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

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

such that for $X\in H$ the object $C\left(X\right)$ 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 $H$ there is naturally (functorially) an adjunction ${f}_{*}:C\left(A\right)\stackrel{←}{\to }C\left(B\right):{f}^{*}$ 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

$\begin{array}{ccccc}& & & & {Y}_{1}{×}_{{X}_{2}}{Y}_{2}\\ & & & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ & & {Y}_{1}& & & & {Y}_{2}\\ & {}^{t}↙& & {↘}^{u}& & {}^{v}↙& & {↘}^{w}\\ {X}_{1}& & & & {X}_{2}& & & & {X}_{4}\end{array}$\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

${w}_{*}{v}^{*}{u}_{*}{t}^{*}\simeq {w}_{*}{{p}_{2}}_{*}{p}_{1}^{*}{t}^{*}\phantom{\rule{thinmathspace}{0ex}}.$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

$\begin{array}{ccccc}& & & & {Y}_{1}{×}_{{X}_{2}}{Y}_{2}\\ & & & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ & & {Y}_{1}& & & & {Y}_{2}\\ & & & {↘}^{u}& & {}^{v}↙& & \\ & & & & {X}_{2}& & & & \end{array}$\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 $X{×}_{Z}Y$ in $H$ that

$C\left(X{×}_{Z}Y\right)\simeq C\left(X\right){\otimes }_{C\left(Z\right)}C\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}.$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

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

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

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 $C\left(X\right)$ is a morphism $\psi :\Psi \to X$ in $H$. A morphism $\left(\psi ,\Psi \right)\to \left(\psi \prime ,\Psi \prime \right)$ is a diagram

$\begin{array}{ccccc}\Psi & & \to & & \Psi \prime \\ & {}_{\psi }↘& & {↙}_{\psi \prime }\\ & & X\end{array}$\array{ \Psi &&\to&& \Psi' \\ & {}_\psi\searrow && \swarrow_{\psi'} \\ && X }

in $H$.

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

${f}_{*}:C\left(X\right)\to C\left(Y\right)$f_* : C(X) \to C(Y)

is simply given by postcomposition with $f$:

${f}_{*}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{c}\Psi \\ {↓}^{\psi }\\ X\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{c}\Psi \\ {↓}^{\psi }\\ X\\ {↓}^{f}\\ Y\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$f_* \;\;:\;\; \left( \array{ \Psi \\ \downarrow^\psi \\ X } \right) \;\; \mapsto \;\; \left( \array{ \Psi \\ \downarrow^\psi \\ X \\ \downarrow^f \\ Y } \right) \,.

The pullback functor

${f}^{*}:C\left(Y\right)\to C\left(X\right)$f^* : C(Y) \to C(X)

is literally given by the (homotopy) pullback

$\begin{array}{ccc}{f}^{*}\Psi & \to & \Psi \\ {↓}^{{f}^{*}\psi }& & {↓}^{\psi }\\ X& \stackrel{f}{\to }& Y\end{array}$\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

${\mathrm{Hom}}_{H/X}\left(\Psi ,{f}^{*}\Phi \right)\simeq {\mathrm{Hom}}_{H/X}\left({f}_{*}\Psi ,\Phi \right)$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

$\begin{array}{ccccc}\Psi & & \to & & \\ & {↘}^{\exists !\overline{k}}& & & {↓}^{k}\\ ↓& & {f}^{*}\Phi & \to & \Phi \\ & ↘& {↓}^{{f}^{*}\psi }& & {↓}^{\psi }\\ & & X& \stackrel{f}{\to }& Y\end{array}$\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 $\overline{k}:\Psi \to {f}^{*}\Phi$.

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

$\begin{array}{ccccc}A& \to & B& \to & C\\ ↓& & ↓& & ↓\\ D& \to & E& \to & F\end{array}$\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 $\left(\begin{array}{c}\Psi \\ {↓}^{\psi }\\ {Y}_{1}\end{array}\right)\in C\left({Y}_{1}\right)$ through a pullback diamond (see the introduction above)

$\begin{array}{ccccc}& & & & {Y}_{1}{×}_{{X}_{2}}{Y}_{2}\\ & & & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ & & {Y}_{1}& & & & {Y}_{2}\\ & & & {↘}^{u}& & {}^{v}↙& & \\ & & & & {X}_{2}& & & & \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}& & {p}_{1}^{*}\Psi \\ & {}^{{q}_{1}}↙& & {↘}^{{p}_{1}^{*}\psi }\\ \Psi & & & & {Y}_{1}{×}_{{X}_{2}}{Y}_{2}\\ & {↘}^{f}& & {}^{{p}_{1}}↙& & {↘}^{{p}_{2}}\\ & & {Y}_{1}& & & & {Y}_{2}\\ & & & {↘}^{u}& & {}^{v}↙& & \\ & & & & {X}_{2}& & & & \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

${v}^{*}{u}_{*}\simeq {{p}_{2}}_{*}{p}_{1}^{*}$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

$\begin{array}{ccc}& & Y\\ & & {↓}^{g}\\ X& \stackrel{f}{\to }& Z\end{array}$\array{ && Y \\ && \downarrow^g \\ X &\stackrel{f}{\to}& Z }

a diagram in $H$, notice that the objects in the fiber product of over categories

$\left(H/X\right){×}_{H/Y}H/Y$(\mathbf{H}/X) \times_{\mathbf{H}/Y} \mathbf{H}/Y

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

$\begin{array}{ccc}\Psi \simeq \Phi & \stackrel{\varphi }{\to }& Y\\ {↓}^{\psi }& & {↓}^{g}\\ X& \stackrel{f}{\to }& Z\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\left(\Psi \simeq \Phi \right)\to X{×}_{Z}Y$(\Psi \simeq \Phi) \to X \times_Z Y

which are precisely the objects of $C\left(X{×}_{Z}Y\right)$. So we get

$C\left(X{×}_{Z}Y\right)\simeq C\left(X\right){×}_{C\left(Z\right)}C\left(Y\right)$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 $A\in H$ may be modeled as a simplicial presheaf. Let ${C}^{\infty }\left(-\right)$ be the map that sends simplicial presheaves to cosimplicial copresheaves as described at ∞-quantity.

Then consider the assignment

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

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

$C\left(X\right)={C}^{\infty }\left(X\right)/\mathrm{CoSCoSh}$C(X) = C^\infty(X)/CoSCoSh

or

$C\left(X\right)={C}_{\mathrm{loc}}^{\infty }\left(X\right)/\mathrm{CoSCoSh}\phantom{\rule{thinmathspace}{0ex}}.$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 $C\left(X\right)$ defines this way as representations of the Lie-∞ algebroid of $X$.

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