Free decomposition spaces Philip Hackneyand Joachim Kock University of Louisiana at Lafayette

2025-05-06 0 0 695.68KB 31 页 10玖币
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Free decomposition spaces
Philip Hackney⋆and Joachim Kock⋆⋆
⋆University of Louisiana at Lafayette
⋆⋆Universitat Aut`onoma de Barcelona and Centre de Recerca Matem`atica;
currently at the University of Copenhagen
Abstract
We introduce the notion of free decomposition spaces: they are sim-
plicial spaces freely generated by inert maps. We show that left Kan
extension along the inclusion
j:∆inert →∆
takes general objects to M¨obius
decomposition spaces and general maps to CULF maps. We establish an
equivalence of
∞
-categories
PrSh
(
∆inert
)
≃Decomp/BN
. Although free
decomposition spaces are rather simple objects, they abound in combina-
torics: it seems that all comultiplications of deconcatenation type arise
from free decomposition spaces. We give an extensive list of examples,
including quasi-symmetric functions.
Contents
Introduction 2
1 Preliminaries 6
1.1 Active and inert maps . . . . . . . . . . . . . . . . . . . 6
1.2 Decomposition spaces and incidence coalgebras . . . . . 7
2 Free decomposition spaces 9
2.1 Left Kan extension along j................ 9
2.2 Two identifications . . . . . . . . . . . . . . . . . . . . . 10
2.3 j!gives decomposition spaces and CULF maps . . . . . 12
3 Main theorem 13
3.1 Untwisting theorem . . . . . . . . . . . . . . . . . . . . 13
3.2
Equivalence between
∆op
inert
-diagrams and decomposition
spaces over BN....................... 14
3.3 Fully faithfullness . . . . . . . . . . . . . . . . . . . . . . 15
4 Miscellaneous results 16
4.1 Remarks about sheaves . . . . . . . . . . . . . . . . . . 16
4.2 Restriction L-species.................... 18
5 Examples in combinatorics 19
5.1 Paths and words . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Further examples of deconcatenations . . . . . . . . . . 23
5.3 Simplices in simplicial sets . . . . . . . . . . . . . . . . . 25
1
arXiv:2210.11192v2 [math.CT] 26 May 2024
Introduction
Background
M¨obius inversion. The motivation for this work comes from the theory
of M¨obius inversion in incidence algebras of posets (Rota [
49
]), which is an
important tool in enumerative combinatorics, as well as in application areas such
as probability theory, algebraic topology, and renormalization, just to mention a
few. The idea is generally to write down counting functions on suitable posets
(or more precisely on their incidence coalgebras), and then express recursive
relations in such a way that the solution is given in terms of convolution with
the M¨obius function. In this work we are not directly concerned with counting
problems or M¨obius functions, but rather with features of the overall framework.
Decomposition spaces. The starting point is the recent discovery that objects
more general than posets admit the construction of incidence algebras and
M¨obius inversion: these are called decomposition spaces by G´alvez, Kock, and
Tonks [
20
], [
21
], and they are the same thing as the 2-Segal spaces of Dyckerhoff
and Kapranov [
16
] who were motivated mainly by representation theory and
homological algebra. (The last ingredient in the equivalence between the two
notions, the fact that a certain unitality condition is automatic, was provided
only recently by Feller et al. [17].)
From the viewpoint of combinatorics, the importance of decomposition spaces
is that many combinatorial coalgebras, bialgebras, and Hopf algebras are not
directly the incidence coalgebra of any poset, but that it is often possible instead
to realize them as incidence coalgebras of a decomposition space (as illustrated
in Section 5 below), and in this way make standard tools available. The price to
pay is that the theory of decomposition spaces is more technical than that of
posets. In particular, a little bit of background in simplicial homotopy theory
and category theory is assumed as a prerequisite for this work, and their basic
vocabulary is used freely. We refer to the introductions and preliminaries sections
of the papers [
20
,
21
,
19
] for background material on the use of simplicial methods
for combinatorial coalgebras.
Briefly (cf. 1.2.1 below), a decomposition space is a simplicial space satisfying
the exactness condition (weaker than the Segal condition) stating that active-
inert pushouts in the simplex category
∆
are sent to homotopy pullbacks. The
active-inert factorization system was already an important ingredient in the
combinatorics of higher algebra, as in Lurie’s book [
45
], where the terminology
originates.
In the present work, we explore further the fundamental relationship between
decomposition spaces and the active-inert factorization system, and single out a
new class of decomposition space, the free decomposition spaces, being simplicial
spaces freely generated by inert maps. We show that most comultiplications in
algebraic combinatorics of deconcatenation type arise from free decomposition
spaces.
2
Active and inert maps for categories and higher categories. The active
maps in the simplex category
∆
are the endpoint-preserving maps; the inert
maps are the distance-preserving maps (cf. 1.1.1 below). The two classes of
maps form a factorization system
∆
= (
∆active,∆inert
) first studied (by Leinster
and Berger [
4
]) to express the interplay between algebra and geometry in the
notion of category: for the nerve of a category, the active maps parametrize the
algebraic operations of composition and identity arrows, while the inert maps
express the bookkeeping that these operations are subject to, namely source and
target. The geometric nature of this background fabric is manifest in the fact
that the category
∆inert
has a natural Grothendieck topology, through which the
gluing conditions are expressed: a simplicial set
X
is the nerve of a category if
and only if
j∗
(
X
), the restriction of
X
along
j:∆inert →∆
, is a sheaf. This is one
form of the classical nerve theorem, which characterizes the essential image of the
nerve functor. In particular, the question whether a simplicial set is the nerve of
a category depends only on the inert part. This viewpoint is the starting point
for the Segal–Rezk approach to
∞
-categories, defined by replacing simplicial sets
by simplicial spaces, and considering the sheaf condition up to homotopy. Many
other developments exploit the active-inert machinery to obtain nerve theorems
in fancier contexts, including Weber’s extensive theory of local right adjoint
monads and monads with arities, with abstract nerve theorems [
56
], [
57
], [
5
],
as well as special nerve theorems for specific operad-like structures (polynomial
monads in terms of trees [
36
], [
24
], properads in terms of directed graphs [
37
],
modular operads and wheeled properads as well as infinity versions [
27
], [
28
], and
so on); see [
25
] for a survey. Recently Chu and Haugseng [
14
] have even developed
a general Segal approach to operad-like structures in terms of algebraic patterns,
where the notion of active-inert factorization system is taken as primitive.
Active and inert maps for decomposition spaces. For decomposition
spaces, the active-inert interplay is more subtle than for categories, and the
exactness condition that characterizes them can no longer be measured on
the inert maps alone. It is now about decomposition of ‘arrows’ rather than
composition. Roughly, the active maps encode all possible ways to decompose
arrows, and the inert maps then separate out the pieces of the decomposition.
The exactness condition characterizing decomposition spaces combines active
and inert maps. It can be interpreted as a locality condition, stating roughly
that the possible decompositions of a local region are not affected by anything
outside the region [23], [19].
Active and inert maps for morphisms. Turning to morphisms, the situation
is more complex for decomposition spaces than for categories. For categories,
the nerve functor is fully faithful, meaning that all simplicial maps are relevant:
the simplicial identities for simplicial maps simply say that source and target,
composition and identities are preserved. For decomposition spaces, this is no
longer the case, as there are different ways in which a simplicial map could
be said to preserve decompositions. The most well-behaved class of simplicial
maps in this respect are the CULF maps [
20
] (standing for ‘conservative’ and
3
‘unique lifting of factorizations’), a class of maps well studied in category theory,
originating with Lawvere’s work on dynamical systems [
41
], and exploited in
computer science in the algebraic semantics of processes [
12
], [
31
], [
11
]. From
the viewpoint of combinatorics the interest in CULF maps is that they preserve
decompositions in a way such as to induce coalgebra homomorphisms between
incidence coalgebras [
43
], [
42
], [
20
]. The formal characterization of CULF maps
is that when interpreted as natural transformations, they are cartesian on active
maps. Independently of the coalgebra interpretation, this pullback condition
interacts very well with the exactness condition characterizing decomposition
spaces.
Contributions of the present paper
In the present work, the focus is not so much on restriction along
j:∆inert →∆
as in the Segal case, but rather on its left adjoint
j!⊣j∗
, left Kan extension
along
j
. Given a presheaf
A:∆op
inert →S
(with values in spaces), we are thus
concerned with the simplicial space
j!
(
A
)
:∆op →S
. It is rarely the case that
j!(A) is Segal. We show instead that j!(A) is always a decomposition space:
Theorem. (Cf. Proposition 2.3.2 and Corollary 2.3.3.) For any
A:∆op
inert →
S
, the left Kan extension
j!
(
A
)
:∆op →S
is a M¨obius decomposition space, and
for any map A′→Aof ∆op
inert-diagrams, we have that j!(A′)→j!(A)is CULF.
The decomposition spaces that arise with
j!
we call free decomposition spaces.
A key to understand the relationship between
∆inert
-presheaves and decompo-
sition spaces is the action of
j!
on the terminal presheaf 1
∈PrSh
(
∆inert
). The
following result is just a calculation:
Lemma 2.3.1. We have
j!
(1)
≃BN
, the classifying space of the natural
numbers.
It follows that free decomposition spaces admit a CULF map to
BN
. In fact
this feature characterizes free decomposition spaces:
Proposition. (Cf. Corollary 3.2.2.) A decomposition space is free if and
only if it admits a CULF map to BN.
We derive this from the following more precise result.
Theorem 3.2.1. The j!functor induces an equivalence of ∞-categories
PrSh(∆inert)≃Decomp/BN.
Here
Decomp
is the
∞
-category of decomposition spaces and CULF maps, and
Decomp/BNits slice over BN.
4
The proof of this result turned out to be quite involved, and ended up
developing into a proof of the following very general result, which is an
∞
-
version of a theorem of Kock and Spivak [40]:
Theorem 3.1.2. For
D
a decomposition space, there is a natural equivalence
of ∞-categories
Decomp/D ≃Rfib(tw D).
Here
tw
(
D
) denotes the edgewise subdivision of the simplicial space
D
— when
D
is a (Rezk complete) decomposition space, this is an
∞
-category [
26
], called the
twisted arrow category of
D
. The right-hand side
Rfib
(
tw D
) is the
∞
-category
of right fibrations over
tw
(
D
), which is equivalent to
PrSh
(
tw D
) under the
basic equivalence between right fibrations and presheaves (see for example [
2
,
Theorem 3.4.6]).
In order to apply the general theorem, take
D
=
BN
, and note the following:
Lemma 2.2.2. There is a natural equivalence of categories
∆inert ≃tw(BN).
Theorem 3.2.1 follows essentially from this observation and the general
theorem, but there is still some work to do to show that in this special case,
the untwisting of the general theorem can actually be identified with left Kan
extension along j, surprisingly.
Since the general theorem is of independent interest, and since the proof is
very long, we have separated it out into a paper on its own [26].
Having characterized free decomposition spaces as those admitting a CULF
map to
BN
, it is interesting to know that such a map is unique if it exists. This
statement is equivalent to the following result.
Proposition 3.3.3. The forgetful functor
Decomp/BN→Decomp
is fully
faithful.
Together with Theorem 3.2.1, this implies a fundamental property of j!:
Corollary 3.3.4. The functor
j!:PrSh
(
∆inert
)
→Decomp
is fully faithful.
Theorem 3.2.1 readily implies the following classical and more special result
due to Street [
54
]: a category admits a CULF functor to
BN
if and only if it is
free on a directed graph.
We also characterize a large class of free decomposition spaces in terms of a
class of species called restriction L-species (Proposition 4.2.3).
Although free decomposition spaces are rather simple, they abound in com-
binatorics. Generally it seems that all comultiplications of deconcatenation type
5
摘要：

FreedecompositionspacesPhilipHackney⋆andJoachimKock⋆⋆⋆UniversityofLouisianaatLafayette⋆⋆UniversitatAut`onomadeBarcelonaandCentredeRecercaMatem`atica;currentlyattheUniversityofCopenhagenAbstractWeintroducethenotionoffreedecompositionspaces:theyaresim-plicialspacesfreelygeneratedbyinertmaps.Weshowthat...

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