$\Phi\colon\ \mathrm{Vect}_{\nabla,\beta}^{\sim}(M)\longrightarrow
1\textrm{-}\mathrm{TFT}(M),$

which is fully faithful and natural in $M$.

Breaking down our general definition to this special case, an object of
$1\textrm{-}\mathrm{Bord}(M)_{n}$ lying over a manifold $S$ is given by a
family $X/S$ of one-dimensional manifolds $X_{s}$, $s\in S$, together with a
map $\gamma\colon X\rightarrow M$ and functions $\rho_{0},\dots,\rho_{n}\colon
X\rightarrow\mathbb{R}$ which cut out codimension-one submanifolds

$X_{a}^{a}=\\{x\in X\mid\rho_{a}(x)=0\\}.$

The properness assumption (O3c) implies that the restrictions $X_{a}^{a}\cap
X_{s}$ to the fibers are compact, i.e., finite collections of points. More
generally, we have the following result.

######  Lemma 4.5.

The submanifold $X_{a}^{a}$ of $X$ is a finite covering of $S$ $($with
possibly empty fibers$)$.

###### Proof.

Let $s\in S$ and $x\in X_{a}^{a}\cap X_{s}$, and denote by $\pi\colon
X\rightarrow S$ the projection. Since ${\rm
d}\rho_{a}|_{X_{s}}(x)\allowbreak\neq 0$, $X_{a}^{a}$ intersects the fiber
$X_{s}$ transversally, that is, ${\rm d}\pi(x)$ is an isomorphism when
restricted to the tangent space $T_{x}X_{a}^{a}$. Therefore,
$\pi|_{X_{a}^{a}}$ is a local diffeomorphism. Furthermore, by assumption,
$X_{a}^{a}$ is proper over $S$, meaning that $\pi|_{X_{a}^{a}}$ is a proper
map. However, a proper local diffeomorphism is a covering map the fibers of
which have at most finitely many points (possibly zero). ∎

To prove Proposition 4.4, we start by constructing a field theory from the
data of a vector bundle with non-degenerate bilinear form and compatible
connection.

Let $(V,\nabla,\beta)\in\mathrm{Vect}_{\nabla,\beta}^{\sim}(M)$. Our goal is
to construct a field theory $Z_{V,\nabla,\beta}$, which will be the value of
$(V,\nabla,\beta)$ under the functor $\Phi$ in Proposition 4.4. Recall that
the smooth functor $Z_{V,\nabla,\beta}$, as a morphism of simplicial objects,
will consist of a sequence of stack maps
$1\textrm{-}\mathrm{Bord}(X)_{n}\to\mathrm{Vect}_{n}$, $n\in\Delta$. At the
$n$th simplicial level, we must have

$Z_{V,\nabla,\beta}(X/S;\rho_{0},\dots,\rho_{n};\gamma)=(W_{0},\dots,W_{n};{f}_{1},\dots,{f}_{n})$

for some vector bundles $W_{a}$ over $S$ and vector bundle maps ${f}_{a}\colon
W_{a-1}\rightarrow W_{a}$. To define $W_{a}$, for each $a=0,\dots,n$, set
first $\tilde{W}_{a}:=(\gamma|_{X_{a}^{a}})^{*}V$, which makes sense since,
for each $a=0,\dots,n$, $X_{a}^{a}$ is a codimension-one submanifold of $X$.
By Lemma 4.5, $X_{a}^{a}$ is a finite covering of $S$, hence any small enough
open $U\subset S$ is covered by $U_{1},\dots,U_{k}\subset X_{a}^{a}$ such that
the projection map $\pi$ provides diffeomorphisms $\pi|_{U_{j}}\colon
U_{j}\rightarrow U$. Hence we can set

$W_{a}|_{U}:=\big{(}\pi|_{U_{1}}^{-1}\big{)}^{*}\tilde{W}_{a}\otimes\cdots\otimes\big{(}\pi|_{U_{k}}^{-1}\big{)}^{*}\tilde{W}_{a},$
(4.3)

where $W_{a}|_{U}=\mathbb{C}$ if $k=0$. These vector bundles glue together to
a vector bundle $W_{a}$ over $S$.

To define ${f}_{a}$ for each $a=1,\dots,n$, consider the subsets
$X_{a-1}^{a}$. Let $Y^{1},\dots,Y^{k}$ be the connected components of
$X_{a-1}^{a}|_{U}$, where $U\subset S$ is a small connected open as above for
both $a-1$ and $a$. The map ${f}_{a}|_{U}$ will be the tensor product of maps
${f}^{j}_{a}$, where ${f}^{j}_{a}$ is determined by the connected component
$Y^{j}$. Each ${f}^{j}$ will be a vector bundle map

${f}^{j}_{a}\colon\
\bigotimes_{Z}\big{(}\pi|_{Z}^{-1}\big{)}^{*}\tilde{W}_{a-1}\longrightarrow\bigotimes_{Z^{\prime}}\big{(}\pi|_{Z^{\prime}}^{-1}\big{)}^{*}\tilde{W}_{a-1},$
(4.4)

where $Z$ runs over the connected components of $Y^{j}\cap X_{a-1}^{a-1}$ and
$Z^{\prime}$ runs over the connected components of $Y^{j}\cap X_{a}^{a}$ (by
possibly making $U$ smaller, we can assume that the projection map is a
diffeomorphism to $U$ when restricted to any one of these sets $Z$ and
$Z^{\prime}$). The tensor product
${f}_{a}|_{U}:={f}^{1}_{a}\otimes\cdots\otimes{f}^{k}_{a}$ is then indeed a
vector bundle map $W_{a-1}|_{U}\rightarrow W_{a}|_{U}$, by definition (4.3).

Now, each $Y^{j}$ is, essentially, either a circle bundle or an interval
bundle over $U$; this only fails to be the case if $\rho_{a-1}(x)=\rho_{a}(x)$
at some points $x\in X_{a-1}^{a}|_{U}$. To address this issue, we let

$U^{j}_{\circ}:=\bigl{\\{}s\in U\mid\rho_{a-1}(y)\neq\rho_{a}(y)\text{ for
each }y\in Y^{j}\cap X_{s}\bigr{\\}},$

which is an open subset of $U$. The complement $U\setminus U^{j}_{\circ}$ is
the set where $Y^{j}$ is a “thin bordism”, in the sense that

$Y^{j}|_{s}=\big{(}X_{a-1}^{a-1}\cap
Y^{j}\big{)}\big{|}_{s}=\big{(}X_{a}^{a}\cap
Y^{j}\big{)}\big{|}_{s}\qquad\text{for}\quad s\in U\setminus U^{j}_{\circ}.$

Note in particular that for $s\in U\setminus U^{j}_{\circ}$, $Y^{j}|_{s}$
consists of finitely many points.

######  Lemma 4.6.

For each $j=1,\dots,k$, write $Y^{j}_{\circ}:=Y_{j}|_{U^{j}_{\circ}}$. Then
$Y^{j}_{\circ}\to U^{j}_{\circ}$ is a fiber bundle whose fibers are compact
one-dimensional manifolds with boundary $($thus, either intervals or
circles$)$. Moreover, if $Y^{j}_{\circ}$ is a circle bundle, then
$U^{j}_{\circ}=U$ and $Y^{j}_{\circ}=Y^{j}$.

###### Proof.

By construction, the total space $Y^{j}_{\circ}$ is a compact manifold and the
projection $\pi|_{Y_{\circ}^{j}}$: $Y^{j}_{\circ}\to U_{\circ}^{j}$ is a
proper submersion, hence a fiber bundle (with possibly empty fibers). The last
statement follows from the fact that circle bundles cannot degenerate, by the
requirements on the functions $\rho_{a}$. ∎

By possibly shrinking $U$ further, we may assume moreover that all these
bundles are trivial. We now define ${f}^{j}$ case by case.

Suppose first that $Y^{j}_{\circ}$ is a circle bundle, so that
$Y^{j}_{\circ}=Y^{j}$ and $U^{j}_{\circ}=U$. In this case, $Y^{j}\cap
X_{a-1}^{a-1}=Y^{j}\cap X_{a}^{a}=\varnothing$, hence we have to produce a
vector bundle map from the trivial line bundle to itself, that is, a function
on $U$. Choose a trivialization $\varphi\colon U\times S^{1}\rightarrow
Y^{j}$. Now set

${f}^{j}:=\operatorname{tr}P(\varphi),$

where $P(\varphi)$ is the parallel transport around the loops
$\varphi_{s}\colon S^{1}\rightarrow Y^{j}\subset X$ given by
$\varphi_{s}(t)=\varphi(s,t)$ with respect to the pullback connection of
$\gamma^{*}V\rightarrow X$. We claim that ${f}^{j}$ is independent of the
choice of $\varphi$. If $\tilde{\varphi}$ is another trivialization of $Y^{j}$
that induces the same orientation on the fibers and agrees with $\varphi$ at
the basepoint $1\in S^{1}$, then it is just a reparametrization of $\varphi$,
and our claim follows from the invariance of parallel transport under
reparametrizations. Without the assumption on basepoints, $P(\varphi_{s})$ and
$P(\tilde{\varphi}_{s})$ are conjugates for each $s\in U$, so the trace
${f}^{j}$ is still independent of the choice of $\varphi$. Finally, if
$\tilde{\varphi}$ induces the opposite orientation, then
$P(\tilde{\varphi}_{s})=P(\varphi_{s})^{-1}$. This yields the same trace,
since $P(\varphi)$ preserves the bilinear form $\beta$; the calculation is

$\displaystyle\operatorname{tr}P(\varphi_{s})=\\!\sum_{i=1}^{n}\varepsilon_{i}\beta\bigl{(}P(\varphi_{s})b_{i},b_{i}\bigr{)}=\\!\sum_{i=1}^{n}\varepsilon_{i}\beta\bigl{(}b_{i},P(\varphi_{s})b_{i}\bigr{)}=\\!\sum_{i=1}^{n}\varepsilon_{i}\beta\bigl{(}P(\varphi_{s})^{-1}b_{i},b_{i}\bigr{)}=\operatorname{tr}P(\tilde{\varphi}_{s}),$

where $b_{1},\dots,b_{n}$ is a generalized orthonormal basis for $\beta$ and
we used the symmetry of $\beta$ (note that $\beta$ is _not_ assumed to be
Hermitian in the complex case).

Suppose now that $Y^{j}_{\circ}$ is a bundle of intervals. In this case, we
have the following lemma.

######  Lemma 4.7.

There exists a smooth map $\varphi\colon U\times[0,1]\rightarrow Y^{j}\subset
X_{a-1}^{a}$ such that $(\pi\circ\varphi)(s,t)=s$ and such that
$\varphi|_{U^{j}_{\circ}\times[0,1]}$ is a trivialization of the interval
bundle $Y^{j}_{\circ}$.

###### Proof.

If $U^{j}_{\circ}=U$, the lemma is clear, because then $Y^{j}$ is an interval
bundle over $U$, which must be necessarily trivial: it admits two nowhere
agreeing sections, given by the zero sets of $\rho_{a-1}$, respectively
$\rho_{a}$. In general, over $U$, our bordism is isomorphic to a bordism of
the form $((\mathbb{R}\times U)/U;\rho_{0},\rho_{1};\gamma)$, where
$\rho_{i}(t,s)=t-\tau_{i}(s)$, $i=0,1$, for smooth functions
$\tau_{0},\tau_{1}\colon U\rightarrow\mathbb{R}$ with $\tau_{0}\leq\tau_{1}$.

In that case, we have $U^{j}_{\circ}=\\{s\in
U\mid\tau_{0}(s)\neq\tau_{1}(s)\\}$, and

$\varphi_{s}(t)=\tau_{0}(s)+\bigl{(}\tau_{1}(s)-\tau_{0}(s)\bigr{)}t$

gives the desired parametrization. ∎

In particular, this means that the paths $\varphi_{s}\colon[0,1]\rightarrow X$
given by $\varphi_{s}(t):=\varphi(s,t)$ map to the fibers $Y^{j}|_{s}$, and,
for $s\in U\setminus U^{j}_{\circ}$, $\varphi_{s}$ is constant (since for such
$s$, $Y^{j}_{\circ}$ is a collection of finitely many points).

Let $P(\varphi)$ be the vector bundle isomorphism between the bundles
$(\gamma\circ\varphi\circ(\mathrm{id}\times i))^{*}V$, $i=0,1$, over $U$ given
over $s\in U$ by parallel translation along the path $\varphi_{s}$. Now notice
that $\varphi\circ(\mathrm{id}\times i)$, $i=0,1$ is a section of $\pi\colon
X\rightarrow S$, with image contained in either $X_{a-1}^{a-1}$ or
$X_{a}^{a}$; hence

$\bigl{(}\gamma\circ\varphi\circ(\mathrm{id}\times
i)\bigr{)}^{*}V=\bigl{(}\gamma\circ\big{(}\pi|_{C}^{-1}\big{)}\bigr{)}^{*}V=\big{(}\pi|_{C}^{-1}\big{)}^{*}\tilde{W}_{b}$

for some connected component $C$ of $Y^{j}\cap X_{b}^{b}$; here either $b=a-1$
or $a$. For all $s\in U$, since $X_{a-1}^{a}|_{s}$ is a (possibly degenerate)
interval, $\big{(}V^{j}\cap X_{a-1}^{a-1}\big{)}|_{s}$ has either zero, one or
two elements. Correspondingly, $\big{(}V^{j}\cap X_{a}^{a}\big{)}|_{s}$ has
two, one or zero elements. In either case, we can use the bilinear form
$\beta$ to turn $P(\varphi)$ into a morphism of the required form (4.4). This
defines ${f}^{j}$ in this case. As before, we use the parametrization
independence as well as the fact that parallel transport preserves $\beta$ in
order to show that this definition is independent of the choice of $\varphi$.

This defines $Z_{V,\nabla,\beta}$ on objects. On morphisms in
$1\textrm{-}\mathrm{Bord}(M)$, we declare that $Z_{V,\nabla,\beta}$ acts by
pullbacks in the obvious way. This concludes the definition of
$Z_{V,\nabla,\beta}$. Of course, there are several things to check in order to
show that this is a field theory. However, all checks can be made _pointwise_
, hence are very similar to the classical arguments outlined above (cf.
Construction 4.1).

###### Proof of Proposition 4.4.

Of course, we set $\Phi(V,\nabla,\beta)=Z_{V,\nabla,\beta}$ for
$(V,\nabla,\beta)\in\mathrm{Vect}^{\sim}_{\nabla,\beta}(M)$, where
$Z_{V,\nabla,\beta}$ is the field theory constructed above. We now discuss how
$\Phi$ acts on morphisms in $\mathrm{Vect}_{\nabla,\beta}^{\sim}(M)$. To this
end, let $(V,\nabla,\beta)$ and $(V^{\prime},\nabla^{\prime},\beta^{\prime})$
be vector bundles on $M$ with connection and a compatible bilinear form, and
let $\alpha\colon V\rightarrow V^{\prime}$ be a vector bundle isomorphism
preserving these additional structures. We now define the smooth natural
transformation

$\eta^{\alpha}:=\Phi(\alpha)\colon\ Z_{V,\nabla,\beta}\longrightarrow
Z_{V^{\prime},\nabla^{\prime},\beta^{\prime}}.$

First, we look at the simplicial level zero. If $X/S=(X/S;\rho_{0};\gamma)$ is
a _single point_ , meaning that $X_{0}^{0}$ has connected fibers (in other
words, $\pi\colon X_{0}^{0}\rightarrow S$ is a diffeomorphism), we have

$Z_{V,\nabla,\beta}(X/S)=\bigl{(}\gamma\circ\big{(}\pi|_{X_{0}^{0}}^{-1}\big{)}\bigr{)}^{*}V,\qquad
Z_{V^{\prime},\nabla^{\prime},\beta^{\prime}}(X/S)=\bigl{(}\gamma\circ\big{(}\pi|_{X_{0}^{0}}^{-1}\big{)}\bigr{)}^{*}V^{\prime}.$

Hence we can set

$\eta^{\alpha}_{X/S}:=\bigl{(}\gamma\circ\big{(}\pi|_{X_{0}^{0}}^{-1}\big{)}\bigr{)}^{*}\alpha$

in this case. Any object in $1\textrm{-}\mathrm{Bord}(M)_{0}$ can, at least
locally, be uniquely decomposed into a union of single-point-objects just
discussed, and hence the requirement that $\eta^{\alpha}$ is symmetric
monoidal determines it on all of $1\textrm{-}\mathrm{Bord}(M)_{0}$.

By Lemma 3.21, any smooth natural transformation $\eta\colon
Z_{V,\nabla,\beta}\rightarrow Z_{V^{\prime},\nabla^{\prime},\beta^{\prime}}$
is determined by its component $\eta_{0}$ at the simplicial level zero;
however, it is not clear that $\eta^{\alpha}$ defined above on simplicial
level zero indeed extends to all higher simplicial levels. Here, again by
Lemma 3.21 it suffices to consider the simplicial level one. To this end, let
$X/S=(X/S;\rho_{0},\rho_{1};\gamma)$ be an object of
$1\textrm{-}\mathrm{Bord}(M)_{1}$ and write
$Z_{V,\nabla,\beta}(X/S)=(W_{0},W_{1};f_{0})$ and
$Z_{V,\nabla,\beta}(X/S)=(W_{0},W_{1};f_{0})$. We have to check that the
diagram

${W_{1}}$${W_{0}}$${W^{\prime}_{1}}$${W_{0}^{\prime}}$$\scriptstyle{\eta^{\alpha}_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{\eta^{\alpha}_{0}}$$\scriptstyle{f_{0}^{\prime}}$

commutes, where $\eta^{\alpha}_{i}=\eta^{\alpha}_{d_{i}^{*}X/S}$, with
$d_{i}\colon[0]\rightarrow[1]$ the usual boundary maps. Since the morphisms
$f_{0}$ and $f^{\prime}_{0}$ are essentially given by parallel transport,
respectively the bilinear form $\beta$, it is now easy to check that this
diagram commutes for all bordisms if and only if $\alpha$ intertwines the
connections and bilinear forms on $V$, respectively $V^{\prime}$.

Finally, we show that $\Phi$ is fully faithful. To this end, let
$\mathrm{pt}_{M}\in 1\textrm{-}\mathrm{Bord}(M)_{0}$ be the _universal point_
, which is the object over $M$ given by

$\mathrm{pt}_{M}=\bigl{(}(\mathbb{R}\times
M)/M;\rho_{0}=\operatorname{pr}_{\mathrm{\mathbb{R}}};\gamma=\operatorname{pr}_{M}\bigr{)}.$
(4.5)

For any natural transformation $\eta\colon Z_{V,\nabla,\beta}\rightarrow
Z_{V^{\prime},\nabla^{\prime},\beta^{\prime}}$, the component
$\eta_{\mathrm{pt}_{M}}$ is a vector bundle isomorphism $V\rightarrow V$. In
particular, if $\alpha\colon V\rightarrow V$ is a vector bundle isomorphism
preserving connections and bilinear forms, tracing through the above
definitions shows that $\eta^{\alpha}_{\mathrm{pt}_{M}}=\alpha$. Hence if
$\eta^{\alpha}=\eta^{\alpha^{\prime}}$, this implies $\alpha=\alpha^{\prime}$;
in other words, $\Phi$ is faithful. Conversely, it is easy to check that
$\Phi\big{(}\eta_{\mathrm{pt}_{M}}\big{)}=\eta$ for any natural transformation
$\eta$, so that $\Phi$ is also full. ∎

###  4.2 Functors from the path category

Let $M$ be a fixed target manifold. In this section, we restrict our attention
to the smooth path category of $M$, a certain subcategory of
$1\textrm{-}\mathrm{Bord}(M)$ which is somewhat easier to describe.

######  Definition 4.8 (smooth path category).

Write $\mathrm{Path}(M)$ for the full smooth subcategory of
$1\textrm{-}\mathrm{Bord}(X)$ consisting of those objects
$(X/S;\rho_{0},\dots,\rho_{n};\gamma)\in 1\textrm{-}\mathrm{Bord}(M)_{n}$ such
that each of the sets $X_{a}^{b}$, $0\leq a\leq b\leq n$, defined in (3.6) has
connected fibers. Let moreover $\mathrm{Path}_{c}(M)$ be the full subcategory
of $\mathrm{Path}(M)$ consisting of those objects
$(X/S;\rho_{0},\dots,\rho_{n};\gamma)$ such that $\gamma$ is fiberwise
constant in a neighborhood of $X_{a}^{a}$ for each $0\leq a\leq n$.

Objects in $\mathrm{Path}(M)_{n}$ over $S\in\mathrm{Man}$ can be thought of as
$S$-families of paths in $M$ with $n+1$ marked points, while the full
subcategory $\mathrm{Path}_{c}(M)_{n}$ consists of those paths that have
sitting instants at the marked points.

######  Notation 4.9 (standard objects).

We denote by

$(\gamma;\tau_{0},\dots,\tau_{n}):=\bigl{(}(\mathbb{R}\times
S)/S;\rho_{0},\dots,\rho_{n};\gamma\bigr{)}\in\mathrm{Path}(M)_{n}$

the object where $\mathbb{R}\times S\rightarrow S$ is the projection onto the
second factor, $\gamma\colon\mathbb{R}\times S\rightarrow M$ is a smooth map,
and the cut functions $\rho_{0},\dots,\rho_{n}$ are given by
$\rho_{j}(x,s)=x-\tau_{j}(s)$ for smooth functions $\tau_{j}\colon
S\rightarrow\mathbb{R}$ satisfying $\tau_{0}\leq\dots\leq\tau_{n}$.

######  Remark 4.10.

Going through the definition of the morphisms in $1\textrm{-}\mathrm{Bord}(M)$
shows that morphisms between standard objects over $S=\mathrm{pt}$ are
(equivalence classes of) diffeomorphisms $F$ of $\mathbb{R}$, which must be
orientation preserving, as they need to preserve the sign of the cut functions
$\rho_{i}$.

######  Remark 4.11.

Denote by $\mathrm{Path}(M)^{\circ}_{n}$ the full subcategory of the fibered
category $\mathrm{Path}(M)_{n}$ consisting of all objects
$(\gamma;\tau_{0},\dots,\tau_{n})$ and the morphisms between them. This forms
a prestack over $\mathrm{Man}$; since any object in $\mathrm{Path}(M)_{n}$ is
locally isomorphic to $(\gamma;\tau_{0},\dots,\tau_{n})$ for suitable $\gamma$
and $\tau_{a}$, the stack $\mathrm{Path}(M)_{n}$ is a stackification of
$\mathrm{Path}(M)^{\circ}_{n}$. In particular, this means that any map from
$\mathrm{Path}(M)_{n}$ to a smooth stack $\mathcal{V}$ is determined on
$\mathrm{Path}(M)^{\circ}_{n}$ (up to unique isomorphism), and, conversely,
any map from $\mathrm{Path}(M)^{\circ}_{n}$ to $\mathcal{V}$ extends, uniquely
up to unique isomorphism, to $\mathrm{Path}(M)_{n}$.

Similar remarks hold for the subcategory
$\mathrm{Path}_{c}(M)_{n}^{\circ}\subseteq\mathrm{Path}_{c}(M)_{n}$; a path
$(\gamma;\tau_{0},\dots,\tau_{n})$ is contained in $\mathrm{Path}_{c}(M)_{n}$
if for any $s\in S$ and each $j=0,\dots,n$, $t\mapsto\gamma(t,s)$ is constant
near $t=\tau_{j}(s)$.

######  Construction 4.12.

Given a vector bundle $V$ with connection $\nabla$, we can define a functor

$Z_{V,\nabla}\colon\ \mathrm{Path}(M)\longrightarrow\mathrm{Vect}$

as follows. For an $S$-family $(\gamma;\tau_{0},\dots,\tau_{n})$ of paths in
$M$, we set

$Z(\gamma;\tau_{0},\dots,\tau_{n})=(W_{0},\dots,W_{n},P_{1},\dots,P_{n}),$

where $W_{a}:=(\gamma\circ(\tau_{a}\times\mathrm{id}))^{*}V$ is a vector
bundle over $S$ and, for each $s\in S$, $P_{j}(s)$ is the parallel transport
via $\nabla$ along the path $t\mapsto\gamma(t,s)$,
$t\in[\tau_{j-1},\tau_{j}]$. To a morphism between two standard objects
$(\gamma;\tau_{0},\dots,\tau_{n})$ and
$(\gamma^{\prime};\tau_{0}^{\prime},\dots,\tau_{n}^{\prime})$ we assign the
identity; this is well-defined because automorphisms in $\mathrm{Path}(M)$ are
reparametrizations of paths that fix the marked points, and parallel transport
is invariant under reparametrizations. This defines the functor on the
subcategory $\mathrm{Path}^{\circ}(M)\subseteq\mathrm{Path}(M)$ of standard
objects; by Remark 4.11, we get a functor on all of $\mathrm{Path}(M)$, unique
up to unique isomorphism.

Now let $(V,\nabla)$ and $(V^{\prime},\nabla^{\prime})$ be two vector bundles
with connection. Clearly, any vector bundle isomorphism $\alpha$ defines a
natural transformation

${\mathrm{Path}(M)_{0}}$${\mathrm{Vect}_{0}}$$\scriptstyle{(Z_{V,\nabla})_{0}}$$\scriptstyle{(Z_{V^{\prime},\nabla^{\prime}})_{0}}$$\scriptstyle{\eta_{0}^{\alpha}}$

at simplicial level zero. As in the proof of Proposition 4.4, it follows that
the condition for $\eta_{0}^{\alpha}$ to extend to higher simplicial levels is
precisely the condition that $\alpha$ preserves connections.

Hence if $\alpha\colon V\rightarrow V^{\prime}$ is a connection-preserving
isomorphism of vector bundles, we get a natural transformation
$\eta^{\alpha}\colon Z_{V,\nabla}\rightarrow Z_{V^{\prime},\nabla^{\prime}}$.
This yields a functor

$\displaystyle\Phi\colon\ \mathrm{Vect}^{\sim}_{\nabla}(M)$
$\displaystyle\longrightarrow\mathrm{Fun}\bigl{(}\mathrm{Path}(M),\mathrm{Vect}\bigr{)},$
$\displaystyle(V,\nabla)$ $\displaystyle\longmapsto Z_{V,\nabla},$
$\displaystyle\alpha$ $\displaystyle\longmapsto\eta^{\alpha}.$

The fundamental result is now the following.

######  Theorem 4.13.

The functor $\Phi$ just constructed is an equivalence of categories.

######  Remark 4.14.

At first sight, there may seem to be a clash of notations with the functor
$\Phi\colon\mathrm{Vect}_{\nabla,\beta}^{\sim}(M)\rightarrow
1\textrm{-}\mathrm{TFT}(M)$ constructed in Section 4.1. However, it is easy to
check that in fact the $\Phi$ just constructed is the composition of the
functor $\Phi$ from before with the restriction functor
$1\textrm{-}\mathrm{TFT}(M)\rightarrow\mathrm{Fun}(\mathrm{Path}(M),\mathrm{Vect})$.
In particular, $Z_{V,\nabla}$ is the restriction of $Z_{V,\nabla,\beta}$ to
$\mathrm{Path}(M)$. Notice that the information about $\beta$ is lost in this
restriction process.

######  Remark 4.15 (simplification).

Let $\gamma,\eta\colon\mathbb{R}\rightarrow M$ be two smooth paths and suppose
that $\gamma(t)=\eta(t)$ for $t$ in some neighborhood of $a\in\mathbb{R}$.
Then the identity map of $\mathbb{R}$ induces an isomorphism
$[\mathrm{id}]_{a}\colon(\gamma,a)\rightarrow(\eta,a)$. If now
$\xi\colon\mathbb{R}\rightarrow M$ is a third path with
$\xi(t)=\gamma(t)=\eta(t)$ for $t$ near $a$, we have the commutative diagram

${(\gamma,a)}$${(\eta,a)}$${(\xi,a)}$$\scriptstyle{[\mathrm{id}]_{a}}$$\scriptstyle{[\mathrm{id}]_{a}}$$\scriptstyle{[\mathrm{id}]_{a}}$

in $\mathrm{Path}(M)_{0}$. A smooth functor
$Z\colon\mathrm{Path}(M)\rightarrow\mathrm{Vect}$ now comes with canonical
coherent isomorphisms between the vector spaces $Z(\gamma,a)$, $Z(\eta,a)$ and
$Z(\xi,a)$, given by the various $Z([\mathrm{id}]_{a})$; this means that we
can (and will) assume in the future that $Z(\gamma,a)$ is _equal to_
$Z(\eta,a)$ for paths that are equal near $a$.

In this section, we will prove the following weaker version of Theorem 4.13,
which states that $\Phi$ is an equivalence when considered as a functor to
$\mathrm{Fun}(\mathrm{Path}_{c}(M),\mathrm{Vect})$. The proof of Theorem 4.13
will then be completed by Proposition 4.21 from Section 4.3, which reduces the
general case to the one just below.

######  Proposition 4.16.

The composition $\operatorname{res}\circ\,\Phi$ is an equivalence of
categories, where

$\operatorname{res}\colon\
\mathrm{Fun}\bigl{(}\mathrm{Path}(M),\mathrm{Vect}\bigr{)}\longrightarrow\mathrm{Fun}\bigl{(}\mathrm{Path}_{c}(M),\mathrm{Vect}\bigr{)}$

is the obvious restriction functor.

We start our preparations for the proof of the above proposition with a couple
of lemmas, for which we fix a smooth functor
$Z\colon\mathrm{Path}(M)\rightarrow\mathrm{Vect}$. The following lemma uses
Notation 4.9.

######  Lemma 4.17 (invertibility).

For all paths $(\gamma;a,b)$ in $\mathrm{Path}(M)_{1}$, the vector bundle map
$Z(\gamma;a,b)$ is invertible.

###### Proof.

Invertibility can be checked pointwise, hence we may assume that $\gamma$ is a
single path (i.e., a family over the point). We have
$Z(\gamma;a,a)=\mathrm{id}$ as $Z(\gamma;a,a)$ is the image of $Z(\gamma;a)$
under the degeneracy $[1]\rightarrow[0]$. Since the set of invertible linear
maps is open and the value of $Z(\gamma;a,b)$ depends smoothly (in particular
continuously) on $a$ and $b$, we have

$b_{0}:=\inf\\{b\mid Z(\gamma;a,b)\text{ is not invertible}\\}>0.$

Suppose that $b_{0}<\infty$. Then, since the set of $b$ such that
$Z(\gamma;a,b)$ is not invertible is a closed set, the infimum is actually a
minimum. Therefore $Z(\gamma;a,b_{0})$ is not invertible, but $Z(\gamma;a,b)$
is invertible for each $b<b_{0}$. Now

$Z(\gamma;a,b_{0})=Z(\gamma;b,b_{0})Z(\gamma;a,b)$

for all $b\in[0,b_{0}]$. If now $b<b_{0}$, then $Z(\gamma;a,b)$ is invertible
by definition of $b_{0}$. On the other hand, since
$Z(\gamma;b_{0},b_{0})=\mathrm{id}$, the linear map $Z(\gamma;b,b_{0})$ is
invertible for $b$ close enough to $b_{0}$. This leads to a contradiction to
the assumption that $Z(\gamma;a,b_{0})$ is not invertible, as for such $b$
close to $b_{0}$, the right hand side is a composition of two invertible maps.
Hence we must have $b_{0}=\infty$, which proves the lemma. ∎

######  Lemma 4.18 (trivial action).

For any path $\gamma$ in $M$ that is constant near $a$ and any orientation-
preserving diffeomorphism $F$ of $\mathbb{R}$ such that $F(a)=a$, the
isomorphism

$Z([F]_{a})\colon\ Z(\gamma;a)\longrightarrow Z(\gamma\circ F;a)=Z(\gamma;a)$

is the identity. Here, the last identity is the one given by Remark 4.15.

###### Proof.

Let $U\ni a$ be a small neighborhood in which $\gamma$ is constant and let
$G\colon\mathbb{R}\rightarrow\mathbb{R}$ be a diffeomorphism satisfying

$G(t)=\begin{cases}F(t),&t\ \text{ near }\ a,\\\ t,&t\not\in U.\end{cases}$

Since $G$ has the same germ at $a$ as $F$, we have $[F]_{a}=[G]_{a}$.
Moreover, $G(U)=U$, so $\gamma=\gamma\circ G$ and therefore $G$ defines an
automorphism $[G]_{[a,a+1]}$ of $(\gamma,a,a+1)$. Thus, the diagram

${Z(\gamma;a)}$${Z(\gamma;a+1)}$${Z(\gamma;a)}$${Z(\gamma;F(a+1))}$$\scriptstyle{Z(\gamma;a,a+1)}$$\scriptstyle{{Z([G]_{a}})}$$\scriptstyle{Z([G]_{a+1})}$$\scriptstyle{Z(\gamma;a,a+1)}$

commutes. However, $Z(\gamma,a,a+1)$ is invertible by Lemma 4.17, and $G$ is
the identity near $t=a+1$, so that $Z([G]_{a+1})=\mathrm{id}$. Hence
$Z([F]_{a})=Z([G]_{a})=\mathrm{id}$. ∎

Lemma 4.18 lets us simplify our analysis as follows. Denote by $T_{a}$ the
translation diffeomorphism given by

$T_{a}(t)=t+a.$ (4.6)

For any path $(\gamma,a)$ which is constant near $t=a$, this induces an
isomorphism $[T_{-a}]_{a}\colon(\gamma,a)\rightarrow(\gamma\circ T_{a},0)$.
Now, let $Z\colon\mathrm{Path}_{c}(M)\rightarrow\mathrm{Vect}$ be a functor
that satisfies the simplifying assumption of Remark 4.15. We obtain a
canonical isomorphism

$\mathcal{T}_{\gamma,a}:=Z([T_{-a}]_{a})\colon\ Z(\gamma,a)\longrightarrow
Z(\gamma\circ T_{a},0)=Z(\gamma(0),0),$ (4.7)

as $\gamma\circ T_{a}$ has a sitting instant at $t=0$; here by abuse of
notation, $\gamma(0)$ denotes the constant path equal to $\gamma(0)$. From an
orientation preserving diffeomorphism $F$, we get a commutative diagram

${Z(\gamma,a)}$${Z(\gamma\circ T_{a},0)}$${Z\big{(}\gamma\circ
F^{-1},F(a)\big{)}}$${Z\big{(}\gamma\circ F^{-1}\circ
T_{F(a)},0\big{)}.}$$\scriptstyle{Z([F]_{a})}$$\scriptstyle{Z([T_{-a}]_{a})}$$\scriptstyle{Z([T_{-F(a)}\circ
F\circ T_{a}]_{0})}$$\scriptstyle{Z([T_{-F(a)}]_{F(a)})}$

Now both $\gamma\circ T_{a}$ and $\gamma\circ F^{-1}\circ T_{F(a)}$ have a
sitting instant at $t=0$, so the assumption from Remark 4.15 on $Z$ tells us
that the two vector spaces in the right column agree; $Z(\gamma\circ
T_{a},0)=Z\big{(}\gamma\circ F^{-1}\circ T_{F(a)},0\big{)}$. A priori,
$Z\big{(}[T_{-F(a)}\circ F\circ T_{a}]_{0}\big{)}$ could be a nontrivial
automorphism of this vector space; however, since $\big{(}T_{-F(a)}\circ
F\circ T_{a}\big{)}(0)=0$, we have $Z\big{(}[T_{-F(a)}\circ F\circ
T_{a}]_{0}\big{)}=\mathrm{id}$ by Lemma 4.18. We see that the isomorphisms
$\mathcal{T}_{\gamma,a}=Z([T_{-a}]_{a})$ satisfy the equivariance property

$\mathcal{T}_{\gamma,a}=\mathcal{T}_{\gamma\circ F^{-1},F(a)}\circ Z([F]_{a})$
(4.8)

for any diffeomorphism $F$ and all $a\in\mathbb{R}$. Using this equivariance
property, we obtain the following lemma.

######  Lemma 4.19 (normalization).

Given a strict functor
${Z}\colon\mathrm{Path}_{c}(M)^{\circ}\rightarrow\mathrm{Vect}$, there exists
a strict functor
$\tilde{Z}\colon\mathrm{Path}_{c}(M)^{\circ}\rightarrow\mathrm{Vect}$ together
with a natural isomorphism $\mathcal{T}\colon Z\rightarrow\tilde{Z}$ such that
$\tilde{Z}(\gamma,a)=\tilde{Z}(\gamma(a),0)$ for all standard objects
$(\gamma,a)$. For such a functor $\tilde{Z}$

$Z([F]_{a})\colon\
\tilde{Z}(\gamma,a)=\tilde{Z}(\gamma(a),0)\longrightarrow\tilde{Z}\big{(}\gamma\circ
F^{-1},F(a)\big{)}=\tilde{Z}(\gamma(a),0)$

acts as the identity for any diffeomorphism $F$ of $\mathbb{R}$.

###### Proof.

We may assume that $Z$ satisfies the simplifying assumptions of Remark 4.15.
Set

$\tilde{Z}(\gamma;\tau_{0},\dots,\tau_{n}):=\bigl{(}W_{0},\dots,W_{n};{f}_{1},\dots,{f}_{n}\bigr{)},$

with $W_{j}:=Z(\gamma(\tau_{j}),0)$, and

${f}_{j}:=\mathcal{T}_{\gamma,\tau_{j}}\circ
Z(\gamma;\tau_{j-1},\tau_{j})\circ\mathcal{T}_{\gamma,\tau_{j-1}}^{-1},$

where the $\mathcal{T}_{\gamma,a}$ are defined as in (4.7). Moreover, for a
diffeomorphism $F$, set
$Z\big{(}[F]_{[\tau_{0},\tau_{n}]}\big{)}:=\mathrm{id}$. By the equivariance
property (4.8), this gives a well-defined functor
$\tilde{Z}\colon\mathrm{Path}_{c}(M)^{\circ}\rightarrow\mathrm{Vect}$.

Finally, let $\mathcal{T}$ be given by

$\mathcal{T}_{(\gamma;\tau_{0},\dots,\tau_{n})}=(\mathcal{T}_{\gamma,\tau_{0}},\dots,\mathcal{T}_{\gamma,\tau_{n}}).$

This clearly defines a natural transformation $Z\rightarrow\tilde{Z}$. That
$\tilde{Z}([F]_{a})$ acts as the identity also follows directly from (4.8). ∎

We are now in a position to prove the main result of this section.

###### Proof of Proposition 4.16.

That $\operatorname{res}\circ\Phi$ is full and faithful is shown just as in
the proof of Proposition 4.4. It therefore remains to show that
$\operatorname{res}\circ\Phi$ is essentially surjective. Moreover, it suffices
to consider the functor on the subcategory of standard objects.

Let $Z\colon\mathrm{Path}_{c}(M)^{\circ}\rightarrow\mathrm{Vect}$ be a strict
smooth functor. We assume moreover that $Z$ is _normalized_ in the sense of
Lemma 4.19; in other words, $Z(\gamma,a)=Z(\gamma(a),0)$ for all paths
$\gamma$ and all $a\in\mathbb{R}$, and $Z([F]_{a})=\mathrm{id}$ for all
diffeomorphisms $F$ on $\mathbb{R}$. In particular, this implies that

$Z(\gamma\circ F;a,b)=Z(\gamma;F(a),F(b))$ (4.9)

as map from $Z(\gamma(a),0)$ to $Z(\gamma(b),0)$.

First let us extract a vector bundle from $Z$. To this end, let
$\mathrm{pt}_{M}$ be the universal point introduced in (4.5). Note that
$\mathrm{pt}_{M}\in\mathrm{Path}_{c}(M)$ and set $V:=Z(\mathrm{pt}_{M})$. For
any path $(\gamma,a)$, our assumption on $Z$ then implies that
$Z(\gamma,a)=V_{\gamma(a)}$. Now in general for objects
$(\gamma;\tau_{0},\dots,\tau_{n})\in\mathrm{Path}_{c}(M)^{\circ}_{n}$, we have

$Z(\gamma;\tau_{0},\dots,\tau_{n})=(W_{0},\dots,W_{n};{f}_{1},\dots,{f}_{n}).$

If $d_{i}\colon[0]\rightarrow[n]$, $i=0,\dots,n$, is the map with image
$i\in[n]$, we have

$W_{i}=d_{i}^{*}Z(\gamma;\tau_{0},\dots,\tau_{n})=Z(d_{i}^{*}(\gamma;\tau_{0},\dots,\tau_{n}))=Z(\gamma;\tau_{i})=V_{\gamma(\tau_{i})},$

hence the $W_{i}$ are already determined by $V$.

It remains to determine the vector bundle maps ${f}_{i}$; we will use
Proposition 4.3 for this. To obtain a section $P$ as in the proposition, we
use modification functions, which are defined as follows.

######  Definition 4.20 (modication function).

A two-sided _modification function_ is a smooth function
$\chi\colon\mathbb{R}\rightarrow[0,1]$ such that

  1. (1)

$\chi$ is nondecreasing,

  2. (2)

$\chi(t)=0$ for $t$ near zero,

  3. (3)

$\chi(t)=1$ for $t$ near one.

By $\chi_{a,b}$, $a\leq b$ we denote the function given by

$\chi_{a,b}(t):=a+(b-a)\chi\bigg{(}\frac{t-a}{b-a}\bigg{)},$

for $t\in\mathbb{R}$. This function is then only nonconstant on $[a,b]$ and
takes values in $[a,b]$.

Later, in Definition 5.5, we will introduce also left and right modification
functions, as well as their family versions. For now, we drop the adjective
“two-sided”. If $\chi$ is a modification function and $\gamma\in
C^{\infty}([0,1],M)$, then $\gamma\circ\chi$ is a path that is defined on all
of $\mathbb{R}$ and which is constant on
$(-\infty,\varepsilon]\cup[1-\varepsilon,\infty)$ for some $\varepsilon>0$.
Hence

$P(\gamma):=Z(\gamma\circ\chi;0,1)$

is well-defined for each $\gamma\in C^{\infty}([0,1],X)$. Note that
$P(\gamma)$ maps $Z(\gamma\circ\chi,0)=Z(\gamma(0);0)=V_{\gamma(0)}$ to
$Z(\gamma\circ\chi;1)=Z(\gamma(1);1)=V_{\gamma(1)}$. This construction works
in families and therefore we get a smooth section $P$ of the bundle
$\mathrm{ev}_{0}^{*}V^{\vee}\otimes\mathrm{ev}_{1}^{*}V$, as required. The
crucial result, which will be shown in Lemma 4.23 below, is then that
$P(\gamma)$ is independent of the choice of modification function.

We need to check that $P$ is multiplicative, in the sense of (4.2). For this,
we must use the next simplicial level. For $a\in[0,1]$ fixed, define
$\xi\colon\mathbb{R}\rightarrow[0,1]$ by

$\displaystyle\xi(t):=\begin{cases}a\chi(t),&t\in[0,1],\\\
a+(1-a)\chi(t-1),&t\in[1,2].\end{cases}$

Then $\xi^{\prime}(t)\geq 0$ everywhere, and $\xi$ is constant near $t=0,1,2$
(with values $0$, $a$, $1$). Hence $t\mapsto\xi_{2}(t):=\xi(2t)$ is a
modification function in the sense of Definition 4.20. Therefore, by
independence of the modification function (Lemma 4.23 below), we have

$P(\gamma)=Z(\gamma\circ\chi;0,1)=Z(\gamma\circ\xi_{2};0,1)=Z(\gamma\circ\xi;0,2).$

Notice that in the last step, we used the parametrization independence (4.9).

Now, since $\gamma\circ\xi$ is constant near $1$, $(\gamma\circ\xi;0,1)$ and
$(\gamma\circ\xi;1,2)$ are also objects of $\mathrm{Path}_{c}(M)$, and we get

$Z(\gamma\circ\xi;0,2)=Z(\gamma\circ\xi;1,2)\circ Z(\gamma\circ\xi;0,1).$

However, by definition, we have

$(\gamma\circ\xi)(t)=\begin{cases}(\mathfrak{s}_{0,a}\gamma\circ\chi)(t),&t\in[0,1],\\\
(\mathfrak{s}_{a,1}\gamma\circ\chi)(t-1),&t\in[1,2].\end{cases}$

Hence, $Z(\gamma\circ\xi;0,1)=P(\mathfrak{s}_{0,a}\gamma)$ and, again by
(4.9),

$Z(\gamma\circ\xi;1,2)=Z\big{(}\mathfrak{s}_{a,1}\gamma\circ\chi\circ
T_{-1};1,2\big{)}=Z(\mathfrak{s}_{a,1}\gamma\circ\chi;0,1)=P(\mathfrak{s}_{a,1}\gamma).$

Therefore,

$P(\gamma)=P(\mathfrak{s}_{a,1}\gamma)P(\mathfrak{s}_{0,a}\gamma),$

that is, $P$ is indeed multiplicative. By Proposition 4.3, there exists a
connection $\nabla$ on $V$ such that $P(\gamma)$ is given by parallel
transport along $\gamma$ with respect to $\nabla$.

To conclude the argument, let $s_{i}\colon[1]\rightarrow[n]$, $i=1,\dots,n$ be
the map with $s_{i}(0)=i-1$ and $s_{i}(1)=i$, and notice that

${f}_{i}=s_{i}^{*}Z(\gamma;\tau_{0},\dots,\tau_{n})=Z\bigl{(}s_{i}^{*}(\gamma;\tau_{0},\dots,\tau_{n})\bigr{)}=Z(\gamma;\tau_{i-1},\tau_{i}).$

Define $\tilde{\gamma}$ by
$\tilde{\gamma}(t):=\gamma(\tau_{i-1}+(\tau_{i}-\tau_{i-1})t)$; then
$Z(\gamma;\tau_{i-1},\tau_{i})=Z(\tilde{\gamma};0,1)$, once more by (4.9).
Since $\tilde{\gamma}$ has sitting instants at $t=0$ and $1$, we have
$\tilde{\gamma}=\tilde{\gamma}\circ\chi$ for a suitable modification function
(just choose $\chi$ in such a way that $\chi(t)=t$ wherever $\tilde{\gamma}$
is not constant). Finally, we get

${f}_{j}=Z(\tilde{\gamma};0,1)=Z(\tilde{\gamma}\circ\chi;0,1)=P(\tilde{\gamma}).$

Since parallel transport is parametrization independent, this coincides with
the parallel transport along $\gamma$ from $\tau_{i-1}$ to $\tau_{i}$, with
respect to the connection $\nabla$. Hence $Z$ coincides with the restriction
of $Z_{V,\nabla}=\Phi(V,\nabla)$ to $\mathrm{Path}_{c}(M)$, where
$Z_{V,\nabla}$ is the functor from Construction 4.12. ∎

###  4.3 General paths

In this section, we finish the proof of Theorem 4.13. Having Proposition 4.16
at hand, this will be achieved by establishing the following result.

######  Proposition 4.21.

The functor $\operatorname{res}$ from the previous section is an equivalence
of categories. More specifically, there is a functor

$\operatorname{ext}\colon\
\mathrm{Fun}\bigl{(}\mathrm{Path}_{c}(M),\mathrm{Vect}\bigr{)}\longrightarrow\mathrm{Fun}\bigl{(}\mathrm{Path}(M),\mathrm{Vect}\bigr{)}$

such that $\operatorname{res}\circ\operatorname{ext}=\mathrm{id}$, together
with a natural isomorphism
$\eta\colon\mathrm{id}\rightarrow\operatorname{ext}\circ\operatorname{res}$.

We need several lemmas, for which we fix a smooth functor
$Z\colon\mathrm{Path}(M)\rightarrow\mathrm{Vect}$. We will assume that $Z$ is
a strict functor (which is possible by Lemma 3.37), and we will also make the
simplifying assumptions discussed in Remark 4.15.

######  Lemma 4.22 (introducing a sitting instant).

Fix numbers $a\leq b$ and let $F\colon\mathbb{R}\rightarrow\mathbb{R}$ be a
smooth monotonically increasing function such that $F(t)=t$ for $t$ near $a$
and near $b$. Then, for all paths $\gamma\colon\mathbb{R}\rightarrow M$, we
have $Z(\gamma;a,b)=Z(\gamma\circ F;a,b)$.

We remark that if we choose $F$ to be constant somewhere in between $a$ and
$b$ (as allowed by the lemma), the path $\gamma\circ F$ will have a sitting
instant somewhere in between $a$ and $b$.

###### Proof.

If $F^{\prime}(x)\neq 0$ for all $x\in[a,b]$, so that $F$ is a diffeomorphism
onto its image in a neighborhood of $[a,b]$, the simplicial structure yields
the commutative diagram

${Z(\gamma;a)}$${Z(\gamma;b)}$${Z(\gamma\circ F;a)}$${Z(\gamma\circ
F;b).}$$\scriptstyle{Z(\gamma;a,b)}$$\scriptstyle{Z([F]_{a})}$$\scriptstyle{Z([F]_{b})}$$\scriptstyle{Z(\gamma\circ
F;a,b)}$

This, together with the fact that $F=\mathrm{id}$ near $a$ and $b$, so that
$Z(\gamma\circ F;a)=Z(\gamma;a)$ and $Z([F]_{a})=\mathrm{id}$, and similarly
for $b$, proves the result in this case.

The general case now follows from the fact that

$F_{\varepsilon}(t):=(1-\varepsilon)F(t)+\varepsilon t$

is a one-parameter family of maps such that $F_{0}=F$ and such that
$F_{\varepsilon}$ is a diffeomorphism whenever $\varepsilon\in(0,1]$.
Therefore, by the observations above, we have $Z(\gamma;a,b)=Z(\gamma\circ
F_{\varepsilon};a,b)$ for all $\varepsilon\in(0,1]$, and, by continuity, the
equality persists for $\varepsilon=0$. ∎

We are now able to prove the following essential lemma, which proves the
independence of the choice of modification function (see Definition 4.20).

######  Lemma 4.23 (independence of $\chi$).

Let $Z\colon\mathrm{Path}_{c}(X)\rightarrow\mathrm{Vect}$ be a functor. Then
for any two modification functions $\chi$, $\tilde{\chi}$, we have

$Z(\gamma\circ\chi_{a,b},a,b)=Z(\gamma\circ\tilde{\chi}_{a,b},a,b)$

as morphisms from $Z(\gamma(a),a)$ to $Z(\gamma(b),b)$.

###### Proof.

Clearly, we may assume for simplicity that $a=0$ and $b=1$. Now, we first
argue that we may furthermore assume that $\chi(t)=\tilde{\chi}(t)=t$ for $t$
in a neighborhood of $\tfrac{1}{2}$. Since $\chi$ is not constant, there
exists some $t_{0}\in(0,1)$ such that $\chi^{\prime}(t_{0})\neq 0$. We may
arrange that $t_{0}=\tfrac{1}{2}$: Choose some diffeomorphism $F$ that is the
identity near $t=0,1$ and sends $\tfrac{1}{2}$ to $t_{0}$. Then
$\xi:=\chi\circ F$ is again a modification function, now satisfying
$\xi^{\prime}(\tfrac{1}{2})\neq 0$. We get that there exists a neighborhood of
$\tfrac{1}{2}$, where $\xi^{\prime}$ is invertible. Hence there exists some
small $\varepsilon>0$ such that we can find a diffeomorphism $G$ of
$\mathbb{R}$ with

$G(t)=\begin{cases}t&\text{if}\quad
t\in(-\infty,\varepsilon]\cup[1-\varepsilon,\infty),\\\\[2.15277pt]
\xi^{-1}(t)&\text{if}\quad
t\in\big{[}\tfrac{1}{2}-\varepsilon,\tfrac{1}{2}+\varepsilon\big{]}.\end{cases}$

Then $\xi\circ G$ is a modification function that is the identity near
$t=\tfrac{1}{2}$, and we have

$Z\bigl{(}\gamma\circ(\xi\circ
G);0,1\bigr{)}=Z\bigl{(}\gamma\circ\chi\circ(F\circ
G);0,1\bigr{)}=Z(\gamma\circ\chi;0,1)$

since $F\circ G$ is the identity near $t=0,\ 1$.

By the above, after replacing $\chi$ and $\tilde{\chi}$ with equivalent
modification functions, we may assume that $\chi(t)=\tilde{\chi}(t)=t$ near
$t=\tfrac{1}{2}$. Now let $F$ and $\tilde{F}$ be monotonic functions on
$\mathbb{R}$ such that

  1. (1)

$F(t)=\tilde{F}(t)=t$ on $\big{(}{-}\infty,-\tfrac{1}{2}+\varepsilon\big{]}$,

  2. (2)

$F(t)=\tilde{F}(t)$ on $(-\infty,0]$,

  3. (3)

$F(t)=\chi(t)$ and $\tilde{F}(t)=\tilde{\chi}(t)$ on $[0,\infty)$.

This is possible since $\chi$ and $\tilde{\chi}$ are both constant near zero.
Now since $F$, $\tilde{F}$ are the identity near $-\tfrac{1}{2}$ and
$\tfrac{1}{2}$, Lemma 4.22 gives

$Z\bigg{(}\gamma\circ
F;-\frac{1}{2},\frac{1}{2}\bigg{)}=Z\bigg{(}\gamma;-\frac{1}{2},\frac{1}{2}\bigg{)}=Z\bigg{(}\gamma\circ\tilde{F};-\frac{1}{2},\frac{1}{2}\bigg{)}.$
(4.10)

On the other hand,

$\displaystyle Z\bigg{(}\gamma\circ
F;-\frac{1}{2},\frac{1}{2}\bigg{)}=Z\bigg{(}\gamma\circ\chi;0,\frac{1}{2}\bigg{)}\circ
Z\bigg{(}\gamma\circ F;-\frac{1}{2},0\bigg{)},$ $\displaystyle
Z\bigg{(}\gamma\circ\tilde{F};-\frac{1}{2},\frac{1}{2}\bigg{)}=Z\bigg{(}\gamma\circ\tilde{\chi};0,\frac{1}{2}\bigg{)}\circ
Z\bigg{(}\gamma\circ F;-\frac{1}{2},0\bigg{)},$ (4.11)

by the construction of $F$ and $\tilde{F}$. As all morphisms involved are
invertible in view of Lemma 4.17, combining (4.10) with (4.11) implies that
$Z\big{(}\gamma\circ\chi;0,\tfrac{1}{2}\big{)}=Z\big{(}\gamma\circ\tilde{\chi};0,\tfrac{1}{2}\big{)}$.
A similar argument shows that
$Z\big{(}\gamma\circ\chi;\tfrac{1}{2},1\big{)}=Z\big{(}\gamma\circ\tilde{\chi};\tfrac{1}{2},1\big{)}$;
combining these observations finishes the proof. ∎

We are now ready to give the proof of the main result of this section.

###### Proof of Proposition 4.21.

We first define the functor $\operatorname{ext}$. For a smooth functor

$Z\colon\ \mathrm{Path}_{c}(M)\allowbreak\rightarrow\mathrm{Vect},$

we set $\operatorname{ext}Z(\gamma,a):=Z(\gamma(a);a)$ and

$\operatorname{ext}Z(\gamma;\tau_{0},\dots,\tau_{n})=Z(\bar{\gamma};\tau_{0},\dots,\tau_{n}),$

where $\bar{\gamma}$ is given on the interval $[\tau_{j-1},\tau_{j}]$ by

$\bar{\gamma}(t)=\big{(}\gamma\circ\chi_{\tau_{i-1},\tau_{i}}\big{)}(t)$

for some modification function $\chi$. By Lemma 4.23, this is independent of
the choice of modification function. Let $F$ be a diffeomorphism of
$\mathbb{R}$, which defines a morphism

$[F]_{a}\colon\ (\gamma\circ F,a)\longrightarrow(\gamma,F(a))$

in $\mathrm{Path}(M)_{0}$. Let $T_{a}$, $a\in\mathbb{R}$ be the translations
defined in (4.6) above; we then set

$\operatorname{ext}Z([F]_{a}):=Z\bigl{(}\big{[}T_{F(a)-a}\big{]}_{a}\colon\bigl{(}\gamma(F(a)),a\bigr{)}\rightarrow\bigl{(}\gamma(F(a)),F(a)\bigr{)}\bigr{)}.$

Using the simplicial structure and Remark 4.11, this determines the functor
$\operatorname{ext}Z$ completely.

In order to show that $\operatorname{ext}Z$ is well defined, we need to check
functoriality. Let $F$ be a diffeomorphism of $\mathbb{R}$, which defines an
automorphism

$[F]_{[a,b]}\colon\ (\gamma\circ
F;a,b)\longrightarrow(\gamma;a^{\prime},b^{\prime}),\qquad\text{where}\quad
a^{\prime}:=F(a),\quad b^{\prime}:=F(b),$

in $\mathrm{Path}(M)_{q}$. We need to show that the square

${Z\bigl{(}(\gamma\circ F)(a),a\bigr{)}}$${Z\bigl{(}(\gamma\circ
F)(b);b\bigr{)}}$${Z\bigl{(}\gamma(a^{\prime}),a^{\prime}\bigr{)}}$${Z\bigl{(}\gamma(b^{\prime}),b^{\prime}\bigr{)}}$$\scriptstyle{{Z([T_{a^{\prime}-a}]_{a}})}$$\scriptstyle{Z(\overline{\gamma\circ
F};a,b)}$$\scriptstyle{{Z([T_{b^{\prime}-b}]_{b}})}$$\scriptstyle{Z(\bar{\gamma};a^{\prime},b^{\prime})}$

commutes. We have

$Z\big{(}\overline{\gamma\circ F};a,b\big{)}=Z\big{(}\gamma\circ
F\circ\chi_{a,b};a,b\big{)}=Z([F]_{b})^{-1}Z\big{(}\gamma\circ
F\circ\chi_{a,b}\circ F^{-1};a^{\prime},b^{\prime}\big{)}Z([F]_{a}).$

Now first notice that $F\circ\chi_{a,b}\circ
F^{-1}=\tilde{\chi}_{a^{\prime},b^{\prime}}$ for some modification function
$\tilde{\chi}$, hence the middle term equals
$Z(\bar{\gamma};a^{\prime},b^{\prime})$ (here, of course, we use Lemma 4.23
again). Secondly,

$Z([F]_{a})=Z\big{(}[T_{a^{\prime}-a}]_{a}\big{)}Z\big{(}[T_{a^{\prime}-a}^{-1}\circ
F]_{a}\big{)},$

with $Z\big{(}[T_{a^{\prime}-a}^{-1}\circ F]_{a}\big{)}=\mathrm{id}$ by Lemma
4.18, as $\big{(}T_{a^{\prime}-a}^{-1}\circ F\big{)}(a)=a$. Using a similar
argument for $Z([F]_{b})$, we obtain

$Z\big{(}\overline{\gamma\circ
F};a,b\big{)}=Z\big{(}[T_{b^{\prime}-b}]_{b}\big{)}^{-1}Z(\bar{\gamma};a^{\prime},b^{\prime})Z\big{(}[T_{a^{\prime}-a}]_{a}\big{)},$

which was the claim.

If $\eta\colon Z\rightarrow Z^{\prime}$ is a smooth natural transformation
between smooth functors
$Z,Z^{\prime}\colon\mathrm{Path}_{c}(M)\rightarrow\mathrm{Vect}$, we set

$(\operatorname{ext}\eta)_{(\gamma,a)}:=\eta_{(\gamma(a),a)}.$

It is straightforward to check that this gives a natural transformation
$\operatorname{ext}\eta\colon\operatorname{ext}Z\rightarrow\operatorname{ext}Z^{\prime}$
and that the assignment $\eta\mapsto\operatorname{ext}\eta$ is functorial in
$\eta$. This finishes the definition of $\operatorname{ext}$.

To see that $\operatorname{res}\circ\operatorname{ext}=\mathrm{id}$, we have
to check that for paths $\gamma$ that already have sitting instants at
$\tau_{0},\dots,\tau_{n}$, one has
$Z(\gamma;\tau_{0},\dots,\tau_{n})=Z(\bar{\gamma};\tau_{0},\dots,\tau_{n})$.
However, for such a path $\gamma$, there exists a modification function $\chi$
such that on each subinterval $[\tau_{j-1},\tau_{j}]$, we have
$\gamma=\gamma\circ\chi_{\tau_{j-1},\tau_{j}}$. Moreover, the composition of
two modification functions is again a modification function, hence the claim
follows once more from Lemma 4.23.

Finally, we construct a natural isomorphism
$\eta\colon\mathrm{id}\rightarrow\operatorname{ext}\circ\operatorname{res}$.
To this end, choose numbers $0<\delta<\ell$ and a modification function $\chi$
such that $\chi(t)=t$ for $t$ near $\delta$ and set

$\eta_{\gamma,a}:=Z(\gamma\circ\chi_{a,a+\ell};a,a+\delta)^{-1}\circ
Z(\gamma;a,a+\delta)\colon\ Z(\gamma;a)\longrightarrow Z(\gamma(a),a).$

We claim that this definition is independent of the modification function
$\chi$ and the choice of $\delta$ and $\ell$. For notational simplicity, let
$a=0$ and suppose $\ell=1$; the case $a\neq 0$ is similar. Now, let
$\tilde{\chi}$ be another modification function also satisfying
$\tilde{\chi}(t)=t$ for $t$ near $\delta$. Since the values of a modification
function on $[\delta,1]$ are irrelevant for the definition of $\eta$, we may
as well assume that $\tilde{\chi}=\chi$ on $[\delta,1]$ (which then implies
that they in fact agree on a neighborhood of $[\delta,1]$). Now, by Lemma
4.23, we have $Z(\gamma\circ\chi;0,1)=Z(\gamma\circ\tilde{\chi};0,1)$. On the
other hand

$Z(\gamma\circ\chi;0,1)=Z(\gamma\circ\chi;\delta,1)\circ
Z(\gamma\circ\chi;0,\delta)=Z(\gamma\circ\tilde{\chi};\delta,1)\circ
Z(\gamma\circ\chi;0,\delta)$

since $\chi$ and $\tilde{\chi}$ agree in a neighborhood of $\delta$. But this
equals

$Z(\gamma\circ\tilde{\chi};0,1)=Z(\gamma\circ\tilde{\chi};\delta,1)\circ
Z(\gamma\circ\tilde{\chi};0,\delta),$

from which we obtain the desired equality
$Z(\gamma\circ\chi;0,\delta)=Z(\gamma\circ\tilde{\chi};0,\delta)$, by virtue
of Lemma 4.17. To see the independence from $\delta$, let $\tilde{\chi}$ be a
modification function with $\tilde{\chi}(t)=t$ near $\tilde{\delta}$. Without
loss of generality, suppose that $\tilde{\delta}<\delta$. By the first step,
we are free to choose the modification function $\chi$ any way we like, under
the constraint that $\chi(t)=t$ near $\delta$. We now choose it in such a way
that in fact $\chi(t)=t$ in a neighborhood of the interval
$\big{[}\tilde{\delta},\delta\big{]}$. Then

$Z(\gamma\circ\chi;0,\delta)=Z\big{(}\gamma\circ\chi;\tilde{\delta},\delta\big{)}\circ
Z\big{(}\gamma\circ\chi;0,\tilde{\delta}\big{)}.$

However, on $\big{[}\delta,\tilde{\delta}\big{]}$, we have
$\gamma\circ\chi=\gamma$, and by the first step, we have
$Z\big{(}\gamma\circ\chi;0,\tilde{\delta}\big{)}=Z\big{(}\gamma\circ\tilde{\chi};0,\tilde{\delta}\big{)}$,
so that

$Z(\gamma\circ\chi;0,\delta)=Z\big{(}\gamma;\tilde{\delta},\delta\big{)}\circ
Z\big{(}\gamma\circ\tilde{\chi};0,\tilde{\delta}\big{)}.$

This finishes the argument that $\eta_{\gamma,a}$ does not depend on $\delta$.
The independence of the choice of $\ell$ now follows immediately.

In order to show that $\eta$ indeed gives rise to a natural transformation, we
have to show that for any diffeomorphism $F$ of $\mathbb{R}$, we have the
equivariance property

$\eta_{\gamma\circ F,a}=\eta_{\gamma,F(a)}\circ Z([F]_{a}).$ (4.12)

First notice that for any path $\xi$ and any diffeomorphism $F$, we have a
commuting square

${Z(\xi\circ
F;a)}$${Z(\xi;F(a))}$${Z(\xi;F(a))}$${Z(\xi;F(b)).}$$\scriptstyle{Z(\xi\circ
F;a\text{,}b)}$$\scriptstyle{Z(\text{[}F\text{]}_{a})}$$\scriptstyle{Z(\text{[}F\text{]}_{b})}$$\scriptstyle{Z(\xi;F(a)\text{,}F(b))}$

Applying this to $\xi=\gamma\circ F\circ\chi_{a,a+1}\circ F^{-1}$, we have

$\displaystyle\eta_{\gamma\circ F,a}$ $\displaystyle=Z(\xi\circ
F;a,a+\delta)^{-1}\circ Z(\gamma\circ F;a,a+\delta)$
$\displaystyle=Z([F]_{a})^{-1}\circ
Z\bigl{(}\xi;F(a),F(a+\delta)\bigr{)}^{-1}\circ
Z\bigl{(}\gamma;F(a),F(a+\delta)\bigr{)}\circ Z([F]_{a})$

Notice that the two appearances of $Z([F]_{a})$ above in fact denote different
things: The one on the right is an isomorphism $Z(\gamma\circ
F;a+\delta)\rightarrow Z(\gamma;F(a+\delta))$, while the one on the left is an
isomorphism $Z(\xi\circ F;a)\rightarrow Z(\xi;F(a))$. The latter is trivial by
Lemma 4.18 since $\xi\circ F$ is constant near $a$ and $\xi$ is constant near
$F(a)$. Finally, it is easy to see that $F\circ\chi_{a,a+1}\circ
F^{-1}=\tilde{\chi}_{F(a),F(a+1)}$ for some modification function
$\tilde{\chi}$. Hence the last expression equals $\eta_{\gamma,F(a)}\circ
Z([F]_{a})$, which finishes the proof of identity (4.12).

Now the functions $\eta_{\gamma,a}$ give a natural transformation
$\eta\colon\mathrm{id}\rightarrow\operatorname{ext}\circ\operatorname{res}$ as
follows. For each functor
$Z\colon\mathrm{Path}_{c}(M)\\!\rightarrow\\!\mathrm{Vect}$, we need to give a
natural transformation $\eta^{Z}\colon
Z\\!\rightarrow\\!\operatorname{ext}\big{(}Z|_{\mathrm{Path}_{c}(M)}\big{)}$.
The components of this natural transformation are given by

$\eta^{Z}_{(\gamma;\tau_{0},\dots,\tau_{n})}=(\eta_{\gamma,\tau_{0}},\dots,\eta_{\gamma,\tau_{n}}),$

with the $\eta_{\gamma,\tau_{j}}$ as constructed above. This gives an
isomorphism from

$Z(\gamma;\tau_{0},\dots,\tau_{n})\longrightarrow\operatorname{ext}Z(\gamma;\tau_{0},\dots,\tau_{n})=Z(\overline{\gamma};\tau_{0},\dots,\tau_{n}),$

as required. The equivariance property (4.12) shows that these isomorphisms
indeed fit together to give a natural transformation.

The last thing to show is that if $\mu\colon Z\rightarrow Y$ is a smooth
natural transformation of smooth functors
$Y,Z\colon\mathrm{Path}_{c}(M)\rightarrow\mathrm{Vect}$, then
$\eta^{Y}\circ\mu=\operatorname{ext}\mu\circ\eta^{Z}$. But this is trivial,
since $\eta^{Z}$ and $\eta^{Y}$ act as the identity on $\mathrm{Path}_{c}(M)$
and $\operatorname{ext}\mu|_{\mathrm{Path}_{c}(M)}=\mu$. ∎

###  4.4 Proof of the classification theorem

We are now in a position to prove Theorem 4.2; more precisely, we will prove
that the functor $\Phi$ from Proposition 4.4 is essentially surjective.

###### Proof of Theorem 4.2.

Let $Z\colon 1\textrm{-}\mathrm{Bord}(M)\rightarrow\mathrm{Vect}$ be a field
theory. By the results of the previous section, we may assume that
$Z|_{\mathrm{Path}(M)}=\Phi(V,\nabla)$, where

$\Phi\colon\
\mathrm{Fun}\bigl{(}\mathrm{Path}(M),\mathrm{Vect}\bigr{)}\longrightarrow\mathrm{Vect}_{\nabla}^{\sim}(M)$

is the equivalence constructed in Section 4.2. This means that
$Z(\gamma,a)=V_{\gamma(a)}$ for all paths $\gamma$ in $M$ and all $a$, and
that $Z(\gamma;a,b)$ is given by parallel transport along $\gamma$ from $a$ to
$b$, with respect to the connection $\nabla$.

To get a bilinear form $\beta$ on $V$, consider the _constant right elbow_ ,
which is the bordism $R=((\mathbb{R}\times
M)/M;\rho_{0},\rho_{1};\gamma_{\mathrm{const}})$, where
$\gamma_{\mathrm{const}}(t,p)=p$, $\rho_{1}\equiv-1$ and
$\rho_{0}(t,p):=t(1-t)$. Then, canonically,

$d_{0}^{*}R\cong(\gamma_{\mathrm{const}},0)\amalg(\gamma_{\mathrm{const}},1)\qquad\text{and}\qquad
d_{1}^{*}R=\varnothing.$

Hence $Z(R)$ is a linear map from $Z(\gamma_{\mathrm{const}},0)\otimes
Z(\gamma_{\mathrm{const}},1)$ to $\mathbb{K}$. However,
$Z(\gamma_{\mathrm{const}},0)=Z(\gamma_{\mathrm{const}},1)=V$, so we get a
bilinear form $\beta:=Z(R)$ on $V$. It is symmetric because the diffeomorphism
$F\colon\mathbb{R}\to\mathbb{R}$,

$F(t)=\frac{1}{2}-t,$

determines an automorphism of $R$ in $1\textrm{-}\mathrm{Bord}(M)$, the
restriction of which to $R_{0}^{0}$ gets mapped to the symmetry isomorphism of
$V\otimes V$ under $Z$. This is because $F$ swaps the two components of
$R^{0}_{0}=(\\{0,1\\}\times M)/M$.

To see that $\beta$ is nondegenerate, let $L$ be the _constant left elbow_ ,
which is the bordism given by $L=(\mathbb{R}\times
M/M;\vartheta_{0},\vartheta_{1};\gamma_{\mathrm{const}})$, where
$\vartheta_{0}(t)\equiv 1$ and $\vartheta_{1}(t)=-\rho_{0}(t)$. Set
$\tau:=Z(L)$, which is a section of the bundle
$\mathrm{Hom}(\mathbb{K},V\otimes V)$, i.e., a section of $V\otimes V$. The
“snake identity”

$(\mathrm{id}\otimes\beta)\circ(\tau\otimes\mathrm{id})=\mathrm{id}$

is satisfied. (To be precise, the left hand side is in fact a map from
$\mathbb{K}\otimes V$ to $V\otimes\mathbb{K}$, but this can be canonically be
identified with an endomorphism of $V$; the requirement is that this
endomorphism be the identity.) We now analyze this identity on each fiber; to
this end, write $\tau_{p}=\sum_{ij}v_{i}\otimes w_{j}$ for some elements
$v_{i},w_{j}\in V_{p}$. Then for any $u\in V_{p}$, we have

$(\mathrm{id}\otimes\beta)\circ(\tau\otimes\mathrm{id})(1\otimes
u)=\sum_{i,j}(\mathrm{id}\otimes\beta)(v_{i}\otimes w_{j}\otimes
u)=\sum_{i,j}\beta(w_{j},u)v_{i}\otimes 1.$

The requirement that this be equal to $u\otimes 1$ implies that $\beta$ must
be nondegenerate, as claimed. Note that it also implies that
$\tau=\sum_{j}\varepsilon_{j}b_{j}\otimes b_{j}$, where $b_{1},\dots,b_{n}$ is
a generalized orthonormal basis for $\beta$; thus, $\epsilon$ is determined by
$\beta$.

We now want to show that $Z=Z_{V,\nabla,\beta}=\Phi(V,\nabla,\beta)$, where
$\Phi$ denotes the functor constructed in Proposition 4.4. We already know
that $Z=Z_{V,\nabla,\beta}$ on simplicial level zero and
$Z(B)=Z_{V,\nabla,\beta}(B)$ on all bordisms $B$ that are (tensor products of)
intervals and/or constant elbows. Using the techniques from the previous
section, we can introduce sitting instants into any non-constant elbow $B$ and
then express $B$ as the composition of two intervals and a constant elbow.
This determines the field theory on all intervals and elbows, as well as on
circles, since any circle can be decomposed into two elbows. Hence we have
$Z=\Phi(V,\nabla,\beta)$. ∎

###  4.5 The oriented case

We now briefly comment on the oriented case. The geometry considered here is
the one considered in Example 3.35, where manifolds are endowed with
orientations. In dimension one, the resulting bordism category will be denoted
$1\textrm{-}\mathrm{Bord}^{\mathrm{or}}(M)$ and we write
$1\textrm{-}\mathrm{TFT}^{\mathrm{or}}(M)$ for the corresponding groupoid of
field theories.

The main difference of $1\textrm{-}\mathrm{Bord}^{\mathrm{or}}(M)$ to the
unoriented bordism category is that there are now two different kinds of
points: Remember that a point is given as the zero set of the cut function
$\rho_{0}$ on a one-dimensional manifold (respectively a family of such). Now
because $\rho_{0}(x)=0$ for $x\in X$, the orientation allows to ask whether
${\rm d}\rho_{0}(x)$ (which is non-zero and hence a basis of $T_{x}X$) is
positively oriented or negatively oriented. This leads to _positive_
respectively _negative points_. Correspondingly, we have one more elementary
bordism, as displayed in Figure 5.

$+$$+$$-$$-$$+$$-$$-$$+$$t$$\tau_{0}$$\tau_{1}$ Figure 5: All possible
connected oriented bordisms. We call them positively and negatively oriented
intervals, left elbow, right elbow, and circle, respectively. The cut
functions are $\rho_{i}=t-\tau_{i}$, so these pictures are read from top to
bottom.

A field theory $Z\in 1\textrm{-}\mathrm{TFT}^{\mathrm{or}}$ will assign vector
spaces $V^{\pm}$ to the two different points. The elbow then provides a
pairing $V^{+}\otimes V^{-}\rightarrow\mathbb{C}$, which must be non-
degenerate due to the snake identity; hence $V^{-}$ can be identified with the
dual space of $V^{+}$ and vice versa. In particular, we do _not_ obtain the
additional datum of a nondegenerate bilinear form on our vector spaces.

Passing from ordinary field theories to oriented field theories over a target
$M$, the result is the following.

######  Theorem 4.24 (classification of $1$-TFTs, oriented case).

There is an equivalence of groupoids

$1\textrm{-}\mathrm{TFT}^{\mathrm{or}}(M)\cong\mathrm{Vect}^{\sim}_{\nabla}(M),$

which is natural in $M$.

Here $\mathrm{Vect}^{\sim}_{\nabla}(M)$ denotes the groupoid of vector bundles
with connection on $M$, with morphisms given by connection-preserving bundle
isomorphisms. The proof is analogous to that of Theorem 4.2.

##  5 Field theories with values in sheaves

In this section, we consider notions of “families of vector spaces” more
general than vector bundles. Let $\mathcal{V}$ denote some stack of
$C^{\infty}$-modules. This means that for each manifold $S$, objects of
$\mathcal{V}(S)$ are modules over $C^{\infty}(S)$, possibly with a bornology
or topology of a particular kind. Below, we will fix suitable conditions
$\mathcal{V}$ should satisfy, but we would like, at a minimum, that the
operation of taking global sections determines an embedding
$\mathrm{Vect}\to\mathcal{V}$, and a symmetric monoidal structure on
$\mathcal{V}$ compatible with this embedding. In line with Definition 3.36, we
write

$1\textrm{-}\mathrm{TFT}(M;\mathcal{V})=\operatorname{Fun}^{\otimes}(1\textrm{-}\mathrm{Bord}(M),\mathcal{V})$

for the groupoid of $1$-dimensional field theories over $M$ taking values in
$\mathcal{V}$, and similarly for the oriented variant. In the TFT case, it is
expected that this introduces no new examples, that is, a field theory with
values in $\mathcal{V}$ automatically takes values in $\mathrm{Vect}$. In this
section, we verify that this is indeed the case.

######  Definition 5.1.

We will call a symmetric monoidal smooth stack $\mathcal{V}$ an _admissible
stack of $C^{\infty}$-modules_ if the following holds.

  1. (C1)

For each $S$, $\mathcal{V}(S)$ is an additive category with kernels, naturally
in $S$, such that $\otimes$ is additive in both variables.

  2. (C2)

There is a linear symmetric monoidal fibered functor
$\mathrm{Vect}\to\mathcal{V}$ which determines, for each $S$, an equivalence
between $\mathrm{Vect}(S)$ and the dualizable objects of $\mathcal{V}(S)$.

  3. (C3)

Let $V\in\mathcal{V}(S\times\mathbb{R})$, $W\in\mathcal{V}(S)$, and
$\operatorname{pr}\colon S\times\mathbb{R}\to S$ be the projection. If
$\sigma\colon V\to\operatorname{pr}^{*}W$ is such that $\sigma|_{U}=0$ for
every open $U\subset S\times\mathbb{R}$ with
$U\cap(S\times\\{0\\})=\varnothing$, then $\sigma=0$.

Condition (C3) is a kind of separation axiom. It is often useful to regard a
morphism $W^{\prime}\to W$ in $\mathcal{V}(S)$ as a generalized element of
$W$. Then a morphism $\sigma\colon V\to\operatorname{pr}^{*}W$ as above can be
interpreted as a “generalized path of elements” of $W$, and the axiom says
that a generalized path vanishes altogether provided it vanishes on
$\mathbb{R}\setminus 0$.

In Section 5.2, we give examples of admissible stacks of $C^{\infty}$-modules.
Before doing this, we state and prove the following theorem, which contains
the previously mentioned result that the target does not matter for
topological field theories.

######  Theorem 5.2.

For any admissible stack of $C^{\infty}$-modules $\mathcal{V}$ and any
manifold $M$, the inclusions

$1\textrm{-}\mathrm{TFT}^{\mathrm{or}}(M)\to
1\textrm{-}\mathrm{TFT}^{\mathrm{or}}(M;\mathcal{V}),\qquad
1\textrm{-}\mathrm{TFT}(M)\to 1\textrm{-}\mathrm{TFT}(M;\mathcal{V})$

are equivalences of groupoids.

######  Remark 5.3.

Combining the above with Theorems 4.2 and 4.24 gives us equivalences of
groupoids

$1\textrm{-}\mathrm{TFT}(M;\mathcal{V})\cong\mathrm{Vect}_{\nabla,\beta}^{\sim}(M),\qquad
1\textrm{-}\mathrm{TFT}^{\mathrm{or}}(M;\mathcal{V})\cong\mathrm{Vect}_{\nabla}^{\sim}(M)$

for any admissible $\mathcal{V}$. This is the exact meaning we intend for
Theorems 1.1 and 1.2, stated in less detail in the introduction.

###  5.1 Proof of Theorem 5.2

We will focus on the unoriented case, the oriented one being similar.
Throughout, we fix a field theory $Z\colon
1\textrm{-}\mathrm{Bord}(M)\to\mathcal{V}$, which we assume to be strict as a
morphism of simplicial stacks.

######  Lemma 5.4.

Let $c\colon M\times\mathbb{R}\to M$ be the tautological $M$-family of
constant paths. Then the object $Z(c;0)\in\mathcal{V}$ is dualizable.

###### Proof.

For $\gamma\colon S\times\mathbb{R}\to M$ and $a<b$, we denote by
$L_{\gamma;a,b}$ the left elbow given by this data, that is, the $S$-family of
bordisms with underlying map $\gamma$ and cut functions
$\rho_{0}(s,t)=(t-a)(t-b)$ and $\rho_{1}(s,t)=-\frac{1}{4}(b-a)^{2}$, where
$t$ is the coordinate of $\mathbb{R}$ (here any $\rho_{1}$ can be chosen such
that it has no zeroes, and such that $\rho_{1}\leq\rho_{0}$ everywhere). We
write similarly $R_{\gamma;a,b}$ for the right elbow. To prove the lemma, we
consider in particular $S=M$ and $\gamma=c$; then for each $\delta>0$, the
images of $L_{c;0,\delta}$ and $R_{c;0,\delta}$ are canonically identified as
maps

$\displaystyle Z(L_{c;0,\delta})\colon\ Z(c;0)\otimes Z(c;0)\to\epsilon_{M},$
$\displaystyle Z(R_{c;0,\delta})\colon\ \epsilon_{M}\to Z(c;0)\otimes Z(c;0),$

where $\epsilon_{M}$ is the monoidal unit of $\mathcal{V}(M)$. We claim that
these morphisms are independent of $\delta$. To see this, pick any
diffeomorphism $\chi\colon\mathbb{R}\to\mathbb{R}$ with $\chi(0)=0$,
$\chi(1)=\delta$ which is affine of slope $1$ near $0$ and $1$. Then $\chi$
determines a morphism in $1\textrm{-}\mathrm{Bord}(M)_{1}$ between
$L_{c;0,\delta}$ and $L_{c\circ\chi;0,1}=L_{c;0,1}$, compatible with the usual
identifications of their boundary components with $(c;0)$. This proves the
claim.

Write $L=Z(L_{c;0,\delta})$, $R=Z(R_{c;0,\delta})$ for this common value.
Then, expressing the interval bordism $(c,0,\delta)$ as a suitable composition
of $L_{c;0,\delta}$ and $R_{c;0,\delta}$, we get

$(\mathrm{id}\otimes L)\circ(R\otimes\mathrm{id})=Z(c;0,\delta)$

for any $\delta>0$. Now consider, for instance, $(c,0,\delta^{2})$ as an
$(M\times\mathbb{R})$-family of intervals, where $\delta$ now denotes the
coordinate on the factor $\mathbb{R}$. Its image under $Z$ is a family of
endomorphisms of $Z(c;0)$ that is constant away from $\delta=0$, and equal to
the identity at $\delta=0$. Thus, from the separation axiom of $\mathcal{V}$,
$Z(c;0,\delta)=Z(c;0,0)=\mathrm{id}_{Z(c;0)}$ for all $\delta$. This proves
that $L$ and $R$ provide the desired evaluation and coevaluation maps. ∎

Recall our notation $\mathrm{Path}_{c}(M)\subset 1\textrm{-}\mathrm{Bord}(M)$
for the subcategory of paths with sitting instants (Definition 4.8). Using the
above lemma and axiom (C2) for $\mathcal{V}$, we find a factorization of
$Z|_{\mathrm{Path}_{c}(M)}$ through $\mathrm{Vect}\hookrightarrow\mathcal{V}$
(essentially uniquely). Proposition 4.16 then gives us a vector bundle with
connection $(V,\nabla)$ on $M$ and an isomorphism

$Z|_{\mathrm{Path}_{c}(M)}\cong Z_{V,\nabla}|_{\mathrm{Path}_{c}(M)}$

of smooth functors, where $Z_{V,\nabla}$ denotes the field theory determined
by parallel transport on $V$. We will eventually show that $Z\cong
Z_{V,\nabla,\beta}$ for some pairing $\beta$.

Lemma 4.22 is easily restated for families of bordisms and reproved for
$\mathcal{V}$-valued field theories, using the separation axiom. We now adapt
some other definitions and lemmas from Section 4 for which special care with
the family aspect is crucial. Let $\gamma\colon S\times\mathbb{R}\to M$ be an
$S$-family of paths and let $a\colon S\to\mathbb{R}$ be any smooth function,
so that we can talk about the $S$-family of objects $(\gamma;a)$ (where its
cut function is $\rho_{a}(s,t)=t-a(s)$). Note that by the implicit function
theorem, any cut function $\rho$ on $S\times\mathbb{R}$ is equivalent to some
$\rho_{a}$: Just take $a$ defined by $\rho(s,a(s))=0$. Similarly, two
functions $a\leq b\colon S\to\mathbb{R}$ determine a family of intervals
$(\gamma;a,b)$. These examples exhaust all possible isomorphism classes of
objects, respectively bordisms.

######  Definition 5.5.

Given functions $a,b\colon S\to\mathbb{R}$ with $a<b$, a _left modification
function_ is a smooth map $\chi\colon S\times\mathbb{R}\to S\times\mathbb{R}$
of the form $\chi(s,t)=(s,\tilde{\chi}(s,t))$ for some $\tilde{\chi}\colon
S\times\mathbb{R}\to\mathbb{R}$ such that

  1. (1)

for each $s\in S$, $t\mapsto\tilde{\chi}(s,t)$ is nondecreasing,

  2. (2)

there exists a neighborhood of $\\{t=b\\}\subset S\times\mathbb{R}$ where
$\chi$ equals the identity,

  3. (3)

there exists a neighborhood of $\\{t=a\\}$, where $\tilde{\chi}(s,t)=a(s)$.

We define _right_ modification functions by swapping the roles of $a$ and $b$,
and _two-sided_ modification functions by requiring fiberwise constancy near
both ends.

######  Lemma 5.6.

For any $a>b\colon S\times\mathbb{R}\to\mathbb{R}$, there exist left, right
and two-sided modification functions on $S\times\mathbb{R}$.

###### Proof.

We can certainly find a nonnegative smooth function $f\colon
S\times\mathbb{R}\to\mathbb{R}$ with support in the open set $\\{(s,t)\mid
a(s)<t<b(s)\\}\subset S\times\mathbb{R}$ and such that
$f|_{\\{s\\}\times\mathbb{R}}$ is not identically zero for any $s\in S$.
Setting

$g(s,t)=\frac{\int_{-\infty}^{t}f(s,u)\,\mathrm{d}u}{\int_{-\infty}^{+\infty}f(s,u)\,\mathrm{d}u}$

gives us a smooth, fiberwise nondecreasing function which is identically $0$
in a neighborhood of $\\{t\leq a\\}$ and identically $1$ in a neighborhood of
$\\{t\geq b\\}$. Then

$\chi_{LR}(s,t)=(s,a(s)+(b(s)-a(s))g(s,t))$

is a two-sided modification function. Next, write

$\kappa(s)=\frac{1}{\int_{-\infty}^{+\infty}f(s,u)\,\mathrm{d}u}\bigg{(}b(s)-a(s)-\int_{-\infty}^{b(s)}g(s,u)\,\mathrm{d}u\bigg{)}.$

Due to the bounds on $g$, this is nonnegative. Finally, set

$h_{s}(t)=a(s)+\int_{-\infty}^{t}(\kappa(s)f(s,u)+g(s,u))\,\mathrm{d}u.$

Clearly, $h_{s}$ is nondecreasing, constant near $t=a(s)$, and affine with
slope $1$ near $t=b(s)$. Moreover, by the choice of $\kappa(s)$, we have
$h_{s}(a(s))=a(s)$ and $h_{s}(b(s))=b(s)$. Thus,

$\chi_{L}(s,t)=(s,h_{s}(t))$

defines a left modification function. Right modification functions can be
constructed similarly. ∎

######  Lemma 5.7.

Let $\gamma\colon S\times\mathbb{R}\to M$ be an $S$-family of paths, fix
smooth functions $a<b\colon S\to\mathbb{R}$, and let $\chi_{0}$, $\chi_{1}$ be
left modification functions. Then
$Z(\gamma\circ\chi_{0};a,b)=Z(\gamma\circ\chi_{1};a,b)$. The same holds for
right modifications and two-sided modifications.

Note that all modification functions agree in a neighborhood of the
boundaries, so the domain and codomain of the maps $Z(\gamma\circ\chi,a,b)$
are canonically identified, as in Remark 4.15.

###### Proof.

The case of two-sided modifications follows immediately from Lemma 4.23, since
$\mathrm{Vect}$ is full in $\mathcal{V}$, so that the maps
$Z(\gamma\circ\chi_{i},a,b)$ are identified with maps of vector bundles.
Suppose now that $\chi_{0}$, $\chi_{1}$ are left modification functions. Let
$U$ be a neighborhood of $\\{t=b\\}$, where $\chi_{0}$ and $\chi_{1}$ both
agree with the identity. Then we can find a function $a^{\prime}$ with
$a>a^{\prime}>b$ and $\\{t=a^{\prime}\\}\subset U$, as well as a map $F\colon
S\times\mathbb{R}\to S\times\mathbb{R}$ over $S$, fiberwise nondecreasing,
which is the identity away from $U$ and near $\\{b=0\\}$, and fiberwise
constant on $\\{t=a^{\prime}\\}$. Thus, using Lemma 4.22, we get

$Z(\gamma\circ\chi_{i},a,b)=Z(\gamma\circ\chi_{i}\circ F,a,b)=Z(\gamma\circ
F,a^{\prime},b)\circ Z(\gamma\circ\chi_{i}\circ F,a,a^{\prime})$

for both $i=0,1$. Finally, notice that

$Z(\gamma\circ\chi_{0}\circ F,a,a^{\prime})=Z(\gamma\circ\chi_{1}\circ
F,a,a^{\prime})$

by the two-sided version of this lemma. This finishes the proof of the left-
sided case; the right-sided case is similar. ∎

######  Lemma 5.8.

The field theory $Z$ determines a stack map $Z^{\prime}\colon
1\textrm{-}\mathrm{Bord}(M)_{0}\to\mathcal{V}$ and a distinguished
decomposition

$Z(\gamma;a)=V_{\gamma(a)}\oplus Z^{\prime}(\gamma;a)$

for each $S$-family of objects $(\gamma,a)$, where $V_{\gamma(a)}$ is the
pullback of the vector bundle $V\to M$ via the map $\gamma(a)\colon S\to M$.

Above, we used the suggestive notation $\gamma(a)$ for the map
$s\mapsto\gamma(s,a(s))$. Note that $V_{\gamma(a)}\cong Z(\gamma(a),a)$ is
what $Z$ assigns to the family of constant germs with the same value as
$\gamma$ on $\\{t=a\\}$.

###### Proof.

Choose smooth functions $a^{\prime},a^{\prime\prime}\colon S\to\mathbb{R}$
with $a^{\prime}<a<a^{\prime\prime}$ and smooth, fiberwise nondecreasing
functions $\chi_{1},\chi_{2}\colon S\times\mathbb{R}\to S\times\mathbb{R}$
such that $\chi_{1}$ is fiberwise constant near $a^{\prime}$ and
$a^{\prime\prime}$ and equal to the identity near $a$, while $\chi_{2}$ is
fiberwise constant near $a^{\prime}$, $a$ and $a^{\prime\prime}$. In other
words, $\chi_{1}$ is a left modification function for $(\gamma;a^{\prime},a)$
and a right modification function for $(\gamma;a,a^{\prime\prime})$, while
$\chi_{2}$ is a two-sided modification function for both
$(\gamma;a^{\prime},a)$ and $(\gamma;a,a^{\prime\prime})$. In particular, both
$\chi_{1}$ and $\chi_{2}$ are two-sided modification functions for
$(\gamma;a^{\prime},a^{\prime\prime})$. The existence of such functions is
clear from Lemma 5.6 and the fact that modification functions can be “patched
together” in the obvious way for neighboring intervals. We obtain the
following diagram of bordisms:

${(\gamma;a)}$${\bigl{(}\gamma(a^{\prime});a^{\prime}\bigr{)}}$${\bigl{(}\gamma(a^{\prime\prime});a^{\prime\prime}\bigr{)}}$${(\gamma(a);a)}$$\scriptstyle{(\gamma\circ\chi_{1};\,a,\,a^{\prime\prime})}$$\scriptstyle{(\gamma\circ\chi_{1};\,a^{\prime}\\!,\,a)}$$\scriptstyle{(\gamma\circ\chi_{2};\,a^{\prime}\\!,\,a)}$$\scriptstyle{(\gamma\circ\chi_{2};\,a,\,a^{\prime\prime})}$

It certainly does not commute in the sense of there being an isomorphism
between the two compositions: the bottom one has a sitting instant at $a$,
while in general the top one does not. However, since the composition of the
upper two bordisms is $(\gamma\circ\chi_{1};a^{\prime},a^{\prime\prime})$
while the lower composition is
$(\gamma\circ\chi_{2};a^{\prime},a^{\prime\prime})$, Lemma 5.7 implies that
the diagram obtained after applying $Z$

${Z(\gamma;a)}$${Z\bigl{(}\gamma(a^{\prime});a^{\prime}\bigr{)}}$${Z\bigl{(}\gamma(a^{\prime\prime});a^{\prime\prime}\bigr{)}}$${Z(\gamma(a);a)}$$\scriptstyle{p_{a}}$$\scriptstyle{Z(\gamma\circ\chi_{1};\,a,\,a^{\prime\prime})}$$\scriptstyle{Z(\gamma\circ\chi_{1};\,a^{\prime}\\!,\,a)}$$\scriptstyle{Z(\gamma\circ\chi_{2};\,a^{\prime}\\!,\,a)}$$\scriptstyle{i_{a}}$$\scriptstyle{Z(\gamma\circ\chi_{2};\,a,\,a^{\prime\prime})}$
(5.1)

commutes. We now define maps $i_{a}$ and $p_{a}$ as indicated in the diagram
by requiring that they make the left, respectively right, triangle commute.
These maps exist and are well-defined as the bottom maps are invertible, by
Lemma 4.17. By commutativity of the diagram, we have $p_{a}i_{a}=\mathrm{id}$.

We now set $Z^{\prime}(\gamma;a)=\operatorname{Ker}p_{a}$. By standard
arguments, $i_{a}$ and the inclusion map $Z^{\prime}(\gamma;a)\to Z(\gamma;a)$
exhibit $Z(\gamma;a)$ as the direct sum

$Z(\gamma;a)\cong V_{\gamma(a)}\oplus Z^{\prime}(\gamma;a).$

This identification does not depend on any of the choices by Lemma 5.7. ∎

######  Lemma 5.9.

For any nowhere thin $S$-family of bordisms $(\gamma;a,b)$, by which we mean
that $a(s)<b(s)$ for all $s\in S$, the map $Z(\gamma;a,b)$ has matrix
representation

$Z(\gamma;a,b)=\begin{pmatrix}P(\gamma;a,b)&0\\\ 0&0\end{pmatrix}$

with respect to the direct sum decompositions of $Z(\gamma;a)$ and
$Z(\gamma;b)$ from Lemma 5.8.

###### Proof.

Choose $a^{\prime}<a<a^{\prime\prime}<b^{\prime}<b<b^{\prime\prime}$ (as
functions $S\to\mathbb{R}$) and consider the commutative diagram

${Z(\gamma;a)}$${Z(\gamma;b)}$${V_{\gamma(a^{\prime})}}$${V_{\gamma(a^{\prime\prime})}}$${V_{\gamma(b^{\prime})}}$${V_{\gamma(b^{\prime\prime})}}$${V_{\gamma(a)}}$${V_{\gamma(b)}}$$\scriptstyle{Z(\gamma;\,a,\,b)}$$\scriptstyle{p_{a}}$$\scriptstyle{p_{b}}$$\scriptstyle{i_{a}}$$\scriptstyle{i_{b}}$

obtained by patching together the corresponding diagrams (5.1) in the obvious
way. In particular, all bottom maps are parallel translations along segments
of $\gamma$.

To prove that the second columns of the matrix representing $Z(\gamma;a,b)$ is
zero, we need to show that this map, restricted to the kernel of $p_{a}$, is
zero. This follows from the diagram, since it shows that $Z(\gamma;a,b)$
factors through $p_{a}$.

A diagram chase shows that the square composed by $Z(\gamma;a,b)$, $i_{a}$,
$i_{b}$, and the parallel translation $P\colon V_{\gamma(a)}\to V_{\gamma(b)}$
commutes. This implies that the first column of $Z(\gamma;a,b)$ is as claimed.
∎

The proof of Theorem 5.2 concludes with the next lemma.

######  Lemma 5.10.

For any $S$-family $(\gamma;a)$, $Z^{\prime}(\gamma;a)=0\in\mathcal{V}(S)$.

###### Proof.

Let $\operatorname{pr}\colon S\times\mathbb{R}\to S$ be the projection and let
$\delta\colon S\times\mathbb{R}\rightarrow\mathbb{R}$ be the coordinate
function on the $\mathbb{R}$-factor. Consider the $(S\times\mathbb{R})$-family
of bordisms
$B=\big{(}(\mathrm{pr}\times\mathrm{id})^{*}{\gamma};\mathrm{pr}^{*}a-\delta^{2},\mathrm{pr}^{*}a\big{)}$.
Then

$Z^{\prime}(B)\colon\
Z^{\prime}\big{(}\operatorname{pr}^{*}\gamma,a-\delta^{2}\big{)}\to
Z^{\prime}(\operatorname{pr}^{*}\gamma,a)\cong\operatorname{pr}^{*}(Z^{\prime}(\gamma,a))$

vanishes away from $S\times 0$ by the previous lemma, and therefore is
identically $0$ by the separation axiom. But this implies that

$\mathrm{id}_{Z^{\prime}(\gamma;a)}=Z^{\prime}(B)|_{S\times 0}=0,$

so $Z^{\prime}(\gamma;a)$ is the zero object of $\mathcal{V}(S)$. ∎

###  5.2 Examples of admissible stacks of $\boldsymbol{C^{\infty}}$-modules

In this section, we give two examples of suitable target stacks $\mathcal{V}$
for TFTs:

  1. 1.

The stack $\mathcal{V}_{\mathrm{alg}}$ of sheaves of $C^{\infty}$-modules with
the algebraic tensor product.

  2. 2.

The stack $\mathcal{V}_{\mathrm{vN}}$ of sheaves of complete bornological
$C^{\infty}$-modules with the completed bornological tensor product,
constructed in Appendix A.5 and briefly reviewed below. Here, $C^{\infty}(S)$
is endowed with its von Neumann bornology (see Example A.2), hence the
notation.

Recall that a _bornology_ on a vector space is a collection of subsets deemed
to be bounded and satisfying appropriate axioms. Bornological vector spaces
and bounded linear maps form a category $\mathrm{Born}$. There is an
appropriate notion of completeness, and thus a full subcategory
$\mathrm{CBorn}\subset\mathrm{Born}$ of _complete_ bornological vector spaces.
We are interested in $\mathrm{CBorn}$ for its pleasant categorical properties.
In particular, there is a completed tensor product $\mathbin{\hat{\otimes}}$
which makes $\mathrm{CBorn}$ into a closed symmetric monoidal category. As is
well known, none of the many possible tensor products on the various
categories of topological vector spaces have this property. See Appendices A.1
to A.3 for a quick overview, or Meyer [8, Chapter 1] for a comprehensive
introduction to the theory of bornological vector spaces.

The goal of this subsection, which will follow immediately by combining
Propositions 5.17 and 5.19 and Corollaries 5.18 and 5.20 below, is to prove
the following.

######  Theorem 5.11.

$\mathcal{V}_{\mathrm{alg}}$ and $\mathcal{V}_{\mathrm{vN}}$ are admissible
stacks of $C^{\infty}$-modules.

######  Remark 5.12.

The purely algebraic $\mathcal{V}_{\mathrm{alg}}$ is familiar and attractive
in its simplicity, but it has the drawback of not admitting a reasonable
“sheaf of sections” functor from any category of infinite-dimensional vector
bundles. Moreover, our proof that it satisfies the separation axiom requires
the corresponding fact for $\mathcal{V}_{\mathrm{vN}}$.

Before proceeding, let us briefly review the definitions of
$\mathcal{V}_{\mathrm{alg}}$ and $\mathcal{V}_{\mathrm{vN}}$. It will be
convenient to use the sheaf-of-categories approach to stacks.

We define
$\mathcal{V}_{\mathrm{alg}}\colon\mathrm{Man}^{\mathrm{op}}\to\mathrm{Cat}^{\otimes}$
to be the functor assigning to each manifold $S$ the symmetric monoidal
category of sheaves of $C^{\infty}_{S}$-modules and $C^{\infty}_{S}$-linear
maps, with the algebraic tensor product. To a map $f\colon T\to S$ is
associated the pullback functor $f^{*}$ defined by

$f^{*}V=C^{\infty}_{T}\otimes_{f^{-1}(C^{\infty}_{S})}f^{-1}(V).$ (5.2)

That this indeed defines a symmetric monoidal stack on the site of manifolds
and satisfies axiom (C1) is a standard fact.

Similar constructions can be carried out in the setting of (complete)
bornological vector spaces. This is outlined in Appendices A.4 and A.5 and is
mostly a formality, except for the fact that sheafification of presheaves with
values in $\mathrm{CBorn}$ requires additional care. In a nutshell, we promote
the sheaf of smooth functions on a manifold $S$ to a sheaf of complete
bornological algebras, denoted by $C^{\infty}_{\mathrm{vN},S}$, and then
define $\mathcal{V}_{\mathrm{vN}}(S)$ to be the category of sheaves of
complete bornological $C^{\infty}_{\mathrm{vN},S}$-modules. We have a
completed tensor product of bornological sheaves, and the pullback functors
$f^{*}\colon\mathcal{V}_{\mathrm{vN}}(S)\to\mathcal{V}_{\mathrm{vN}}(T)$ are
defined similarly to (5.2), using this completed tensor product. Thus
$\mathcal{V}_{\mathrm{vN}}$ is a symmetric monoidal stack satisfying axiom
(C1).

Next, we construct a functor
$i\colon\mathcal{V}_{\mathrm{alg}}\to\mathcal{V}_{\mathrm{vN}}$. Let $S$ be a
manifold and $V\in\mathcal{V}_{\mathrm{alg}}(S)$. Given $U\subset S$ open, we
say that a subset $B\subset V(U)$ is $\mathrm{vN}$-bounded if each point $x\in
U$ has an open neighborhood $W\subset U$ such that

$B|_{W}\subset C_{1}b_{1}+\dots+C_{n}b_{n}$

for $\mathrm{vN}$-bounded subsets $C_{1},\dots,C_{n}\subset C^{\infty}(U)$ and
$b_{1},\dots,b_{n}\in V(W)$. It is routine to check that this indeed satisfies
the axioms of a (vector) bornology on $V(U)$. We denote the resulting presheaf
of bornological vector spaces by $V_{\mathrm{vN}}$.

######  Example 5.13.

If $V=\Gamma_{E}$ is the sheaf of sections of a finite-dimensional vector
bundle over $E\to S$, then the bornology constructed above coincides with the
bornology associated to its Fréchet topology, where a set of sections is
bounded if all derivatives of a given order are uniformly bounded on compact
sets.

######  Proposition 5.14.

For any sheaf $V$ of $C^{\infty}_{S}$-modules, $V_{\mathrm{vN}}$ is a sheaf of
complete bornological $C^{\infty}_{\mathrm{vN},S}$-modules.

###### Proof.

It follows directly from the definition of the $\mathrm{vN}$-bornology that
for each open $U\subseteq S$, the bornological vector space
$V_{\mathrm{vN}}(U)$ is a bornological $C^{\infty}_{\mathrm{vN}}(U)$-module
(i.e., the module action map is bounded), and that restriction maps are
bounded. We have to show that $V_{\mathrm{vN}}$ satisfies descent. Let
$U\subset S$ be open and let $\\{U_{i}\\}_{i\in I}$ be an open cover of $U$.
Since $V$ is an (algebraic) sheaf, we know that the map

$V_{\mathrm{vN}}(U)\stackrel{{\scriptstyle
r}}{{\longrightarrow}}\operatorname{Ker}\bigg{(}\prod\limits_{i\in
I}V_{\mathrm{vN}}(U_{i})\longrightarrow\prod\limits_{i,j\in
I}V_{\mathrm{vN}}(U_{i}\cap U_{j})\bigg{)}$

is a vector space isomorphism. We have to show that it is an isomorphism of
bornological vector spaces. Since we already know that restriction maps are
bounded, we have to show that the inverse image under $r$ of bounded sets in
$\prod_{i\in I}V_{\mathrm{vN}}(U_{i})$ is bounded. It suffices to show this
for sets of the form $\prod_{i\in I}B_{i}$ with $B_{i}\subset
V_{\mathrm{vN}}(U_{i})$ bounded, as these generate the product bornology. Here
we have

$r^{-1}\bigg{(}\prod\limits_{i\in I}B_{i}\bigg{)}=\\{v\in V(U)\mid
v|_{U_{i}}\in B_{i}\\}.$

That this is $\mathrm{vN}$-bounded follows again directly from the definition
of the bornology, as boundedness can be checked locally. This shows that
$V_{\mathrm{vN}}$ is a sheaf of bornological
$C^{\infty}_{\mathrm{vN},S}$-modules.

We now show that $V_{\mathrm{vN}}(U)$ is complete for each open $U\subseteq
S$. To this end, we have to show that each bounded subset of
$V_{\mathrm{vN}}(U)$ is contained in a complete bounded disc. Here a bounded
disc is complete if the normed space $V(U)_{B}=\mathrm{span}(B)\subset V(U)$
is complete, where the norm is the Minkowski functional defined by $B$. Let
$\\{W_{i}\\}_{i\in I}$ be an open cover of $U$, let
$b_{1}^{i},\dots,b_{n_{i}}^{i}\in V(W_{i})$ and
$C_{1}^{i},\dots,C_{n_{i}}^{i}\subset C^{\infty}(W_{i})$ be complete
$\mathrm{vN}$-bounded discs. Then the sets
$B_{i}:=C_{1}^{i}b_{1}^{i}+\dots+C_{n_{i}}^{i}b_{n_{i}}^{i}\subset V(W_{i})$
and

$B:=\big{\\{}v\in V(U)\mid v|_{W_{i}}\in B_{i}\big{\\}}$

are $\mathrm{vN}$-bounded discs. Clearly, any $\mathrm{vN}$-bounded set in
$V(U)$ is contained in such a disc, for some choice of $\\{W_{i}\\}$,
$\\{b_{i}\\}$ and $\\{C_{k}^{i}\\}$, hence it suffices to show that $V(U)_{B}$
is a Banach space.

We first show that each of the normed spaces $V(W_{i})_{B_{i}}$ are complete.
To this end, consider the linear map

$\pi\colon\quad X\longrightarrow V(W_{i})_{B_{i}},\qquad
X:=\prod_{k=1}^{n_{i}}C^{\infty}(W_{i})_{C_{k}^{i}},$

sending $(f_{1},\dots,f_{n_{i}})$ to $f_{1}b_{1}+\dots+f_{n_{i}}b_{n_{i}}$. It
is surjective by definition of $B_{i}$, and it is bounded since any bounded
set $C\subset X$ is contained in $\lambda(C_{1}^{i}\times\cdots\times
C_{n_{i}}^{i})$ for some $\lambda>0$, hence
$\pi(C)\subset\lambda\pi(C_{1}^{i}\times\cdots\times C_{n_{i}}^{i})=\lambda
B$. Notice that $X$ is a Banach space as the discs $C_{k}^{i}$ are complete.
Because $\pi$ is bounded, its kernel is closed, so that the quotient
$X/\operatorname{Ker}(\pi)$ is a Banach space. The quotient map
$\bar{\pi}\colon X/\operatorname{Ker}(\pi)\to V(W_{i})$ is a bounded vector
space isomorphism. To see that it is a homeomorphism, it remains to show that
the inverse is bounded. To this end, let $v\in V(W_{i})_{B_{i}}$. Let
$\lambda>0$ be such that $v\in\lambda B_{i}$. Because
$\pi\big{(}C_{1}^{i}\times\cdots\times C_{n_{i}}^{i}\big{)}=B_{i}$, we can
choose a preimage $v^{\prime}\in\lambda(C_{1}^{i}\times\cdots\times
C_{n_{i}}^{i})\subset X$. Then

$\displaystyle\big{\|}\bar{\pi}^{-1}(v)\big{\|}_{X/\operatorname{Ker}(\pi)}$
$\displaystyle=\inf_{w\in\operatorname{Ker}(\pi)}\|v^{\prime}+w\|_{X}\leq\|v^{\prime}\|_{X}$
$\displaystyle=\inf\big{\\{}\mu>0\mid
v^{\prime}\in\mu\big{(}C_{1}^{i}\times\cdots\times
C_{n_{i}}^{i}\big{)}\big{\\}}\leq\lambda.$

This holds for any $\lambda$ such that $v\in\lambda B_{i}$. Therefore, we get

$\big{\|}\bar{\pi}^{-1}(v)\big{\|}_{X/\operatorname{Ker}(\pi)}\leq\inf\\{\lambda>0\mid
v\in\lambda B_{i}\\}=\|v\|_{B_{i}}.$

This shows that $\bar{\pi}^{-1}$ is bounded, hence $\bar{\pi}\colon
X/\operatorname{Ker}(\pi)\to V(W_{i})_{B_{i}}$ is a homeomorphism. Since
$X/\operatorname{Ker}(\pi)$ is complete, so is $V(W_{i})_{B_{i}}$.

Now consider the map

$V(U)_{B}\stackrel{{\scriptstyle r}}{{\longrightarrow}}\prod_{i\in
I}V(W_{i})_{B_{i}},\qquad v\mapsto(v|_{W_{i}})_{i\in I},$

where the right hand side is a product of Banach spaces, endowed with the norm
obtained by taking the supremum over all the norms of the $V(W_{i})_{B_{i}}$.
In other words, the unit ball of the product is $\prod_{i\in I}B_{i}$. Observe
that this map is well-defined since if $v\in\lambda B$, then
$v|_{W_{i}}\in\lambda B_{i}$ and $(v|_{W_{i}})\in\lambda\prod_{i\in I}B_{i}$.
The map is bornologically proper, since $r^{-1}\big{(}\prod_{i\in
I}B_{i}\big{)}=B$. On the other hand, its image is the kernel of the bounded
map

$\prod_{i\in I}V(W_{i})_{B_{i}}\longrightarrow\prod_{i,j\in I}V(W_{i}\cap
W_{j})_{B_{ij}},\qquad(v_{i})_{i\in I}\longmapsto(v_{i}|_{W_{i}\cap
W_{j}}-v_{j}|_{W_{i}\cap W_{j}})_{i,j\in I},$

hence closed. Here $B_{ij}=B_{i}|_{W_{i}\cap W_{j}}+B_{j}|_{W_{i}\cap W_{j}}$.
This shows that $V(U)_{B}$ is isomorphic to a closed subspace of a Banach
space, hence complete. ∎

To promote the assignment $V\mapsto V_{\mathrm{vN}}$ to a functor, we have to
deal with morphisms. Here we have the following lemma.

######  Lemma 5.15.

Let $V$ and $V^{\prime}$ be $($algebraic$)$ $C^{\infty}_{S}$-modules and let
$\Phi\colon V\to V^{\prime}$ be a morphism of sheaves. Then, for each
$U\subseteq S$, $\Phi_{U}\colon V(U)\to V^{\prime}(U)$ is
$\mathrm{vN}$-bounded.

###### Proof.

Let $B\subset V(U)$ be $\mathrm{vN}$-bounded. We have to show that also
$\Phi_{U}(B)$ is $\mathrm{vN}$-bounded. To this end let $x\in U$ and choose
$W$ such that $B|_{W}\subseteq C_{1}b_{1}+\cdots+C_{n}b_{n}$ for elements
$b_{1},\dots,b_{n}\allowbreak\in V(W)$ and $\mathrm{vN}$-bounded subsets
$C_{1},\dots,C_{n}\subset C^{\infty}(W)$. Then since $\Phi$ is a morphism of
sheaves and $C^{\infty}_{S}$-linear,

$\displaystyle\Phi_{U}(B)=\Phi_{W}(B|_{W})\subseteq\Phi(C_{1}b_{1}+\cdots+C_{n}b_{n})=C_{1}\Phi(b_{1})+\cdots+C_{n}\Phi(b_{n}).$

Hence $\Phi_{U}(B)$ is $\mathrm{vN}$-bounded. ∎

We therefore obtain a functor
$i\colon\mathcal{V}_{\mathrm{alg}}\to\mathcal{V}_{\mathrm{vN}}$ sending $V$ to
$V_{\mathrm{vN}}$ and which sends morphisms $V\to V^{\prime}$ of sheaves to
the corresponding morphism $V_{\mathrm{vN}}\to V^{\prime}_{\mathrm{vN}}$,
which exists by the previous lemma. Since conversely, each morphism of sheaves
$V_{\mathrm{vN}}\to V^{\prime}_{\mathrm{vN}}$ gives a morphism of the
underlying algebraic sheaves of $C^{\infty}_{S}$-modules, we obtain the
following corollary.

######  Corollary 5.16.

The functor $i\colon\mathcal{V}_{\mathrm{alg}}\to\mathcal{V}_{\mathrm{vN}}$
described above is fully faithful.

######  Proposition 5.17.

$\mathcal{V}_{\mathrm{alg}}$ satisfies axiom (C2).

###### Proof.

For each $S$, there is a “sheaf of sections” functor
$\Gamma\colon\mathrm{Vect}(S)\to\mathcal{V}_{\mathrm{alg}}(S)$. To see that
these fit together to a map of stacks, we need to show that for each vector
bundle $V\in\mathrm{Vect}(S)$ and each map $f\colon T\to S$, the natural maps
$f^{*}(\Gamma_{V})\to\Gamma_{f^{*}V}$ are isomorphisms. On the stalk at $x\in
T$, this yields a map

$f^{*}(\Gamma_{V})_{x}\cong
C^{\infty}_{T,x}\otimes_{C^{\infty}_{S,f(x)}}\Gamma_{V,f(x)}\to\Gamma_{f^{*}V,x}$

compatible with the identifications $\Gamma_{V,f(x)}\cong
C^{\infty}_{S,f(x)}\otimes V_{f(x)}$ and $\Gamma_{f^{*}V,x}\cong
C^{\infty}_{T,x}\otimes V_{f(x)}$. Here, $V_{f(x)}$ is the fiber of the vector
bundle $V$; all other subscripts denote stalks at the given point. Thus, the
above is an isomorphism on stalks and hence an isomorphism of sheaves. That
$\Gamma$ has all dualizable objects as essential image is the content of the
Serre–Swan theorem. ∎

######  Corollary 5.18.

$\mathcal{V}_{\mathrm{vN}}$ satisfies axiom (C2).

###### Proof.

Composition with $i$ gives the map
$\Gamma\colon\mathrm{Vect}\to\mathcal{V}_{\mathrm{vN}}$. Since every
dualizable $C^{\infty}_{\mathrm{vN},S}$-module has finite rank, any dualizable
object in $\mathcal{V}_{\mathrm{vN}}(S)$ is already dualizable in
$\mathcal{V}_{\mathrm{alg}}(S)$. ∎

######  Proposition 5.19.

$\mathcal{V}_{\mathrm{vN}}$ satisfies the separation axiom (C3).

###### Proof.

Fix a $C^{\infty}_{\mathrm{vN},S}$-module $W$, and denote by $V$ its pullback
to $S\times\mathbb{R}$. It suffices to show that if a section $\sigma\in V(U)$
is such that $\sigma|_{U\setminus(S\times 0)}$ vanishes, then in fact
$\sigma=0$. By definition, $V$ is the sheafification of the presheaf

$V^{\prime}\colon\ U\mapsto
C^{\infty}_{\mathrm{vN}}(U)\mathbin{\hat{\otimes}}_{\left(\operatorname{pr}^{-1}C^{\infty}_{\mathrm{vN},S}\right)(U)}\big{(}\operatorname{pr}^{-1}W\big{)}(U),$

so we know that $\sigma$ is determined by a coherent collection of sections
$\sigma_{i}\in V^{\prime}(S_{i}\times T_{i})$ for some collection of opens
$S_{i}\subset S$, $T_{i}\subset\mathbb{R}$, since products $S_{i}\times T_{i}$
form a basis for the topology of $U\subset S\times\mathbb{R}$. Our question is
reduced to showing that if $\sigma_{i}$ vanishes on
$S_{i}\times(T_{i}\setminus 0)$, then it vanishes on $S_{i}\times T_{i}$. Now,

$\displaystyle V^{\prime}(S_{i}\times T_{i})$ $\displaystyle\cong
C^{\infty}_{\mathrm{vN}}(T_{i}\times
S_{i})\mathbin{\hat{\otimes}}_{C^{\infty}_{\mathrm{vN}}(S_{i})}V(S_{i})$
$\displaystyle\cong(C^{\infty}(T_{i})\mathbin{\hat{\otimes}}C^{\infty}(S_{i}))\mathbin{\hat{\otimes}}_{C^{\infty}(S_{i})}V(S_{i})$
$\displaystyle\cong
C^{\infty}_{\mathrm{vN}}(T_{i})\mathbin{\hat{\otimes}}V(S_{i})$
$\displaystyle\cong C^{\infty}_{\mathrm{vN}}(T_{i};V(S_{i})).$

The first identification is by definition, the second follows from Proposition
A.14, the third from the fact that
$C^{\infty}(T_{i})\mathbin{\hat{\otimes}}\textrm{---}$ commutes with the
colimit defining the tensor product over $C^{\infty}(S_{i})$, and the fourth
from Proposition A.13. Thus, $\sigma_{i}$ gets identified with a smooth
function $f\colon T_{i}\to V(S_{i})$ which, by definition, is a smooth
function with values in some Banach space $V_{B}\subset V(S_{i})$ (cf.
Appendix A.3). Hence it vanishes identically if it vanishes away from $0$. ∎

######  Corollary 5.20.

$\mathcal{V}_{\mathrm{alg}}$ satisfies the separation axiom (C3).

######  Remark 5.21.

One might wish for a more elementary proof of the corollary, and we would like
to note that the bornological tensor product is handy in this case as well.
Arguing as in the proof of Proposition 5.19, one reduces property (C3) for
$\mathcal{V}_{\mathrm{alg}}$ to the claim that if $V$ is a
$C^{\infty}(S)$-module and

$\sigma\in C^{\infty}(\mathbb{R}\times S)\otimes_{C^{\infty}(S)}V$

maps to $0$ in $C^{\infty}((\mathbb{R}\setminus 0)\times
S)\otimes_{C^{\infty}(S)}V$, then $\sigma=0$. This is not immediately clear.
Notice, however, that if we give $V$ the fine bornology, Proposition A.11
allows us to regard $\sigma$ as an element of

$C^{\infty}_{\mathrm{vN}}(\mathbb{R}\times
S)\mathbin{\hat{\otimes}}_{C^{\infty}(S)}V\cong
C^{\infty}_{\mathrm{vN}}(\mathbb{R})\mathbin{\hat{\otimes}}C^{\infty}_{\mathrm{vN}}(S)\mathbin{\hat{\otimes}}_{C^{\infty}(S)}V,$

and therefore as a smooth function from $\mathbb{R}$ to some Banach space. The
claim follows.

##  Appendix A Bornological sheaves

In this appendix, we construct the symmetric monoidal stack
$\mathcal{V}_{\mathrm{vN}}$ of sheaves of complete bornological
$C^{\infty}$-modules on the site $\mathrm{Man}$ of smooth manifolds. Formally,
this construction is very similar to that of its well-known algebraic
counterpart $\mathcal{V}_{\mathrm{alg}}$, and our main goal here is to
highlight the differences. We start recalling several basic facts about
bornological vector spaces, providing proofs for the bits that are not easily
located in the literature. For this, our main reference is Meyer [8, Chapter
1]. The basic facts about sheaves of (complete) bornological vector spaces and
modules are quoted from Houzel [5].

###  A.1 Bornological vector spaces

A (convex) _bornology_ on a vector space $V$ over $\mathbb{K}=\mathbb{R}$ or
$\mathbb{C}$ is a collection $\mathfrak{S}$ of subsets of $V$ deemed to be
bounded. These have to satisfy appropriate axioms which we will not repeat
here. For any collection $\mathfrak{S}^{\prime}$ of subsets of $V$, there is a
smallest bornology containing $\mathfrak{S}^{\prime}$, the bornology
_generated by_ $\mathfrak{S}^{\prime}$. A linear map $f\colon V\to W$ is
_bounded_ if it sends bounded subsets to bounded subsets.

A _disk_ in a vector space $V$ is a convex, balanced subset $B\subset V$. We
denote by $V_{B}=\mathbb{R}\cdot B\subset V$ the subspace spanned by $B$. The
closed ball of a seminorm on $V_{B}$ is an absorbing disk, and conversely a
disk $B\subset V$ determines a unique seminorm on $V_{B}$. We say that $B$ is
_norming_ respectively _complete_ if $V_{B}$ is a normed respectively a Banach
space. A bornological vector space is _separated_ if every bounded disk is
norming, and _complete_ if every bounded subset is contained in a complete
bounded disk.

######  Example A.1.

The smallest possible bornology on a vector space $V$ is the one generated by
convex hulls of finite subsets. This is called the _fine bornology_. It is
always complete. We denote by $\mathrm{Fin}(V)$ this bornological vector
space. With this bornology, we have the relation
$V\cong\operatorname*{colim}V_{\alpha},$ where $V_{\alpha}$ runs through all
finite-dimensional subspaces of $V$, endowed with their fine bornology.

######  Example A.2.

Let $V$ be a locally convex topological vector space. Traditionally, a subset
of $B\subset V$ is called bounded if it is absorbed by any neighborhood of the
origin. This defines a bornology on $V$, called the _von Neumann bornology_.
It is complete if $V$ is complete as a topological vector space. We denote by
$\mathrm{vN}(V)$ this bornological vector space.

We are interested in bornological vector spaces for their convenient
categorical properties. We denote by

$\mathrm{CBorn}\subset\mathrm{Born}$

the category of bornological vector spaces and its full subcategory of
complete bornological vector spaces.111Meyer [8] denotes $\mathrm{Born}$ by
$\mathfrak{Born}^{1/2}$, reserving the notation $\mathfrak{Born}$ for the full
subcategory of separated bornological spaces.

######  Theorem A.3 ([8, Proposition 1.126]).

$\mathrm{Born}$ and $\mathrm{CBorn}$ are additive categories and admit all
limits and colimits.

The forgetful functor $\mathrm{Born}\to\mathrm{Vect}$ preserves limits and
colimits; in other words, limits and colimits in $\mathrm{Born}$ are obtained
by putting appropriate bornologies on the corresponding constructions with
plain vector spaces. The bornology on a direct sum $\bigoplus_{i\in I}V_{i}$
is generated by images of bounded subsets of each $V_{i}$ via the standard
inclusions; a subset $B\subset\prod_{i\in I}V_{i}$ is bounded if and only if
each projection $p_{i}(B)\subset V_{i}$ is bounded; kernels and cokernels are
endowed with the subspace and quotient bornology, respectively.

The inclusion $i\colon\mathrm{CBorn}\to\mathrm{Born}$ has a left adjoint, the
_completion_ functor $\mathrm{Born}\to\mathrm{CBorn}$, $V\mapsto V^{c}$. This
means that there are natural isomorphisms

$\operatorname{Hom}(V,W)\cong\operatorname{Hom}\big{(}V^{c},W\big{)}$

for $W$ complete. In particular, limits in $\mathrm{CBorn}$ are calculated as
limits in $\mathrm{Born}$. This is also true for direct sums. However,
completion does not preserve colimits in general: the cokernel of a map
$f\colon V\to W$ in $\mathrm{CBorn}$ is obtained by modding out the
bornological closure of the image, $\operatorname{Coker}f=W/\overline{f(V)}$
[8, Section 1.3.3]. We may write $\operatorname{sep}\operatorname*{colim}$,
$\operatorname{sep}\operatorname{Coker}$, etc. to emphasize that a
construction is taken inside $\mathrm{CBorn}$.

The space $\operatorname{Hom}(V,W)$ of all bounded linear maps has a natural
bornology generated by the uniformly bounded subsets, i.e., those
$L\subset\operatorname{Hom}(V,W)$ such that

$L(B)=\\{f(x)\mid f\in L,\ x\in B\\}\subset W$

is bounded for all bounded $B\subset V$. This is complete if $W$ is.

We equip the algebraic tensor product $V_{1}\otimes V_{2}$ with the bornology
generated by subsets of the form $B_{1}\otimes B_{2}$, with $B_{1}\subset
V_{1}$, $B_{2}\subset V_{2}$ bounded. We define the _completed bornological
tensor product_ by

$V_{1}\mathbin{\hat{\otimes}}V_{2}=(V_{1}\otimes V_{2})^{c}.$

######  Theorem A.4 ([8, Proposition 1.111]).

With the completed tensor product and the bornology on hom-sets defined above,
$\mathrm{CBorn}$ becomes a closed symmetric monoidal category. This means that
$\mathbin{\hat{\otimes}}$ is unital, commutative and associative in the
appropriate sense, and there are natural isomorphisms

$\operatorname{Hom}\big{(}V_{1}\mathbin{\hat{\otimes}}V_{2},W\big{)}\cong\operatorname{Hom}(V_{1},\operatorname{Hom}(V_{2},W))$

for all $V_{1},V_{2},W\in\mathrm{CBorn}$. Likewise, $\mathrm{Born}$ is closed
symmetric monoidal with its uncompleted tensor product $\otimes$.

For $V\in\mathrm{Born}$, denote by $\mathfrak{S}_{d}(V)$ the set of bounded
disks of $V$, with the partial order relation of being absorbed: $B_{1}\leq
B_{2}$ if $B_{1}\subset cB_{2}$ for some constant $c>0$. This implies that we
have an injective, continuous linear map $V_{B_{1}}\to V_{B_{2}}$ of
seminormed spaces. Denote by $\mathfrak{S}_{c}(V)$ the subset of complete
bounded disks. By definition, $\mathfrak{S}_{c}(V)$ is cofinal in
$\mathfrak{S}_{d}(V)$ if and only if $V$ is complete.

######  Proposition A.5.

Let $V$ be a complete bornological vector space. Then there is a canonical
isomorphism

$V\cong\operatorname*{colim}_{B\in\mathfrak{S}_{c}(V)}V_{B}$

in $\mathrm{CBorn}$. If $V\in\mathrm{Born}$, replacing $\mathfrak{S}_{c}(V)$
with $\mathfrak{S}_{d}(V)$ gives a similar isomorphism in $\mathrm{Born}$.

######  Remark A.6.

Complete bornological vector spaces are directly related to the category
$\mathrm{Ind}(\mathrm{Ban})$ of inductive systems of Banach spaces. The
assignment of the directed system $\\{V_{B}\\}_{B\in\mathfrak{S}_{c}(V)}$ to
$V\in\mathrm{CBorn}$ defines the _dissection_ functor
$\mathrm{diss}\colon\mathrm{Born}\to\mathrm{Ind}(\mathrm{Ban})$. There are
also a versions of dissection for non-complete and non-separated spaces.

In sheaf theory, filtered colimits and their commutation properties with
certain kinds of limits play an important role. Thus, we analyze now such
colimits. Recall that the underlying set of a filtered colimit of vector
spaces (and hence also of bornological vector spaces) is the filtered colimit
of the underlying diagram of sets.

######  Lemma A.7.

Let $V=\operatorname*{colim}_{i\in I}V_{i}$ be a filtered colimit in
$\mathrm{Born}$, and denote by $j_{i}\colon V_{i}\to V$ the standard map. Then
a subset of $V$ is a bounded disk if and only if it is of the form $j_{i}(B)$
for some $i\in I$ and bounded disk $B\subset V_{i}$.

###### Proof.

Expressing $V$ in terms of coproducts and a coequalizer, we find a map

$p\colon\ \bigoplus\nolimits_{i\in I}V_{i}\to V$

such that bounded subsets of $V$ are precisely those of the form $p(B)$ with
$B$ bounded. A bounded disk $B$ in the direct sum is given by the convex hull
of a finite collection $B_{i_{1}},\dots,B_{i_{n}}$ of bounded disks
$B_{i_{j}}\subset V_{i_{j}}$. Pick an upper bound $i$ for the collection
$\\{i_{1},\dots,i_{n}\\}\subset I$, and let $B_{i}\subset V_{i}$ be the
smallest bounded disk containing the images in $V_{i}$ of each
$B_{i_{j}}\subset V_{i_{j}}$. Then clearly $p(B)=p(B_{i})$, which finishes the
proof. ∎

######  Proposition A.8.

Small filtered colimits commute with finite limits in $\mathrm{Born}$.

###### Proof.

Let $I$ be a finite indexing category, and $J$ filtered. Fix a diagram
$V\colon I\times J\to\mathrm{Born}$. We want to show that the natural bounded
linear map

$\phi\colon\ \operatorname*{colim}_{j\in J}\lim_{i\in I}V(i,j)\to\lim_{i\in
I}\operatorname*{colim}_{j\in J}V(i,j)$

is an isomorphism. Since filtered colimits of vector spaces commute with
finite limits, we know that $\phi$ is a linear isomorphism. It remains to show
$\phi$ that has a bounded inverse, that is, every bounded disk $B$ in the
codomain is contained in the image of a bounded disk.

By Lemma A.7, each of the bounded disks
$B_{i}=p_{i}(B)\subset\operatorname*{colim}_{j\in J}V(i,j)$ is the image,
through the standard map $\iota_{i,k_{i}}\colon
V(i,k_{i})\to\operatorname*{colim}_{j\in J}V(i,j)$, of a bounded disk
$B_{i,k_{i}}\subset V(i,k_{i})$, for some $k_{i}\in J$. Since $I$ is finite,
we can pick an upper bound $k$ for the collection $\\{k_{i}\\}_{i\in I}$, and
we can write $B_{i}=j_{k}(B_{i,k})$ for suitable bounded disks $B_{i,k}\subset
V(i,k)$.

Now, $\lim_{i\in I}V(i,k)$ is a subspace of the direct product $\prod_{i\in
I}V(i,k)$, which, again by finiteness, is also a direct sum. The bounded disks
$B_{i,k}\subset V(i,k)$ span a bounded disk in that direct sum, and by
restriction, a bounded disk $B_{k}\subset\lim_{i\in I}V(i,k)$. This in turns
determines a bounded disk $B^{\prime}=\iota_{k}(B_{k})$ in the domain of
$\phi$, and $\phi(B^{\prime})=B$. ∎

In general, a filtered colimit of separated, or even complete, bornological
spaces need not be separated, as the following example shows.

######  Example A.9.

Let $X$ be a manifold and $x\in X$. Consider the space of germs of smooth
functions around $x$,

$C^{\infty}_{X,x}=\operatorname*{colim}_{U\ni x}C^{\infty}(U).$

Endowing $C^{\infty}(U)$ with its usual von Neumann bornology and taking the
colimit in $\mathrm{Born}$ gives $C^{\infty}_{X,x}$ a bornology. Then the germ
of a function $f\in C^{\infty}(X)$ which vanishes to infinite order at $0$ is
in the closure of $\\{0\\}$ in $C^{\infty}_{X,x}$. In fact, we can find a
sequence of functions $f_{n}\in C^{\infty}(X)$ with trivial germ at $0$
converging to $f$ in the Fréchet sense, so (equivalently) in the bornological
sense. Therefore, the same holds for their images in $C^{\infty}_{X,x}$.

On the other hand, filtered colimits $V=\operatorname*{colim}V_{i}$ of
separated or complete bornological spaces remain separated or complete for
diagrams with only injective morphisms or, more generally, if every bounded
$B_{i}\subset V_{i}$ with $j_{i}(B_{i})=0\subset V$ is already in the kernel
of some morphism $V_{i}\to V_{j}$ of the diagram. Such inductive systems are
called _stable_ by Houzel [5].

######  Proposition A.10.

Let $\\{V_{i}\\}_{i\in I}$ be a stable filtered diagram of separated or
complete bornological spaces. Then $V=\operatorname*{colim}_{i\in I}V_{i}$ is
separated or complete, respectively.

###### Proof.

Let $B\subset V$ be a bounded disk. Then there exists $i\in I$ and a bounded
disk $B_{i}\subset V_{i}$ such that $j_{i}(B_{i})=B$. Now,
$B_{i}\cap\operatorname{Ker}j_{i}$ is a bounded disk with trivial image in
$V$, so it has trivial image already in $V_{k}$ for some $k\in I$. Let $B_{k}$
be the image of $B_{i}$ in $V_{k}$, so that $B=j_{k}(B_{k})$. Then
$(V_{k})_{B_{k}}\to V_{B}$ is injective, so $V_{B}$ is a normed space. ∎

######  Proposition A.11.

Let $V,\,W\in\mathrm{CBorn}$ and assume $V$ has the fine bornology. Then
$V\otimes W$ is already complete. If both $V$ and $W$ have the fine bornology,
the same is true of their tensor product.

###### Proof.

The first assertion is clear if $V$ is finite-dimensional. Otherwise, we have,
by Proposition A.5, $V\cong\operatorname*{colim}V_{\alpha}$, where
$V_{\alpha}$ ranges through the collection of finite-dimensional subspaces of
$V$, endowed with their fine bornologies and partially ordered by inclusion.
Since, like any left adjoint, tensor product commutes with colimits, we have

$V\otimes W\cong\operatorname*{colim}(V_{\alpha}\otimes W).$

This is a stable filtered colimit of complete bornological spaces, hence
complete. If also $W$ is fine, then

$V\otimes W\cong\operatorname*{colim}(V_{\alpha}\otimes W_{\beta}),$

where $V_{\alpha}\subset V$ and $W_{\beta}\subset W$ range through all finite-
dimensional subspaces. This is a cofinal subcollection of the finite-
dimensional subspaces of $V\otimes W$, and therefore induces the fine
bornology. ∎

###  A.2 Bornological algebras and modules

It is straightforward to define the notion of bounded bilinear maps
$V_{1}\times V_{2}\to W$ between complete bornological vector spaces, and a
bornology on the space $\operatorname{Hom}^{(2)}(V_{1},V_{2};W)$ of such.
Moreover, there is a natural isomorphism

$\operatorname{Hom}^{(2)}(V_{1},V_{2};W)\cong\operatorname{Hom}\big{(}V_{1}\mathbin{\hat{\otimes}}V_{2},W\big{)}.$

A bornological algebra is a bornological vector space $A$ with a bounded,
associative bilinear product $A\times A\to A$. A bornological module is a
bornological vector space $M$ with a bounded, associative, bilinear action map
$A\times M\to M$. Any of those is called complete if the underlying bornology
is complete.

In this paper we only consider unital, commutative bornological algebras and
unital modules, so we drop the extra adjectives. Moreover, we focus on the
complete case. Thus we get a category $\mathrm{CAlg}$ of complete bornological
algebras and, for each $A\in\mathrm{CAlg}$, a category $\mathrm{CMod}(A)$ of
complete bornological modules. Module categories are enriched in themselves,
that is, for $M,N\in\mathrm{CMod}(A)$, the space of $A$-linear maps
$\operatorname{Hom}_{A}(M,N)$, with its subspace bornology, is naturally a
complete $A$-module.

By Proposition A.11, the functor
$\mathrm{Fin}\colon\mathrm{Vect}\to\mathrm{CBorn}$ sends algebras and modules
over them to complete bornological algebras and modules. Similarly, a Fréchet
algebra $A$ induces a complete bornological algebra structure on
$\mathrm{vN}(A)$, and a Fréchet $A$-module $M$ gives a complete
$\mathrm{vN}(A)$-module structure on $\mathrm{vN}(M)$ [8, Theorem 1.29].

Given modules $M$ and $N$ over the bornological algebra $A$, we set

$M\mathbin{\hat{\otimes}}_{A}N=\operatorname{Coker}\big{(}M\mathbin{\hat{\otimes}}A\mathbin{\hat{\otimes}}N\to
M\mathbin{\hat{\otimes}}N\big{)},$

where the map is specified by $(m,a,n)\mapsto am\otimes n-m\otimes an$.

The usual adjunction between extension and restriction of scalars carries over
to the bornological setting. Below, we use $j$ to regard $B$ and $N$ as
$A$-modules.

######  Proposition A.12.

Let $j\colon A\to B$ be a map of complete bornological algebras and let $M$,
$N$ be complete modules over $A$ and $B$ respectively. Then there are natural
isomorphisms

$\operatorname{Hom}_{B}\big{(}B\mathbin{\hat{\otimes}}_{A}M,N\big{)}\cong\operatorname{Hom}_{A}(M,N).$

Moreover, the extension of scalars functor
$B\mathbin{\hat{\otimes}}_{A}\textrm{---}\colon\mathrm{CMod}(A)\to\mathrm{CMod}(B)$
is monoidal.

###### Proof.

Since limits in $\mathrm{CBorn}$ commute with the forgetful functor to
$\mathrm{Vect}$, we can identify the left-hand side with those bounded
bilinear maps $f\colon B\times M\to N$ such that

$f(bb^{\prime},m)=bf(b^{\prime},m),\qquad f(bj(a),m)=f(b,am)$

for all $a\in A$, $b,b^{\prime}\in B$ and $m\in M$. With this observation, the
usual algebraic manipulations give a linear bijection between the hom spaces
in question, and it is easy to check that everything is compatible with the
bornologies.

Letting $j=\mathrm{id}\colon A\to A$, we see that
$A\mathbin{\hat{\otimes}}_{A}M\cong M$. From this, and the fact that tensor
products over $A$ and $B$ are associative, one deduces that
$B\mathbin{\hat{\otimes}}_{A}\textrm{---}$ is monoidal. ∎

###  A.3 Smooth functions

Given a smooth manifold $S$, we denote by $C^{\infty}_{\mathrm{vN}}(S)$ the
space of smooth functions with the von Neumann bornology associated to its
usual Fréchet topology. This can be described as the bornology of uniform
boundedness of all derivatives of each given order on compacts. If $V$ is a
complete bornological space and $f\colon S\to V$ a function, we say that $f$
is _smooth_ if there exists a complete bounded disk $B\subset V$ such that $f$
takes values in $V_{B}$, and $f\colon S\to V_{B}$ is smooth as a function with
values in a Banach space.

######  Proposition A.13.

For any smooth manifold $S$ and complete bornological vector space $V$, there
is a natural isomorphism of vector spaces

$C^{\infty}(S;V)\cong C^{\infty}_{\mathrm{vN}}(S)\mathbin{\hat{\otimes}}V.$

This defines a bornology on the left-hand side, which we denote
$C^{\infty}_{\mathrm{vN}}(S;V)$.

###### Proof.

Using the dissection isomorphism $V=\operatorname*{colim}_{B}V_{B}$, where $B$
ranges over all complete bounded disks in $V$ (Proposition A.5), we get

$C^{\infty}(S)\mathbin{\hat{\otimes}}V\cong\operatorname*{colim}_{B}C^{\infty}(S)\mathbin{\hat{\otimes}}V_{B}.$

On the other hand, we have
$C^{\infty}(S;V)=\operatorname*{colim}_{B}C^{\infty}(S,V_{B})$ by definition.
This reduces the proposition to the case $V=\mathrm{vN}(V_{\nu})$ for a Banach
space $V_{\nu}$.

The rest of the argument is functional analysis. We have
$C^{\infty}(S;\mathrm{vN}(V_{\nu}))=\mathrm{vN}(C^{\infty}(S;V_{\nu}))$ by [7,
Corollary 3.9]. It is well known [14, Theorem 44.1] that
$C^{\infty}(S;V_{\nu})\cong
C^{\infty}(S)\mathbin{\hat{\otimes}}_{\pi}V_{\nu}$, where we used the
completed projective tensor product. Finally,
$\mathrm{vN}(V)\mathbin{\hat{\otimes}}\mathrm{vN}(W)\cong\mathrm{vN}(V\mathbin{\hat{\otimes}}_{\pi}W)$
if $V$ is nuclear [8, Theorem 1.91], which finishes the proof, since
$C^{\infty}(S)$ is nuclear for any manifold $S$ (see, e.g., the corollary of
Theorem 51.5 in [14]). ∎

######  Proposition A.14.

For any smooth manifolds $S$ and $T$, we have

$C^{\infty}_{\mathrm{vN}}(S\times T)\cong
C^{\infty}_{\mathrm{vN}}(S)\mathbin{\hat{\otimes}}C^{\infty}_{\mathrm{vN}}(T).$

###### Proof.

This follows from the corresponding statement in the Fréchet setting, using
the completed projective tensor product [14, Theorem 51.6], and the fact that
$\mathrm{vN}(V)\mathbin{\hat{\otimes}}\mathrm{vN}(W)\cong\mathrm{vN}(V\mathbin{\hat{\otimes}}_{\pi}W)$
if $V$ is nuclear. ∎

###  A.4 Bornological sheaves

In this section, we outline the theory of sheaves of bornological vector
spaces. We follow Houzel [5, Section 2].

Let $\mathcal{C}$ be a category with limits, and denote by
$\mathrm{PSh}_{\mathcal{C}}(X)$ and $\mathrm{Sh}_{\mathcal{C}}(X)$ the
categories of presheaves and sheaves with values in $\mathcal{C}$ on a
topological space (or, more generally, a Grothendieck site) $X$. We are
interested in the cases $\mathcal{C}=\mathrm{CBorn}$ and, as a preliminary
step, $\mathcal{C}=\mathrm{Born}$. We will say that a (pre)sheaf with values
in $\mathrm{Born}$ is _complete_ if it takes values in $\mathrm{CBorn}$.

The first item in our wish list is to have a left adjoint to the inclusion
$\mathrm{Sh}_{\mathcal{C}}(X)\to\mathrm{PSh}_{\mathcal{C}}(X)$, which we call
the _sheafification_ functor. Let $F\in\mathrm{PSh}_{\mathcal{C}}(X)$,
$U\subset X$ open, and $\mathfrak{U}=\\{U_{i}\subset U\\}_{i\in I}$ be an open
cover. As usual, we write

$F(\mathfrak{U})=\lim\bigg{(}\prod_{i\in
I}F(U_{i})\rightrightarrows\prod_{i,j\in I}F(U_{i}\cap U_{j})\bigg{)}$

and say that $F$ is a sheaf if the canonical map $F(U)\to F(\mathfrak{U})$ an
isomorphism for all $\mathfrak{U}$. We will call $F$ a _semisheaf_ if the maps
$F(U)\to F(\mathfrak{U})$ are always monomorphisms. Since the inclusion
$\mathrm{CBorn}\subset\mathrm{Born}$ preserves limits, the condition of being
a (semi)sheaf in one of these categories is equivalent to being a (semi)sheaf
in the other. Set

$F^{+}(U)=\operatorname*{colim}_{\mathfrak{U}}F(\mathfrak{U}),$

where $\mathfrak{U}$ runs through all open covers of $X$. For the traditional
choices $\mathcal{C}=\mathrm{Set}$, $\mathrm{Vect}$, etc., $F^{+}$ is a
semisheaf, and is a sheaf if $F$ is already a semisheaf. This defines the
sheafification functor $F\mapsto F^{++}$. When $\mathcal{C}=\mathrm{Born}$,
$F^{++}$ is obviously a sheaf when regarded as taking values in
$\mathrm{Vect}$, and it can be checked directly that it is in fact a
bornological sheaf.

If $F\in\mathrm{PSh}_{\mathrm{Born}}(X)$ is a complete semisheaf, then the
inductive system defining $F^{+}$ is stable, so it follows, from Proposition
A.10, that $F^{+}$ is a complete sheaf. However, if $F$ fails to be a
semisheaf, then $F^{+}$ may not be complete even if $F$ is. This issue is
resolved in Houzel [5, p. 32] as follows. Let
$F\in\mathrm{PSh}_{\mathrm{Born}}$ be arbitrary and let $E$ be the smallest
sub-presheaf of $U\mapsto\widehat{F(U)}$ such that $F^{\prime}\colon
U\mapsto\widehat{F(U)}/E(U)$ is a complete semisheaf. Then $(F^{\prime})^{+}$
is a complete sheaf, and this construction gives a left adjoint to the
inclusion $\mathrm{Sh}_{\mathrm{CBorn}}(X)\to\mathrm{PSh}_{\mathrm{Born}}(X)$.
By restriction, this left adjoint gives us both a sheafification functor
$\mathrm{PSh}_{\mathrm{CBorn}}(X)\to\mathrm{Sh}_{\mathrm{CBorn}}$ and a
completion functor
$\mathrm{Sh}_{\mathrm{Born}}\to\mathrm{Sh}_{\mathrm{CBorn}}$.

The second item in our wish list is to get tensor products and internal homs.
If $E,F\in\mathrm{PSh}_{\mathrm{Born}}(X)$, then $\operatorname{Hom}(E,F)$
gets a bornology as a subspace of the product $\prod_{U\subset
X}\operatorname{Hom}(E(U),\allowbreak F(U))$, $U\subset X$ open, and we get
also a $\mathrm{Born}$-valued presheaf $\underline{\operatorname{Hom}}(E,F)$.
It is a sheaf if so is $F$, and it is complete if so is $F$. We define
$E\otimes F$ as the sheafification of the presheaf $U\mapsto E(U)\otimes
F(U)$, and $E\mathbin{\hat{\otimes}}F$ as the associated complete sheaf.

The following statement summarizes this discussion.

######  Theorem A.15.

Let $\mathcal{C}=\mathrm{Born}$ or $\mathrm{CBorn}$ and let $X$ be a
topological space. Then the inclusion
$\mathrm{Sh}_{\mathcal{C}}(X)\to\mathrm{PSh}_{\mathcal{C}}(X)$ admits a left
adjoint. Moreover, with the hom-sheaf and tensor product defined above,
$\mathrm{Sh}_{\mathcal{C}}(X)$ is a closed symmetric monoidal category.

If $f\colon X\to Y$ is a continuous map and
$E\in\mathrm{PSh}_{\mathrm{Born}}(X)$, then the direct image $f_{*}E$ is the
presheaf $U\mapsto E(f^{-1}(U))$, $U\subset Y$ open. It is a sheaf if $E$ is.
For $F\in\mathrm{Sh}_{\mathrm{Born}}(Y)$, let $f^{-1}(F)$ is the
sheafification of the presheaf $U\mapsto\operatorname*{colim}_{V\supset
f(U)}F(V)$. This defines a left adjoint

$f^{-1}\colon\
\mathrm{Sh}_{\mathrm{Born}}(Y)\to\mathrm{Sh}_{\mathrm{Born}}(X).$

to $f_{*}$. If $g\colon Y\to Z$ is another map, then $g_{*}\circ
f_{*}\cong(g\circ f)_{*}$ and $f^{-1}\circ g^{-1}\cong(g\circ f)^{-1}$. The
first fact is obvious and the second follows because $f^{-1}$ is adjoint to
$f_{*}$.

Note also that if $f$ is an open map and $F$ takes values in $\mathrm{CBorn}$,
then so does $f^{-1}(F)$.

Finally, we turn to sheaves of algebras and modules. Let $A$ be a sheaf of
complete bornological algebras over $X$, and denote by $\mathrm{CMod}(A)$ the
category of sheaves of complete $A$-modules. If
$M,M^{\prime}\in\mathrm{CMod}(A)$, we denote by
$\underline{\operatorname{Hom}}_{A}(M,M^{\prime})$ the subsheaf of
$\underline{\operatorname{Hom}}(M,M^{\prime})$ consisting of $A$-linear
morphisms. It is complete if $M^{\prime}$ is. The tensor product
$M\mathbin{\hat{\otimes}}_{A}M^{\prime}$ is the complete sheaf associated to
the presheaf $U\mapsto M(U)\otimes_{A(U)}M^{\prime}(U)$. Assume now we are
given a map $f\colon X\to Y$. Then $f_{*}A$ is naturally a sheaf of
bornological algebras, and $f_{*}M$ an $f_{*}A$-module. Also, if $B$ is a
sheaf of bornological algebras on $Y$ and $N\in\mathrm{CMod}(B)$ is a module,
then $f^{-1}(B)$ is naturally a sheaf of bornological algebras, and
$f^{-1}(N)$ an $f^{-1}(B)$-module.

###  A.5 The stack $\boldsymbol{\mathcal{V}_{\mathrm{vN}}}$

The assignment $U\mapsto C^{\infty}_{\mathrm{vN}}(U)$ defines a sheaf of
complete bornological algebras on the site $\mathrm{Man}$ of smooth manifolds.
Its restriction to the small site of a manifold $S$ will be denoted
$C^{\infty}_{\mathrm{vN},S}$.

We denote by
$\mathcal{V}_{\mathrm{vN}}(S)=\mathrm{CMod}(C^{\infty}_{\mathrm{vN},S})$ the
category of sheaves of complete $C^{\infty}_{\mathrm{vN},S}$-modules. As
observed in the previous subsection, this is a closed symmetric monoidal
category.

Any smooth map $f\colon T\to S$ induces a homomorphism
$C^{\infty}_{\mathrm{vN},S}\to f_{*}C^{\infty}_{\mathrm{vN},T}$ of sheaves of
complete bornological algebras. We denote by $f^{\sharp}\colon
f^{-1}(C^{\infty}_{\mathrm{vN},S})\to C^{\infty}_{\mathrm{vN},T}$ its adjoint.
This makes $C^{\infty}_{\mathrm{vN},T}$ into a sheaf of
$f^{-1}(C^{\infty}_{\mathrm{vN},S})$-modules. Given
$M\in\mathcal{V}_{\mathrm{vN}}(S)$, we set

$\hat{f}^{*}M=C^{\infty}_{\mathrm{vN},T}\mathbin{\hat{\otimes}}_{f^{-1}(C^{\infty}_{\mathrm{vN},S})}f^{-1}(M).$

This defines a symmetric monoidal functor

$\hat{f}^{*}\colon\
\mathcal{V}_{\mathrm{vN}}(S)\to\mathcal{V}_{\mathrm{vN}}(T),$

and the assignments $S\to\mathcal{V}_{\mathrm{vN}}(S)$ and
$f\mapsto\hat{f}^{*}$ determine a prestack of symmetric monoidal categories

$\mathcal{V}_{\mathrm{vN}}\colon\
\mathrm{Man}^{\mathrm{op}}\to\mathrm{Cat}^{\otimes}.$

Finally, we need to show that $\mathcal{V}_{\mathrm{vN}}$ is in fact a stack.
This is the case because the pretopology of all open covers of a manifold $S$
and the pretopology of open covers subordinated to a given open cover generate
the same Grothendieck site. More concretely, if $\mathfrak{U}=\\{U_{i}\subset
S\\}$ is an open cover and $V_{i}\in\mathcal{V}_{\mathrm{vN}}(U_{i})$ is a
collection of objects with coherent isomorphisms between their restrictions to
overlaps $U_{i}\cap U_{j}$, we can construct a presheaf of
$C^{\infty}_{\mathrm{vN},S}$-modules $V^{\prime}$ which agrees with $V_{i}$ on
$U_{i}$ and assigns $0$ to any open not contained in some $U_{i}$.
Sheafification then gives the desired “glued together” object
$V\in\mathcal{V}_{\mathrm{vN}}(S)$.

### Acknowledgments

We thank Dmitri Pavlov, Peter Teichner, Konrad Waldorf and, especially,
Stephan Stolz for helpful discussions. We are further indebted to the Max
Planck Institute for Mathematics in Bonn and the University of Greifswald,
where part of this research was conducted. The first-named author was
partially supported by the ARC Discovery Project grant FL170100020 under Chief
Investigator and Australian Laureate Fellow Mathai Varghese.

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