bornological Lie algebra chains
$C_{\bullet}^{\operatorname{\mathrm{born}}}(\Gamma_{c}(\mathcal{K}))$. Much of
the material within this section can be read up on in more detail in [40]. We
assume the reader is familiar with the basic notions of sheaves on topological
spaces.

######  Definition C.1.

[40, Chapter V.1] Let $M$ be a topological space.

  * i)

A _precosheaf_ (of abelian groups) $\mathcal{P}$ on $M$ is a covariant functor
from the category of open sets of $M$, morphisms given by inclusions, into the
category of abelian groups.
Given an inclusion $U\subset V$ of open sets, we denote the associated mapping
$\mathcal{P}(U)\to\mathcal{P}(V)$ by $\iota_{U}^{V}$, called the _extension
map_ from $U$ to $V$ of the precosheaf $\mathcal{P}$.

  * ii)

A _cosheaf_ is a precosheaf $\mathcal{P}$ with the property that for every
open cover $\mathcal{U}$ of an open set $U\subset M$, the sequence

$\displaystyle\bigoplus_{i,j}\mathcal{P}(U_{i}\cap
U_{j})\to\bigoplus_{i}\mathcal{P}(U_{i})\to\mathcal{P}(U)\to 0$

is exact, where the maps are given by

$\displaystyle(a_{ij})_{i,j}\mapsto\left(\sum_{j}\iota_{U_{i}\cap
U_{j}}^{U_{i}}(a_{ij}-a_{ji})\right)_{i},\quad(b_{i})_{i}\mapsto\sum_{i}\iota_{U_{i}}^{U}b_{i}.$

We call a cosheaf $\mathcal{P}$ _flabby_ if all extension maps $\iota_{U}^{V}$
are injective.

  * iii)

A _morphism of (pre-)cosheaves_ is a natural transformation between the
functors defining the (pre-)cosheaves.

Most practical examples of cosheaves arise from some notion of compactly
supported objects, since compactly supported objects on some open set can
always extended by zero to bigger open sets. The following proposition
formalizes this:

######  Proposition C.2.

[40, Proposition V.1.6] Let $\mathcal{S}$ be a sheaf a topological space $M$,
and consider the precosheaf $\mathcal{S}_{c}$ which associates to an open
$U\subset M$ the set

$\displaystyle\mathcal{S}_{c}(U):=\\{s\in\mathcal{S}(M):\operatorname{supp}s\subset
U\\}$

and whose extension maps are given by extending by zero. If $\mathcal{S}$ is
soft, then $\mathcal{S}_{c}$ is a flabby cosheaf.

Completely dually to sheaf theory, one can define the _Čech complex_
$\check{C}_{\bullet}(\mathcal{U};\mathcal{P})$ of a (pre-)cosheaf
$\mathcal{P}$ associated to an open cover $\mathcal{U}$ of $M$, which is then
given as a chain complex

$\displaystyle\dots\to\bigoplus_{i,j}\mathcal{P}(U_{i}\cap
U_{j})\to\bigoplus_{i}\mathcal{P}(U_{i})\to 0,$

its differential being a skew-symmetric linear combination of the extension
maps $\iota_{U}^{V}$. Its construction is fully dual to the standard, sheaf-
theoretic Čech cochain complex, and we direct the reader to [40, Chapter VI.4]
for details. We denote its homology by
$\check{H}_{\bullet}(\mathcal{U};\mathcal{P})$. The defining properties of a
cosheaf $\mathcal{P}$ directly imply

$\displaystyle\check{H}_{0}(\mathcal{U};\mathcal{P})=\mathcal{P}(M)$

independently of the choice of $\mathcal{U}$. Refinements of open covers
induce on the associated Čech complexes the structure of a inverse system, and
as such we may define the Čech homology of a cosheaf $\mathcal{P}$ as the
inverse limit of the Čech homologies of its associated open covers:

$\displaystyle\check{H}_{\bullet}(M;\mathcal{P}):=\varprojlim\check{H}_{\bullet}(\mathcal{U};\mathcal{P}).$

Just like sheaf cohomology can be calculated in terms of resolutions, we shall
calculate Čech homology in terms of _coresolutions:_

######  Definition C.3.

[40, Chapter VI.7]

  * i)

A precosheaf $\mathcal{P}$ on $M$ is called _locally zero_ if for every $x\in
M$ and every open neighbourhood $U$ of $x$ there is an open neighbourhood
$V\subset U$ so that $\iota_{U}^{V}=0$.

  * ii)

A sequence of precosheaves

$\displaystyle\mathcal{P}_{1}\stackrel{{\scriptstyle
f}}{{\to}}\mathcal{P}_{2}\stackrel{{\scriptstyle g}}{{\to}}\mathcal{P}_{3}$

is called _locally exact_ ifthe precosheaf

$\displaystyle U\mapsto\operatorname{Im}f(\mathcal{P}_{1}(U))/\ker
g(\mathcal{P}_{2}(U))$

is locally zero.

  * iii)

A _coresolution_ of a cosheaf $\mathcal{P}$ is a locally exact sequence of
cosheaves

$\displaystyle\dots\to\mathcal{P}_{2}\to\mathcal{P}_{1}\to\mathcal{P}_{0}\to\mathcal{P}\to
0.$

The coresolution is called _flabby_ if the
$\mathcal{P}_{0},\mathcal{P}_{1},\dots$ (but not necessarily $\mathcal{P}$)
are flabby.

To calculate Čech homology of cosheaves, the following result will be helpful:

######  Proposition C.4.

[40, Thms VI.7.2, VI.13.1] Let $\mathcal{P}$ be a cosheaf on $M$ with flabby
coresolution

$\displaystyle\dots\to\mathcal{P}_{2}\to\mathcal{P}_{1}\to\mathcal{P}_{0}\to\mathcal{P}\to
0.$

  * i)

The Čech homology $\check{H}_{\bullet}(M;\mathcal{P})$ is equal to the
homology of the complex

$\displaystyle\dots\to\mathcal{P}_{2}(M)\to\mathcal{P}_{1}(M)\to\mathcal{P}_{0}(M)\to
0.$

  * ii)

If $\mathcal{U}$ is an open cover of $M$ with the property that

$\displaystyle\dots\to\mathcal{P}_{2}(U)\to\mathcal{P}_{1}(U)\to\mathcal{P}_{0}(U)\to\mathcal{P}(U)\to
0$

is exact whenever $U$ is a finite intersection of elements of $\mathcal{U}$,
then

$\displaystyle\check{H}_{\bullet}(\mathcal{U};\mathcal{P})=\check{H}_{\bullet}(M;\mathcal{P}).$

######  Corollary C.5.

For every flabby cosheaf $\mathcal{P}$ on $M$, and every open cover
$\mathcal{U}$ of $M$, we have

$\displaystyle
H_{r}(M,\mathcal{P})=H_{r}(\mathcal{U},\mathcal{P})=\begin{cases}\mathcal{P}(M)&\text{
if }r=0,\\\ 0&\text{ else.}\end{cases}$

###### Proof.

Consider the flabby coresolution
$0\to\mathcal{P}\stackrel{{\scriptstyle\operatorname{id}}}{{\to}}\mathcal{P}\to
0$ and apply Proposition C.4. ∎

One concept which one might hope for in the theory of cosheaves is a dual
version of the well-known concept of sheafification, in other words, a way to
universally assign to every precosheaf an appropriate cosheaf. For sheaves,
one speaks of a left-adjoint functor to the inclusion of presheaves into
sheaves, and sheafification exists for presheaves in most standard categories,
e.g. the category of sets or abelian groups. Since sheafification respects
stalks, locally, the original presheaf and its associated sheafification carry
the same information. Surprisingly, the dual concept of “cosheafification” is
a lot more involved, and even existence of this concept in most standard
categories is a difficult question, let alone constructing it explicitly, see
for example [50]. Instead, we will consider the concept of a _cosheaf on a
base_. While the dual notion of sheaves on a base is well-studied, we are not
aware of any mention in the literature of the cosheaf-theoretic version
thereof.

######  Definition C.6.

Let $\mathcal{B}$ be a topological base of $M$. In the following, view
$\mathcal{B}$ as a subcategory of the category of open sets of $M$.

  * i)

A _precosheaf $S$ on _ $\mathcal{B}$ is a covariant functor from $\mathcal{B}$
to the category of abelian groups. We denote the image of $U\in\mathcal{B}$ as
$S(U)$ and the arising extension maps for $U\subset V\in\mathcal{B}$ by
$\iota_{U}^{V}$.

  * ii)

Choose for any $U\in\mathcal{B}$ an open cover $\\{U_{i}\\}_{i\in I}$ by
elements in $\mathcal{B}$, and for every $i,j\in I$ an open cover
$\\{V_{ij,k}\\}_{k\in K}$ of $U_{i}\cap U_{j}$ by elements in $\mathcal{B}$.
We call a precosheaf $S$ on $\mathcal{B}$ a _cosheaf on $\mathcal{B}$_ if, for
all such choices, the following sequence is exact:

$\displaystyle 0\leftarrow
P(U)\leftarrow\bigoplus_{i}P(U_{i})\leftarrow\bigoplus_{ijk}P(V_{ij,k}).$

  * iii)

A morphism of (pre-)cosheaves on $\mathcal{B}$ is a natural transformation of
the functors defining the (pre-)cosheaves.

The sequence is the analogue of the cosheaf condition, but rather than working
with all open sets, we just work with elements of a topological base
$\mathcal{B}$. If $\mathcal{B}$ is chosen as the topology of $M$, then this
definition is equivalent to the definition of a cosheaf on $M$. This is
precisely the dual of the well-studied concept of sheaves on a base, by
viewing $\mathrm{Ab}$-valued cosheaves as $\text{Ab}^{\text{op}}$-valued
sheaves.

######  Theorem C.7.

Given a topological space $M$ and a topological base $\mathcal{B}$ of $M$. An
$\mathrm{Ab}$-valued cosheaf on $\mathcal{B}$ extends, up to cosheaf
isomorphism, uniquely to a cosheaf on $M$. A morphism between two cosheaves on
$\mathcal{B}$ of $M$ extends uniquely to a morphism between the induced
cosheaves on $M$.

###### Proof.

The following proof is due to [51]. The analogue statement for
$\mathcal{C}$-valued sheaves is true whenever $\mathcal{C}$ is a complete
category (see [52] or [53, Lemma 2.2.7]). However, since Ab is a cocomplete
category, $\text{Ab}^{\text{op}}$ is a complete category. This proves the
statement. ∎

It is known that the setwise cokernels of cosheaf morphisms are again
cokernels [40, Prop VI.1.2], the proof being a simple diagram chase. This
straightforwardly extends to cosheaves on a base:

######  Proposition C.8.

Let $\mathcal{B}$ be a topological base of $M$. Let further
$\phi:\mathcal{P}\to\mathcal{S}$ be a morphism of cosheaves on $\mathcal{B}$,
and define a precosheaf $\operatorname{coker}\phi$ by assigning to
$B\in\mathcal{B}$

$\displaystyle\operatorname{coker}\phi(B):=\mathcal{S}(B)/\phi(\mathcal{P}(B)),$

with extension maps induced by the cosheaf maps of $\mathcal{S}$. Then
$\operatorname{coker}\phi$ defines a cosheaf on $\mathcal{B}$.

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