The n-Category Café

Suppose we have fixed some $(\infty,1)$-category $\hat C$ whose objects we regard as generalized spaces ($\infty$-stacks), whose morphisms we regard as cocycles and whose 2-morphisms as coboundaries, etc, so that for $X$, $A$ in $\hat C$ we express the cohomology on $X$ with coefficients in $A$ as $H(X,A) := \pi_0(\hat C(X;A)) = Ho_C(X,A)$.

A cocycle $g \in \hat C(X,A)$ is interpreted as classifying an $A$-principal bundle on $X$, whose total “space” $P \to X$ is the homotopy pullback of $pt \to A$ along $g$.

Suppose that $\hat C$ sits inside an $(\infty,2)$-category $\hat C'$ which hosts also the corresponding associated bundles (higher vector bundles) which may have non-invertible morphisms between them (which are 2-morphisms between cocycles=1-morphisms in $\hat C'$). A representation of $A$ is supposed to be some 1-morphism $\rho : A \to \infty Vect$ in $\hat C'$ with $\infty Vect$ not in $\hat C$ and the pullback along the composition of that with the cocycle $g$ gives the total space $E$ of the associated bundle.

Ordinary sections of $E$ are in $Hom_{\hat C'(X,Vec)}(pt, \rho \circ g)$ and one can see that under homotopy pullback of the point every such section canonically gives rise to a span of total spaces of the form

$$\array{ && Q \\ & \swarrow && \searrow \\ \Omega_{pt} \infty Vect &&&& E }$$

where on the left we have the based loop object of $\infty Vect$ at the point at which we work.

Not all spans of this form arise from ordinary sections of $E$ this way: if we allow $Q$ here to be arbitrary such a span encodes general spaces $U \to E$ over $E$ equipped with an $\Omega \infty Vect$-valued cocycle on them.

To better see where we are, suppose we look at $n=2$ where $E$ has as fibers 2-vector spaces and set $\infty-Vect := 2Vect := Ch(Vect)-Mod$ in the above, equipped with the canonical point. Then $\Omega \infty Vect = Ch(Vect)$ and an $\Omega \infty-Vect$-cocycle is a complex of vector bundles.

So in this case generalized section spans as above arrange themselves naturally into a structure $C(E)$ whose

- objects are pairs consisting of a space $U \to E$ and a complex of vector bundles $V \to U$ on $U$;

- morphisms are pairs consisting of a commuting triangle $$\array{ U &&\stackrel{f}{\to}&& U' \\ & \searrow && \swarrow \\ && E }$$

together with a morphism $V \to f^* v'$ of the corresponding complexes of vector bundles. For other choices of $\infty Vect$ we get accordingly other structure than complexes of vector bundles on the spaces $U$, $U'$.

We may also forget for the time being the way we obtained the space $E$ here as the total space of some associated bundle and consider this construction $C(X)$ for all objects (spaces) $X$ in $\hat C$.

Now I am coming to my question: it seems that this gadget $C(X)$ wants to play the role of of the right notion of nice geometric functions on $X$:

- whatever it is ($(\infty,1)$-category or the like), it is monoidal, thanks to the fact that it consists of maps to the (in general directed) loop space object $\Omega \infty Vect$.

- it is naturally the thing acted on via pull(-tensor-)push by bi-branes, i.e. by those spans in $\hat C'$

$$\array{ && Q \\ & \swarrow && \searrow \\ E_1 &&&& E_2 }$$

which in the pull-push realization of QFT are supposed to act on them.

In fact, since $C(X)$ should really be just the thing of span-morphism from $\Omega \infty Vect$ into $X$, the action of further spans on this is much like the action of the category of spans on its under-category under $\Omega \infty Vect$, if you see what I mean.

In any case, be that as it may: I am wondering how the “functions” $C(X)$ for the special case spelled out above, i.e. for $\infty Vect := Ch(Vect)-Mod$ would relate to the concept of “function” given by complexes of coherent sheaves.

From one point of view, $C(X)$ is really the fibred category associated with the stack of complexes of vector bundles on the over-category $\hat C/X$. There is a canonical morphism from that to sheaves on $X$. What is the image of that map when we are working over the algebraic site?