, the push-pull operation v *u *v^* u_* is encoded in the pullback property of the total outer rectangle. On the other hand, the pull-push operation p 2 *p 1 *{p_2}_* p_1^* is determined by the pullback property of the upper square. By the above fact both properties are equivalent. This means that indeed

v *u *p 2 *p 1 * v^* u_* \simeq {p_2}_* p_1^*

and hence that the pull-push operations defined by over-categories are compatible with composition of spans.

Finally, there is a simple observation on the cartesian product on over categories:

for

Y g X f Z \array{ && Y \\ && \downarrow^g \\ X &\stackrel{f}{\to}& Z }

a diagram in H\mathbf{H}, notice that the objects in the fiber product of over categories

(H/X)× H/YH/Y (\mathbf{H}/X) \times_{\mathbf{H}/Y} \mathbf{H}/Y

are those pairs ψ:ΨX\psi : \Psi \to X and ϕ:ΦY\phi : \Phi \to Y such that we get a (homotopy) commutative diagram

ΨΦ ϕ Y ψ g X f Z. \array{ \Psi \simeq \Phi &\stackrel{\phi}{\to}& Y \\ \downarrow^\psi && \downarrow^g \\ X &\stackrel{f}{\to}& Z } \,.

Again by the universal property of the pullback this is the same as maps

(ΨΦ)X× ZY (\Psi \simeq \Phi) \to X \times_Z Y

which are precisely the objects of C(X× ZY)C(X \times_Z Y). So we get

C(X× ZY)C(X)× C(Z)C(Y) C(X \times_Z Y) \simeq C(X) \times_{C(Z)} C(Y)

remarks

  • This is – more or less implicitly – the notion of geometric ∞-functions that underlies John Baez‘ notion of groupoidification as well as the generalized sections that appear at these sigma-model notes.

  • The definition seems to be disturbingly non-linearized, but this should be viewed in light of the possible nature of the XXs considered here. If X=EX = E is, for instance, the groupoid incarnation of the total space of the vector bundle associated to a GG-principal bundle, then a choice of groupoid over EE picks a bunch of vectors in that bundle, hence picks a “distributional section” of that bundle.

under-categories of \infty-quantities

By essentially simply applying Isbell duality for the case that the underlying site is CartSp to the above example one obtains the following example.

Tentative.

Recall the notion of ∞-quantity. Notice that by the discussion at models for ∞-stack (∞,1)-toposes every object AHA \in \mathbf{H} may be modeled as a simplicial presheaf. Let C ()C^\infty(-) be the map that sends simplicial presheaves to cosimplicial copresheaves as described at ∞-quantity.

Then consider the assignment

C():H(,1)Cat C(-) : \mathbf{H} \to (\infty,1)Cat

that sends every XX to the (,1)(\infty,1)-category of cosimplicial copresheaves to the under category

C(X)=C (X)/CoSCoSh C(X) = C^\infty(X)/CoSCoSh

or

C(X)=C loc (X)/CoSCoSh. C(X) = C^\infty_{loc}(X)/CoSCoSh \,.

From the discussion at ∞-quantity and Lie-∞ algebroid representation we see that we can think of objects in C(X)C(X) defines this way as representations of the Lie-∞ algebroid of XX.

Now pullback is left adjoint and push-forward is right adjoint.

quasicoherent sheaves

The choice C(X)=C(X) = the stable (∞,1)-category of quasicoherent sheaves on a derived stack XX is discussed at

Last revised on October 15, 2009 at 16:18:02. See the history of this page for a list of all contributions to it.