The n-Category Café

Suppose we have fixed some $(\mathrm{\infty }\mathrm{,1})$-category $\hat{C}$ whose objects we regard as generalized spaces ($\mathrm{\infty }$-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 $H(X,A):={\pi }_0(\hat{C}(X;A))={\mathrm{Ho}}_C(X,A)$.

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 $\mathrm{pt}\to A$ along $g$.

Suppose that $\hat{C}$ sits inside an $(\mathrm{\infty }\mathrm{,2})$-category $\hat{C}\prime$ 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 $\hat{C}\prime$). A representation of $A$ is supposed to be some 1-morphism $\rho :A\to \mathrm{\infty }\mathrm{Vect}$ in $\hat{C}\prime$ with $\mathrm{\infty }\mathrm{Vect}$ 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 ${\mathrm{Hom}}_{\hat{C}\prime (X,\mathrm{Vec})}(\mathrm{pt},\rho \circ g)$ 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

$$\begin{array}{ccc}& & Q\\ & \swarrow & & \searrow \\ {\Omega }_{\mathrm{pt}}\mathrm{\infty }\mathrm{Vect}& & & & E\end{array}$$

where on the left we have the based loop object of $\mathrm{\infty }\mathrm{Vect}$ 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 $\Omega \mathrm{\infty }\mathrm{Vect}$-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 $\mathrm{\infty }-\mathrm{Vect}:=2\mathrm{Vect}:=\mathrm{Ch}(\mathrm{Vect})-\mathrm{Mod}$ in the above, equipped with the canonical point. Then $\Omega \mathrm{\infty }\mathrm{Vect}=\mathrm{Ch}(\mathrm{Vect})$ and an $\Omega \mathrm{\infty }-\mathrm{Vect}$-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 $$\begin{array}{ccccc}U& & \stackrel{f}{\to }& & U\prime \\ & \searrow & & \swarrow \\ & & E\end{array}$$

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

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

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

$$\begin{array}{ccc}& & Q\\ & \swarrow & & \searrow \\ E_1& & & & E_2\end{array}$$

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 $\Omega \mathrm{\infty }\mathrm{Vect}$ into $X$, the action of further spans on this is much like the action of the category of spans on its under-category under $\Omega \mathrm{\infty }\mathrm{Vect}$, 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 $\mathrm{\infty }\mathrm{Vect}:=\mathrm{Ch}(\mathrm{Vect})-\mathrm{Mod}$ 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?