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
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
is a composite of two spans, then the pull-push through both spans seperately should be equivalent to that through the total span
Of course this just means that the two ways to pull-push through the pullback diamond
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
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
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
in $\mathbf{H}$.
For $f : X \to Y$ a morphism in $\mathbf{H}$ the push-forward functor
is simply given by postcomposition with $f$:
The pullback functor
is literally given by the (homotopy) pullback
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
which is established by the essential uniqueness of the universal morphism into the pullback
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
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)
This is described by the diagram
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
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
a diagram in $\mathbf{H}$, notice that the objects in the fiber product of over categories
are those pairs $\psi : \Psi \to X$ and $\phi : \Phi \to Y$ such that we get a (homotopy) commutative diagram
Again by the universal property of the pullback this is the same as maps
which are precisely the objects of $C(X \times_Z Y)$. So we get
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.
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
that sends every $X$ to the $(\infty,1)$-category of cosimplicial copresheaves to the under category
or
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 $X$.
Now pullback is left adjoint and push-forward is right adjoint.
The choice $C(X) =$ the stable (∞,1)-category of quasicoherent sheaves on a derived stack $X$ is discussed at