Culf maps and edgewise subdivision Philip Hackney1andJoachim Kock2 with an appendix coauthored with Jan Steinebrunner3

2025-05-06 0 0 769.49KB 53 页 10玖币
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Culf maps and edgewise subdivision
Philip Hackney1and Joachim Kock2
with an appendix coauthored with Jan Steinebrunner3
1University of Louisiana at Lafayette
2Universitat Aut`onoma de Barcelona and Centre de Recerca Matem`atica;
currently at the University of Copenhagen
3University of Copenhagen
Abstract
We show that, for any simplicial space
X
, the
∞
-category of culf maps
over
X
is equivalent to the
∞
-category of right fibrations over
Sd
(
X
), the
edgewise subdivision of
X
(when
X
is a Rezk complete Segal space or
2-Segal space, this is the twisted arrow category of
X
). We give two proofs
of independent interest; one exploiting comprehensive factorization and the
natural transformation from the edgewise subdivision to the nerve of the
category of elements, and another exploiting a new factorization system
of ambifinal and culf maps, together with the right adjoint to edgewise
subdivision. Using this main theorem, we show that the
∞
-category of
decomposition spaces and culf maps is locally an ∞-topos.
Contents
1 Introduction 2
2 Comprehensive Factorization 8
3 Culf maps and ambifinal maps 13
4 The last-vertex map 16
5 Edgewise subdivision and the natural transformation λ21
6 Culfy and righteous maps 26
7 Main theorem via pullback along λ28
8 Main theorem via right Kan extension 30
9 Decomposition spaces and Rezk completeness 35
Appendix A: Relative complete maps and Rezk completion 40
1
arXiv:2210.11191v1 [math.AT] 20 Oct 2022
1 Introduction
Background
1.1. Decomposition spaces (
2
-Segal spaces).
Decomposition spaces [
25
,
26
,
27
] (the same thing as 2-Segal spaces [
18
]; see [
21
]) are simplicial
∞
-groupoids
(simplicial spaces) subject to an exactness condition weaker than the Segal
condition. Technically the condition says that certain simplicial identities are
pullback squares; equivalently, a simplicial space is a decomposition space when
every slice and every coslice is a Segal space.
Where the Segal condition expresses composition, the weaker condition ex-
presses decomposition. The motivation of G´alvez–Kock–Tonks [
25
,
26
,
27
] for
introducing and studying decomposition spaces was that they have incidence
coalgebras and M¨obius inversion. The motivation of Dyckerhoff and Kapra-
nov [
18
] came rather from homological algebra and representation theory. In
both lines of development, an important example of a decomposition space is
Waldhausen’s S-construction [
55
], of an abelian category
A
, say. Recall that
S
(
A
) is a simplicial groupoid which is contractible in degree 0, has the objects
of
A
in degree 1, and short exact sequences in degree 2, etc. Wide-ranging gen-
eralizations of Waldhausen’s construction resulted from the decomposition-space
viewpoint; see [
6
,
8
,
9
], culminating with the discovery that every decomposition
space arises from a certain generalized Waldhausen construction, which takes as
input certain double Segal spaces.
1.2. Edgewise subdivision.
The edgewise subdivision of a simplicial space
X
, first introduced by Segal [
48
], is a new simplicial space
Sd
(
X
) (of the same
homotopy type) with (
Sd X
)
n
=
X2n+1
. Formally (cf. 5.2 below),
Sd :
=
Q∗
,
for
Q:∆→∆
given by [
n
]
7→
[
n
]
op ?
[
n
] = [2
n+
1]. When
X
is the nerve of a
category,
Sd
(
X
) is the nerve of the twisted arrow category. A significant example
of edgewise subdivision is the fact (due to Waldhausen [
55
]) that the edgewise
subdivision of the Waldhausen S-construction is the Quillen Q-construction [
42
],
in this way relating the two main approaches to K-theory of categories.
Decomposition spaces can be characterized in terms of edgewise subdivision,
by a theorem of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer [
7
]:
X
is decomposition if and only if
Sd
(
X
)is Segal. In this paper we explore similar
viewpoints, not just on simplicial spaces but also on simplicial maps.
1.3. Culf maps.
The most important class of simplicial maps for decomposition
spaces — those that induce coalgebra homomorphisms — are the culf maps
(standing for “conservative” and “unique-lifting-of-factorization”). The culf
condition is weaker than being a right (or left) fibration. For
∞
-categories, the
culf maps are the same thing as the conservative exponentiable fibrations studied
by Ayala and Francis [
4
]. For 1-categories, culf functors are also called discrete
Conduch´e fibrations [31].
A technically convenient formulation of the culf condition states that certain
squares are pullbacks (cf. 3.2 below). While that condition will feature in all
our proofs, it is useful to know (cf. 5.3) that a simplicial map
p
is culf if and
2
only if
Sd
(
p
) is a right fibration. (For 1-categories, where edgewise subdivision
is just the twisted arrow category, this result goes back to Lamarche and Bunge–
Niefield [14].)
Further interpretations can be given in analogy with right (or left) fibrations.
Recall that a functor
p:E→B
is a right fibration when for every object
x∈E
,
the induced functor on slices
px:E/x →B/px
is an equivalence. Similarly,
p:E→B
is a left fibration if every induced map on coslices is an equivalence.
The culf condition is weaker:
p
is culf when for every
x∈E
the induced map on
coslices is a right fibration, or equivalently, the induced map on slices is a left
fibration.
1.4. Interval preservation, and culf maps in combinatorics.
The data
over which to slice and then coslice, or coslice and then slice, is just a 1-simplex
f:x→y
. The slice of the coslice (or the coslice of the slice) is then precisely
Lawvere’s notion of interval of
f
, denoted
I
(
f
). Intuitively, the interval of an
arrow
f
is the category of its factorizations. Yet another characterization of culf
maps is that they are the maps that induce equivalences on all intervals (cf. 3.9).
This is the original viewpoint on culf maps of Lawvere [36].
The notion of interval of a 1-simplex is central to the combinatorial theory of
decomposition spaces [
27
], [
24
], [
23
], since it generalizes the notion of intervals in
a poset, which form the basis for the incidence coalgebra of the poset. Just as the
comultiplication map in classical incidence coalgebras splits poset intervals, the
general notion of incidence coalgebra of decomposition spaces is about splitting
decomposition-space intervals, or equivalently, summing over factorizations. The
interpretation of the culf condition from the viewpoint of combinatorics is thus
to preserve interval structure, or to preserve decomposition structure, loosely
speaking.
1.5. Culf maps in dynamical systems and process algebra.
Lawvere’s
original motivation, both for the notion of interval and the notion of culf map,
came from dynamical systems and the general theory of processes [
36
] (part of
his long-time effort to understand continuum mechanics categorically). In this
theory, the general role of culf maps is to express abstract notions of duration and
synchonization, but depending on the situation they are also given interpretation
in terms of “response” and “control.” The interval of an arrow, thought of
as a process, is then the space of trajectories, or executions, of the process.
It is important that the culf condition is weaker than left fibrations (discrete
opfibrations) or right fibrations (discrete fibrations): where left or right fibrations
express determinism, namely unique evolution forward or backward from a given
state (object) (see [
57
] for a development of this viewpoint in computer science),
the culf condition only expresses synchronization of a given process, or control
of it, by a scheduling.
Brown and Yetter [
12
] interpretated the culf condition as preservation of
more abstract notions of dynamics in the theory of
C∗
-algebras. Melli`es [
40
] and
Eberhart–Hirschowitz–Laouar [
19
] exploit similar viewpoints in game semantics.
1.6. Lamarche conjecture.
Working on abstract notions of processes in
computer science, at a time when presheaf semantics was gaining importance
3
to model concurrency (see for example Cattani–Winskel [
16
]), Lamarche (1996)
made the conjecture that for any category
C
, the category
Catculf/C
of culf maps
over
C
is a topos. It was soon discovered, though, that the conjecture is false in
general, by famous counterexamples by Johnstone [
31
], Bunge–Niefield [
14
], and
Bunge–Fiore [13]. (For the interesting history of this conjecture, see [35].)
The categories
C
for which
Catculf/C
is a topos are very special, expressing
a certain local linear time evolution [
13
] (see Fiore [
22
] for further analysis).
This includes the nonnegative reals, the monoid
N
, and more generally free
categories on a graph — these were the examples of importance to Lawvere [
36
]
for dynamical systems. From the viewpoint of computer science the condition
expresses a strict interleaving property (covering models such as labeled transition
systems and synchronization trees [
57
]), but comes short in capturing more
general notions of concurrency.
1.7. Kock–Spivak theorem.
Decomposition spaces were first considered in
connection with process algebra when Kock and Spivak [
35
] discovered that
Lamarche’s conjecture is actually true in general, if just categories are replaced by
decomposition spaces: they showed that for any discrete decomposition space
D
(i.e. a simplicial set rather than a simplicial space), there is a natural equivalence
of categories
Decomp/D 'PrSh(Sd D).
This result shows that not only are culf maps natural to consider in connection
with decomposition spaces, but that also decomposition spaces are a natural
setting for culf maps: even if the base
D
is actually a category, the nicely behaved
class of culf maps into it is from decomposition spaces rather than from categories.
From the viewpoint of processes, the lack of composability is something that
occurs naturally in applications: Schultz and Spivak [
46
] observe that even if
time intervals compose, processes over them do not necessarily compose, since
constraints (called “contracts”) may not extend over time.
Contributions of this paper
One version of our main theorem is the following
∞
-version of the Kock–Spivak
result:
Theorem D
(Theorem 9.3)
.
The
∞
-category of decomposition spaces and culf
maps is locally an
∞
-topos. More precisely, for
X
a decomposition space, we
have an equivalence
Decomp/X 'RFib(Sd X)'RFib([
Sd X)'PrSh([
Sd X).
Here
d
(−)
denotes the Rezk completion of a Segal space. We also will explain
in Proposition 9.12 that if
X
itself is Rezk complete as a decomposition space,
then
Sd
(
X
) is Rezk complete as a Segal space, and we can write
Decomp/X '
PrSh(Sd X) directly.
The substantial part of the result is the first equivalence in the display, which
we establish as a special case of the following general theorem:
4
Theorem C
(Theorem 7.1 & Theorem 8.12)
.
For any simplicial space
X
, there
is a natural equivalence
Culf(X)∼
→RFib(Sd X).
Theorem D follows from this since anything culf over a decomposition space is
again a decomposition space, so for
X
a decomposition space, we have
Culf
(
X
)
'Decomp/X .
We give two proofs of Theorem C. The first uses the ideas of the proof of
the Kock–Spivak theorem in the discrete case, but develops these ideas into
more formal and conceptual arguments (as often required when upgrading a
1-categorical argument to
∞
-categories). In particular we (prove and) exploit
the comprehensive factorization system (final, right-fibration) in the
∞
-category
of simplicial spaces, extending the one for ∞-categories.
We show that Waldhausen’s last-vertex map
Nel
(
X
)
→X
from the nerve of
the
∞
-category of elements back to a simplicial space
X
is final (Lemma 4.5).
This was shown by Lurie and Cisinski for simplicial sets by combinatorial
constructions. Here we give a conceptual high-level proof.
We then exploit the natural transformation
λ: Nel ⇒Sd
first studied by
Thomason [52], and show that it is cartesian on culf maps (Lemma 5.11).
With these preparations, we can exhibit an inverse to the displayed equiva-
lence: it is given essentially by pullback along
λ
(modulo some identifications
involving Nel).
The second proof is completely new, and involves the right adjoint to edgewise
subdivision. It also involves a new factorization system of ambifinal maps and
culf maps. This factorization system restricts to the stretched-culf factorization
system on the
∞
-category of intervals of [
27
], which in turn restricts to the
active-inert factorization system on
∆
. Indeed, the class of ambifinal maps is
the saturation of the class of active maps between representables.
The second proof of Theorem C follows from several small lemmas of inde-
pendent interest:
First we study the
Q!aQ∗
adjunction, and show that its unit is final on
representables (Corollary 8.2) while its counit is ambifinal on representables
(Proposition 8.3).
Moving on to the
Q∗aQ∗
adjunction, we show that just as
Q∗
takes culf
maps to right fibrations (Lemma 5.3), its right adjoint
Q∗
takes right fibrations
to culf maps (Proposition 8.4). The key properties are now that the unit for the
Q∗aQ∗
adjunction is cartesian on culf maps (Lemma 8.7) and that the counit
is cartesian on right fibrations (Lemma 8.8).
After these preparations, the inverse to the equivalence displayed in Theo-
rem C is shown to be given by first applying
Q∗
to get a culf map, and then
pullback along the unit η0of the Q∗aQ∗adjunction.
Lemma 5.3 together with the theorem of Bergner et al. [
7
] shows that edgewise
subdivision is a key aspect of decomposition spaces and culf maps. The lemmas
just quoted show that conversely, the classical notion of edgewise subdivision
inevitably leads to culf maps and ambifinal maps, which are much more recent
notions.
5
摘要：

CulfmapsandedgewisesubdivisionPhilipHackney1andJoachimKock2withanappendixcoauthoredwithJanSteinebrunner31UniversityofLouisianaatLafayette2UniversitatAutonomadeBarcelonaandCentredeRecercaMatematica;currentlyattheUniversityofCopenhagen3UniversityofCopenhagenAbstractWeshowthat,foranysimplicialspaceX,...

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