PreprintPDF Available

Tracelet Hopf algebras and decomposition spaces

Authors:
  • Université de Paris CNRS
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.
Content may be subject to copyright.
Submitted to:
ACT 2021
© N. Behr and J. Kock
This work is licensed under the
Creative Commons Attribution License.
Tracelet Hopf algebras and decomposition spaces
Nicolas Behr
Universit´
e de Paris, CNRS, IRIF
nicolas.behr@irif.fr
Joachim Kock
Universitat Aut`
onoma de Barcelona &
Centre de Recerca Matem`
atica
kock@mat.uab.cat
Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this
work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommu-
tative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic
aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic
rewriting systems such as chemical reaction networks.
1 Introduction
Double-Pushout (DPO) [13] and more generally compositional categorical rewriting systems [1, 7] pro-
vide a versatile and mathematically sound framework for modeling complex transition systems, with
a paradigmatic example the modeling of reaction systems in biochemistry [10] and in organic chem-
istry [6]. The specification of an individual rewriting operation (direct derivation) in essence amounts to
providing a rewrite rule, i.e., a span of monomorphisms, that acts as a sort of template for the operation,
together with a match, which permits to specify the location within a host object where the local replace-
ment operation is to be performed. In practical applications, it is often the case that the rewriting rules
themselves involve only comparatively small graph-like objects. In contrast, the host objects to which
the rewrites are applied could easily be several orders of magnitude larger, so that an enormous number
of matches may be possible for a given rule and a given host object.
A natural and powerful approach to overcome this fundamental problem consists in focusing on the
combinatorial, statistical and structural properties of interactions of rewriting rules within derivation
traces, and to aim for a classification of traces in terms of “interaction patterns”. Unlike in compositional
diagrammatic calculi such as in particular the theory of string diagrams, the key obstacle for such a
type of analysis in rewriting theories resides in the fact that two given rules may in general interact in
a multitude of ways, i.e., there does not exist a notion of deterministic rule composition. Instead, as
first demonstrated in [4] and further developed in [1, 9, 6, 7], it is necessary to define a notion of non-
deterministic rule composition via a form of recursive application of the concurrency theorem, which
then indeed permits to define tractable methods to reason statically about classes of rule compositions.
Taking inspiration from the notion of pathways in chemical reaction systems, this approach was then
further refined in [2] to the notion of tracelets, which in essence act as the carriers of causal information
in derivation sequences.
The main objective of the present paper consists in establishing a principled mathematical approach
to formalize the combinatorics of tracelets. Generalizing results of [5] on rewriting systems over directed
multi-graphs to the categorical rewriting theory setting, we will demonstrate that it is indeed the notion
of combinatorial Hopf algebras that naturally captures the rich structure of tracelets. Apart from a
rewriting-theoretic construction of the Hopf algebras (Section 5), we report on our original discovery
that this at first sight seemingly ad hoc construction is in fact interpretable in terms of the theory of
decomposition spaces (Sections 3 and 4). Our motivation for this approach has been the analogy between
arXiv:2105.06186v2 [cs.LO] 13 Jul 2021
2Tracelet Hopf algebras and decomposition spaces
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X0
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X1
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X2
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X3
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X4
(a) A direct derivation sequence of length 5 (with edge creation/deletion rules, and where“wires” indicate matches).
Nicolas Behr, CAP’20, IHÉS, December 3, 2020
X0
(b) Tracelet and shift equivalence example. (c) Defining property of tracelets (here of length 3).
Figure 1: An illustration of graph rewriting sequences (top) and of the tracelet picture (bottom).
the inductive definition of tracelets and the decomposition-space axioms in homotopy combinatorics.
The benefit of this ‘detour’ is to situate the tracelet Hopf algebra in a general framework covering most
combinatorial Hopf algebras, thereby exhibiting the constructions and proofs as general ideas.
Decomposition spaces were introduced in combinatorics by G´
alvez, Kock and Tonks [16], [17] as
a far-reaching homotopical generalization of posets for the purpose of incidence algebras and M¨
obius
inversion, and independently by Dyckerhoff and Kapranov [12] in homological algebra and representa-
tion theory, motivated mainly by the Waldhausen S-construction and Hall algebras. Classically, since
the work of Rota [22] in the 1960s, incidence algebras were defined from posets through the process of
decomposing intervals. The construction can fruitfully be formulated in terms of the nerve of a poset.
However, a great many combinatorial co- and Hopf algebras are not of incidence type, meaning that they
are not the incidence coalgebra of a poset. The basic observation of [16] is that simplicial objects more
general than nerves of posets admit the incidence coalgebra construction, allowing most of the basic
features of the theory to carry over — and that most combinatorial Hopf algebras do arise as incidence
Hopf algebras of (monoidal) decomposition spaces.
2 Double-Pushout rewriting and tracelet theory
Consider a rewriting system of undirected multi-graphs such as the one depicted in Figure 1(a), with
some elementary rules that link and unlink vertices with edges. Starting from some initial graph X0,
each rewriting step (direct derivation) consists in choosing a rewriting rule together with an occurrence
(match) of the input motif (drawn at the bottom of the rule diagrams) within the graph that is rewritten.
It is intuitively clear that sequences of rewrites (derivation traces) have a rich intrinsic structure, since
the graph-like nature of the input and output interfaces of the rules as well as the internal structure of the
N. Behr and J. Kock 3
rules entail that the nature of interactions between sequential rewriting steps is not easily encapsulated in
any simple form of composition structure. To wit, consider the diagram in Figure 1(b), which provides
a sort of movie-script depiction of the five-step derivation trace depicted in the top part of the figure.
For clarity, red wires are used to indicate inputs to rules that are present in the original configuration X0,
while blue wires indicate inputs to rules that have originated from outputs of preceding rules.
The main purpose of the theory of tracelets [2] then consists in rendering mathematically precise the
meaning of this intuitive picture. In particular, according to the Tracelet Characterization Theorem [2,
Thm. 2], each derivation trace of length nis uniquely characterized by a tracelet of length n(cf. the
sub-diagram consisting of the five rules and all blue wires in Figure 1(b)) and a match of the tracelet
into the initial configuration X0(depicted as red “wires” in Figure 1(b)). Crucially, the compositional
structure of tracelets offers a form of statics causal analysis via algebraic relations such as commutator
relations. This type of analysis takes advantage of algebraic relations such as shift equivalence, which in
the example of Figure 1(b) amounts to the observation that the rules in the boxes highlighted in orange
may be freely moved “along the wires” so as to exchange their order (i.e., without changing the overall
effect of the rewriting sequence). Finally, as sketched via the highlighted triangle pattern that is produced
via the rewriting sequence in the example, tracelets permit to statically reason about the combinatorics of
pattern-counting problems in an efficient manner (cf. [3] for a prototype of such an analysis in the setting
of counting patterns in planar rooted binary trees).
As depicted in Figure 1(c), a tracelet of length 3 exhibits already quite a non-trivial compositional
structure, in that as sketched the internal structure of partial overlaps of rule inputs and outputs in such
a tracelet is not inherently of a purely sequential nature; to wit, the diagram encodes a special kind
of trace of length 3, with the defining property that it may be equivalently (up to isomorphisms) be
obtained via nested composition operations. It is in this particular sense that tracelets offer a minimal
causal presentation of the structure of rewriting sequences, since via the equivalences to nested pairwise
composition operations, they permit to efficiently express n-step sequences as just a special type of
derivation sequences.
2.1. Categorical rewriting theory. For simplicity, we will focus here on the variant of tracelet theory
for so-called double-pushout (DPO) rewriting, and for rewriting rules without application conditions
(referring to [2] for the general1theory). Throughout this paper, let Cdenote an adhesive category [20]
that is finitary (in the sense of [11], i.e., with only finitely many subobjects for each object up to iso-
morphisms), and that possesses a strict initial object obj(C).Rewriting rules are defined as spans of
monomorphisms r= (OoKiI), also denoted for brevity as r= (OI). In the tradition of
rewriting theory, we refer to such rules as linear rules (with “linear” referring to the nature of the span
as a span of monos), and denote the class of all such rules as Lin(C). For every rLin(C)and object
Xobj(C), let Mr(X)denote the set of (DPO-admissible) matches of r into X, where mMr(X)iff m
is a monomorphism and the pushout complement marked POC in the diagram below left exists:
O I O K I O I O K I
YXY K X V WVV W
r
mm
rm
:=mm
r
rm
r
n
r
n
:=
r
r
n
n
DPO POCPO DPOPOC PO
(1)
Note that in an adhesive category, pushouts along monomorphisms (here marked PO) are guaranteed to
exist; in contrast, pushout complements (here marked POC) may fail to exist, since not every composable
1Available generalizations include the type of rewriting (with Sesqui-Pushout semantics an alternative option), the choice of
base-categories (with M-adhesive categories [14] a more general option) as well as the inclusion of constraints and application
conditions into the compositional rewriting semantics (cf. also [7]).
4Tracelet Hopf algebras and decomposition spaces
pair of arrow can be completed into a pushout square. Moreover, rmas well as Y=rm(X)are evidently
only defined up to universal isomorphisms. It is customary to refer to the data of the aforementioned
diagram as a direct derivation. For later convenience, taking advantage of the symmetry of the definition,
we mark by DPOdiagrams that arise as DPO-type direct derivations in the “opposite direction”, i.e.,
“against” the direction of rules (here, from left to right; cf. the diagram above right).
2.2. Tracelets. The class of tracelets Tfor DPO-type type rewriting over Cis defined recursively:
Tracelets of length 1: for every rule r= (OI)Lin(C), define T(r)T1as a diagram
O I O K I
O I O K I
r
r
:=
oI
oI
r
r
.(2)
Tracelets of length n +1:denoting by Tn(for n>1) the class of trecelets of length n, the class
Tn+1is defined to consist of diagrams as below, where rk= (OkIk)Lin(C)are linear rules
(for k=1,...,n+1), where the top right part of the diagram encodes a tracelet TTnof length
n, where µ= (In+1MOn···1)is a span of monos, with the cospan In+1Y(n+1)
n+1,nOn···1its
pushout, and such that the direct derivations marked DPO and DPOexist:
(3)
Then this data defines a tracelet T(rn+1)µ
Tof length n+1 (the tracelet composition of T(rn+1)
with Talong µ) uniquely up to universal isomorphisms (see comments below) as
T(rn+1)µ
T:=
On+1In+1O1I1
On+1···1Y(n+1)
n+1,n·· · Y(n+1)
2,1In+1···1
rn+1r1
.(4)
For later convenience, we will sometimes speak of the commutative square below rule riin a tracelet as
the i-th plaquette. Let T:=n1Tndenote the class of all (finite length) tracelets. For later convenience,
we also introduce the notations in(T):=In···1(“input interface” of TTn), out(T):=On···1(“output
interface” of TTn), MTT(rn+1)(T)(for “matches”, i.e., admissible partial overlaps µof the length 1
tracelet T(rn+1)with the tracelet TTn) and [[T]] for the so-called evaluation of the tracelet T, which
for TTnis defined with notations as in the top right of (4) as
[[T]] := (On···1In···1) = (On···1Y(n)
n,n1)◦ · · · ◦ (Y(n)
2,1In···1).(5)
Here, denotes the operation of span composition (considered up to span-isomorphisms).
Up to this point, one might say that DPO-type tracelets are some form of data structure that encodes
a certain form of sequential compositions of rewriting rules. This point of view is augmented via the
following definition, which finally reveals tracelets as a particular notion of compositional diagrams.
N. Behr and J. Kock 5
2.3. Tracelet composition. Let TTmand T0Tnbe two tracelets of length mand n, respectively (for
m,n>0). Let µ:= (Im···1MO0
n···1)be a partial overlap (of the “input interface” Im···1of Twith
the “output interface” O0
n···1of T0) whose pushout Im···1Y(m+n)
n+1,nO0
n···1satisfies that in the diagram
below, all direct derivations marked DPO and DPO, respectively, exist:
OmImO1I1MO
nI
nO
1I
1
Om···1Y(m)
m,m1·· · Y(m)
2,1Im···1O
n···1Y(n)
n,n1.. . Y(n)
2,1I
n···1
Om+n···1Y(m+n)
m+n,m+n1·· · Y(m+n)
n+2,n+1Y(m+n)
n+1,nY(m+n)
n,n1·· · Y(m+n)
2,1Im+n···1
r1
DPO
r
nr
1
DPODPO
rm
DPO
PO
(6)
In this case, we write µMTT(T0)to say that µis a(n DPO-admissible) match of Tinto T0, and we
denote the composition of Twith T0along µas
Tµ
T0:=
OmImO
1I
1
Om+n···1Y(m+n)
n+1,n·· · Y(m+n)
2,1Im+n···1
rmr
1
.(7)
The definition of tracelets and their composition might appear somewhat ad hoc at first sight, yet
it is very natural if viewed in diagrammatic form. To this end, consider the example of a tracelet of
length 3 such as in Figure 1(c). The “wires” in the schematic diagram that link individual length 1
tracelets encode the partial overlaps; as indicated, the tracelet of length 3 may be realized recursively
via either determining the partial overlap of the first and the second sub-tracelet, composing, and then
determining the resulting overlap of the composite tracelet of length 2 with the third tracelet of length 1,
or (equivalently as it will turn out) computing the composition of the third and second tracelets of length
1, and of that composite with the first tracelet of length 1. This so-called associativity property of tracelet
compositions is at the heart of the algebraic properties of tracelets. It will be further illustrated when we
now pass to discuss tracelets in the framework of decomposition spaces.
3 The decomposition space of rewrite rules
In this section we describe a decomposition space Xof rewrite rules (for a fixed rewrite system in a
fixed adhesive category Cas above), whose incidence algebra is the rule algebra.
3.1. Decomposition spaces. A decomposition space [12], [16] is a simplicial groupoid X:op Grpd
satisfying a certain exactness property designed precisely to allow the incidence coalgebra construction,
classically defined for posets. (The nerve of a poset or a category is an example of a decomposition
space. Where categories encode composition, decomposition spaces owe their name to encoding de-
composition.) Many situations where compositionality is hard to achieve can be dealt with instead with
decompositions, as is often the case in combinatorics, where combinatorial structures can be split into
smaller ones without the ability to compose [18], [15]. Often non-deterministic composition structures
can be turned around and constitute instead a decomposition.
There are different ways to formulate the decomposition-space axioms. One (simplified) version
states that for any endpoint-preserving monotone map α:[2]\[n], defining a decomposition of any
6Tracelet Hopf algebras and decomposition spaces
n-simplex into an n1-simplex and an n2-simplex, the natural square
XnXn1×Xn2
X2X1×X1
long edges
short edges
(8)
is a pullback. It says that an n-simplex can be reconstructed from the two smaller simplices of the
decomposition together with the information of a gluing of the the long edges of the two simplices onto
the short edges of a base 2-simplex.
This condition is considerably weaker than the Segal condition (which characterizes categories,
hence composition rather than just decomposition), which says that a single-vertex overlap between the
two smaller simplices is enough to glue. In the decomposition-space case, the base 2-simplex is required
as a kind of context for the gluing.
3.2. Groupoids of tracelets. An isomorphism between two tracelets of length nis by definition a family
of object-wise isomorphisms between the involved objects in Cmaking all squares commute. We denote
by Xnthe groupoid of all tracelets of length n. (In particular, the only tracelet of length 0 is the empty
one (which evaluates to the trivial rule), so X0={∗}.)
Theorem 3.3. The groupoids Xnassemble into a simplicial groupoid X:op Grpd (whose face and
degeneracy maps we proceed to describe below).
To describe the simplicial structure, we give the standard presentation in terms of face and degeneracy
maps, so that the data is a diagram in the category of groupoids of shape
{∗} X1X2X3···
d0
d1s0d1
d2
d0
s1
s0
d3
d2
d1
d0
s2
s1
s0
(9)
3.4. Description of the face maps. The top face map dn:XnXn1(resp. the bottom face map
d0:XnXn1) is defined via (1) performing tracelet surgery (cf. Corollary A.7) to exhibit the tracelet
as a composition of the first (the last) rule with an (n1)-tracelet, followed by (2) extracting the (n1)-
length tracelet. This is illustrated in Figure 1(c) for the case of n=3, with d0(d3) defined to return the
tracelet shaded in light blue (in light yellow).
Recall from Section 2.2 that a plaquette in position i of a given tracelet is defined as the i-th direct
derivation, i.e., the commutative subdiagram of the tracelet involving the i-th rule (read from the right).
Then the inner face maps di:XnXn1(for 0 <i<n) replace the two plaquettes piand pi+1in the
chain with a single new plaquette p0with the same starting point as piand the same endpoint as pi+1, by
applying the “synthesis” part of the concurrency theorem (cf. Theorem A.2) to convert the sub-sequence
of plaquettes pi+1after piinto a one-step direct derivation along the composite rule p0
i+1,i. The result
of this operation is guaranteed to be a tracelet of length n1. Referring once again Figure 1(c) for an
illustration of the case n=3, d1(d2) are defined to return the tracelet shaded in green (in pink).
The degeneracy maps si:XnXn+1(for 0 in) insert a copy of the trivial rule /0 /0 /0 in the
tracelet at position i.
In the form stated, Xis only a pseudo-simplicial groupoid. This means that the simplicial identities
only hold up to (specified) isomorphism, and that there are coherence issues to deal with. The reason
for this pseudo-ness is that composition of rules and tracelets, as involved in the face maps, is only well
N. Behr and J. Kock 7
defined up to isomorphism, relying as it does on pushouts and pullbacks. To actually get well-defined
face maps, it is necessary to make choices of these universal constructions, and these choices screw up
the strict simplicial identities. (A well-known example of this phenomenon is how composition of spans
by means of pullbacks defines a bicategory, not an ordinary category.)
This pseudo-ness is not at all a problem for the sake of decomposition-space theory, designed to be
up to homotopy, and it does not affect the incidence algebra we construct from this decomposition space
(which in any case is spanned by iso-classes of rewrite rules). Nevertheless it is very fruitful to provide
also a strict model of X. The standard technique for constructing this (which goes back to insight from
algebraic topology from the 1970s (notably Quillen,2Waldhausen, and Segal)) is to beef up the groupoid
of n-simplices to something equivalent that contains all the (redundant) data involved in the face maps.
Specifically, a 2-simplex should not just be a 2-tracelet, but rather a 2-tracelet together with a choice
of composite rule. In this way the middle face map d1:X2X1does not have to compute any composite
by means of choices; it can simply return the choice already built in. The fact that these choices are
unique up to universal isomorphisms say precisely that this bigger groupoid is equivalent to the original,
and hence that the homotopy properties of the bigger simplicial groupoids are the same. In Figure 2(b) we
see such a fully specified 2-simplex. The two short edges (01 and 12) are the two rules in a 2-tracelet, and
the squares marked PO and POC are the plaquettes constituting altogether the 2-tracelet. The pullback
square (blue, marked as PB) is not part of the data of the tracelet, but it is included in the fully specified
notion of 2-simplex.
In degree 3 we arrive at the first point where there is an interesting simplicial identity to establish,
namely commutativity of the square
X3X2
X2X1,
d2
d1
d1
d1
(10)
which in essence states that a sequential composition of three rules may be recovered equivalently from
two steps of pairwise rule compositions in either of the nesting orders.
For the groupoids of bare tracelets, this simplicial identity cannot be strict, due to the choices of
pushouts and pullbacks involved in composition of rules and tracelets. That the equation holds up to
natural isomorphism is a nontrivial statement which involves the concurrency theorem (in the particular
form called associativity theorem [9, 4, 5]). We explain how the same theorem implies the strict equation
for the fully specified 3-simplices. This exhibits the beautiful geometry inherent in the associativity
theorem. As always, the idea is that a fully specified 3-simplex should contain all information about all
choices. In particular (in order for the four face maps to be forgetful) it should contain four 2-simplices
of the form of Figure 2(b). A full picture of such a subdivided tetrahedron is given in Figure 2(e). One
can chase through how this is built up from composition of tracelets, over specified overlaps: Consider
the diagram depicted in Figure 2(c), which is formed by (1) a 2-simplex encoding a composition of
two rules r21 and r10 into some rule r20, and (2) another 2-simplex of which one “short edge” is the
rule r20, and which contains another rule r32 and the data of the composition of r32 with r20 into some
rule r30. Upon closer inspection, it is possible (via a number of somewhat intricate steps) to construct
2Historical remark: B´
enabou (1963) had described a bicategory spans. Quillen used the techniques of big redundant n-
simplices to exhibit the same structure as a strict simplicial groupoid, now called Quillen’s Q-construction. Instead of having
simply chains of ncomposable spans in degree n, he defined it to be diagrams of shape , that is composable spans,
together with all the relevant pullbacks. Similar constructions were given in related situation by Waldhausen and Segal, and
today the technique is standard in algebraic topology.
8Tracelet Hopf algebras and decomposition spaces
from this data the interior and the other two faces of a tetrahedron. To this end, one first invokes the
“analysis” part of the DPO-type concurrency theorem (cf. Theorem A.2 in Appendix A) in order to obtain
from the sub-diagram that encodes the one-step direct derivation of the object I03 along the composite
rule O20 K20 K20 the data of a sequence of two direct derivations along the “constituent” rules
O21 K21 I21 after O10 K10 I10. This construction in particular delivers an object Zlocated in
the interior of the tetrahedron. Over several further steps (involving pushout and pullback operations), it
is then possible to fill the remaining two faces of the 3-simplex with the structure of two sequential rule
compositions, ultimately resulting in the diagram of Figure 2(e). The fact that all these constructions are
given by universal properties (pushouts and pullbacks, together with the axioms of adhesive categories),
ensures that the groupoid of such fully specified 3-simplices is equivalent to the groupoid of bare 3-
tracelets. The face maps are now obvious (or even tautological) and all the simplicial identities are
clearly strict for this reason: they merely return data already contained in (the beefed-up version of) X3.
The higher simplices are increasingly cumbersome to describe, due to our limited vision of geometry
in dimension higher than 3, but the principle is easy to follow: just include all information about all
possible composites, and the overall geometric shape is always a geometric n-simplex whose edges
are rules, whose 2-dimensional faces are as in Figure 2(b) and whose 3-dimensional faces are as in
Figure 2(e).
The fact that in each dimension the bare tracelets contain information necessary and sufficient to
reconstruct the full specified simplex is an expression of the the central result of [2] that it is indeed
tracelets that provide the minimal carriers of causal information in sequential rule compositions.
We proceed to establish that Xis a decomposition space. Since this is a homotopy invariant property,
we may work with the simple version of groupoids of n-tracelets. Before the check, let us just note that
Xis not a Segal space (a category), because of the non-deterministic nature of composition. Specifically,
a 2-simplex cannot be reconstructed from knowing its two short edges.
Theorem 3.5. Xis a decomposition space. This means that for all 0<i<n the two squares
Xn+1Xn
XnXn1
dn+1
didi
dn
Xn+1Xn
XnXn1
d0
di+1di
d0
(11)
are (homotopy) pullbacks.
To check this, it is enough to show that the fibers of the maps pictured vertically are equivalent. We
shall see that indeed all fibers of inner face maps are canonically identified with the fiber of d1:X2X1).
3.6. Fiber calculations. Consider d1:X2X1which sends a pair of composable rules with minimal
gluing (r2,w,r1)to the composite rule r0. The fiber over r0X1is thus the groupoid of all (r2,w,r1)that
compose to r0. We denote this groupoid (X2)r0. Notice that the objects of X2are composable pairs of
plaquettes with the property that the intermediate point between the two plaquettes is a minimal gluing
(of the output of rule r1with the input of rule r2; said otherwise, the middle cospan in the two-step direct
derivation sequence is a pushout of its own pullback).
Lemma 3.7. The (homotopy) fiber of di:XnXn1(for 0<i<n) over a tracelet which in position i
has a plaquette with rule r0is equivalent to the groupoid (X2)r0. In particular, it does not depend on the
whole plaquette p0under r0, and it does not depend on the context in any way. (Proof: cf. Appendix B.1)
One can now unpack the general construction of incidence algebras of decomposition spaces (cf.
Appendix C.4) to establish:
N. Behr and J. Kock 9
Proposition 3.8. The incidence algebra is the rule algebra of [9].
This algebra is not our main focus in this work. Rather do we regard the decomposition space Xas
a stepping stone towards more interesting decomposition spaces and Hopf algebras, notably the tracelet
Hopf algebra.
4 Decomposition spaces of tracelets
So far we have defined the decomposition space Xof rules, whose incidence algebra is the rule algebra
of [9]. We now proceed towards Hopf algebras spanned by tracelets.
The Hopf algebra of tracelets should be spanned by iso-classes of tracelets. Furthermore, we now
impose one more restriction on tracelets, namely that they should be non-degenerate as simplices of X.
This means that the rules involved are not allowed to be the trivial rule.3The non-degenerate simplices
of Xdo not form a simplicial object, since inner faces of non-degenerate simplices are not always non-
degenerate, but the outer face maps survive (as a consequence of the decomposition-space axioms), so
as to define a presheaf
X:op
inert Grpd.
Left Kan extension along the inclusion functor j:inert defines a new simplicial groupoid:
Y:=j!
X:op Grpd.
This is a general construction that makes sense for any (complete) decomposition space, and by a result
of Hackney and Kock [19] it always produces a decomposition space again. Furthermore, one can expand
explicitly what its simplices are:
Yk=
α:[k]\[n]
Xn.
(The sum is over active maps, cf. Appendix C.2.) In particular
Y0=X0and Y1=
nN
Xn.
So the new 1-simplices are the non-degenerate tracelets of any length. The higher simplices are ‘subdi-
vided tracelets’. To see this, recall that the decomposition space axioms can be written (cf. [16, Prop.
6.9]) as saying that for any active map α:[k]\[n]the canonical square
XnXn1× · · · × Xnk
XkX1× · · · × X1
(12)
is a pullback. Here the vertical maps are active and the horizontal maps are combinations of inert maps.
What the condition says is that it is possible to glue together ksimplices (of different dimensions ni) if
just one has available a ‘mould’ to glue them together in, namely a k-simplex whose kprincipal edges
match the long edges of the ksimplices. (This is also the essence of the very definition of tracelet.)
3By imposing this condition, we account directly for an equivalence relation imposed in [2] called ‘equivalence up to trivial
tracelets’ (cf. Definition 5.2).
10 Tracelet Hopf algebras and decomposition spaces
An example of such a composition is depicted in Figure 2(c), in which a length 2 tracelet (depicted
as the 2-simplex 012) is composed along the short edge 02 of the 2-simplex 023 with a tracelet of length
1 (here depicted as the edge 23). Figure 2(d) then depicts the method for computing the resulting tracelet
of length 3, which itself is depicted in Figure 2(f).
Since non-degeneracy in a decomposition space can be measured on principal edges (cf. [17]), we
also have the pullback
Xn
Xn1× · · · ×
Xnk
XkX1× · · · × X1.
(13)
We see that a k-simplex in Yis the data of a tracelet τof length k(not necessarily non-degenerate) to-
gether with a non-degenerate tracelet σiglued onto each of the principal edges of this base tracelet along
their evaluation. (That is, the rule given by evaluating the tracelet σimust match the rule corresponding
to the ith principal edge of τ.)
The corresponding algebra, given by the standard incidence algebra construction (cf. Appendix C.4),
is spanned by isomorphism classes of non-degenerate tracelets, and the product of two tracelets is given
by summing over all possible tracelet composites.
We now proceed to extend this structure into a Hopf algebra. This is not straightforward, because it
is easy to see that the decomposition space Yis not monoidal under sum. A monoidal structure exists
in degree 1, by declaring the product of two tracelets to be the composite along trivial overlap:
TT0:=T/0
T0.
But this definition is not compatible with higher simplices.
Our task is now to explain how this is fixed in a canonical way. The solution amounts to imposing the
so-called shift equivalence relation on tracelets, an equivalence relation already important in the theory.
In the graphical interpretation it is about saying that for tracelets that are not connected, it should make
no difference in which order they are applied. After passing to this equivalence relation, the monoidal
structure will be well defined in all simplicial degrees. This final symmetric monoidal decomposition
space of tracelets up to shift-equivalence will be denoted Z. We shall go deeper into the notion of shift
equivalence in Section 5 (and interested reader are referred to [8, 1, 7] for the full background information
and details). Here we just state the following proposition, which gives an alternative approach to shift
equivalence.
Asplitting vertex of a tracelet is an inner vertex for which the corresponding rule overlap is trivial.
This property is invariant under precomposition with active maps. (That is, if σ0=g(σ)for gan active
map not eliminating vertex v, then vis splitting for σ0if and only if it is splitting for σ.) Second, there
are expected transitive properties in connection with ‘stages’ in the sense of higher-order simplices of
Y. Note that this transitive property does not imply that irreducibility is compatible with inert maps
(outer face maps).
A non-degenerate tracelet TY1=n
Xnis primitive if it does not admit any splitting. (A higher-
dimensional simplex σYk(that is a subdivided tracelet) is primitive if its long edge is primitive in
Y1=n
Xn(that is, its underlying tracelet is primitive).)
Lemma 4.1. Every maximal splitting of a given simplex has, up to isomorphism and permutation, the
same primitive pieces.
N. Behr and J. Kock 11
Proposition 4.2. Tracelets are shift-equivalent in the restricted sense of trivial overlaps if and only if
they have the same factorization into primitives.
Proposition 4.3. Shift-equivalence is compatible with the simplicial structure. This defines a simplicial
groupoid Zwith Zk=Yk/. This simplicial groupoid is a (locally finite) decomposition space.
Note that if e
Yndenotes the groupoid of shift-equivalence classes, then we have
Zk=
[k]\[n]
Yk×Xk
1e
Yn1× · · · × e
Ynk
This makes sense: in the fiber product, the maps from the factors e
Ynireturn the long edge, which is
invariant under shift-equivalence.
Theorem 4.4. There is a level-wise equivalence of groupoids
Zk'S(Yirr
k),
assembling into an equivalence of simplicial groupoids. In particular, Zis symmetric monoidal under
.
Note that the primitive tracelets themselves do not form a simplicial groupoid. Only the active maps
are well defined. The outer face maps applied to an primitive tracelet is not necessarily primitive. (Said
in another way, a splitting vertex of a face might not be splitting for the whole simplex.) But after we
apply S(the free-symmetric-monoidal-category monad), which is just a fancy way of saying ‘monomials
of’ or ‘families of’, it does work.4
The upshot is now that the standard incidence algebra construction (cf. Appendix C.4) yields a Hopf
algebra Hof tracelets up to shift equivalence. This is the Hopf algebra we are really interested in, and
towards which the previous ones were preliminary constructions. By Poincar´
e–Birkhoff–Witt, His the
enveloping algebra of the Lie algebra of primitive tracelets. In the next section we spell out the structure
maps of this Hopf algebra in details.
5 The Hopf algebra of tracelets
Independently of the constructions of decomposition spaces presented in the preceding sections, it is
possible to construct the tracelet Hopf algebra via an extension of the rule diagram Hopf algebra con-
struction of [5] (which was based upon relational calculus) into the category-theoretical setting provided
by tracelet theory. The readers more familiar with rewriting theory might appreciate however that the
constructions presented in this section, while perhaps somewhat ad hoc at first sight, have a clear inter-
pretation from decomposition space theory. Throughout this section, we fix a field Kthat will typically
be chosen as either Ror C(or, possibly, Q). An essential prerequisite for our Hopf algebra construction
is given by the following notions of equivalence relations.
5.1. Shift equivalence (cf. [2]). Let Sdenote the equivalence relation on Tdefined as the reflexive
symmetric transitive closure of the relation on pairwise composition operations on tracelets: let T=
TBµ
TA(for some admissible match µ= (IBMOA)), and denote by [[TB]] = (OBKBIB)
and [[TA]] = (OAKAIA)the evaluations of TBand TA, respectively. Suppose [[TB]] and [[TA]] are
sequentially independent in the composition along µ, which entails that Mis isomorphic to both the
pullbacks of the cospans KBIBMand MOAKA, respectively. In this situation we define the
composite tracelet T=TAµ
TB(for µ=IAMOB) to be shift equivalent to the tracelet T=TBµ
TA.
4Note how this is analogous to the decomposition space of forests [18]: every forest is of course a disjoint union of trees,
but the top face of a tree is not in general a tree, only a forest.
12 Tracelet Hopf algebras and decomposition spaces
5.2. Normal form equivalence (cf. [2]). Let Adenote an equivalence relation on T(so-called abstrac-
tion equivalence) whereby TAT0iff Tand T0are tracelets of the same length, and if moreover there
exists an isomorphism T
=
T0(induced from isomorphisms on objects so that the resulting diagram
commutes). Let Tbe defined as the reflexive symmetric transitive closure of a relation whereby for any
TT, we let TTT]TTT]T(with T:=T()T1, and where ]:=µ
denotes
tracelet composition along trivial overlap). Then we define the tracelet normal form equivalence relation
as N:=rst(A∪ ≡T∪ ≡S), i.e., as the reflexive transitive closure of the union of the aforementioned
three relations.
Definition 5.3 (Primitive tracelets).Denote by Prim(TN)the set of primitive tracelets, defined as
Prim(TN):={[T]N|T6=T∧ 6 ∃TA,TB6=T:TNTA]TB}.(14)
Primitive tracelets play a central role in our construction, since they are in a certain sense the smallest
“indecomposable” building blocks of tracelets with respect to (de-)composition (just as primitive rule
diagrams in [5]).
Proposition 5.4 (Tracelet normal form).Every tracelet T Tis N-equivalent to a tracelet normal form
in the sense that TNT, and5T6=T:TNUiITi, where TiPrim(TN)for all i I, and with
I a (finite) index set. (Proof: cf. Appendix B.2)
Definition 5.5 (Tracelet K-vector space ˆ
T).Let ˆ
Tbe the K-vector space spanned by a basis indexed by
N-equivalence classes, in the sense that there exists an isomorphism δ:TN
basis(ˆ
T)from the set6
of N-equivalence classes of tracelets TN:=TNto the set of basis vectors of basis(ˆ
T). We will use
the notation ˆ
T:=δ(T)for the basis vector associated to some class TTN. We denote by Prim(ˆ
T)ˆ
T
the sub-vector space of ˆ
Tspanned by basis vectors indexed by primitive tracelets.
Definition 5.6 (Tracelet algebra product and unit).Let ⊗≡⊗Kbe the tensor product operation on the
K-vector space ˆ
T. Then the multiplication map µand the unit map η:Kˆ
Tare defined via their action
on basis vectors of ˆ
Tas follows:
µ:ˆ
Tˆ
Tˆ
T:ˆ
Tˆ
T07→ ˆ
Tˆ
T0,ˆ
Tˆ
T0:=
µMTT(T0)
δTµ
T0N(15)
η:Kˆ
T:k7→ k·ˆ
T.(16)
Both definitions are suitably extended by (bi-)linearity to generic (pairs of) elements of ˆ
T.
Proposition 5.7. The morphisms µand ηgive rise to an associative, unital K-algebra (ˆ
T,µ,η), which
we refer to as tracelet algebra. (Proof: cf. Appendix B.3)
Definition 5.8 (Tracelet coproduct and counit).Fixing the notational convention ]i/0 Ti:=Tfor later
convenience, let TN]iITibe the tracelet normal form for a given tracelet TT(where TiPrim(TN)
for all iIif T6=T). Then the tracelet coproduct and tracelet counit εare defined via their action
on basis vectors ˆ
T=δ(T)of ˆ
Tas
:ˆ
Tˆ
Tˆ
T:ˆ
T7→ (ˆ
T):=
XI
δ[]xXTx]Nδ]yI\XTxN(17)
and ε:ˆ
TK:ˆ
T7→ coe f f ˆ
T(ˆ
T). Both definitions are extended by linearity to generic elements of ˆ
T.
5We chose to make the case distinction explicit in order to emphasize that the normal form of a non-trivial tracelet T6=T
does itself not contain trivial sub-tracelets, such that manifestly TiPrim(TN)in TN]iITi. This is clearly the case, since
invoking Ton ]iITiwould in effect remove any trivial constituent Ti=T.
6Here, we tacitly assume that the N-equivalence classes indeed form a proper set, which is in all known applications the
case since abstraction equivalence Ais part of the definition of N. For example, it is well known that isomorphism classes
of finite directed multigraphs indeed form a set.
N. Behr and J. Kock 13
Proposition 5.9. The data (ˆ
T,,ε)defines a coassociative, cocommutative and counital coalgebra.
Proof. Since the construction of and εis the standard construction for a deconcatenation coalgebra
(cf. e.g. [21]), the proof is omitted here for brevity.
The algebra and coalgebra structures on ˆ
Tare compatible in the following sense:
Theorem 5.10 (Bialgebra structure).The data (ˆ
T,µ,η,,ε)defines a bialgebra. (Proof: cf. Appendix B.4)
By virtue of the definition of the tracelet normal form, it is evident that both composition and decom-
position of tracelets is compatible with a filtration structure given by number of “connected components”
in the following sense:
Theorem 5.11. The tracelet bialgebra (ˆ
T,µ,η,,ε)is connected and filtered, with connected compo-
nent ˆ
T(0):=spanK{ˆ
T}, and with the higher components of the filtration given by the subspaces
n>0 : ˆ
T(n):=spanKˆ
T1]. . . ]ˆ
Tnˆ
T1,..., ˆ
TnPrim(ˆ
T)o,(18)
where in a slight abuse of notations ˆ
T1]. . . ]ˆ
Tn:=δ(T1]. .. ]Tn)(Proof: cf. Appendix B.6)
Utilizing results from the general theory of Hopf algebras (cf. e.g. [21] for an excellent review), we
finally obtain one of the central results of the present paper:
Theorem 5.12 (Compare [5], Sec. 3.4 and Thm. 3.2).The tracelet bialgebra (ˆ
T,µ,η,,ε)admits the
structure of a Hopf algebra, where the antipode S, which is to say the endomorphism of ˆ
Tthat makes the
diagram below commute,
ˆ
Tˆ
Tˆ
Tˆ
T
ˆ
TKˆ
T
ˆ
Tˆ
Tˆ
Tˆ
T
εη
SId
µ
IdS
µ
e
(19)
is given by S :=Id1. The latter denotes the inverse of the identity morphism Id :ˆ
Tˆ
Tunder the
convolution product of linear endomorphisms on ˆ
T. More concretely, letting e :=ηεdenote the unit
for the convolution product ,
S(ˆ
T) = Id1(ˆ
T) = (e(eId))1=e(ˆ
T) +
k1
(eId)k(ˆ
T).(20)
It might be instructive to compute the action of the coproduct and of the antipode on tracelets of low
filtration degree. To this end, in the equations below let ˆ
TiPrim(ˆ
T)denote primitive elements of ˆ
T.
(ˆ
T) = ˆ
Tˆ
TS(ˆ
T) = ˆ
T
(ˆ
T1) = ˆ
Tˆ
T1+ˆ
T1ˆ
TS(ˆ
T1) = ˆ
T1
(ˆ
T=ˆ
T1]ˆ
T2) = ˆ
Tˆ
T+ˆ
Tˆ
T+ˆ
T1ˆ
T2+ˆ
T2ˆ
T1S(ˆ
T1]ˆ
T2) = ˆ
T1ˆ
T2+ˆ
T2ˆ
T1ˆ
T1]ˆ
T2
(21)
Finally, yet again taking inspiration from [5], one may demonstrate that the tracelet Hopf algebra is
isomorphic to a Hopf algebra that is well-known in the setting of the Heisenberg-Weyl diagram Hopf
algebra and the Poincar´
e-Birkhoff-Witt theorem for “normal-ordering” of elements of the Hopf algebra:
Theorem 5.13. Let LT:= (Prim(ˆ
T),[.,.])denote the tracelet Lie algebra, where [ˆ
T,ˆ
T0]:=ˆ
Tˆ
T0
ˆ
T0ˆ
T is the commutator operation (w.r.t. ). Then the tracelet Hopf algebra is isomorphic (in the sense
of Hopf algebra isomorphisms) to the universal enveloping algebra of LT.
14 Tracelet Hopf algebras and decomposition spaces
(a) 1-simplices (b) 2-simplices
(c) From adjacent 2-simplices. . . (d) . . . via the Concurrency Theorem. . .
(e) . .. to 3-simplices. (f) Tracelets of length 3 and evaluation
Figure 2: Elements of Tracelet Decomposition Space theory (Note: in order to allow for a more in-detail
inspection, the figures are hyperlinked to on-line interactive 3D views of the respective diagrams).
N. Behr and J. Kock 15
References
[1] Nicolas Behr (2019): Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework.
In: Proceedings of GCM 2019,EPTCS 309, pp. 23–52, doi:10.4204/eptcs.309.2.
[2] Nicolas Behr (2020): Tracelets and Tracelet Analysis Of Compositional Rewriting Systems. In: Proceedings
of ACT 2019,EPTCS 323, pp. 44–71, doi:10.4204/EPTCS.323.4.
[3] Nicolas Behr (2021): On Stochastic Rewriting and Combinatorics via Rule-Algebraic Methods. In: Proceed-
ings of TERMGRAPH 2020, 334, pp. 11–28, doi:10.4204/eptcs.334.2.
[4] Nicolas Behr, Vincent Danos & Ilias Garnier (2016): Stochastic mechanics of graph rewriting. In: Proceed-
ings of LICS ’16, doi:10.1145/2933575.2934537.
[5] Nicolas Behr, Vincent Danos, Ilias Garnier & Tobias Heindel (2016): The algebras of graph rewriting.arXiv
preprint 1612.06240.
[6] Nicolas Behr & Jean Krivine (2020): Rewriting theory for the life sciences: A unifying framework for CTMC
semantics. In: Graph Transformation (ICGT 2020),TCS 12150, doi:10.1007/978-3-030-51372-6.
[7] Nicolas Behr & Jean Krivine (2021): Compositionality of Rewriting Rules with Conditions.Compositionality
3, doi:10.32408/compositionality-3-2.
[8] Nicolas Behr & Pawel Sobocinski (2018): Rule Algebras for Adhesive Categories. In: Proceedings of (CSL
2018),LIPIcs 119, pp. 11:1–11:21, doi:10.4230/LIPIcs.CSL.2018.11.
[9] Nicolas Behr & Pawel Sobocinski (2020): Rule Algebras for Adhesive Categories (extended journal version).
LMCS Volume 16, Issue 3. Available at https://lmcs.episciences.org/6615.
[10] Pierre Boutillier et al. (2018): The Kappa platform for rule-based modeling.Bioinformatics 34(13), pp.
i583–i592, doi:10.1093/bioinformatics/bty272.
[11] Benjamin Braatz, Hartmut Ehrig, Karsten Gabriel & Ulrike Golas (2014): Finitary M-adhesive categories.
Mathematical Structures in Computer Science 24(4), pp. 240403–240443, doi:10.1017/S0960129512000321.
[12] Tobias Dyckerhoff & Mikhail Kapranov (2019): Higher Segal spaces.Lecture Notes in Mathematics 2244,
Springer-Verlag.
[13] H. Ehrig, K. Ehrig, U. Prange & G. Taentzer (2006): Fundamentals of Algebraic Graph Transformation.
Monographs in Theoretical Computer Science. An EATCS Series, doi:10.1007/3-540-31188-2.
[14] Hartmut Ehrig et al. (2014): M-adhesive transformation systems with nested application conditions. Part 1:
parallelism, concurrency and amalgamation.MSCS 24(04), doi:10.1017/s0960129512000357.
[15] Imma G´
alvez-Carrillo, Joachim Kock & Andrew Tonks (2016): Decomposition spaces in combinatorics.
Preprint, arXiv:1612.09225.
[16] Imma G´
alvez-Carrillo, Joachim Kock & Andrew Tonks (2018): Decomposition spaces, incidence algebras
and M¨
obius inversion I: Basic theory.Adv. Math. 331, pp. 952–1015, doi:10.1016/j.aim.2018.03.016.
[17] Imma G´
alvez-Carrillo, Joachim Kock & Andrew Tonks (2018): Decomposition spaces, incidence algebras
and M¨
obius inversion II: Completeness, length filtration, and finiteness.Adv. Math. 333, pp. 1242–1292,
doi:10.1016/j.aim.2018.03.017.
[18] Imma G´
alvez-Carrillo, Joachim Kock & Andrew Tonks (2020): Decomposition spaces and restriction
species.Int. Math. Res. Notices 2020(21), pp. 7558–7616, doi:10.1093/imrn/rny089.
[19] Philip Hackney & Joachim Kock (2021): Free decomposition spaces.In preparation.
[20] Stephen Lack & Paweł Soboci´
nski (2004): Adhesive Categories. In: Proceedings of FoSSaCS 2004,LNCS
2987, pp. 273–288, doi:10.1007/978-3-540-24727-2 20.
[21] Dominique Manchon (2008): Hopf algebras in renormalisation.Handbook of algebra 5, pp. 365–427.
[22] Gian-Carlo Rota (1964): On the foundations of combinatorial theory. I. Theory of M ¨
obius functions.Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368 (1964). Available at https://www.maths.
ed.ac.uk/~v1ranick/papers/rota1.pdf.
16 Tracelet Hopf algebras and decomposition spaces
A Background material
A.1. Double-Pushout (DPO) rewriting theory. An important ingredient in analyzing rewriting theories
based upon DPO semantics over adhesive categories is the following theorem, which is a central result
of categorical rewriting theory [13], and which we state here in the variant used in [2]. Note that while
the following statement is formulated for rules, which in the present context represent the special case
of length 1 tracelet, applying the theorem repeatedly leads precisely to the notion of tracelets of arbitrary
length (compare Section 2.2). Referring to [8] for the full details of the categorical rule-based calculus,
suffice it here to quote the theorem, and inviting the readers to inspect the relevant commutative diagrams
in (24) and (25) below:
Theorem A.2 (Concurrency theorem [8]).There exists a bijection ϕ:A
=
B on pairs of DPO-admissible
matches between the sets A and B,
A={(m2,m1)|m1MR1(X0),;m2MR2(X1)}
=B={(µ21,m21 )|µ21 MR2(R1),m21 MR21 (X0)},(22)
where X1=R1m1(X0)and R21 =R2µ21R1such that for each corresponding pair (m2,m1)A and
(µ21,m21 )B, it holds that
R21m21 (X0)
=R2m2(R1m1(X0)).(23)
The importance of this theorem with regard to the rule decomposition spaces constructed in the
present paper resides in the fact that from any two-step sequence of direct derivations, one can con-
struct (uniquely up to universal isomorphisms) a one-step direct derivation along a composite rule, which
amounts to the “synthesis” part of the concurrency theorem:
O2K2I2O1K1I1
O21 K2J21 K1I21
X2¯
X1X1¯
X0X0
K21
¯
K21
x1
¯x1
m
2
o2i2
j2j1
j21
m2m
1
¯m2
¯x0
o1
¯m1
i1
x0
m1
!
(5)
(6)
PB
PB
(72)(71)
(22)
PO
(12)
PO
(11)
(21)
PO
(32)
(42)
PO
(31)
(41)
PO PO
PO PO
(24)
Conversely, given a direct derivation along a composite rule, one may construct (again uniquely up
to isomorphisms) a two-step sequence of direct derivations along the constituent rules, which amounts
N. Behr and J. Kock 17
to the “analysis” part of the theorem:
O2K2I2O1K1I1
O21 K2J21 K1I21
X2¯
X1X1¯
X0X0
K21
¯
K21
m
2
o2i2
j2j1
o1i1
m1
(5)
PB
(82)(81)
PO PO
!!
PO
(92)PO
(91)
!
(12)
(22)(21)
PO
(32)
PO
(31)
PO PO
(131)
PO
(10)
(11)(132)
(25)
The concurrency theorem may be used to derive the following technical result, which is useful for
interpreting the notion of tracelet composition as well as the internal structure of three-simplices as
depicted in Figure 2(d):
Corollary A.3 (Cf. [2], Cor. 2).Let rn···1= (On···1In···1)be a span of monomorphisms, and let
(Y(n)
j+1,jY(n)
j,j1)be n spans of monomorphisms (for 0jn) with Y (n)
n+1,n=On···1, Y (n)
1,0=In···1, and
such that
(On···1In···1) = (On···1Y(n)
n,n1). . . (Y(n)
2,1In···1).
Then for each object X0and for each DPO-admissible match (In···1X0)MRn···1(X0)(for Rn···1=
(rn···1), the direct derivation of X0by Rn···1to X0along this match
On···1In···1
XnX0
DPO
(26a)
uniquely (up to isomorphisms) encodes an n-step DPO-type derivation sequence of the following form,
and vice versa:
On···1Y(n)
n,n1··· Y(n)
2,1In···1
XnXn1·· · X1X0
DPO
DPO
(26b)
A.4. Partial overlaps and minimal gluings. An interesting property of adhesive categories concerns
gluings of objects:
Lemma A.5. For every cospan of monomorphisms A a
Xb
B, there exist a span A α
Iβ
B of
monomorphisms, a cospan A a0
X0B of monomorphisms and a monomorphism X0x
X such that
[α,β]is a pullback of ]a,b[
]a0,b0[is a pushout of [α,β], and in particular a jointly epimorphic cospan
a=xa0and b =xb0
18 Tracelet Hopf algebras and decomposition spaces
We refer to the span [α,β]as a partial overlap and to the cospan ]a0,b0[as a minimal gluing of A and B.
Partial overlaps and minimal gluings are unique up to isomorphisms in X0.
Proof. The statement follows from the property of so-called effective binary unions [20], i.e., in an
adhesive category binary unions of subobjects are computed as pushouts of their intersection (which is
itself computed via pullback). Uniqueness up to isomorphisms then follows from the universal property
of pushouts.
Minimal gluings of objects are a key concept for constructing tracelets, as they model partial inter-
actions of rules in sequences of rewriting steps.
A.6. Tracelet surgery. For several of the constructions presented in this paper (such as in particular for
the definition of face maps in tracelet decomposition spaces), it is necessary to consider the following
type of operation that permits to extract information from tracelets. The basis for this type of reasoning
is once again the concurrency theorem (cf. Theorem A.2).
Corollary A.7 (Tracelet surgery; [2], Cor. 1).Let T Tnbe a tracelet of length n, so that it consists of n
sequential direct derivations tn,...,t1(written in the following as T tn|. . . |t1). Then for any consecutive
direct derivations tj|tj1in T, one may uniquely (up to isomorphisms) construct a diagram t(j|j1)and a
tracelet T(j|j1)of length 2as follows (where amounts to an application of the “analysis” part of the
concurrency theorem):
OjIjOj1Ij1
Y(n)
j+1,jY(n)
j,j1Y(n)
j1,j2
rjrj1
DPO DPO
Mj|j1
OjIjOj1Ij1
Oj|j1Y(2)
j|j1Ij|j1
Y(n)
j+1,jY(n)
j,j1Y(n)
j1,j2
rjrj1
PO
DPO
DPO
DPO DPO
(27a)
t(j|j1):=
Oj|j1Ij|j1
Y(n)
j+1,jY(n)
j1,j2
DPO
,T(j|j1):=T(rj)µ
T(rj1)(27b)
Here, µ:= (IjMOj1)is the span of monomorphisms obtained by taking the pullback of the
cospan (IjY(n)
j,j1Oj1), and this span is (by virtue of the concurrency theorem) an admissible
match. By associativity of the tracelet composition, this extends to consecutive sequences tj|...|tjkof
adjacent direct derivations in T inducing diagrams t(j|...|jk)and tracelets of length 1, denoted T(j|...|jk),
where for k =0, t(j)=tjand T(j)=T(rj).
B Proofs
B.1. Proof of Lemma 3.7. We already noted that dioperates entirely within the pair of consecutive
plaquettes, and that it does not modify the outer feet. The fiber thus consists of pairs of composable
plaquettes (pi+1,pi)with some middle foot y, but with fixed outer feet (the same as the feet of p0). By
definition of di, we should first construct was the pushout of the pullback and then compose the two
rules along w, and this is required to be r0. That is precisely the description of the fiber (X2)r0. It remains
to see that the rest of the data is uniquely reconstructible. But this follows via the “analysis” part of
the concurrency theorem, whereby if r0is realized as a composition of rule ri+1with rule rialong some
N. Behr and J. Kock 19
admissible match into rule r0, the direct derivation along r0may be equivalently expressed as a two-step
sequence of direct derivations along rifollowed by ri+1(cf. Theorem A.2 of Appendix A.1).
B.2. Proof of Proposition 5.4. For the case T=T, since is by assumption a strict initial object,
there exists precisely one inhabitant in the isomorphism class of T(i.e. Titself). Moreover, as Tis a
tracelet of length 1, neither Snor Tmay be invoked non-trivially, thus proving the claim in this case.
For the case T6=T, invoking Nrepeatedly in order to compute the N-equivalence class of Tin effect
removes any occurrence of sub-tracelets containing the trivial rule; since by assumption T6=Tand
since Tis of finite length, the process returns an N-irreducible class of minimal representatives T0T
with T06=T. Such a T0could then itself be obtainable by repeated operations ]on shorter tracelets (all
non-trivial), which by invoking Aand Syield manifestly equivalent “permutations” of ]composants,
thus proving the claim.
B.3. Proof of Proposition 5.7. Due to linearity, it suffices to verify the claim on basis vectors of ˆ
T.
Associativity of µfollows from the associativity of tracelet composition (cf. [2, Thm. 1]):
µ(µid)[ ˆ
Tˆ
T0ˆ
T00] = ( ˆ
Tˆ
T0)ˆ
T00 =ˆ
T(ˆ
T0ˆ
T00) = µ(id µ)[ ˆ
Tˆ
T0ˆ
T00].
Unitality is established via
µ(ηid)[ ˆ
T] = µ[ˆ
Tˆ
T] = ˆ
T=µ[ˆ
Tˆ
T] = µ(id η)[ ˆ
T].
More precisely, we have invoked the natural isomorphism 1ˆ
T=ˆ
T=ˆ
T1 in several steps.
B.4. Proof of Theorem 5.10. It is necessary to provide proofs of all four axioms of bialgebras in order
to verify the claim. Due to linearity, one may again focus on calculations in terms of basis vectors only.
Axiom I:
εη[1K] = ε[ˆ
T] = 1K=id[1K].(28)
Axiom II:
η[1K] = [ˆ
T] = ˆ
Tˆ
T= (ηη)[1K].(29)
Axiom III: The verification of the following property
εµ[ˆ
Tˆ
T0] = ε[ˆ
Tˆ
T0]!
= (εε)[ ˆ
Tˆ
T0](30)
requires a little additional work. We have to prove that ˆ
Tˆ
T0contains the summand ˆ
T(with coefficient
1) if and only if ˆ
T=ˆ
T0=ˆ
T. The proof follows from the fact that ˆ
Tis the unit element for – for all
ˆ
T6=ˆ
T,
ˆ
Tˆ
T=ˆ
Tˆ
T=ˆ
T6=ˆ
T,(31)
which proves the claim for all cases but for the case ˆ
T6=ˆ
Tand ˆ
T06=ˆ
T. However, it is straightforward
to verify from the precise definition of tracelets and of the tracelet composition operations that it is not
possible to obtain a tracelet N-equivalent to Tby composition of tracelets not themselves equal to T,
which completes the proof of the Axiom III property.
Axiom IV: The most difficult to prove property is the compatibility of product and coproduct, which
must be verified on arbitrary basis vectors ˆ
T,ˆ
T0ˆ
T:
µ[ˆ
Tˆ
T0]!
= (µµ)(id τid)()[ ˆ
Tˆ
T0].(32)
Here, τ:ˆ
Tˆ
Tˆ
Tˆ
Tis the interchange morphism, defined to act as τ[ˆ
Tˆ
T0]:=ˆ
T0ˆ
T. While the
property is straightforward to prove for the case where either both or one of ˆ
Tor ˆ
T0is equal to ˆ
T, the
general case requires an intricate case study. Let us first introduce some auxiliary notations.
20 Tracelet Hopf algebras and decomposition spaces
Definition B.5. Given two spans si:= (Ai
ai
Xi
bi
Bi)(i=1,2), denote by
si]sj:= (A1+A2
[a1,a2]
X1+X2
[b1,b2]
B1+B2)
their disjoint union (i.e., the pushout of s1ss2over the span s:= ()in the category
span(C)of spans over C). Given a span s=kA0
kXmBm(where all objects of the span are
finite disjoint unions of objects), a minimal presentation of sis defined as a disjoint union of spans of the
form
s=]
γ
sα,sα= (AγXγBγ)(33)
which satisfies that sα6=sfor all indices α. We furthermore define the format of s, denoted Φ(s), as
Φ(s):=]γ{(|Aγ|,|Bγ|)},(34)
where |Aγ|and |Bγ|denote the number of connected components of Aγand Bγ, respectively. Finally, we
let Fm,ndenote the set of all formats with the sum of first entries equal to mand the sum of second entries
to n.
Note that with the above definitions, the format of a span of the form
s= (]αAα)(]βBβ)(35)
evaluates to
Φ(s) = ]
α
{(1,0)}!]
]
β
{(0,1)}
.(36)
Let ˆ
TX:=UxXˆ
Txand ˆ
T0
Y:=UyYˆ
T0
y(for some finite index sets Xand Y, and so that ˆ
Tx6=ˆ
Tand
ˆ
T0
y6=ˆ
Tfor all indices xand y) denote two tracelets in normal form. Equipped with the notion of
formats of spans (which in our applications encode partial overlaps), we may refine the formula for the
composition of the two tracelets as follows:
ˆ
TXˆ
T0
Y=
µMTTX(T0
Y)
δTXµ
T0
Y
=
FF|in(TX)|,|out(T0
Y)|
µMTTX(T0
Y)
Φ(µ)=F
δTXµ
T0
Y
FF|in(TX)|,|out(T0
Y)|
ˆ
TX
Fˆ
T0
Y(37)
Here, we made use of the fact that MTTX(T0
Y) = M[[TX]]([[T0
Y]]) (i.e., matches of tracelets are determined
by matches of their evaluations), as well as of the fact that each admissible match µis in fact a span of
the form in(TX)Mout(T0
Y), and thus a span of the format Φ(µ)F|in(TX)|,|out (T
Y)|.
To proceed, note that for a given class of contributions ˆ
TX
Fˆ
T0
Y, the number of connected compo-
nents is given by nXnX
F+nYnY
F+|F|(where nXand nYdenote the number of connected components
of TXand TY, respectively, where nX
Fand nY
Fdenote the sums of first/second entries in the elements of
F, and where |F|denotes the total number of pairs of integers F). Moreover, if we let πΠFdenote a
partition of F(into a part πand a part F\π), one may verify that for a given admissible match µwith
N. Behr and J. Kock 21
Φ(µ) = F, if we let µ|πand µ|F\πdenote the restrictions of µto the sub-formats πand F\π, respec-
tively, then µ|πand µ|F\πare again admissible matches (of their respective domains and codomains).
We thus obtain the following intermediate result:
(ˆ
TXˆ
T0
Y) =
FF|in(TX)|,|out(T0
Y)|
ˆ
TX
Fˆ
T0
Y
=
FF|in(TX)|,|out(T0
Y)|
πΠF
VX
WY
(ˆ
TV
πˆ
T0
W)(ˆ
TX\V
F\πˆ
T0
Y\W)
(38)
Note that in the last step, we took advantage of the definition of .
F., in that the factors ˆ
TV
πˆ
T0
Wcover
indeed all possibilities for connected components arising from composing some connected components
ˆ
TVof ˆ
TXwith some connected components ˆ
TWof ˆ
TYin the format π(and analogously for the other
tensor factors with compositions of the format F\π). It is now straightforward to verify that since
we are summing over all possible formats F, and since .
F.by definition only retains the non-zero
contributions (i.e., those that come from admissible matches µof format F), it follows that we are in
fact evaluating all possible formats F0F|in(V)|,|out(W)|for the left and F00 F|in(X\V)|,|out(Y\W)|for the right
tensor factors, respectively, which finally leads to the proof of axiom IV:
(ˆ
TXˆ
T0
Y) =
VX
WY
F0F|in(T
V)|,|out(T0
W)|
F00F|in(TX\V)|,|out(T0
Y\W)|
(ˆ
TV
F0ˆ
T0
W)(ˆ
TX\V
F00 ˆ
T0
Y\W)
=
VX
WY
(ˆ
TVˆ
T0
W)(ˆ
TX\Vˆ
T0
Y\W) = (µµ)(Id τId)(ˆ
TX)(ˆ
T0
Y).
(39)
B.6. Proof of Theorem 5.11. By definition, the coproduct satisfies that for all n0 and ˆ
TT(n),
(ˆ
T)
n
m=0
ˆ
T(m)ˆ
T(nm).(40)
Note in particular that ˆ
Tis indeed the only basis element for which the coproduct is an element of
ˆ
T(0)ˆ
T(0). Finally, by definition of tracelet superposition and of tracelet composition, it is clear that for
all m,n0 and basis elements ˆ
TT(m)and ˆ
T0T(n),
TT0
m+n
r=0
T(r),(41)
since the trivial overlap µis always an admissible match (hence realizing ˆ
T]ˆ
T0T(m+n), i.e., the
contribution to TT0with the highest possible filtration degree), and since non-trivial overlaps will
reduce the overall number of connected components in general.
C Glossary
C.1. Simplicial groupoids. Let denote the category whose objects are the nonempty finite orders
[n]:={01≤ ··· ≤ n}
22 Tracelet Hopf algebras and decomposition spaces
and whose arrow are the monotone maps. A simplicial groupoid is a functor X:op Grpd . The value
of Xon [n]is denoted Xn. Exploiting the generators-and-relations description of , one can describe a
simplicial set as a diagram
X0X1X2X3· · ·
d0
d1s0d1
d2
d0
s1
s0
d3
d2
d1
d0
s2
s1
s0
(42)
subject to the simplicial identities:disi=di+1si=1 and
didj=dj1di,dj+1si=sidj,disj=sj1di,sjsi=sisj1,(i<j).
The indexing convention is that dideletes the ith vertex and sirepeats the ith vertex.
C.2. Active and inert maps. Inside the simplex category , we have the subcategory inert of inert
maps: these are the maps that preserve distance, meaning φ:[m][n]such that φ(i+1) = φ(i) + 1.
While is generated by face maps and degeneracy maps, inert is generated only by the outer face maps.
We also have the subcategory of active maps, which are precisely the endpoint-preserving maps in .
These two classes of maps form a factorization system on : every map factors uniquely as an active
map followed by an inert map.
For a simplicial groupoid X:op Grpd, we use the same terminology for simplicial operators
induced by active and inert maps in . For example, the inert face maps are precisely the outer face
maps d0:XnXn1and dn:XnXn1(for all n>0), and the active maps are the inner face maps
di:XnXn1for all 0 <i<n. (All degeneracy maps are active.)
C.3. Decomposition spaces. It is a fact that active and inert maps in admit pushouts along each
other and that the resulting new maps are again active and inert. A simplicial simplicial groupoid X:
op Grpd is called a decomposition space [16] when it sends active-inert pushouts in to (homotopy)
pullbacks in Grpd. It turns out it is enough to check the following squares to be (homotopy) pullbacks
(for all 0 <i<n):
Xn+1Xn
XnXn1
dn+1
didi
dn
Xn+1Xn
XnXn1
d0
di+1di
d0
(43)
Particularly important are the first two instances of this:
X3X2
X2X1
d3
d1d1
d2
X3X2
X2X1
d0
d2d1
d0
(44)
These conditions can be read as saying that a 3-simplex can be reconstructed by gluing two 2-simplices
along a 1-simplex: the long edge of one along a short edge of the other.
An equivalent way of stating the decomposition-space axioms is that for every active map α:[k]\
[n], the square
XnXn1× · · · × Xnk
XkX1× · · · × X1
(45)
N. Behr and J. Kock 23
is a (homotopy) pullback (cf. [16, Prop. 6.9]). This time the condition says that an n-simplex can
be reconstructed by gluing ksimplices (of dimensions n1,...,nk) onto the principal edges of a base
k-simplex, in close analogy with the tracelet axioms.
C.4. Incidence coalgebras. The motivating property of decomposition space is that they allow the
incidence-coalgebra construction (and with it many algebraic constructions from combinatorics, such as
M¨
obius inversion). The incidence coalgebra of Xhas as underlying vector space Qπ0X1(where π0X1is
the set of iso-classes of 1-simplices), and the comultiplication law is defined as
(f) =
σX2|d1(σ)= f
d2(σ)d0(σ)
The decomposition-space axiom is designed to ensure that this comultiplication law is coassociative.
(The 3-simplices enter the proof of coassociativity, and the higher simplices are useful for various pur-
poses, as in the present case where tracelets are higher simplices.) If Xis monoidal (for this notion, see
[16]) then the incidence coalgebra becomes a bialgebra.
In the present paper, we are interested in the incidence algebra instead of the incidence coalgebra. It
can generally be derived from the coalgebra as a convolution algebra. Even if the sum defining the inci-
dence coalgebra is infinite, the corresponding sum in the convolution product becomes finite if restricted
to finitely-supported functions [17].
C.5. About groupoids and homotopy pullbacks. One technicality that complicates the framework of
decomposition spaces and combinatorial Hopf algebra is the necessity to work with groupoids instead of
sets. Even if ultimately the incidence bialgebra will be generated by iso-classes, it does not work with
decomposition ‘sets’ of iso-classes of objects (such as tracelets). The problem is that taking iso-classes
too early kills important symmetries which must be respected in order for the constructions to work.
This is explained in detail in [15]. Once this is understood it is actually a big simplification to work with
groupoids, because with sets of iso-classes there are many pitfalls in connection with symmetries. With
the groupoid formalism, all these issues are taken care of automatically.
In conclusion, one should always work with the naturally defined groupoids of objects, without trying
to take iso-classes of pick representatives. A large amount of conditions take the form of pullbacks
conditions. In the setting of groupoids, it is crucial to work with homotopy pullbacks instead of ordinary
pullbacks, as the latter are not invariant under homotopy equivalence. An ordinary pullback of sets
X×SY Y
X S
q
p
(46)
is given by X×SY={(x,y)|px =qy}. For the homotopy pullback of groupoids, an explicit isomorphism
is included as data:
X×SY={(x,y,σ)|xX,yY,σ:px 'qy}
(These are the objects; the arrows are pairs of arrows compatible with the specified isos.) This can com-
plicate certain calculations, but very often it is actually universal properties and standard manipulations
that are important, and at that level of abstraction, homotopy pullbacks work very much like ordinary
pullbacks do for sets.
In spite of their complicated appearance, homotopy pullbacks are often much closer to actual math-
ematical practice. For example, when we say ‘glue together two 2-simplices along the long edge of one
24 Tracelet Hopf algebras and decomposition spaces
and the short edge of the other’, it is unrealistic to assume that the long edge of one is literally equal
to the short edge of the other. What actually happens is that they are isomorphic, and that a specific
identification is employed (and must be referenced) in the constructions.
(We stress that homotopy pullbacks are used for groupoids, whereas inside ordinary categories, such
as our fixed adhesive category C, the notion of pullback is the ordinary 1-categorical one.)
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Building upon the rule-algebraic stochastic mechanics framework, we present new results on the relationship of stochastic rewriting systems described in terms of continuous-time Markov chains, their embedded discrete-time Markov chains and certain types of generating function expressions in combinatorics. We introduce a number of generating function techniques that permit a novel form of static analysis for rewriting systems based upon marginalizing distributions over the states of the rewriting systems via pattern-counting observables.
Article
Full-text available
Taking advantage of a recently discovered associativity property of rule compositions, we extend the classical concurrency theory for rewriting systems over adhesive categories. We introduce the notion of tracelets, which are defined as minimal derivation traces that universally encode sequential compositions of rewriting rules. Tracelets are compositional, capture the causality of equivalence classes of traditional derivation traces, and intrinsically suggest a clean mathematical framework for the definition of various notions of abstractions of traces. We illustrate these features by introducing a first prototype for a framework of tracelet analysis, which as a key application permits to formulate a first-of-its-kind algorithm for the static generation of minimal derivation traces with prescribed terminal events.
Article
Full-text available
Sesqui-pushout (SqPO) rewriting is a variant of transformations of graph-like and other types of structures that fit into the framework of adhesive categories where deletion in unknown context may be implemented. We provide the first account of a concurrency theorem for this important type of rewriting, and we demonstrate the additional mathematical property of a form of associativity for these theories. Associativity may then be exploited to construct so-called rule algebras (of SqPO type), based upon which in particular a universal framework of continuous-time Markov chains for stochastic SqPO rewriting systems may be realized.
Article
Full-text available
Motivation: We present an overview of the Kappa platform, an integrated suite of analysis and visualization techniques for building and interactively exploring rule-based models. The main components of the platform are the Kappa Simulator, the Kappa Static Analyzer and the Kappa Story Extractor. In addition to these components, we describe the Kappa User Interface, which includes a range of interactive visualization tools for rule-based models needed to make sense of the complexity of biological systems. We argue that, in this approach, modeling is akin to programming and can likewise benefit from an integrated development environment. Our platform is a step in this direction. Results: We discuss details about the computation and rendering of static, dynamic, and causal views of a model, which include the contact map (CM), snaphots at different resolutions, the dynamic influence network (DIN) and causal compression. We provide use cases illustrating how these concepts generate insight. Specifically, we show how the CM and snapshots provide information about systems capable of polymerization, such as Wnt signaling. A well-understood model of the KaiABC oscillator, translated into Kappa from the literature, is deployed to demonstrate the DIN and its use in understanding systems dynamics. Finally, we discuss how pathways might be discovered or recovered from a rule-based model by means of causal compression, as exemplified for early events in EGF signaling. Availability and implementation: The Kappa platform is available via the project website at kappalanguage.org. All components of the platform are open source and freely available through the authors' code repositories.
Article
Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church-Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of M-adhesive categories, where M-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.
Article
This chapter focuses on the Hopf algebras in renormalization. The chapter presents the Birkhoff decomposition. This chapter also explains the BCH approach to Birkhoff decomposition. This chapter discusses renormalized multiple zeta values, which is an application to number theory. The chapter explains the connected graded Hopf algebras. It concludes with a brief discussion on the renormalization group and the beta function.
Rule Algebras for Adhesive Categories
Nicolas Behr & Pawel Sobocinski (2018): Rule Algebras for Adhesive Categories. In: Proceedings of (CSL 2018), LIPIcs 119, pp. 11:1-11:21, doi:10.4230/LIPIcs.CSL.2018.11.
Rule Algebras for Adhesive Categories (extended journal version)
Nicolas Behr & Pawel Sobocinski (2020): Rule Algebras for Adhesive Categories (extended journal version). LMCS Volume 16, Issue 3. Available at https://lmcs.episciences.org/6615.