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 of ∞-stacks – in the simplest case just Top or ∞-Grpd – a geometric function theory is some kind of assignment
such that for the object behaves to some useful extent like a collection of “functions on ”.
More concretely, this will usually be taken to mean that satisfies properties of the following kind:
existence of pull-push – For every morphism in there is naturally (functorially) an adjunction with playing the role of push-forward of functions along and playing the role of pullback of functions along ;
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
respect for fiber products – With respect to some suitable tensor product of geometric functions one has for each (homotopy) fiber product in that
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
that sends each object to its over category .
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 is a morphism in . A morphism is a diagram
For a morphism in the push-forward functor
is simply given by postcomposition with :
The pullback functor
is literally given by the (homotopy) pullback
of a morphism along .
A quick way to check that pushforward and pullback 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 . The universal property of the pullback says that this essentially uniquely corresponds to the adjunct morphism .
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 through a pullback diamond (see the introduction above)
This is described by the diagram
By the above definitions, the push-pull operation is encoded in the pullback property of the total outer rectangle. On the other hand, the pull-push operation 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:
a diagram in , notice that the objects in the fiber product of over categories
are those pairs and 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 . 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 s considered here. If is, for instance, the groupoid incarnation of the total space of the vector bundle associated to a -principal bundle, then a choice of groupoid over picks a bunch of vectors in that bundle, hence picks a “distributional section” of that bundle.
under-categories of -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.
Recall the notion of ∞-quantity. Notice that by the discussion at models for ∞-stack (∞,1)-toposes every object may be modeled as a simplicial presheaf. Let be the map that sends simplicial presheaves to cosimplicial copresheaves as described at ∞-quantity.
Then consider the assignment
that sends every to the -category of cosimplicial copresheaves to the under category
From the discussion at ∞-quantity and Lie-∞ algebroid representation we see that we can think of objects in defines this way as representations of the Lie-∞ algebroid of .
Now pullback is left adjoint and push-forward is right adjoint.
The choice the stable (∞,1)-category of quasicoherent sheaves on a derived stack is discussed at