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 $\mathbf{H}$ of ∞-stacks – in the simplest case just Top or ∞-Grpd – a geometric function theory is some kind of assignment

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

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

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

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

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

$\array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & {}^t\swarrow && \searrow^u && {}^v\swarrow && \searrow^w \\ X_1 &&&& X_2 &&&& X_4 }$

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

$w_* v^* u_* t^* \simeq {w}_* {p_2}_* p_1^* t^* \,.$

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

$\array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& }$

should coincide.

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

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

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

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

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

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 $C(X)$ is a morphism $\psi : \Psi \to X$ in $\mathbf{H}$. A morphism $(\psi,\Psi) \to (\psi',\Psi')$ is a diagram

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

in $\mathbf{H}$.

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

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

is simply given by postcomposition with $f$:

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

The pullback functor

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

is literally given by the (homotopy) pullback

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

of a morphism $\psi : \Psi \to Y$ along $f$.

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

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

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

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

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

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

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

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 $\left(\array{ \Psi \\ \downarrow^{\psi} \\ Y_1}\right) \in C(Y_1)$ through a pullback diamond (see the introduction above)

$\array{ &&&& Y_1 \times_{X_2} Y_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& } \,.$

This is described by the diagram

$\array{ && p_1^* \Psi \\ & {}^{q_1}\swarrow && \searrow^{p_1^* \psi} \\ \Psi&&&& Y_1 \times_{X_2} Y_2 \\ &\searrow^f && {}^{p_1}\swarrow && \searrow^{p_2} \\ && Y_1 &&&& Y_2 \\ & && \searrow^u && {}^v\swarrow && \\ &&&& X_2 &&&& } \,.$

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

$v^* u_* \simeq {p_2}_* p_1^*$

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

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

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

$(\mathbf{H}/X) \times_{\mathbf{H}/Y} \mathbf{H}/Y$

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

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

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

$(\Psi \simeq \Phi) \to X \times_Z Y$

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

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

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 $X$s considered here. If $X = E$ is, for instance, the groupoid incarnation of the total space of the vector bundle associated to a $G$-principal bundle, then a choice of groupoid over $E$ 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 $A \in \mathbf{H}$ may be modeled as a simplicial presheaf. Let $C^\infty(-)$ be the map that sends simplicial presheaves to cosimplicial copresheaves as described at ∞-quantity.

Then consider the assignment

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

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

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

or

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

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

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