## Abstract

We construct, for a locally compact metric space *X*, a space of polylipschitz forms \({\overline{\Gamma }}^*_c(X)\), which is a pre-dual for the space of metric currents \({\mathscr {D}}_*(X)\) of Ambrosio and Kirchheim. These polylipschitz forms may be seen as an analog of differential forms in the metric setting.

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## Acknowledgements

We thank Rami Luisto and Stefan Wenger for discussions on the topics of the manuscript.

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### Publisher's Note

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P.P. was supported in part by the Academy of Finland project #297258. E.S. was partly supported by the Vilho, Yrjö ja Kalle Väisalä Foundation (postdoc pool) and by the Swiss National Science Foundation Grant 182423.

Communicated by L. Ambrosio.

## Appendix A. Cohomorphisms and their associated linear maps

### Appendix A. Cohomorphisms and their associated linear maps

In this appendix, we define cohomomorphisms between presheaves and describe a general construction yielding a linear map associated to a given cohomomorphism.

Let \(f:X\rightarrow Y\) be a continuous map between paracompact Hausdorff spaces and let \(A=\{A(U);\rho ^A_{U,V}\}_U\) and \(B=\{B(U);\rho ^B_{U,V}\}_U\) be presheaves on *X* and *Y*, respectively. A collection

of linear maps for each open \(U\subset Y\), satisfying

is called an *f*-*cohomomorphism of presheaves*; cf. [2, Chapter I.4]. For \(f=\mathrm {id}:X\rightarrow X\), condition (A.1) becomes (5.2) and thus \(\mathrm {id}\)-cohomomorphisms are simply presheaf homomorphisms.

An *f*-cohomomorphism \(\varphi :B\rightarrow A\) between presheaves induces a natural linear map

*the linear map (on sections) associated to*\(\varphi \). Given a global section \(\omega :Y\rightarrow {\mathcal {B}}(Y)\), the section \(\varphi ^*(\omega ):X\rightarrow {\mathcal {A}}(X)\) is defined as follows: for \(x\in X\),

where *U* is a neighborhood of *f*(*x*) and \(g_U\in B(U)\) satisfies \(\omega (f(x))=[g_U]_{f(x)}\).

To see that \(\varphi ^*(\omega )(x)\) is well-defined, suppose that \(\omega (f(x))=[g_U]_{f(x)}=[g'_V]_{f(x)}\), i.e., that there is a neighborhood \(D\subset U\cap V\) of *f*(*x*) for which

By (A.1) we have that

in particular \([\varphi _U(g_U)]_x=[\varphi _V(g'_V)]_x\).

### Remark A.1

Let \(\omega \in {\mathscr {G}}({\mathcal {B}}(Y))\) be compatible with \(\{ g_U \}_{\mathcal {U}}\). Suppose \(U_V\) satisfies \(V=f^{-1}U_V\) for each \(V\in f^{-1}{\mathcal {U}}\). Then, by (A.1) and the fact that \(\{g_U\}_{\mathcal {U}}\) is compatible \(\omega \), we have that the collection \(\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}\) is compatible with \(\varphi ^*\omega \).

If \(\omega \in \Gamma ({\mathcal {B}}(Y))\) and \(\{g_U \}_{\mathcal {U}}\) represents \(\omega \), then \(\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}\) represents \(\omega \), and if \(\{g_U \}_{\mathcal {U}}\) satisfies the overlap condition (5.5) then \(\{ \varphi _V(g_{U_V}) \}_{f^{-1}{\mathcal {U}}}\) also satisfies the overlap condition.

We collect some fundamental properties of linear maps associated to cohomorphisms in the next proposition.

### Proposition A.2

Let \(f:X\rightarrow Y\) and \(g:Y\rightarrow Z\) be continuous maps between paracompact Hausdorff spaces and let \(A=\{A(U)\}_U\), \(B=\{B(U)\}_U\) and \(C=\{C(U)\}_U\) be presheaves on *X*, *Y* and *Z* respectively. Suppose

are *f*-cohomomorphisms, and

is an *g*-cohomomorphism.

- (1)
For \(\omega \in {\mathscr {G}}({\mathcal {B}}(Y))\) we have

$$\begin{aligned} {\text {spt}}(\varphi ^*\omega )\subset f^{-1}({\text {spt}}(\omega )). \end{aligned}$$ - (2)
The associated linear map \(\varphi ^*:{\mathscr {G}}({\mathcal {B}}(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X))\) satisfies

$$\begin{aligned} \varphi ^*(\Gamma ({\mathcal {B}}(Y)))\subset \Gamma ({\mathcal {A}}(X)). \end{aligned}$$ - (3)
The collection \(\{ \varphi _U+\varphi '_U:B(U)\rightarrow A(f^{-1}U) \}_U\) is an

*f*-cohomomorphism and$$\begin{aligned} (\varphi +\varphi ')^*=\varphi ^*+\varphi '^*:{\mathscr {G}}({\mathcal {B}}(Y))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)). \end{aligned}$$ - (4)
The collection \(\{\varphi _{g^{-1}U}\circ \psi _U:C(U)\rightarrow A((g\circ f)^{-1}U)\}_U\) is an \((g\circ f)\)-cohomomorphism and

$$\begin{aligned} (\varphi \circ \psi )^*=\varphi ^*\circ \psi ^*:{\mathscr {G}}({\mathcal {C}}(Z))\rightarrow {\mathscr {G}}({\mathcal {A}}(X)). \end{aligned}$$

### Remark A.3

Given presheaves

and bilinear maps \(\{ \varphi _U;B_1(U)\times B_2(U)\rightarrow A(f^{-1}U) \}\) an analogous construction gives an associated bilinear map

The induced bi-linear map satisfies (2) and (3) and also

- (1’)
For each \((\omega ,\sigma ) \in {\mathscr {G}}({\mathcal {B}}_1(Y))\times {\mathscr {G}}({\mathcal {B}}_2(Y))\), we have

$$\begin{aligned} {\text {spt}}(\varphi ^*(\omega ,\sigma ))\subset f^{-1}({\text {spt}}\omega \cap {\text {spt}}\sigma ). \end{aligned}$$

We will need this only for the case \(\mathrm {id}:X\rightarrow X\) in the construction of cup products. The details are similar as above and we omit them.

### Proof of Proposition A.2

The proofs are straightforward and we merely sketch them.

If \(\varphi ^*\omega (x)\ne 0\) then, since \(\varphi _U\) is linear, (A.2) implies that \(\omega (f(x))=[g_U]_{f(x)}\ne 0\), proving (1). Claim (2) follows directly from Remark A.1.

To prove (3) we observe that from (A.2) it is easy to see that, if \(\varphi ':B\rightarrow A\) is another *f*-cohomorphism between presheaves, then \(\varphi +\varphi '\) is an *f*-cohomomorphism and we have

To prove (4), note that condition (A.1) follows for \(\phi \circ \psi \) from the fact that it holds for \(\varphi \) and \(\psi \). Using (A.2) (and the same notation) we see that

\(\square \)

Proposition A.2 has the following immediate corollary.

### Corollary A.4

If \(f:X\rightarrow Y\) is a proper continuous map and \(\varphi :B\rightarrow A\) an *f*-cohomomorphism between presheaves *B* on *Y* and *A* on *X*, then

In particular presheaf homomorphisms always have this property.

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Pankka, P., Soultanis, E. Metric currents and polylipschitz forms.
*Calc. Var.* **59, **76 (2020). https://doi.org/10.1007/s00526-020-01741-5

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### Mathematics Subject Classification

- 49Q15
- 53C23
- 30L99