open cover $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ of an open set $U\subset X$, the
sequence

(A.1) $\displaystyle\bigoplus_{i,j}\mathcal{P}(U_{i}\cap
U_{j})\to\bigoplus_{i}\mathcal{P}(U_{i})\to\mathcal{P}(U)\to 0$

is exact, where the maps are given by

(A.2) $\displaystyle(a_{ij})_{i,j}\mapsto\left(\sum_{j}\iota_{U_{i}\cap
U_{j}}^{U_{i}}(a_{ij}-a_{ji})\right)_{i},\quad(b_{i})_{i}\mapsto\sum_{i}\iota_{U_{i}}^{U}b_{i}.$

  * iii)

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

Unless mentioned otherwise, all our (pre-)cosheaves take values in the
category of $\mathbb{R}$-vector spaces.

######  Definition A.2.

Let $S$ be a precosheaf over a topological space $X$, and
$\mathcal{U}=\\{U_{\alpha}\\}$ an open cover of $X$. Write
$\alpha:=(\alpha_{1},\dots,\alpha_{p+1})$ for the $p$-simplex defined by a
collection of indices $\alpha_{i}$, and write

(A.3) $\displaystyle U_{\alpha}:=U_{\alpha_{1}}\cap\dots\cap
U_{\alpha_{p+1}}.$

Define for a $p$-simplex $\alpha$ and a number $i\in\\{1,\dots,p+1\\}$ the
$(p-1)$-simplex $\alpha^{(i)}$ as the simplex which arises from removing the
$i$-th index from $\alpha$. For all $p\geq 0$, we define the space of _Čech
$p$-chains for $S$ associated to the cover $\mathcal{U}$_ as

(A.4)
$\displaystyle\check{C}_{p}(\mathcal{U}_{X};S):=\bigoplus_{\alpha=(\alpha_{1},\dots,\alpha_{p+1})}S(U_{\alpha}).$

We may then express elements $c\in\check{C}_{p}(\mathcal{U}_{X};S)$ as formal
linear combinations

(A.5) $\displaystyle c=\sum_{\alpha}c_{\alpha},\quad c_{\alpha}\in
S(U_{\alpha_{1}}\cap\dots\cap U_{\alpha_{p+1}}),$

so that only finitely many $c_{\alpha}$ are nonzero.
The _Čech differential_
$\check{\partial}:\check{C}_{p}(\mathcal{U}_{X};S)\to\check{C}_{p-1}(\mathcal{U}_{X};S)$
via

(A.6)
$\displaystyle\check{\partial}(c_{\alpha}\alpha):=\sum_{i=1}^{p+1}(-1)^{i-1}\left(\iota_{U_{\alpha}}^{U_{\alpha^{(i)}}}c_{\alpha}\right)\alpha^{(i)}.$

The _Čech homology associated to the cover $\mathcal{U}$ and the precosheaf
$S$_, which is denoted by $\check{H}_{\bullet}(\mathcal{U}_{X};S)$, is defined
as the homology of the chain complex
$\check{C}_{\bullet}(\mathcal{U}_{X};P):=\bigoplus_{p\geq
0}\check{C}_{p}(\mathcal{U}_{X};S)$.
The _Čech homology_ of $S$ is defined as

(A.7)
$\displaystyle\check{H}_{\bullet}(X;S):=\varprojlim\check{H}_{\bullet}(\mathcal{U}_{X};S)$

where the inverse limit is taken with respect to refinement of covers.

######  Remark A.3.

The symmetric group $\Sigma_{p}$ acts on multiindices $\alpha$ of length $p$
by permutation of the entries, and we denote this permutation by
$\sigma\cdot\alpha$. Recall now the notation from (A.5). If
$c=\sum_{\alpha}c_{\alpha}\cdot\alpha$ and
$\alpha=(\alpha_{1},\dots,\alpha_{p})$ is one of the multiindices, we call $c$
_skew-symmetric_ if

(A.8) $\displaystyle
c_{\sigma\cdot\alpha}=\operatorname{sign}(\sigma)c_{\alpha}\quad\forall\alpha.$

The _skew-symmetrized_ Čech complex is defined as the subcomplex

(A.9)
$\displaystyle\check{C}_{\bullet}^{a}(\mathcal{U}_{X};S)\subset\check{C}_{\bullet}(\mathcal{U}_{X};S)$

of skew-symmetric cochains. Dualizing the corresponding results for Čech
cohomology of a sheaf [32, Section 3.8], one finds that the inclusion
$\check{C}_{\bullet}^{a}(\mathcal{U}_{X};S)\hookrightarrow\check{C}_{\bullet}(\mathcal{U}_{X};S)$
is a quasi-isomorphism.

We want to remark on a special class of cosheaves on which Čech homology is
trivial.

######  Definition A.4.

A cosheaf is called _flabby_ if all its extension maps are injective.

######  Proposition A.5 ([25], Chapter V, Prop. 1.6).

Let $P$ be a soft sheaf over a topological space $X$. Then compactly supported
sections of $P$ admit the structure of a flabby cosheaf over $X$, where the
extension maps extend sections by zero.

######  Proposition A.6 ([25], Chapter VI, Cor. 4.5).

Let $S$ be a flabby cosheaf over $X$ and $\mathcal{U}$ an open cover of $X$.
Then

(A.10) $\displaystyle\check{H}_{p}(\mathcal{U}_{X};S)=\begin{cases}S(X)&\text{
if }p=0,\\\ 0&\text{ else.}\end{cases}$

##  Appendix B The Hochschild-Serre Spectral Sequence for locally convex Lie
algebras

In the finite-dimensional setting, the Hochschild-Serre spectral sequence is
standard and a proof is laid out in [10, Chapter 1.5.1] and [20]. For general
locally convex Lie algebras and continuous cohomology, one generally needs a
number of topological assumptions. For example, restriction maps of continuous
cochains like $C^{q}(\mathfrak{g})\to
C^{r}(\mathfrak{h},\Lambda^{q-r}(\mathfrak{g}/\mathfrak{h})^{*})$ are not
necessarily surjective if the subspace ${\operatorname{\mathfrak{h}}}$ is not
complemented. We formulate some assumptions which suffice for the setting in
this paper:

######  Theorem B.1.

Let $\mathfrak{g}$ be a complete, barreled, locally convex, nuclear Lie
algebra whose strong dual space ${\operatorname{\mathfrak{g}}}^{*}$ is
complete, $\mathfrak{h}\subset\mathfrak{g}$ a finite-dimensional Lie
subalgebra, and $A$ a complete, locally convex space on which $\mathfrak{g}$
acts continuously. There is a cohomological spectral sequence
$\\{E^{p,q}_{r},d_{r}\\}$ converging to continuous cohomology
$H^{\bullet}(\mathfrak{g})$ with

(B.1) $\displaystyle
E^{p,q}_{1}=H^{q}\left(\mathfrak{h},C^{p}\left(\mathfrak{g}/\mathfrak{h},A\right)\right),$

where $C^{p}(X,Y)$ denotes skew-symmetric, jointly continuous, multilinear
maps

(B.2) $\displaystyle\underbrace{X\times\dots\times X}_{p\text{ times }}\to Y,$

and cohomology is taken with respect to continuous cochains. This spectral
sequence is contravariantly functorial, in the sense that a diagram of
continuous Lie algebra morphisms

${\mathfrak{h}}$${\mathfrak{g}}$${\tilde{\mathfrak{h}}}$${\tilde{\mathfrak{g}}}$

induces linear maps

(B.3) $\displaystyle E_{r}^{p,q}(\tilde{\mathfrak{g}},\tilde{\mathfrak{h}})\to
E_{r}^{p,q}(\mathfrak{g},\mathfrak{h})$

compatible with the differentials for all $p,q,r\geq 0$.

###### Proof.

We define on the continuous cochains $C^{\bullet}(\mathfrak{g})$ the
filtration

(B.4) $\displaystyle F^{p}C^{p+q}(\mathfrak{g};A):=\\{c\in
C^{p+q}(\mathfrak{g},A):c(X_{1},\dots,X_{p+q})=0\text{ when
}X_{1},\dots,X_{q+1}\in\mathfrak{h}\\}.$

This is an ascending filtration with

(B.5) $\displaystyle
C^{r}(\mathfrak{g},A)=F^{0}C^{r}(\mathfrak{g},A)\supset\dots\supset
F^{r}C^{r}(\mathfrak{g},A)\supset F^{r+1}C^{r}(\mathfrak{g},A)=0,$

and

(B.6) $\displaystyle dF^{p}C^{p+q}(\mathfrak{g};A)\subset
F^{p}C^{p+q+1}(\mathfrak{g};A).$

Denote by $\hat{\Lambda}^{q}$ the functor assigning to a locally convex vector
space $X$ the closure of the skew-symmetric tensors in its iterated projective
tensor product $X^{\mathbin{\widehat{\otimes}}^{q}}$, see for example [33,
Chapter III.7, IV.9]. We have a well-defined map

(B.7) $\begin{gathered}F^{p}C^{p+q}(\mathfrak{g},A)\to
L(\hat{\Lambda}^{q}\mathfrak{h}\mathbin{\widehat{\otimes}}\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A),\quad
c\mapsto\tilde{c},\\\ \tilde{c}((h_{1}\wedge\dots\wedge
h_{q})\otimes[g_{1}]\wedge\dots\wedge[g_{p}])):=c(h_{1},\dots,h_{q},g_{1},\dots,g_{p}).\end{gathered}$

This map is independent of the choices of representatives $g_{i}$ by
definition of the filtration and it is surjective because finite-dimensional
subspaces are always complemented, so
$\mathfrak{g}\cong\mathfrak{h}\oplus\mathfrak{g}/\mathfrak{h}$ as a direct sum
of locally convex vector spaces. The kernel of this map equals
$F^{p+1}C^{p+q}(\mathfrak{g},A)$. The image of this map is also indeed
contained in the continuous linear maps by continuity of elements in the
domain. Since $\mathfrak{h}$ is finite-dimensional, we trivially have

(B.8)
$\displaystyle(\mathfrak{h}\mathbin{\widehat{\otimes}}\mathfrak{g}/\mathfrak{h})^{*}\cong\mathfrak{h}^{*}\mathbin{\widehat{\otimes}}(\mathfrak{g}/\mathfrak{h})^{*}.$

By the assumptions on ${\operatorname{\mathfrak{g}}}$ and $A$, we may apply
[34, Proposition 50.5] twice to find

(B.9) $\displaystyle
L(\hat{\Lambda}^{q}\mathfrak{h}\mathbin{\widehat{\otimes}}\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A)\cong
L(\hat{\Lambda}^{q}\mathfrak{h},L(\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A))\cong
C^{q}(\mathfrak{h},L(\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A)).$

Hence we get an isomorphism of vector spaces

(B.10) $\displaystyle
F^{p}C^{p+q}(\mathfrak{g},A)/F^{p+1}C^{p+q}(\mathfrak{g},A)\cong
C^{q}\left(\mathfrak{h},L(\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A)\right).$

The differential of $C^{\bullet}(\mathfrak{g},A)$ descends to the differential
of this complex like in the purely algebraic case, so the spectral sequence
associated to this filtration indeed has first page:

(B.11) $\displaystyle
E^{p,q}_{1}=H^{q}\left(\mathfrak{h},L(\hat{\Lambda}^{p}\mathfrak{g}/\mathfrak{h},A)\right).$

The functoriality with respect to Lie algebra pairs
$(\mathfrak{g},\mathfrak{h})$ is analogous to the purely algebraic setting. ∎

######  Remark B.2.

This spectral sequence in the algebraic setting is generally also phrased with
information about the second page if $\mathfrak{h}$ is an ideal. Adapting this
to the continuous setting would require stronger assumptions, since this in
particular requires commuting the projective tensor product with the
cohomology.

## References

  * [1] I. M. Gel’fand and D. B. Fuks.  Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold.  Funkcional. Anal. i Priložen., 3(3):32–52, 1969.
  * [2] I. M. Gel’fand and D. B. Fuks.  Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold. II.  Funkcional. Anal. i Priložen., 4(2):23–31, 1970.
  * [3] I. M. Gel’fand and D. B. Fuks.  Cohomologies of the Lie algebra of formal vector fields.  Izv. Akad. Nauk SSSR Ser. Mat., 34:322–337, 1970.
  * [4] R. Bott and G. Segal.  The cohomology of the vector fields on a manifold.  Topology, 16(4):285–298, 1977.
  * [5] D. B. Fuks.  Characteristic classes of foliations.  Uspehi Mat. Nauk, 28(2(170)):3–17, 1973.
  * [6] Miguel Angel Virasoro.  Subsidiary conditions and ghosts in dual-resonance models.  Physical Review D, 1(10):2933, 1970.
  * [7] Bas Janssens and Cornelia Vizman.  Universal central extension of the Lie algebra of Hamiltonian vector fields.  Int. Math. Res. Not. IMRN, (16):4996–5047, 2016.
  * [8] Bas Janssens and Cornelia Vizman.  Integrability of central extensions of the Poisson Lie algebra via prequantization.  J. Symplectic Geom., 16(5):1351–1375, 2018.
  * [9] Pedro Boavida de Brito and Michael Weiss.  Manifold calculus and homotopy sheaves.  Homology Homotopy Appl., 15(2):361–383, 2013.
  * [10] D. B. Fuks.  Cohomology of infinite-dimensional Lie algebras.  Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986.  Translated from the Russian by A. B. Sosinskiĭ.
  * [11] B. Hennion and M. Kapranov.  Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras.  arXiv preprint arXiv:1811.05032, 2018.
  * [12] Lukas Miaskiwskyi.  Continuous cohomology of gauge algebras and bornological Loday-Quillen-Tsygan theorems.  To appear.
  * [13] R. Bott.  Lectures on Gelfand-Fuks cohomology and characteristic classes.  Notes by M. Mostow and J. Perchik, New Mexico State University, 1973\.
  * [14] Toru Tsujishita.  Continuous cohomology of the Lie algebra of vector fields, volume 34.  1981\.
  * [15] Victor Guillemin and Steven Shnider.  Some stable results on the cohomology of the classical infinite-dimensional Lie algebras.  Trans. Amer. Math. Soc., 179:275–280, 1973.
  * [16] Sorin Dăscălescu, Constantin Năstăsescu, and Şerban Raianu.  Hopf algebras, volume 235 of Monographs and Textbooks in Pure and Applied Mathematics.  Marcel Dekker, Inc., New York, 2001.  An introduction.
  * [17] Ivan Kolář, Peter W. Michor, and Jan Slovák.  Natural operations in differential geometry.  Springer-Verlag, Berlin, 1993.
  * [18] Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [19] Brian Hall.  Lie groups, Lie algebras, and representations, volume 222 of Graduate Texts in Mathematics.  Springer, Cham, second edition, 2015.  An elementary introduction.
  * [20] G. Hochschild and J.-P. Serre.  Cohomology of Lie algebras.  Ann. of Math. (2), 57:591–603, 1953.
  * [21] Armand Borel.  Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.  Ann. of Math. (2), 57:115–207, 1953.
  * [22] Mamoru Mimura and Hirosi Toda.  Topology of Lie groups. I, II, volume 91 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1991.  Translated from the 1978 Japanese edition by the authors.
  * [23] F. Gourdeau, Z. A. Lykova, and M. C. White.  A Künneth formula in topological homology and its applications to the simplicial cohomology of $l^{1}({\mathbb{Z}}^{k}_{+})$.  Studia Math., 166(1):29–54, 2005.
  * [24] Kevin Costello and Owen Gwilliam.  Factorization algebras in quantum field theory. Vol. 1, volume 31 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2017.
  * [25] Glen E. Bredon.  Sheaf theory, volume 170 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1997.
  * [26] Raoul Bott.  On the Gelfand-Fuks cohomology.  In Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973), pages 357–364, 1975.
  * [27] Lars Hörmander.  The analysis of linear partial differential operators. I.  Classics in Mathematics. Springer-Verlag, Berlin, 2003.  Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)].
  * [28] Grégory Ginot.  Notes on factorization algebras, factorization homology and applications.  In Mathematical aspects of quantum field theories, Math. Phys. Stud., pages 429–552. Springer, Cham, 2015.
  * [29] Henri Cartan and Samuel Eilenberg.  Homological algebra.  Princeton University Press, Princeton, N. J., 1956.
  * [30] Samuel Eilenberg and Norman Steenrod.  Foundations of algebraic topology.  Princeton University Press, Princeton, New Jersey, 1952.
  * [31] Robert Clifford Gunning and Hugo Rossi.  Analytic functions of several complex variables, volume 368.  American Mathematical Soc., 2009.
  * [32] Roger Godement.  Topologie algébrique et théorie des faisceaux.  Publications de, 1, 1958.
  * [33] Helmut H. Schaefer.  Topological vector spaces.  Graduate Texts in Mathematics, Vol. 3, pages xi+294. Springer-Verlag, New York-Berlin, 1971.  Third printing corrected.
  * [34] François Trèves.  Topological vector spaces, distributions and kernels.  Academic Press, New York-London, 1967.