$\operatorname{Aut}(*_{S})\simeq\operatorname{Aut}(S)$. This means that a
general object is a forest or aggregate of such corollas. The connected
version $\mathfrak{G}^{\rm ctd}$ has morphisms generated by simple edge
contractions:
$\,{\vphantom{\circ}}_{s}{\circ}_{t}\colon*_{S}\amalg*_{T}\to*_{S\setminus{\sigma}\amalg
T\setminus\\{t\\}}$ and simple loop contractions
$\circ_{ss^{\prime}}\colon*_{S}\to*_{S\setminus\\{s,s^{\prime}\\}}$. These
satisfy quadratic relations and are crossed with respect to the isomorphisms.
For any $\phi$ and any isomorphism $\sigma$ there are unique $\phi^{\prime}$
and $\sigma^{\prime}$ such that
$\phi\circ\sigma=\sigma^{\prime}\circ\phi^{\prime}$. Each morphism has an
underlying graph $\mathbbnew{\Gamma}(\phi)$ which is the source corolla, but
with one edge for each operation $\,{\vphantom{\circ}}_{s}{\circ}_{t}$ or
$\circ_{st}$. The edges are glued from the half edges $s$ and $t$. It is
defined in a way such that
$\mathbbnew{\Gamma}(\phi\circ\sigma)=\mathbbnew{\Gamma}(\phi^{\prime})$, in
the notation above, and its automorphisms are induced from those of the
source, see [5, 41]. The isomorphism class of this graph gives the isomorphism
class of $\phi$: $[\mathbbnew{\Gamma}(\phi)]=[\phi]$.

$\mathfrak{G}^{\rm ctd}\subset\mathfrak{G}$ is a sub-Feynman category whose
basic morphisms have a connected ghost graph. A morphism $\phi$ in
$\mathfrak{G}^{\rm ctd}$ is basic if and only if $\mathbbnew{\Gamma}(\phi)$ is
connected; see Figure 2.

Figure 2: The basic morphisms: Edge contraction loop contraction and merger,
whose ghost graph is not connected. The shading is extra data not captured by
the ghost graph.

For the non-connected version, there is one more generator
$\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}\colon*_{S}\amalg*_{T}\to*_{S\amalg
T}$. In this case, there is a non-homogeneous relation
$\,{\vphantom{\circ}}_{s}{\circ}_{t}=\circ_{s,t}\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}$
which allows one to replace the edge contractions and only retain loop
contractions and mergers as generators. The basic morphisms now need not to
have connected underlying graph anymore.

Each corolla $*_{S}$ is isomorphic to $*_{{|S|}}$ –$n$ is again short hand for
the set $\\{1,\dots,n\\}$ – and hence
$[{\mathrm{id}}_{*_{S}}]=[{\mathrm{id}}_{{|S|}}]$. Furthermore fixing
isomorphisms $\sigma\colon S\to{S}$, with $\sigma(s)=1$,
$\sigma(s^{\prime})=2$ and $\sigma^{\prime}\colon T\to{T}$ with
$\sigma^{\prime}(t)=1$,
$[\,{\vphantom{\circ}}_{s}{\circ}_{t}]=[\,{\vphantom{\circ}}_{0}{\circ}_{1}]$,
$[\circ_{s,s^{\prime}}]=[\circ_{1,2}]$ and
$[\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}]=\,{\vphantom{\boxminus}}_{{|S|}\,}{\boxminus}_{\,{|T|}}$.
Thus there is only one isomorphism class per basic morphism of generating
type.

Beside the natural grading $|-|$, which is $1$ for an edge contraction or a
merger and $0$ for an isomorphism or a loop contraction, there is an
additional degree given by $\operatorname{deg}(*_{S})=|S|$ which is additively
continued $\operatorname{deg}(*_{S}\amalg*_{T})=|S|+|T|$, to count the total
number of legs in the forest. This yields
$\operatorname{deg}(\phi):=\frac{1}{2}(\operatorname{deg}(s(\phi))-\operatorname{deg}(t(\phi))$,
which is integer, since both loop and edge contractions reduce the set of legs
by an even number, while isomorphisms and mergers keep this number fixed. The
degree of an isomorphism or a merger is hence $0$ and that of a simple loop or
edge contraction is $1$. In this way
$\operatorname{deg}(\phi)=|E(\mathbbnew{\Gamma}(\phi))|$. The sum
$wt(\phi)=\operatorname{deg}(\phi)+|\phi|$ is $1$ for the mergers and loop
contractions and $2$ for the edge contractions. This is also the degree
$|{-}|_{\rm max}$ for the presentation in terms of the
$\,{\vphantom{\circ}}_{s}{\circ}_{t}$, $\circ_{ss^{\prime}}$ and
$\,{\vphantom{\boxminus}}_{v}{\boxminus}_{w}$ and hence is a proper degree
function.

######  Remark 5.21 (loop number and Euler characteristic).

In the graphical interpretation for a basic morphism
$wt(\phi)=|E(\mathbbnew{\Gamma}(\phi))|+b_{0}(\phi)$, where $b_{0}$ is the
number of components of $\mathbbnew{\Gamma}(\phi)$. This is related to the
Euler characteristic as follows
$\chi(\mathbbnew{\Gamma}(\phi))=wt(\phi)-|\phi|$, where
$\chi(\mathbbnew{\Gamma}(\phi))=b_{0}(\mathbbnew{\Gamma}(\phi))-b_{1}(\mathbbnew{\Gamma}(\phi))=|V(\mathbbnew{\Gamma}(\phi))|-|E(\mathbbnew{\Gamma}(\phi))|$
and $b_{1}$ is the loop number, aka. first Betti number.

The coproduct in $B^{\rm iso}$ is given by [25, Section 3.6].

$\displaystyle\Delta{[\phi}]=\Delta[\mathbbnew{\Gamma}(\phi)]=\sum_{\gamma\subset\mathbbnew{\Gamma}(\phi)}[\gamma]\otimes[\mathbbnew{\Gamma}(\phi)/\gamma].$
(5.3)

This is a version of the core Hopf algebra [44], but with legs and at a
bialgebra level. See Figure 3 for an example of a summand.

Figure 3: A summand of the coproduct (5.3) in terms of morphisms of
aggregates. Note in the actual morphisms all flags and vertices are labeled.
In the figure only the vertices are tracked to avoid clutter. For the
coproduct only the isomorphism classes of the morphism are involved.

######  Theorem 5.22.

Both $B^{\rm iso}(\mathfrak{G}^{\rm ctd})$ and $B^{\rm iso}(\mathfrak{G})$ are
color connected and hence pathlike. They are graded by $deg,|-|$ and $wt$,
with $wt$ being a proper degree function on $\mathfrak{G}$ and $deg$ being
proper on $\mathfrak{G}^{\rm ctd}$. For $B^{\rm iso}(\mathfrak{G}^{\rm ctd})$
the bivariate Quillen index is given by $deg$ and for $B^{\rm
iso}(\mathfrak{G})$ the bivariate Quillen index is given by $wt$.

More precisely, the grouplike elements are $[{\mathrm{id}}]$ and
$[{\mathrm{id}}_{X}]$ which are generated by the classes
$[{\mathrm{id}}_{*_{S}}]=[{\mathrm{id}}_{{|S|}}]$. In the connected case, the
skew primitive elements are generated by the
$[\,{\vphantom{\circ}}_{s}{\circ}_{t}]$ and the $[\circ_{s,s^{\prime}}]$ while
in the connected case these are generated by the $[\circ_{s,s^{\prime}}]$ and
the $[\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}]$ via products with
grouplikes.

The antipodes for the skew primitives in the localization and the single
parameter Laurent polynomials are given as follows:

$\displaystyle
S(\,{\vphantom{\boxminus}}_{n}{\boxminus}_{m})=-[{\mathrm{id}}_{n}]^{-1}[{\mathrm{id}}_{m}]^{-1}[{\mathrm{id}}_{n+m}]^{-1}\,{\vphantom{\boxminus}}_{n}{\boxminus}_{m}S(\,{\vphantom{\boxminus}}_{n}{\boxminus}_{m})=-q^{-2(n+m)}\,{\vphantom{\boxminus}}_{n}{\boxminus}_{m},$
$\displaystyle
S([\,{\vphantom{\circ}}_{s}{\circ}_{t}])=[{\mathrm{id}}_{|S|}]^{-1}[{\mathrm{id}}_{|T|}]^{-1}[{\mathrm{id}}_{|S|+|T|-2}]^{-1}[\,{\vphantom{\circ}}_{s}{\circ}_{t}]S[\,{\vphantom{\circ}}_{s}{\circ}_{t}]=-q^{-2(|S|+|T|+1)}\,{\vphantom{\circ}}_{s}{\circ}_{t},$
$\displaystyle
S([\circ_{s,s^{\prime}}])=[{\mathrm{id}}_{|S|}]^{-1}[{\mathrm{id}}_{|T|}^{-1}][{\mathrm{id}}_{|S|+|T|-2}]^{-1}[\circ_{s,s^{\prime}}]S[\,{\vphantom{\circ}}_{s}{\circ}_{t}]=-q^{-2(|S|+|T|+1)}[\circ_{s,s^{\prime}}].$

The connected Hopf quotient identifies all the objects with and hence all
grouplikes ${\mathrm{id}}_{X}$ with $1={\mathrm{id}}$. The Brown coaction
again sets the $q$ variables on the right to $1$, which halves the $q$ degrees
in the above formulae.

###### Proof.

The first assertions follows from Theorem 5.12 and the rest is straightforward
from the definitions. ∎

######  Remark 5.23.

Besides the core Hopf algebra of all graphs, one can restrict and decorate as
is done in [8, 44]. For instance Figure 3 only contains 3-valent vertices,
which corresponds to $\phi^{3}$ theory. It is also possible to retract the
legs, cf. [24, 25].

##  6 Conclusion and outlook

###  6.1 Conclusion

We have shown that pathlike coalgebras in addition to quiver and incidence
coalgebras capture the two types of bialgebras stemming from Feynman
categories, such as those of Baues, Goncharov and those of Connes–Kreimer.
This realization provides several outcomes.

The structure of path-like coalgebras exactly answers the question of what the
harbinger of the antipode in the various constructions is. It precisely
identifies the grouplike elements as the obstruction to the existence of the
antipode. For characters, their values on theses elements determines their
$\star$-invertibility. This also explains the appearance of quotients. These
correspond to restrictions on the characters. The most common, which says that
all grouplike elements have character value 1, leads to the connected
quotient. Other restrictions, like grouplike invertible or grouplike central,
yield different quotients and explain the universality of the quantum
deformation, which in turn explain the naturally of Brown’s coaction.

This sets the stage for further work.

###  6.2 Outlook

Our analysis is based on using filtrations to reduce to an initial part, where
the antipode or convolution inverse is known. The notion of pathlike
coalgebras codifies this situation if the initial part is setlike. Section
2.2.4, however, can be taken as a clue that there could be some other
structures, which do not necessitate only grouplike elements, but allows for
more information about the isomorphisms. The following example of the
Drinfel’d double leads the way to a new research in this direction. Physically
this corresponds to a (finite) gauge group. In the infinite case, one has to
take the precautions spelled out in Appendix A.

In particular, we present a double categorical interpretation in which the
antipode basically switches the horizontal and vertical composition. The
appearance of double categories is natural in view of [5, 36]. It opens the
way to a different version of connectedness, whose lowest degree is of
Drinfel’d double type rather than setlike, but keeps a link between the two
approaches. We will consider the generalization of the bialgebra and Hopf
structures along these lines in [42]. There is a general construction, see
[32, Section IX.4] of a Drinfel’d double for a finite dimensional Hopf algebra
with invertible antipode; we will concentrate on the case $k[G]$. This is an
interesting Hopf algebra whose algebra part comes from a category.

######  Definition 6.1.

For a finite group $G$ the Drinfel’d double $D(k[G])$ is the quasi-triangular
quasi-Hopf algebra whose underlying vector space has the basis
$\,g\underset{\displaystyle x}{\boxvoid}$ with $x,g\in G$ $D(k[G])=\bigoplus
k\,g\underset{\displaystyle x}{\boxvoid}$ with bialgebra structure and
antipode given by

$\displaystyle\,g\underset{\displaystyle
x}{\boxvoid}\,h\underset{\displaystyle
y}{\boxvoid}=\delta_{g,xhx^{-1}}\,g\underset{\displaystyle
xy}{\boxvoid},\qquad\Delta(\,g\underset{\displaystyle
x}{\boxvoid})=\sum_{g_{1}g_{2}=g}\,g_{1}\underset{\displaystyle
x}{\boxvoid}\otimes\,g_{2}\underset{\displaystyle x}{\boxvoid}\,,\qquad
S(\,g\underset{\displaystyle
x}{\boxvoid})=\,x^{-1}g^{-1}x\underset{\displaystyle x^{-1}}{\boxvoid}\\!.$

The unit is $\sum_{g\in G}\,g\underset{\displaystyle e}{\boxvoid}$ and the
counit is defined by $\epsilon(\,e\underset{\displaystyle x}{\boxvoid})=1$ and
$\epsilon(\,g\underset{\displaystyle x}{\boxvoid})=0$ if $g\neq e$.

The algebra structure is the algebra of morphisms of the so-called loop
groupoid, $\Lambda\underline{G}$. This is the category which has the object
set $G$ and the morphism set $\,g\underset{\displaystyle
x}{\boxvoid}\in\operatorname{Hom}(g,x^{-1}gx)$, see, e.g., [40, 61]. It is
hence a colored monoid with colors $G$ with $s(\,g\underset{\displaystyle
x}{\boxvoid})=g$ and $t\big{(}\,g\underset{\displaystyle
x}{\boxvoid}=x^{-1}gx\big{)}$.

The coalgebra structure is not the dual, but the categorical coalgebra for the
groupoid $\amalg_{x\in G}\underline{G}$ which has $X=G$ as objects and has
morphisms $\,g\underset{\displaystyle x}{\boxvoid}\colon x\to x$. Accordingly,
the coalgebra is simply colored for the comodule structures for the set-like
coalgebra $k[G]\colon\lambda(\,g\underset{\displaystyle
x}{\boxvoid})=x\otimes\,g\underset{\displaystyle x}{\boxvoid}$,
$\rho(\,g\underset{\displaystyle x}{\boxvoid})=\,g\underset{\displaystyle
x}{\boxvoid}\otimes x$.

The notation $\,g\underset{\displaystyle x}{\boxvoid}$ goes back to [16],
where these are partition functions on a torus for a field theory with finite
gauge group. The two indices stand for the monodromies around the fundamental
cycles, cf. [33, 40].

######  Lemma 6.2.

The dual of $D(k[G])$ is a Hopf algebra, whose coproduct is the coalgebra
structure coming from the morphisms of a category. In particular, let
$\delta_{\,g\underset{\displaystyle x}{\boxvoid}}$ be the dual basis then

$\displaystyle\delta_{\,g\underset{\displaystyle
x}{\boxvoid}}\delta_{\,h\underset{\displaystyle
y}{\boxvoid}}=\delta_{x,y}\,gh\underset{\displaystyle
x}{\boxvoid},\qquad\Delta(\delta_{\,g\underset{\displaystyle
x}{\boxvoid}})=\sum_{y}\delta_{\,g\underset{\displaystyle
y}{\boxvoid}}\otimes\delta_{\,y^{-1}gy\underset{\displaystyle
y^{-1}x}{\boxvoid}},\qquad S(\delta_{\,g\underset{\displaystyle
x}{\boxvoid}})=\delta_{\,xg^{-1}x^{-1}\underset{\displaystyle
x^{-1}}{\boxvoid}}.$

###### Proof.

Straightforward computation using the dual basis
$\delta_{\,g\underset{\displaystyle x}{\boxvoid}}$. ∎

The product structure uses the underlying monoid structure of the
objects/colors. This is not quite a monoidal structure on the category
$\Lambda\underline{G}$. The definition of the antipode uses that the objects
(as colors) are invertible. But note, that $\,g\underset{\displaystyle
x}{\boxvoid}\otimes\,h\underset{\displaystyle y}{\boxvoid}$ would need to be a
morphism from $gh$ to $x^{-1}gxy^{-1}hy$ and if $x=y$ indeed
$\,gh\underset{\displaystyle x}{\boxvoid}$ is such a morphism. In general, if
$G$ is not Abelian no such morphism needs to exist. The correct notion is that
the Drinfel’d double is a subcategory of a natural double category structure
on $\operatorname{Iso}({\mathcal{C}})$, see [36, Appendix C], where in this
particular case ${\mathcal{C}}=\underline{G}$. This is commensurate with the
dictum: “A monoidal category is a two category with one object”. The
composability of horizontal 2-cells is then exactly the condition that $x=y$
in the above calculation. In particular, the symbols
$\,g\underset{\displaystyle x}{\boxvoid}$ are two-morphisms with the following
horizontal and vertical composition:

$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{x}$$\scriptstyle{\Downarrow\,g\underset{\displaystyle
x}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xgx^{-1}}$$\textstyle{\ast}$,$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{x}$$\scriptstyle{\Downarrow\,g\underset{\displaystyle
x}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xgx^{-1}}$$\scriptstyle{y}$$\scriptstyle{\Downarrow\atop\,xgx^{-1}\underset{\displaystyle
y}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xygy^{-1}x^{-1}}$$\textstyle{\ast}$=$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{xy}$$\scriptstyle{\Downarrow\,g\underset{\displaystyle
xy}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xy}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xyg(xy)^{-1}}$$\textstyle{\ast}$,$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{x}$$\scriptstyle{\Downarrow\,g\underset{\displaystyle
x}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{h}$$\scriptstyle{\Downarrow\,k\underset{\displaystyle
x}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xgx^{-1}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{hxx^{-1}}$$\textstyle{\ast}$=$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{gh}$$\scriptstyle{x}$$\scriptstyle{\Downarrow\,g\underset{\displaystyle
x}{\boxvoid}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{xghx^{-1}}$$\textstyle{\ast}$

The algebra structure is given by the categorical algebra for the vertical
composition, viz. $\circ_{h}$, as the colored multiplication with
$s_{v}(\,g\underset{\displaystyle
x}{\boxvoid})=t_{v}(\,g\underset{\displaystyle x}{\boxvoid})=x$:

$\displaystyle\,g\underset{\displaystyle
x}{\boxvoid}\cdot\,h\underset{\displaystyle
y}{\boxvoid}=\delta_{s(\,g\underset{\displaystyle
x}{\boxvoid})=t(\,h\underset{\displaystyle
y}{\boxvoid})}\,g\underset{\displaystyle
x}{\boxvoid}\circ_{v}\,h\underset{\displaystyle y}{\boxvoid}$

and $\Delta$ will be given as the dual of the horizontal composition with
$s_{h}(\,g\underset{\displaystyle
x}{\boxvoid})=g,t_{h}(\,g\underset{\displaystyle x}{\boxvoid})=xgx^{-1}$

$\displaystyle\Delta(\,g\underset{\displaystyle
x}{\boxvoid})=\sum_{(\,h\underset{\displaystyle
y}{\boxvoid},\,k\underset{\displaystyle
z}{\boxvoid})\colon\,h\underset{\displaystyle
y}{\boxvoid}\circ_{h}\,k\underset{\displaystyle
z}{\boxvoid}=\,g\underset{\displaystyle
x}{\boxvoid}}\,h\underset{\displaystyle
y}{\boxvoid}\otimes\,k\underset{\displaystyle z}{\boxvoid}\,,$

where we dropped the dual notation $\delta$.

Another direction of further research is the connection to the cubical
structure of the coproduct, whose analysis in this context appeared in [24,
Section 4.5]. Cubical complexes more generally appear for cubical Feynman
categories, [6]. These are intimately related to Cutkosky rules [45] and the
work [46]. To illustrate the cubical structure, we recall, cf. [24, equation
(4.29)], that simplicial objects have a coproduct whose summands are

$x_{(0,i_{1},\dots,i_{k-1},n)}\times
x_{(0,1,\dots,i_{1})}x_{(i_{1},i_{1}+1,\dots,i_{2})}\cdots
x_{(i_{k-1},i_{k-1}+1,\dots,n)}.$ (6.1)

This has a cubical structure, which can be understood in terms of the Serre
diagonal:

$\displaystyle\delta\colon\
P=[0,1]^{n-1}\xrightarrow{\;\;\cong\;\;}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\bigcup_{K\cup
L=\\{1,\dots,n-1\\}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{K}^{-}[0,1]^{n-1}\times\partial_{L}^{+}[0,1]^{n-1}\xrightarrow{\;\;\subset\;\;}P\times
P.$

The term given in (6.1) is identified with the subset
$L=\\{i_{1},\dots,i_{k-1}\\}$. In view of [46] one has an additional
interpretation. The right hand side can be seen as cuts (or stops) in the path
from $0$ to $n$ on the quiver $[n]=0\to 1\to\dots\to n$ and these can be
viewed as cuts of a linear graph. This is in line with the simplicial
structure of iterated integrals.

Furthermore, there is a symmetry in the Serre diagonal of the cube given by
interchanging $0$ and $1$ in the standard presentation. This switches the
coproduct to its opposite. These two observations should generalize to the
categorical setting, and when applied to graphs may explain some of the
results of [46]. A last aspect is functoriality of the construction, see [25,
Section 1.7], which is likely related the structure of cointeracting
coalgebras [23, 46] in the present context.

Furthermore, we identified the skew primitive elements (as indecomposables) as
being of central importance. This parallels the role of primitives in the
usual connected case. In particular, there should be $B_{+}$ operators
associated to them. These are indeed explicitly present and key elements for
$E_{2}$-operads and hence for Gerstenhaber brackets and Deligne’s conjecture
[43]. Moreover, there is a connection between these elements and the terms of
the master equations [38] and [41, Section 7.5]. The general structure of
$B_{+}$ operators will be taken up in [37], where we will also make the
connection to Hochschild complexes.

##  Appendix A Duals, actions and colored algebras

###  A.1 Duals

The dual of an $R$ module $M^{*}$ is the $R$-module
$\operatorname{Hom}_{R}(M,R)$. If $M$ is graded, $M=\bigoplus_{x\in X}M_{x}$
then the graded dual is $\check{M}=\bigoplus_{x\in X}M_{x}^{*}$.

The dual of a coalgebra $C$ is the algebra $C^{*}:=\operatorname{Hom}(C,R)$
with product $fg(c)=(f\otimes g)(\Delta c)$. If $C$ has a counit
$\epsilon_{C}$ as a coalgebra then $\epsilon_{C}\in C^{*}$ is a unit for the
product by definition.

The dual of a finite dimensional algebra $A^{*}=\operatorname{Hom}_{R}(A,R)$
is a coalgebra with the comultiplication $\Delta(f)(a\otimes b):=f(ab)$. In
general, $\Delta^{*}$ is a morphism $A^{*}\to(A\otimes A)^{*}$ and there is an
injection $A^{*}\otimes A^{*}\to(A\otimes A)^{*}$, which is an isomorphism in
the case of a finite dimensional algebra. Therefore, one restricts to the so-
called finite dual $A^{\circ}$ which is the maximal subspace of $A^{*}$ such
that $\Delta(A^{\circ})\subset A^{\circ}\otimes A^{\circ}$. For a Dedekind
ring, this is [11]:

$\displaystyle A^{\circ}=\\{f\in A^{*}\,|\,\operatorname{Ker}(f)\mbox{
contains an ideal $I$ for which $A/I$ is finitely generated}\\}.$

The Hopf-dual of a Hopf algebra over a field is defined to be $H^{\circ}$ with
the induced structures. For a graded algebra $A=\bigoplus_{x\in X}A_{x}$, we
can consider the graded finite dual $\check{A}^{\circ}=\bigoplus_{x\in
X}X_{x}^{\circ}$.

###  A.2 Colored coalgebras and algebras

If $C[X]$ is the setlike coalgebra on $X$, $C[X]^{*}$ has multiplication
$fg(x)=f(x)g(x)$ and the constant function $u(x)\equiv 1$ is the unit. In
particular, for the dual basis $\delta_{x}\delta_{y}=\delta_{x,y}\delta_{x}$
and $u=\sum_{x\in X}\delta_{x}$. This is well defined as the application to
any $c\in C[X]$ yields only a finite sum. Define the algebra $A[X]$ as the $R$
algebra on the monoid $X$ defined by $xy=\delta_{x,y}y$. This is a subalgebra
of the dual algebra $A[X]\subset C^{*}[X]$ using the identification
$x\leftrightarrow\delta_{x}$. Generators are the functions with finite support
on $X$. $A[X]$ is not unital unless $X$ is finite, but it is almost unital in
the sense that $u=\sum_{x}x\in C[X]^{*}$ acts as a unit, and when $u$ is
restricted to $A[X]$, it acts locally as a finite sum.

If $X$ is finite, this yields a bialgebra structure on $R^{(X)}$. This is a
Hopf algebra if and only if $X$ only has one element, as the putative antipode
must satisfy $S(x)=x^{-1}$, but the $x$ act as projectors.

######  Lemma A.1.

If $R$ is a Dedekind ring then $A[X]^{\circ}=\check{A}[X]^{\circ}=C[X]$ under
the identification $x\leftrightarrow\delta_{x}$. The coproduct is given by
decomposition:

$\displaystyle\Delta(\delta_{x})=\sum_{(x_{1},x_{2})\colon
x_{1}x_{2}=x}\delta_{x_{1}}\otimes\delta_{x_{2}}.$

###### Proof.

By [11] $f\in A[X]^{\circ}$ if and only if $A[X]f$ is finitely generated.
Since $xf(y)=f(yx)=\delta_{x,y}f(x)$, we see that $f(x)\delta_{x}\in A[X]f$.
As the $\delta_{x}$ are a basis, $A[X]f$ being finitely generated is
equivalent to almost all $f(x)=0$. The coproduct on $\check{A}[X]^{\circ}$ is
given by $\Delta(\delta_{x})=\delta_{x}\otimes\delta_{x}$ and the
identification is again by $x\leftrightarrow\delta_{x}$. ∎

Linearizing the construction above yields the following definition whose
unital version is equivalent to a category enriched in $R$-Mod.

######  Definition A.2.

A colored algebra is double graded $R$ module $A=\bigoplus_{(x,y)\in X\times
X}A_{x,y}$ with $R$-bilinear associative multiplications $A_{x,z}\otimes
A_{z,y}\to A_{x,y}$. It is unital if there are elements ${\mathrm{id}}_{x}\in
A_{x,x}$ which are left and right units under the multiplication. It is split
unital if $A_{x,x}={\rm Rid}_{x}\oplus\bar{A}_{x,x}$.

For a colored monoid the associated colored algebra is given by
$A_{x,y}=R^{(M_{x,y})}$ with the induced multiplication. The unital structure
is the induced structure.

Suppose $M$ is graded, $M=\bigoplus_{x\in X}M_{x}$, and hence has a right
coaction by $C[X]$. Specializing the general formula for an algebra action
induced from a right coalgebra coaction, $f\cdot m=({\mathrm{id}}\otimes
f)(\rho(m))$, the coaction corresponding to a grading in turn defines an
action of $A[X]$ in which the $\delta_{x}$ act by a complete set of orthogonal
projectors $\pi_{x}\colon M\to M_{x}$, that is $\delta_{x}m=\pi_{x}(m)$. The
operation $u=\sum_{x\in X}\delta_{x}$ acts as unit. This sum is finite when
acting on any $m\in M$ as $M$ is the direct sum of its graded components.

######  Lemma A.3.

A colored algebra is an algebra in the category of $A[x]$-bimodules.

###### Proof.

The multiplication respects the outer grading and taking the product
$\otimes_{A[X]}$ amounts to having the inner gradings coincide. ∎

######  Remark A.4.

Note that if one defines a colored algebra as an algebra in $A[x]$-bimodules
then one does not obtain finiteness. Indeed $C^{*}[X]$ is an $A[X]$ bialgebra.
To remedy this one can postulate that $u$ acts locally finitely. The question
that remains is where the finiteness assumptions should be kept and where they
could be relaxed.

##  Appendix B Rota–Baxter operators and algebras

######  Definition B.1.

A Rota–Baxter (RB) operator of weight $\lambda$ on an associative $R$-algebra
$A$ a is a linear operator $T\colon A\to A$ that satisfies the equation

$T(X)T(Y)=T(T(X)(Y))+T(XT(Y))+\lambda T(XY).$ (B.1)

A Rota–Baxter (RB) algebra is a unital associative algebra with a choice of
Rota–Baxter operator.

Note that if $T$ is an RB-operator, so is $(1-T)$, furthermore (B.1) says that
${\rm Im}(T)$, and similarly ${\rm Im}(1-T)$, are subalgebras of $A$, which
will be called $A_{+}$ and $A_{-}$. As $(1-T)+T=1$, we see that
$A=A_{+}+A_{-}$.

NB: This sum does not need to be direct.

The pair $(A_{+},A_{-})$ is called the Birkhoff decomposition of $A$ defined
by $T$.

######  Remark B.2 (scaling).

Scaling the RB operator by $\mu$, we see that $\mu T$ is a RB algebra of
weight $\lambda\mu$, so that if $\lambda\in R^{*}$ is invertible, we can
always scale to have $\lambda=0$ or $\lambda=-1$.

######  Example B.3.

The standard example are Laurent series $k((x))=k[[x]]\big{[}x^{-1}\big{]}$,
with $T$ being the projection onto the pole part, is an RB-algebra with
$\lambda=-1$.

A generalization of this uses projectors, which are operators with $T^{2}=1$.
In this case, $1-T$ is also a projector and the two projectors are orthogonal
$T(1-T)=0$. For a projector $T$ on an algebra $A$, set $(1-T)A=A_{+}$ and
$T(A)=A_{-}$; then $A=A_{+}\oplus A_{-}$.

######  Proposition B.4 (projectors and subalgebras).

Given a projector $T$, if $A_{+}$ and $A_{-}$ are subalgebras, then $T$ is an
RB-operator with weight $\lambda=-1$. Furthermore, $T(1)$ is an idempotent,
$T(1)^{2}=T(1)$, and multiplication by $T(1)$ is a projector onto a subspace
of $A_{-}$.

In particular, if $T(1)=0$, then $A_{+}$ is a unital subalgebra, and, if
$T(1)$ is regular, then $T(1)=1$ and $A_{-}$ is a unital subalgebra.

###### Proof.

The conditions means that $(1-T)$ and $T$ are the projectors onto
$A=A_{+}\oplus A_{-}$. Since $A_{-}$ is a subalgebra, we have that

$0=(1-T)[T(X)T(Y)]=T(X)T(Y)-T(T(X)T(Y)).$ (B.2)

Since $A_{+}$ we have that

$\displaystyle
0=T[(1\\!-\\!T)(X)(1\\!-\\!T)(Y))]=T(XY)\\!-\\!T(T(X)Y)\\!-\\!T(XT(Y))+T(T(X)T(Y)).$
(B.3)

Plugging (B.2) into (B.3), we obtain (B.1) for $\lambda=-1$. Plugging in
$X=Y=1$ and using that $T^{2}(1)=T(1)$ we obtain that $T(1)^{2}=T(1)$ and
hence multiplication by $T(1)$ is a projector. Plugging in $Y=1$ we obtain
$T(1)(1-T)(X)=0$. ∎

If $A$ is regular, e.g., a domain, then there are only two possibilities
$T(1)=0$ or $T(1)=1$. Thus after possibly switching $T$ and $1-T$, we have
that $A=A_{+}\oplus A_{-}$ with $A_{+}$ a unital subalgebra and $A_{-}$ a non-
unital subalgebra. Vice-versa an RB-operator for $\lambda=-1$ comes from a
subdirect sum.

######  Definition B.5 ([3]).

Two subalgebras $(A_{-},A_{+})$ form a subdirect sum of $A$ if there exists a
subalgebra $C\in A\times B$ which satisfies that for any $a\in A$ there is a
unique representation $a=a_{-}+a_{+}$, with $(a_{-},a_{+})\in C$.

Such a decomposition defines $T(a)=a_{-}$ which satisfies (B.1) for
$\lambda=-1$. Vice-versa given an RB $T$ with $\lambda=-1$, setting
$A_{-}=T(A),A_{+}=(1-T)(A)$, $C=\\{(T(a),(1-T)(a))\colon a\in A\\}\subset
A_{-}\times A_{+}$ gives the subdirect sum.

######  Theorem B.6 ([3]).

The set of RB-operators $T$ with weight $\lambda=-1$ is in bijection with the
subdirect sum decomposition of $A$ under the above correspondence.

##  Appendix C Set version of graphs

We give a purely set based, non-graphical version here. A reference for the
graphical version is [41, Section 2.1, Appendix A]. The basic objects will be
${\mathcal{V}}=\operatorname{Iso}(\mathcal{F}{\rm in}\mathcal{S}{\rm et})$.
This fixes the $\operatorname{Iso}({\mathcal{V}}^{\otimes})$ to have as
objects finite tuples of sets $(S_{1},\dots,S_{n})$, where the empty set is
allowed both as a tuple and in the entries. The morphisms are componentwise or
inner isomorphisms together with outer permutations, which act as follows: A
permutation $p\in\SS_{n}$ acts by
$p(S_{1},\dots,S_{n})=(S_{p(1)},\dots,S_{p(n)})$. The inner isomorphisms
$\sigma\in\operatorname{Aut}(S_{i})$ act as
$({\mathrm{id}},\dots,{\mathrm{id}},\sigma,{\mathrm{id}},\dots,{\mathrm{id}})$.
The monoidal product is the joining of tuples. In graphical form, these are
aggregates, that is a disjoint unions of corollas. The outer permutations
permute the corollas and the inner isomorphisms the flags of the individual
corollas.

There are two versions of the category: the connected and the non-connected
version. To write the relations in a concise fashion, we abuse notation and
write $\phi$ for
$({\mathrm{id}},\dots,{\mathrm{id}},\phi,{\mathrm{id}},\dots,{\mathrm{id}})$.

For the connected version the morphisms are generated under the monoidal
product and concatenation by the isomorphisms and the following morphisms,
which called simple edge contractions and simple loop contractions:

$\displaystyle\,{\vphantom{\circ}}_{s}{\circ}_{t}\colon\
(S,T)\to(S\setminus{\sigma}\amalg
T\setminus\\{t\\}),\qquad\circ_{ss^{\prime}}\colon\
(S)\to(S\setminus\\{s,s^{\prime}\\}).$

These satisfy the following relations.

  1. 1.

The basic morphisms are equivariant with respect to isomorphisms in the sense
that: For $\sigma\in\operatorname{Aut}(S)$,
$\sigma^{\prime}\in\operatorname{Aut}(T)$:

$\displaystyle\,{\vphantom{\circ}}_{\sigma(s)}{\circ}_{\sigma^{\prime}(t)}(\sigma,\sigma^{\prime})=\sigma\amalg\sigma^{\prime}|_{S\setminus\\{s\\}\amalg
T\setminus\\{t\\}}\,{\vphantom{\circ}}_{s}{\circ}_{t},\qquad\circ_{\sigma(s),\sigma(s^{\prime})}\sigma=\sigma|_{S\setminus\\{s,s^{\prime}\\}}\circ_{s,s^{\prime}}\\!.$

For an outer permutation:

$\displaystyle p\,\circ_{s,s^{\prime}}=\circ_{s,s^{\prime}}p,\qquad
p^{\prime}\,{\vphantom{\circ}}_{p(s)}{\circ}_{p(t)}=\,{\vphantom{\circ}}_{s}{\circ}_{t}p,$

where $p^{\prime}$ is the image of the permutation in the embedding
$\SS_{n-1}\to\SS_{n}$ given by the induced block permutation.

  2. 2.

Loop contractions commute:

$\displaystyle\circ_{s_{1},s_{1}^{\prime}}\circ_{s_{2},s_{2}^{\prime}}=\circ_{s_{2},s_{2}^{\prime}}\circ_{s_{1},s_{1}^{\prime}}\\!.$

  3. 3.

Loop and edge contractions commute:

$\displaystyle\,{\vphantom{\circ}}_{s_{1}}{\circ}_{t_{1}}\circ_{s_{2}s_{2}^{\prime}}=\circ_{s_{2}s_{2}^{\prime}}\,{\vphantom{\circ}}_{s_{1}}{\circ}_{t_{1}}.$

  4. 4.

Edge contractions commute if the “edges” do not form a cycle. For $s_{1}\in
S_{1}$, $t_{1}\in T_{1}$, $s_{2}\in S_{2}$, $t_{2}\in T_{2}$ and
$|\\{S_{1},T_{1},S_{2},T_{2}\\}|\geq 3$

$\displaystyle\,{\vphantom{\circ}}_{s_{1}}{\circ}_{t_{1}}\,{\vphantom{\circ}}_{s_{2}}{\circ}_{t_{2}}=\,{\vphantom{\circ}}_{s_{1}}{\circ}_{t_{1}}\,{\vphantom{\circ}}_{s_{2}}{\circ}_{t_{2}}.$

  5. 5.

Edge contractions commute even if they form a cycle, only that after one edge
contraction, the second edge contraction is a loop contraction. For
$s_{1},s_{2}\in S$, $t_{1},t_{2}\in T$,

$\displaystyle\circ_{s_{2},t_{2}}\,{\vphantom{\circ}}_{s_{1}}{\circ}_{t_{1}}=\circ_{s_{1},t_{1}}\,{\vphantom{\circ}}_{s_{2}}{\circ}_{t_{2}}.$

In the non-connected case, there is an additional generating morphism called
simple merger:

$\displaystyle\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}\colon\
(S,T)\to(S\amalg T).$

These satisfy the relations:

  1. 1.

Mergers are equivariant with respect to the internal action

$\displaystyle\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}(\sigma,{\mathrm{id}})=(\sigma\amalg{\mathrm{id}})\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}.$

  2. 2.

Mergers are equivariant with respect to the outer action. On
$(S_{1},\dots,S_{n})$:

$\displaystyle\,{\vphantom{\boxminus}}_{S_{i+1}}{\boxminus}_{S_{i}}=\,{\vphantom{\boxminus}}_{S_{i}}{\boxminus}_{S_{i+1}}\tau_{i,i+1}\qquad\mbox{and}\qquad
p\,{\vphantom{\boxminus}}_{S_{i}}{\boxminus}_{S_{i+1}}=\,{\vphantom{\boxminus}}_{S_{p^{\prime}(i)}}{\boxminus}_{S_{p^{\prime}(i+1}}p^{\prime},$

where $p^{\prime}$ is the image of the permutation $p$ under the embedding
$\SS_{n-1}\to\SS_{n}$ given by the block permutation with block $(i,i+1)$.

  3. 3.

Simple mergers commute

$\displaystyle\,{\vphantom{\boxminus}}_{S_{1}}{\boxminus}_{T_{1}}\,{\vphantom{\boxminus}}_{S_{2}}{\boxminus}_{T_{2}}=\,{\vphantom{\boxminus}}_{S_{2}}{\boxminus}_{T_{2}}\,{\vphantom{\boxminus}}_{S_{1}}{\boxminus}_{T_{1}}.$

  4. 4.

Mergers and edge/loop contractions on different sets commute.

  5. 5.

If $s\in S$ and $t\in T$, then

$\displaystyle\,{\vphantom{\circ}}_{s}{\circ}_{t}=\circ_{st}\,{\vphantom{\boxminus}}_{S}{\boxminus}_{T}.$

### Acknowledgements

This paper is the outgrowth of a long lasting conversations with Dirk Kreimer
whom we would like to congratulate and profoundly thank for his interest and
support. Much of the impetus for this work has come through interactions with
him and his group, and it would not have come into existence without Dirk. RK
wishes to thank the Humboldt University, the KMPB, and the Humboldt Foundation
for making these visits possible.

We furthermore would like to thank F. Brown, K. Yeats and C. Berger for
helpful discussions and the organizers of the conference Algebraic Structures
in Perturbative Quantum Field Theory at the IHES, in November 2020. The
wonderful interactions that took place there led to part of the results of
this paper. We also wish to thank the referees for their input which was very
beneficial for the paper.

## References

  * [1] Andruskiewitsch N., Ferrer Santos W., The beginnings of the theory of Hopf algebras, Acta Appl. Math. 108 (2009), 3–17.
  * [2] Artin M., Noncommutaive rings. Course notes, 1999, available at http://www-math.mit.edu/~etingof/artinnotes.pdf.
  * [3] Atkinson F.V., Some aspects of Baxter’s functional equation, J. Math. Anal. Appl. 7 (1963), 1–30.
  * [4] Baues H.J., The double bar and cobar constructions, Compositio Math. 43 (1981), 331–341.
  * [5] Berger C., Kaufmann R.M., Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra, arXiv:2201.10537.
  * [6] Berger C., Kaufmann R.M., Derived decorated Feynman categories, in preparation.
  * [7] Brown F., Motivic periods and the projective line minus three points, arXiv:1407.5165.
  * [8] Brown F., Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Number Theory Phys. 11 (2017), 453–556, arXiv:1512.06409.
  * [9] Butcher J.C., An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106.
  * [10] Cartier P., A primer of Hopf algebras, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 537–615.
  * [11] Chen C.Y., Nichols W.D., A duality theorem for Hopf module algebras over Dedekind rings, Comm. Algebra 18 (1990), 3209–3221.
  * [12] Chen K.-T., Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217–246.
  * [13] Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203–242, arXiv:hep-th/9808042.
  * [14] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann–Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249–273, arXiv:hep-th/9912092.
  * [15] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), 215–241, arXiv:hep-th/0003188.
  * [16] Dijkgraaf R., Pasquier V., Roche P., Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18 (1990), 60–72.
  * [17] Duffaut Espinosa L.A., Ebrahimi-Fard K., Gray W.S., A combinatorial Hopf algebra for nonlinear output feedback control systems, J. Algebra 453 (2016), 609–643, arXiv:1406.5396.
  * [18] Dür A., Möbius functions, incidence algebras and power series representations, Lecture Notes in Math., Vol. 1202, Springer-Verlag, Berlin, 1986.
  * [19] Ebrahimi-Fard K., Guo L., Kreimer D., Integrable renormalization. II. The general case, Ann. Henri Poincaré 6 (2005), 369–395, arXiv:hep-th/0403118.
  * [20] Ebrahimi-Fard K., Kreimer D., The Hopf algebra approach to Feynman diagram calculations, J. Phys. A: Math. Gen. 38 (2005), R385–R407, arXiv:hep-th/0510202.
  * [21] Fiedorowicz Z., Loday J.-L., Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), 57–87.
  * [22] Foissy L., Les algèbres de Hopf des arbres enracinés décorés. II, Bull. Sci. Math. 126 (2002), 249–288, arXiv:math.QA/0105212.
  * [23] Foissy L., Cointeracting bialgebras, Talk at Algebraic Structures in Perturbative Quantum Field Theory, IHES, November 2020.
  * [24] Gálvez-Carrillo I., Kaufmann R.M., Tonks A., Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects, Commun. Number Theory Phys. 14 (2020), 1–90, arXiv:1607.00196.
  * [25] Gálvez-Carrillo I., Kaufmann R.M., Tonks A., Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation, Commun. Number Theory Phys. 14 (2020), 91–169, arXiv:1607.00196.
  * [26] Gelfand S.I., Manin Yu.I., Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
  * [27] Goncharov A.B., Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005), 209–284, arXiv:math.AG/0208144.
  * [28] Guo L., An introduction to Rota–Baxter algebra, Surveys of Modern Mathematics, Vol. 4, International Press, Somerville, MA, Higher Education Press, Beijing, 2012.
  * [29] Joni S.A., Rota G.-C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93–139.
  * [30] Joyal A., On disks, dualities, and $\Theta$ categories, Preprint, 1997.
  * [31] Joyal A., Street R., Braided tensor categories, Adv. Math. 102 (1993), 20–78.
  * [32] Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  * [33] Kaufmann R.M., Orbifolding Frobenius algebras, Internat. J. Math. 14 (2003), 573–617, arXiv:math.AG/0107163.
  * [34] Kaufmann R.M., On spineless cacti, Deligne’s conjecture and Connes–Kreimer’s Hopf algebra, Topology 46 (2007), 39–88.
  * [35] Kaufmann R.M., Lectures on Feynman categories, in 2016 MATRIX Annals, MATRIX Book Ser., Vol. 1, Springer, Cham, 2018, 375–438, arXiv:1702.06843.
  * [36] Kaufmann R.M., Feynman categories and representation theory, in Representations of Algebras, Geometry and Physics, Contemp. Math., Vol. 769, Amer. Math. Soc., Providence, RI, 2021, 11–84, arXiv:1911.10169.
  * [37] Kaufmann R.M., B-plus operators in field theory, Feynman categories and Hopf algebras, in preparation.
  * [38] Kaufmann R.M., Manin Yu., Zagier D., Higher Weil–Petersson volumes of moduli spaces of stable $n$-pointed curves, Comm. Math. Phys. 181 (1996), 763–787, arXiv:alg-geom/9604001.
  * [39] Kaufmann R.M., Medina-Mardones A.M., Cochain level May–Steenrod operations, Forum Math. 33 (2021), 1507–1526, arXiv:2010.02571.
  * [40] Kaufmann R.M., Pham D., The Drinfel’d double and twisting in stringy orbifold theory, Internat. J. Math. 20 (2009), 623–657, arXiv:0708.4006.
  * [41] Kaufmann R.M., Ward B.C., Feynman categories, Astérisque 387 (2017), vii+161 pages, arXiv:1312.1269.
  * [42] Kaufmann R.M., Yang M., Higher categorical Hopf algebras, in preparation.
  * [43] Kaufmann R.M., Zhang Y., Permutohedral structures on $E_{2}$-operads, Forum Math. 29 (2017), 1371–1411.
  * [44] Kreimer D., The core Hopf algebra, in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 313–321, arXiv:0902.1223.
  * [45] Kreimer D., Cutkosky rules from outer space, arXiv:1607.04861.
  * [46] Kreimer D., Yeats K., Algebraic interplay between renormalization and monodromy, arXiv:2105.05948.
  * [47] Leroux P., Les catégories de Möbius, Cah. Topol. Géom. Différ. 16 (1975), 280–282.
  * [48] Loday J.-L., Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften, Vol. 301, Springer-Verlag, Berlin, 1998.
  * [49] Mac Lane S., Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1978.
  * [50] Marcolli M., Ni X., Rota–Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications, J. Singul. 15 (2016), 80–117, arXiv:1408.3754.
  * [51] May J.P., Ponto K., More concise algebraic topology. Localization, completion, and model categories, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012.
  * [52] Moerdijk I., On the Connes–Kreimer construction of Hopf algebras, in Homotopy Methods in Algebraic Topology (Boulder, CO, 1999), Contemp. Math., Vol. 271, Amer. Math. Soc., Providence, RI, 2001, 311–321, arXiv:math-ph/9907010.
  * [53] Montgomery S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993\.
  * [54] Munthe-Kaas H.Z., Wright W.M., On the Hopf algebraic structure of Lie group integrators, Found. Comput. Math. 8 (2008), 227–257, arXiv:math.AC/0603023.
  * [55] Nichols W., Sweedler M., Hopf algebras and combinatorics, in Umbral Calculus and Hopf Algebras (Norman, Okla., 1978), Contemp. Math., Vol. 6, Amer. Math. Soc., Providence, R.I., 1982, 49–84.
  * [56] Quillen D.G., Homotopical algebra, Lecture Notes in Math., Vol. 43, Springer-Verlag, Berlin – New York, 1967.
  * [57] Rota G.-C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.
  * [58] Schmitt W.R., Antipodes and incidence coalgebras, J. Combin. Theory Ser. A 46 (1987), 264–290.
  * [59] Taft E.J., Wilson R.L., On antipodes in pointed Hopf algebras, J. Algebra 29 (1974), 27–32.
  * [60] Takeuchi M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561–582.
  * [61] Willerton S., The twisted Drinfeld double of a finite group via gerbes and finite groupoids, Algebr. Geom. Topol. 8 (2008), 1419–1457, arXiv:math.QA/0503266.


# Releasing Atmospheric Martian Dust in Sand Grain Impacts

T. Becker  University of Duisburg-Essen, Faculty of Physics, Lotharstr. 1,
47057 Duisburg, Germany  J. Teiser  University of Duisburg-Essen, Faculty of
Physics, Lotharstr. 1, 47057 Duisburg, Germany  T. Jardiel  Instituto de
Cerámica y Vidrio C/Kelsen 5, Campus Cantoblanco 28049 Madrid, Spain  M.
Peiteado  Instituto de Cerámica y Vidrio C/Kelsen 5, Campus Cantoblanco 28049
Madrid, Spain  O. Munoz  Instituto de Astrofísica de Andalucía, CSIC Glorieta
de la Astronomía s/n 18008 Granada, Spain  J. Martikainen  Instituto de
Astrofísica de Andalucía, CSIC Glorieta de la Astronomía s/n 18008 Granada,
Spain  J.C. Gomez Martin  Instituto de Astrofísica de Andalucía, CSIC Glorieta
de la Astronomía s/n 18008 Granada, Spain  G. Wurm  University of Duisburg-
Essen, Faculty of Physics, Lotharstr. 1, 47057 Duisburg, Germany

###### Abstract

Emission of dust up to a few micrometer in size by impacts of sand grains
during saltation is thought to be one source of dust within the Martian
atmosphere. To study this dust fraction, we carried out laboratory impact
experiments. Small numbers of particles of about 200µm in diameter impacted a
simulated Martian soil (bimodal Mars Global Simulant). Impacts occurred at
angles of $\sim 18^{\circ}$ in vacuum with an impact speed of $\sim 1\rm m/s$.
Ejected dust was captured on adjacent microscope slides and the emitted
particle size distribution (PSD) was found to be related to the soil PSD. We
find that the ejection of clay sized dust gets increasingly harder the smaller
these grains are. However, in spite of strong cohesive forces, individual
impacts emit dust of 1µm and less, i.e. dust in the size range that can be
suspended in the Martian atmosphere. More generally, the probability of
ejecting dust of a given size can be characterized by a power law in the size
range between 0.5µm and 5µm (diameter).

Mars, Exoplanet atmospheres

##  1 Introduction

Mars has a non-negligible but thin atmosphere with an average surface pressure
of about 6 hPa (Zurek, 2017). While this is too little to suspend grains as
large as Earth’s atmosphere is capable of, small dust particles can persist.
Airborne dust has been observed and modeled to be on the order of 1µm in
effective radius during clear skies and temporarily up to a few µm during dust
storms (Clancy et al., 1995, 2010; Määttänen et al., 2013; Lemmon et al.,
2019; Chen-Chen et al., 2019; D’Aversa et al., 2022).

Micrometer dust in the atmosphere is a fundamental component of the planet’s
climate system e.g. increasing the water content at high altitude (Neary et
al., 2020; Vandaele et al., 2019; Holmes et al., 2021). However, dust
entrained by the atmosphere is not suspended permanently. There is a
continuous sedimentation of dust as evident in the degradation of solar panels
on Martian landers (Lorenz et al., 2021). Therefore, atmospheric dust has to
be replenished continuously.

With dust devils and local dust storms being observed (Stanzel et al., 2008;
Reiss et al., 2016; El-Said et al., 2020; Baker et al., 2021; Favaro et al.,
2021), gusts of wind picking up dust at rover sites (Newman et al., 2021), and
dunes migrating in some places (Banks et al., 2018), it is obvious that at
least some fraction of the entrained dust is related to wind driven
activities. Plenty of work has been carried out on wind-driven saltation,
where sand is moving by means of short-trajectory hops along the surface,
aiding further lifting with each impact (Greeley et al., 1976, 1980; Iversen
et al., 1976; Iversen & White, 1982; Merrison, 2012; Kok et al., 2014;
Musiolik et al., 2018; Swann et al., 2020; Kruss et al., 2020). This implies a
kind of bimodal transport behaviour. While micrometer dust can be injected
into the atmosphere, typically, sand grains of about 100µm are the easiest to
be moved by wind on the surface.

This existence of a preferred particle size in motion is a consequence of the
underlying force balance between the acting forces - gravity, cohesion, and
gas drag. For large particles, gravity, going with the volume of a particle,
dominates over cohesion which itself linearly depends on the grain size.
Therefore, at the large particle size end, gas drag (depending on the
particle’s cross section) decreases compared to gravity and large grains are
ever harder to be moved.

On the small size end, cohesion dominates, eventually, and the ratio between
gas drag force and cohesion decreases towards smaller grain size, also making
it ever harder to lift smaller grains. Interpolating between both extremes,
there is a sweet spot for particles being most susceptible to wind drag at
about 100 µm under Martian conditions (Iversen et al., 1976; Greeley et al.,
1980; Shao & Lu, 2000). A corresponding plot, in which different measurements
have been combined by Neakrase et al. (2016), shows the threshold friction
velocity corresponding to grain size (fig. 5).

These findings are consistent with observations on Mars. For Gale crater,
Weitz et al. (2018) find that, volumentrically, most particles in aeolian
bedforms are in the size range of $50\rm\mu m$ to $150\rm\mu m$. The grains
found by the Phoenix lander are in the range between $20\rm\mu m$ to
$100\rm\mu m$ (Goetz et al., 2010). Noting that Goetz et al. (2010) give
number size distributions, these values are comparable. We considered these
size constraints to set up our simulated Martian soil.

Having listed a number of different size fractions by now, we note that not
all of these sizes fall in single well defined classification schemes of
sediments. The large grains observed on Mars mentioned above span a range from
medium silt to fine sand. Also ”dust” is rather unspecified with respect to
size so far. To make this somewhat more quantitative and in view of the
application to particles within the Martian atmosphere, we refer to ”dust” as
particles which can be suspended for long time in the atmosphere, i.e.
essentially being restricted to clay size particles up to a few micrometer on
Mars, reaching slightly into the very fine silt region.

Dust can be lifted during aeolian activity by different mechanisms, with the
simplest case being direct dust suspension due to gas drag. For Earth, direct
dust entrainment by wind has e.g. been discussed by Loosmore & Hunt (2000).
However, if potential saltators are present they are picked up first. Martian
friction velocities are always just on the lower limit of what is needed to
move these saltators (Neakrase et al., 2016), implying they are regularly much
too low to pick up dust directly. Wind driven dust fluxes are therefore rather
regularly observed in combination with saltation of larger sand grains (Shao
et al., 1993).

With the 100 µm fraction being moved first (fig. 5). it is only natural that
wind driven dust flux can only be observed together with a movement of larger
sand grains. As those are too heavy to be picked up by the wind they move
along ballistic trajectories by steadily bouncing off and reimpacting the
surface after being liberated by the wind. This idea is visualized in fig. 1,
which, in principle, is not new but similar to many standard sketches of this
kind. In our choice of sketch, we use an image from a wind tunnel experiment
by Kruss et al. (2020) as underlying background. We do so in order to point
out the major aspects for the experiments reported here. The wind tunnel
experiment was carried out at the threshold of saltation under Martian gravity
on a parabolic flight. The size of the observed saltating grains was on the
order of 100µm.

Figure 1: Principle of dust suspension by impact splash of saltating grains.
The underlying image is from Kruss et al. (2020) and shows a superposition of
images of saltating grains under Martian gravity with a time interval of about
2 ms. The ”suspended dust grains” inset is only a sketch but refers to the
dust fraction sampled in the experiments.

Furthermore, impact velocities at the threshold are of the order of 1 m/s and
impacts occur inclined. Therefore, this sketch already includes some
quantitative aspects. However, the dust fraction being present in these
experiments could not be detected.

Additional to direct dust entrainment or liberation of dust by saltating
impacts of sand grains, other mechanisms are deemed to have an effect on
ejecting dust into the Martian atmosphere, as well. Electrostatic forces are
among those mechanisms. It has e.g. been shown that electrostatic forces can
directly aid lifting (Renno & Kok, 2008; Holstein-Rathlou et al., 2010). Kok &
Renno (2008) showed, that saltating particles can charge by repeated bouncing
from the surface. Those charged particles then build an electrical field
between them and the ground (Schmidt et al., 1998; Zheng et al., 2003). This
further aids lifting of charged dust grains emitted upon saltating impacts. In
fact, field measurements by Esposito et al. (2016) showed that the amount of
lifted dust can increase by a factor of 10 in view of high electric fields.

As far as airborne dust size distributions are concerned, there have been wind
tunnel experiments where the dust fraction has been observed downstream
(Gillette et al., 1974; Alfaro et al., 1998). As dust emission is of utmost
importance for life on Earth, there are also frequent field measurements from
various arid regions as discussed by Mahowald et al. (2014).

Some of these dust size distributions are shown to be in agreement with a
model by Kok (2011), who considered the particle size distribution (PSD) to be
the result of fragmentation of brittle dust aggregates. Other distributions
have larger deviations in comparison to that model and may better be
quantified by a simple power law. One goal of our work is to connect the
observed dust size distribution to the size distribution of the initial
reservoir of grains being present if an impact ejects dust.

There are not many further analytic models to explain dust size distributions
beside the brittle fragmentation model by Kok (2011). One difficulty is that
experimental or field data which are compared against each other, mostly refer
to data collected in well developed saltation cascades. These cascades are
rather complex as many parameters such as impactor sizes, collision velocities
and soil properties determine the outcome. Additionally, the effect that
individual saltator impacts have on overall dust emission is difficult to
evaluate under continuous saltation conditions. In that regard, we aim to
provide experimental data on what PSD a limited number of only a few
independent impacts generate from a well characterized soil.

We are mostly interested in dust particles with sizes of a few µm, which can
be suspended on Mars. With grains of this size, there are a couple of studies
on ejection related to planet formation. In these studies, dust aggregates and
their collisions are a fundamental process (Wurm & Teiser, 2021).

It might be worth noting at this point that not only the size of a free
particle plays a role, but also its composition. If it is an aggregate
consisting of smaller grains for example, its stability upon collisions is
greatly reduced. Furthermore, as gas drag is dependent on the surface to mass
ratio, aerodynamic properties are influenced by particle composition as well.
However, not every non-spherical particle has to be an aggregate in this sense
if its individual grains are rather cemented together.

Fragmentation of dust aggregates consisting of micron-sized grains usually
creates power-law size distributions of fragments, where the number of
fragments steeply decreases toward larger grain sizes (Wurm et al., 2005;
Deckers & Teiser, 2014). This is in stark contrast to the size distributions
of aeolian dust mentioned above (Gillette et al., 1974; Alfaro et al., 1998;
Mahowald et al., 2014). Another study by Krijt & Kama (2014) gives a model for
the size distribution of aggregate fragments. According to it, an aggregate
purely consisting of 1µm particles, would break into fragments (also
aggregates) with a minimum size substantially larger than 1µm. Following this
model, even a soil containing fine 1µm dust, would not necessarily release
individual grains upon impact of a saltating particle. Dominik & Tielens
(1997) studied the collisions of micron-sized dust aggregates (including
aggregates consisting of particles of different sizes) and quantify at which
speed or impact energy individual (sub)micron-sized grains and aggregates are
ejected. However, they give no size distribution and their study would not
account for a wide range of particle sizes being present in the soil.

This shows that there is not a simple model to account for all details of dust
ejection in an arrangement of brittle, granular and cohesive matter. The
outcome of saltation on dust emission depends on the soil and especially the
behaviour of the cohesive 1µm size fraction is not well understood.

With this in mind, we study the effect of individual impacts into well
prepared Martian analog soil. Our major goals are to determine (1) if the most
cohesive fraction of small µm-sized dust particles can directly be released
upon impact of a (saltating) sand grain, (2) if there is a generic size
distribution of ejecta, and (3) if we can pin down in more detail how much
mass of dust in the suspension size range can be ejected in a single impact,
i.e. to give an absolute value to scale the size distribution from point 2.

##  2 Experiments

To achieve these goals, a number of different aspects have to be considered
for impact experiments. First, a suitable, well-characterized Martian analog
soil has to be set up. Second, an experiment has to be set up to generate
impacts with a limited number of grains of suitable size into this soil. These
impacts have to be independent of each other. The impact parameters (angle,
speed) should be appropriate for impacts of sand grains at the saltation
threshold on Mars. Finally, these impacts and especially their microscopic
ejecta have to be observed to allow a quantification of size and, in the best
case, absolute ejecta mass or volume determinations.

###  2.1 Soil: Martian simulant

There are several soils currently in use to serve as Martian analogs with
respect to different parameters. We choose the Mars Global Simulant (MGS) here
(Cannon et al., 2019) as it is based on a mineralogical composition which we
consider appropriate for simulating cohesion and impact dynamics. We
conditioned a sample according to the following rationale: The top layer of
Martian soil is strongly dependent on the sedimentation of airborne dust.
Ongoing sedimentation of airborne dust provides a thick top layer of µm-sized
dust. However, if saltation occurs periodically - which after all is the
premise of the dust liberation studied here - there would be a continuous
mixing of large saltators and dust. In an idealized case, we therefore have
two important size scales: the large saltating grains and the atmospheric
dust. With the measured size distributions on Mars for the large (silt to sand
size) grains and general scale of saltating grains in mind, we prepared the
coarse fraction of particles with sizes between 20 and 180µm by sieving the
original MGS sample. The size distribution of this coarse fraction is shown in
fig. 2.

Figure 2: Volume size distributions of the MGS sample; The sample is a mix of
two populations of equal total volumes.

To yield a fine fraction of suspendable dust ($\sim$ 1µm) the original MGS
sample was first milled for one hour using a Retsch PM100 planetary ball mill.
The milled powder was then suspended in a liquid medium and subjected to a
high-performance dispersion process for a few minutes using a high-shear
stirring device (Ultra-Turrax). The dispersed suspension was finally poured
onto a 20µm metal sieve for a wet sieving separation of the fine silt to clay
particles.

Figure 3: SEM image of the MGS sample. Dust of the fine fraction is sticking
to the surface of a particle of the coarse fraction.

This is the fraction that is important for us as the suspended dust is drawn
from this reservoir. We therefore consider the volume size distribution of
this fraction in comparison to our measurements. The size distribution of this
fraction can also be found in fig. 2.

We prepared the soil by mixing the dust fraction with the coarse 20-180µm
particle fraction. For the mixing ratio we picked 1 to 1 mass fractions. This
might be somewhat arbitrary, however, if the sample is stirred afterwards we
expect some self-regulating in how much small dust sticks to individual large
grains of the coarse fraction, though this may be debatable. Certainly, the
influence of the mixing ratio should be studied in the future but that is
beyond the scope of this study. The attained sample is then used as the top
layer of our dust bed. Its thickness is at least $2$mm, covering the filling
material, which only consists of the coarse grain fraction. An SEM image of a
particle from the coarse fraction, covered with smaller silt and clay grains
can be seen in fig. 3. Even though the experiment is performed under
terrestrial, rather than Martian gravity, we deem that of minor importance, as
efficiency of splashing events only shows a weak dependence on gravity (Durán
et al., 2011).

###  2.2 Impact setup

The soil prepared this way is used for impact experiments. A sketch of the
setup is shown in fig. 4.

Figure 4: Sketch of the impact setup; a) accelerator and impact bed with
microscope slides to the side, trapping airborne dust and wires to provide an
inhomogeneous electrical field. Camera on the right (distance bed-camera not
to scale); b) cross section of the accelerator unit; About 20-50 impactors are
released from a particle reservoir by rotating (gear motor) a portioning
wheel. They fall in the path of the plunger and are launched when it is
accelerated by the solenoid; the aperture guides the impactors to the soil.

The impactors are drawn from the same material (MGS) as the dust and coarse
fractions. We used a size fraction between 180 to 250µm for the first setup.
Smaller sized particles are not only more difficult to ration, but also pose a
problem for any moving parts of the dispensing mechanism. According to the
threshold curve in fig. 5, particles of 100µm would have been most desirable;
however, even though 100µm particles are the fraction with lowest threshold
velocity according to the plot, larger particles certainly are moved by wind
as well, as observations have shown (Greeley, 2002). Additionally, fig. 5
shows that the threshold windspeed for 250µm sand particles is still lower
than eg. for 20µm grains. A Study performed by Swann et al. (2020) even
suggests, that 200µm particles require equal or even lower threshold wind
velocities than 100µm particles to be moved under Martian conditions. Thus, we
are confident that we are within a reasonable size range to simulate saltators
on the Martian surface.

Figure 5: Curves of Threshold friction velocities depending on grain size for
different models as well as experimental results. (taken from Neakrase et al.
(2016))

These impactors are accelerated by a plunger and impact the soil at angles of
$18.8^{\circ}\pm 2.5^{\circ}$. This value fits well into the range suggested
by several studies (Bagnold, 1941; Chepil, 1945; White & Schulz, 1977; Jensen
& Sorensen, 1986). Impact velocities are in the range of $1.04\pm 0.2$m/s and
thus in agreement to the shear velocity at the threshold of saltation (Swann
et al., 2020). Due to the grain size being slightly larger than particles at
the saltation threshold, however, impact energies are somewhat higher. This
should not have too big of an impact, though. A study conducted by Bogdan et
al. (2020) showed, that there is no correlation between impact energy and
energy dissipated to the ejected particles for small ($\pm 50\%$) variation in
impact energy. Even though their setup was slightly different from the one
used in this study, as impacts occured in a $90^{\circ}$ angle and the ratio
of impactor to target particle size was smaller, we still believe there to be
enough parallels for their findings to be somewhat relevant to our case.

Each run, about 20-50 particles impact the model soil. The impacts are well
separated, so that we do not expect one impact to influence another (Bogdan et
al., 2020). We therefore see the sum of the outcome of a small number of
individual impacts. An exemplary image of such impact is shown in fig. 6
including the splash and the impactor trajectory, implied by a white sphere.
The footage of the impacts, taken with a NAC MEMRECAM HX-3 at 5000 fps, can be
used to analyze impact angles and speeds. The camera was mounted 15cm to the
side of the dust bed with the lens being in line with the bed’s surface (fig.
4).

Figure 6: Image of an impact, showing the impactor as white circle as well as
a cloud of ejecta.

The important small dust fraction is not visible with this method. Instead, it
is captured on two microscope slides placed next to the soil as seen in the
sketch in fig. 4. The capture efficiency is increased by electrostatic
attraction as two electrodes are placed behind each slide with a voltage of
about $1.6\rm kV$ applied between them.

As the micrometer dust is susceptible to gas drag - this is why it is
suspended even at hPa Martian pressure in the first place - it would easily be
missed in the experiments if it couples to a convective flow. Therefore, the
experiment is placed within a vacuum chamber and operated at a pressure of
$2\times 10^{-2}\rm mbar$.

The microscope images taken with an AxioCam ICc 1 mounted on a Zeiss
AxioImager.M2, are our main raw data. As the contrast is high in these images,
we segmented each image by setting a simple threshold brightness. There is no
large sensitivity to this threshold, as the small dust grains are well focused
and thus have a clear outline on the images.

##  3 Results

In total the data analyzed and presented here are from two runs of impact
experiments with electrical field applied and two without electrical field for
comparison. Each run consists of 7 individual launches, firing between 20 to
50 impactors each time. Afterwards, we analyzed 600 images from the slides on
both sides. For each identified particle, we measured the cross section and
derived the diameter $s$ of an equivalent area sphere. In fig. 7, we show the
retrieved dust size distributions together with the initial size distribution
of the soil.

For the two runs with electric field turned on, we count a total of 4738 and
4921 particles, respectively. For the two runs without field the counts are
1115 and 3018. The increase in captured dust with an electrical field being
present shows that there must be some sort of charge or dipole moment present
on individual dust grains after the impact. If that was inherent or due to
triboelectrical processes during the impact cannot be determined. In view of
electrostatic lifting mechansims mentioned above, it cannot be ruled out that
electrostatics add a lifting force here.

The electrical field also seems to remove some variability in the size
distributions. This can be explained by variations of trajectories of ejected
material. Due to the electric field, many particles with trajectories not
aimed at the slides are being captured as well, thus, homogenizing the number
of detected grains.

Overall, we see a significant decrease (40-80%) in particle count when doing
experiments without the electrical field, which might be due to a different
lifting efficiency or capture efficiency. Most important though, as seen in
fig. 7 the slope of the size distribution without field is the same as with
field for the fraction $<$5µm. This implies that electrostatic attraction does
not induce a size bias in that range. Above 5µm the size distributions with
and without field show less similarity.

Figure 7: Volume size distributions of ejected dust grains for the 2 runs with
electric field applied and one run without. Overplotted is a power law fitted
to the data (black solid line). The same fit, shifted by half an order of
magnitude, is overplotted for the experiment without field (black dashed
line). Also seen is the initial size distribution used in the sample (dust
bed). It is approximated with a dashed horizontal line.

Overall, the data has a lower cut-off due to the microscope resolution but
also due to the smallest grain size present in the sample. Due to the
resolution, we consider the data below 0.5µm to be only of a qualitative
nature, showing that also dust of this small size is ejected. In fact, all
particle sizes present in the initial size distribution are lifted. We
consider it an important finding that impact emission really includes the
smallest grains which are very cohesive and are not easily removed by direct
wind lifting.

The detection of larger particles holds a bias as well. They are less
susceptible to the electrical field due to their higher mass. Additionally,
with increasing size the probability of sticking to the glass slides greatly
decreases, lowering the number of emitted particles that are being captured.
We therefore only consider the data below 5µm to give a quantitatively
consistent size distribution. This is not a limitation to our task, as
airborne Martian dust is restricted to even smaller grains under fair weather
conditions.

In the unbiased size range from $s=0.5-5$µm the volume size distribution can
well be approximated by a power law

$\frac{dV}{d\left(\log\left(\frac{s}{\mathrm{\mu
m}}\right)\right)}=a\cdot\left(\frac{s}{\mathrm{\mu m}}\right)^{b}$ (1)

Consistent power laws are found for the two experiment runs analyzed in detail
(with electric field). We get $a=77.2\rm\mu m^{3}\pm 4.5\,\rm\mu m^{3}$ and
$b=3.72\pm 0.1$, which are the averages of the two values and the range
measured. We do not have a physical model, and as such we cannot assess
whether these results are within expectations. However, in order to give a
simple analytic description of the size distribution, we take this as an
empirical finding that helps us in quantifying dust fluxes.

It also has to be noted that the size distribution of the two runs are not
scaled in any way so the data show that not only the power law matches well
but also the absolute values are very robust with a variation for a given size
bin of typically less than 10 % in the size range between 0.5 to 5µm. This
implies that the average absolute value of ejected dust, in principle, can be
quantified on this level.

###  3.1 Total volumes

While the size distribution is somewhat uncertain for sizes of 1µm or less, as
outlined above, this is of minor importance for the total volume or mass of
grains ejected up to a certain size, as the volume size distribution increases
rather steeply with size. If we choose a hard upper cut-off diameter
$s_{cut}$, motivated by larger dust not being capable of being suspended, this
total volume can be estimated by integration of eq. 1 and gives

$V_{capt}=\frac{a}{b}\cdot\left(\frac{s_{cut}}{\mathrm{\mu m}}\right)^{b}$ (2)

This is an absolute value measured but only gives the total volume of dust
actually captured. It thus only refers to the sample area of the slides that
has been imaged and analyzed. To get an estimate for the total amount of dust,
we have to introduce a correction factor $f_{capt}$ which relates the true
volume of ejected dust $V_{true}$ up to $s_{cut}$ to the captured volume
$V_{capt}$

$V_{true}=\frac{V_{capt}}{N\cdot f_{capt}}$ (3)

Here, $N$ is the number of impactors per run. It varies between 20 and 50 due
to the dispension mechanisms inaccuracy. On average, a number of 27$\pm$12.7
particles impacts the dust bed every launch, summing up to $N=189\pm$38 over
all seven runs of the experiment.

####  3.1.1 Estimate of $f_{capt}$

As fig. 7 shows, the applied electrical field strongly increases the amount of
captured dust. It also keeps the power of the size distribution the same. As
electric forces on tribocharged grains are size dependent, but as we do not
see a size fractionation in the discussed size range with electric field
turned on, this suggests that essentially all free dust grains are attracted
by the electrodes and detected by the slides. This is certainly a
simplification but the easiest one to chose due to lack of information on the
grains.

We obtained our data by taking a series of images of the dust trapped on the
microscope slides. Overall, only $\sim 1\%$ of each slide was imaged, as about
36000 individual images would be necessary to cover the entirety of a slide.
However, we only acquire 300 images per slide. Not every region of the
microscope slides is covered with dust. As those areas have no quantitative
value for us, there aren´t any images taken of them, as would be when just
picking 300 random locations on the slide. Thus, the area of the 300 images
taken corresponds to a slightly larger area that has been scanned for dust.
Taking this biased sampling into account, a reasonable factor from only
analyzing parts of the slide would then be $f_{capt}=1/100$. Certainly, this
is only a rough estimate. However as a first benchmark, it might suffice.

###  3.2 Ejection probability

We want to relate the ejected as well as the detected mass (volume) and size
distribution to the initial size distribution of the soil. We show the initial
arbitrarily scaled size distribution in fig. 7.

The size distribution of the initial sample has been determined by light
scattering with a Mastersizer 3000, thus, possibly resulting in a small
difference in the definition of size with respect to the microscopic size
determination. Both kind of distributions might therefore be shifted slightly
relative to each other. Fortunately though, the original volume size
distribution is essentially flat in the studied range, noting some difference
at grain sizes of 1µm and below. Compared to the steep power law of ejected
dust we consider this to be of minor importance. To first order, we can
therefore approximate the initial size distribution as constant.

Relatively larger particles imaged on microscope slides could be aggregates of
smaller sized grains but this is not discernible in our data. Due to their
higher porosity or surface over mass ratio, aggregates are more susceptible to
electrical fields and also picked up by the wind more easily. In that regard
it would be of interest to know whether or not and, if, what fraction of the
particles we capture are aggregates. While the optical analysis methods of
this study are not sufficient to bring deeper insight into this matter, we are
currently setting up another experiment with the goal of studying just that
matter. That being said, concerning this study we settle with just analyzing
the sizes of captured particles and set aside any other properties these
particles might have. As such, with the flat initial distribution, the power
law found for the ejected dust can be assumed to be a direct probability
density $\rho$ of how much dust of a given size is ejected, or

$\rho=\frac{dV_{true}/dlogs}{dV_{init}/dlogs}\sim s^{b}.$ (4)

###  3.3 Absolute emitted dust masses

The probability distribution quantifies the size distribution of ejected
grains and may be adapted to more complex initial size distributions. Besides
the size distribution, the absolute masses or volumes ejected are important.
Our data provide some insight on this as well. For the given parameters of (1)
flat and slow impacts of $\sim$ 1 m/s, (2) flat initial fine dust
distribution, (3) a mass ratio of 1 to 1 of fine to coarse grains, and (4) an
average impactor size of $\sim 200$µm (volume of $V_{imp}=4\cdot 10^{6}\rm\mu
m^{3}$), we find an absolute ejected dust mass according to eq. 1 and 2.

Therefore, in our special setting, a ratio of

$R=\frac{V_{true}}{V_{impactor}}=c\cdot\left(\frac{s_{cut}}{1\mathrm{\mu
m}}\right)^{b}$ (5)

is ejected as dust with $c=a/(b\cdot N\cdot f_{capt}\cdot V_{imp})=3\cdot
10^{-6}$ $(a=77\mathrm{\mu m^{3}},b=3.72,N=189,V_{imp}=4\cdot
10^{6}\mathrm{\mu m^{3}})$. That e.g. gives an ejected mass of $R=2\cdot
10^{-4}$ times the impactor mass ejected in dust below 3µm per single impact.

##  4 Caveats

We note that these results and mass estimates are based on the kind of soil we
prepared in the laboratory. For a thick dust layer, fluffy or compact or
weathered soils with quite different particle sizes and sticking properties
between grains, this might and will be different.

The results are also tied to the specific impact velocity we used. While this
was motivated from other experiments on wind driven matter at low gravity just
beyond the threshold, other conditions are thinkable. Small variations in
impact velocity might not change the results much (Bogdan et al., 2020).
However, for strong increase there will be an increase in ejecta mass (Colwell
et al., 2008). How this would change the size distribution of the fine dust
fraction, we cannot say at this stage.

Our upper cut-off of 5 µm is the largest size for which we would argue that
our size distribution follows the power law given. Everything beyond could be
extrapolated but with some uncertainty. If these particles on the order of 10
µm would turn out to be high porosity aggregates with low sedimentation
speeds, they might become important.

The initial size distribution shows some deviation below 1 µm from the flat
distribution we assumed. While this small size fraction is small in volume
compared to the volume fraction of larger grains due to the steep power law
and while the different techniques of size measurements might account for
this, it should be kept in mind that our ejection probability might be less
accurate below 1 µm.

##  5 Conclusions

In laboratory experiments, we quantified the amount and size of grains ejected
after individual impacts of sand-sized grains into a Martian analog soil at
low pressure. We find that the probability of releasing dust grains of a given
size from the soil in the range from 0.5µm to 5µm is a steep power law with
power $3.72$. Thus, even though the probability of ejection increases with
grain size in this range, there is no hard cut off. Meaning, even the smallest
sized dust of this fraction is released upon impact. An estimate using the
given parameters shows: a fraction of $2\cdot 10^{-4}$ of an impactor’s mass
is released in suspendable dust below 3µm in an impact. We conclude that
saltation is a source of dust grains in the Martian atmosphere.

##  6 Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreement No 101004052. We
appreciate two very helpful reviews by two anonymous referees.

## References

  * Alfaro et al. (1998) Alfaro, S. C., Gaudichet, A., Gomes, L., & Maillé, M. 1998, Geophys. Res. Lett., 25, 991, doi: 10.1029/98GL00502
  * Bagnold (1941) Bagnold, R. 1941, The physics of blown sand and desert dunes (Meuthen)
  * Baker et al. (2021) Baker, M., Newman, C., Charalambous, C., et al. 2021, Journal of Geophysical Research (Planets), 126, e06514, doi: 10.1029/2020JE006514
  * Banks et al. (2018) Banks, M. E., Fenton, L. K., Bridges, N. T., et al. 2018, Journal of Geophysical Research (Planets), 123, 3205, doi: 10.1029/2018JE005747
  * Bogdan et al. (2020) Bogdan, T., Kollmer, J. E., Teiser, J., Kruss, M., & Wurm, G. 2020, Icarus, 341, 113646, doi: 10.1016/j.icarus.2020.113646
  * Cannon et al. (2019) Cannon, K. M., Britt, D. T., Smith, T. M., Fritsche, R. F., & Batcheldor, D. 2019, Icarus, 317, 470, doi: 10.1016/j.icarus.2018.08.019
  * Chen-Chen et al. (2019) Chen-Chen, H., Pérez-Hoyos, S., & Sánchez-Lavega, A. 2019, Icarus, 319, 43, doi: 10.1016/j.icarus.2018.09.010
  * Chepil (1945) Chepil, W. S. 1945, Soil Science, 60, 305
  * Clancy et al. (1995) Clancy, R. T., Lee, S. W., Gladstone, G. R., McMillan, W. W., & Rousch, T. 1995, J. Geophys. Res., 100, 5251, doi: 10.1029/94JE01885
  * Clancy et al. (2010) Clancy, R. T., Wolff, M. J., Whitney, B. A., et al. 2010, Icarus, 207, 98, doi: 10.1016/j.icarus.2009.10.011
  * Colwell et al. (2008) Colwell, J. E., Sture, S., Cintala, M., et al. 2008, Icarus, 195, 908, doi: 10.1016/j.icarus.2007.12.019
  * D’Aversa et al. (2022) D’Aversa, E., Oliva, F., Altieri, F., et al. 2022, Icarus, 371, 114702, doi: 10.1016/j.icarus.2021.114702
  * Deckers & Teiser (2014) Deckers, J., & Teiser, J. 2014, ApJ, 796, 99, doi: 10.1088/0004-637X/796/2/99
  * Dominik & Tielens (1997) Dominik, C., & Tielens, A. G. G. M. 1997, ApJ, 480, 647, doi: 10.1086/303996
  * Durán et al. (2011) Durán, O., Claudin, P., & Andreotti, B. 2011, Aeolian Research, 3, 243 , doi: https://doi.org/10.1016/j.aeolia.2011.07.006
  * El-Said et al. (2020) El-Said, A., Lewis, S. R., & Patel, M. R. 2020, Icarus, 336, 113470, doi: 10.1016/j.icarus.2019.113470
  * Esposito et al. (2016) Esposito, F., Molinaro, R., Popa, C. I., et al. 2016, Geophys. Res. Lett., 43, 5501, doi: 10.1002/2016GL068463
  * Favaro et al. (2021) Favaro, E. A., Balme, M. R., Davis, J. M., et al. 2021, Journal of Geophysical Research (Planets), 126, e6723, doi: 10.1029/2020JE006723
  * Gillette et al. (1974) Gillette, D. A., Blifford, Irving H., J., & Fryrear, D. W. 1974, J. Geophys. Res., 79, 4068, doi: 10.1029/JC079i027p04068
  * Goetz et al. (2010) Goetz, W., Pike, W. T., Hviid, S. F., et al. 2010, Journal of Geophysical Research (Planets), 115, E00E22, doi: 10.1029/2009JE003437
  * Greeley (2002) Greeley, R. 2002, Planetary and Space Science, 50, 151, doi: https://doi.org/10.1016/S0032-0633(01)00127-1
  * Greeley et al. (1980) Greeley, R., Leach, R., White, B., Iversen, J., & Pollack, J. 1980, Geophysical Research Letters, 7, 121, doi: 10.1029/GL007i002p00121
  * Greeley et al. (1976) Greeley, R., White, B., Leach, R., Iversen, J., & Pollack, J. 1976, Geophysical Research Letters, 3, 417, doi: 10.1029/GL003i008p00417
  * Holmes et al. (2021) Holmes, J. A., Lewis, S. R., Patel, M. R., et al. 2021, Earth and Planetary Science Letters, 571, 117109, doi: 10.1016/j.epsl.2021.117109
  * Holstein-Rathlou et al. (2010) Holstein-Rathlou, C., Gunnlaugsson, H. P., Merrison, J. P., et al. 2010, Journal of Geophysical Research: Planets, 115, doi: 10.1029/2009JE003411
  * Iversen et al. (1976) Iversen, J. D., Pollack, J. B., Greeley, R., & White, B. R. 1976, Icarus, 29, 381, doi: 10.1016/0019-1035(76)90140-8
  * Iversen & White (1982) Iversen, J. D., & White, B. R. 1982, Sedimentology, 29, 111, doi: 10.1111/j.1365-3091.1982.tb01713.x
  * Jensen & Sorensen (1986) Jensen, J. L., & Sorensen, M. 1986, Sedimentology, 33, 547, doi: https://doi.org/10.1111/j.1365-3091.1986.tb00760.x
  * Kok (2011) Kok, J. F. 2011, Proceedings of the National Academy of Science, 108, 1016, doi: 10.1073/pnas.1014798108
  * Kok & Renno (2008) Kok, J. F., & Renno, N. O. 2008, Phys. Rev. Lett., 100, 014501, doi: 10.1103/PhysRevLett.100.014501
  * Kok et al. (2014) Kok, J. F., Mahowald, N. M., Fratini, G., et al. 2014, Atmospheric Chemistry & Physics, 14, 13023, doi: 10.5194/acp-14-13023-2014
  * Krijt & Kama (2014) Krijt, S., & Kama, M. 2014, A&A, 566, L2, doi: 10.1051/0004-6361/201423862
  * Kruss et al. (2020) Kruss, M., Musiolik, G., Demirci, T., Wurm, G., & Teiser, J. 2020, Icarus, 337, 113438, doi: 10.1016/j.icarus.2019.113438
  * Lemmon et al. (2019) Lemmon, M. T., Guzewich, S. D., McConnochie, T., et al. 2019, Geophys. Res. Lett., 46, 9448, doi: 10.1029/2019GL084407
  * Loosmore & Hunt (2000) Loosmore, G. A., & Hunt, J. R. 2000, J. Geophys. Res., 105, 20,663, doi: 10.1029/2000JD900271
  * Lorenz et al. (2021) Lorenz, R. D., Martínez, G. M., Spiga, A., et al. 2021, Planet. Space Sci., 207, 105337, doi: 10.1016/j.pss.2021.105337
  * Määttänen et al. (2013) Määttänen, A., Listowski, C., Montmessin, F., et al. 2013, Icarus, 223, 892, doi: 10.1016/j.icarus.2012.12.001
  * Mahowald et al. (2014) Mahowald, N., Albani, S., Kok, J. F., et al. 2014, Aeolian Research, 15, 53, doi: 10.1016/j.aeolia.2013.09.002
  * Merrison (2012) Merrison, J. P. 2012, Aeolian Research, 4, 1, doi: 10.1016/j.aeolia.2011.12.003
  * Musiolik et al. (2018) Musiolik, G., Kruss, M., Demirci, T., et al. 2018, Icarus, 306, 25, doi: 10.1016/j.icarus.2018.01.007
  * Neakrase et al. (2016) Neakrase, L. D. V., Balme, M. R., Esposito, F., et al. 2016, Space Sci. Rev., 203, 347, doi: 10.1007/s11214-016-0296-6
  * Neary et al. (2020) Neary, L., Daerden, F., Aoki, S., et al. 2020, Geophys. Res. Lett., 47, e84354, doi: 10.1029/2019GL084354
  * Newman et al. (2021) Newman, C., Apestigue, V., Arruego, I., et al. 2021, in AAS/Division for Planetary Sciences Meeting Abstracts, Vol. 53, AAS/Division for Planetary Sciences Meeting Abstracts, 206.06
  * Reiss et al. (2016) Reiss, D., Fenton, L., Neakrase, L., et al. 2016, Space Sci. Rev., 203, 143, doi: 10.1007/s11214-016-0308-6
  * Renno & Kok (2008) Renno, N. O., & Kok, J. F. 2008, Space Sci. Rev., 137, 419, doi: 10.1007/s11214-008-9377-5
  * Schmidt et al. (1998) Schmidt, D. S., Schmidt, R. A., & Dent, J. D. 1998, Journal of Geophysical Research: Atmospheres, 103, 8997, doi: https://doi.org/10.1029/98JD00278
  * Shao & Lu (2000) Shao, Y., & Lu, H. 2000, Journal of Geophysical Research: Atmospheres, 105, 22437, doi: 10.1029/2000JD900304
  * Shao et al. (1993) Shao, Y., Raupach, M. R., & Findlater, P. A. 1993, J. Geophys. Res., 98, 12,719, doi: 10.1029/93JD00396
  * Stanzel et al. (2008) Stanzel, C., Pätzold, M., Williams, D. A., et al. 2008, Icarus, 197, 39, doi: 10.1016/j.icarus.2008.04.017
  * Swann et al. (2020) Swann, C., Sherman, D. J., & Ewing, R. C. 2020, Geophys. Res. Lett., 47, e84484, doi: 10.1029/2019GL084484
  * Vandaele et al. (2019) Vandaele, A. C., Korablev, O., Daerden, F., et al. 2019, Nature, 568, 521, doi: 10.1038/s41586-019-1097-3
  * Weitz et al. (2018) Weitz, C. M., Sullivan, R. J., Lapotre, M. G. A., et al. 2018, Geophys. Res. Lett., 45, 9471, doi: 10.1029/2018GL078972
  * White & Schulz (1977) White, B. R., & Schulz, J. C. 1977, Journal of Fluid Mechanics, 81, 497–512, doi: 10.1017/S0022112077002183
  * Wurm et al. (2005) Wurm, G., Paraskov, G., & Krauss, O. 2005, Icarus, 178, 253, doi: 10.1016/j.icarus.2005.04.002
  * Wurm & Teiser (2021) Wurm, G., & Teiser, J. 2021, Nature Reviews Physics, doi: 10.1038/s42254-021-00312-7
  * Zheng et al. (2003) Zheng, X. J., Huang, N., & Zhou, Y.-H. 2003, Journal of Geophysical Research: Atmospheres, 108, doi: https://doi.org/10.1029/2002JD002572
  * Zurek (2017) Zurek, R. W. 2017, Understanding Mars and Its Atmosphere, Cambridge Planetary Science (Cambridge University Press), 3â€“19, doi: 10.1017/9781139060172.002