A pasting theorem for iterated Segal spaces_2

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arXiv:2210.04549v2 [math.CT] 23 May 2024

A PASTING THEOREM FOR ITERATED SEGAL SPACES

JACO RUIT

Abstract. We introduce a novel notion of pasting shapes for iterated Segal

spaces which classify particular arrangements of composing cells in d-uple

Segal spaces. Using this formalism, we then continue to prove a pasting theo-

rem for these iterated Segal spaces.

Contents

1. Introduction 1

2. Preliminaries 5

3. Pasting shapes and their nerves 9

4. Proof of the pasting theorem 29

5. Outlook: an (∞,d)-categorical pasting theorem 40

References 42

1. Introduction

Ad-uple category is a generalization of a d-category, introduced by Ehres-

mann [Ehr63], which has k-cells in d

kdiﬀerent directions for 0 ≤k≤d. In

such a structure, the compatible k-cells that point in the same direction may be

composed, while cells with diﬀerent directions may be related by (k+ 1)-cells.

For d= 2 and d= 3, these are also known as double categories [Ehr63, Deﬁnition

10] and intercategories [GP15, Section 1], respectively. These d-uple categories

are the natural place to study mathematical structures that allow for multiple

sorts of morphisms (1-cells) between them.

For instance, in the realm of algebra, one can consider not only the usual

maps between rings but also bimodules between rings. These are the two di-

rections of 1-cells in the Morita double category ([Shu11, Example 2.3]). Since

its introduction, the theory of double categories has found a wide range of ap-

plications throughout category theory. For instance, the theory of 2-categorical

limits admits a neat description using double categories [GP99]. Pseudo 2-

functors that form a proarrow equipment are better understood as being double

categories with additional properties [Ver92], [Shu08]. These proarrow equip-

ments give rise to formal category theories. In [Shu11], we see an application

of the theory of double categories to homotopy theory.

Whereas in a usual category, one may consider strings of compatible mor-

phisms and take their composites, there are now many conﬁgurations of com-

patible cells in a d-uple category. For instance, one may start with compatible

2 JACO RUIT

2-cells v1,v2,v3,v4,v5,v6in a double category D:

x00 x10 x20 x40

x01 x11 x21 x31 x41

x32 x42

x03 x23 x33 x43,

and wonder: does there exist a unique composite 2-cell vin D? It has been

shown by Dawson and Par´

e [DP93] that (in particular) this arrangement of 2-

cells admits such a composite v. However, not every compatible arrangement of

2-cells in a general double category admits a composite. The authors of [DP93]

established that there is an arrangement that does not have a composite in a

general double category, which is called the pinwheel (we will see this arrange-

ment again in Subsection 3.2), and which is in a particular sense, the canonical

example of such an inadmissible arrangement (see [Daw95]). The procedure

of obtaining new cells by composing compatible arrangements is also called

pasting, which was ﬁrst introduced by B´

enabou [B´

en67] in the context of 2-

categories.

The practice of pasting cells occurs in the context of many diﬀerent categor-

ical structures. Hence, one would like to have access to a pasting theorem that

asserts the existence and uniqueness of composites for certain conﬁgurations of

cells in the categorical structures one considers. A famous pasting theorem for

2-categories was formulated and proven by Power [Pow91]. Nowadays, a wide

range of pasting theorems for (strict) ω-categories are available. Forest [For22]

has recently uniﬁed the main pasting theorems in this context to a more gen-

eral pasting theorem for ω-categories. There is a pasting theorem for double

categories due to Dawson and Par´

e [DP93].

The emergence of (weak) ∞-category theory has created a need for variants

of pasting theorems in the weaker setting. In this context, it is no longer natural

to ask for unique composites, but instead require that the space1of composites

is contractible. Lately, Hackney, Ozornova, Riehl and Rovelli [HORR21] have

proven such a pasting theorem for (∞,2)-categories that generalizes Power’s

pasting theorem for 2-categories. There is an ∞-analog for double categories

as well, so called double ∞-categories, introduced by Haugseng [Hau13] and

further studied by Moser [Mos20]. Thus, one may now ask: does there exist a

pasting theorem for these double ∞-categories?

The goal of this paper is to answer this question aﬃrmatively. We will treat a

more general pasting problem for d-uple categories in the weak ∞-categorical

setting. Note that this would already be interesting from the strict perspective:

the pasting theorem particularly yields a novel pasting theorem for d-uple cat-

egories. The author does not know of an existing similar result for d > 2.

1By space, we will always mean an ∞-groupoid in this paper.

A PASTING THEOREM FOR ITERATED SEGAL SPACES 3

We will consider an ∞-categorical variant of d-uple categories that are called

d-uple Segal spaces [Hau17]. Ordinary (1-uple) Segal spaces were ﬁrst intro-

duced by Rezk [Rez01] as a model for ∞-categories. The d-uple Segal spaces

are iterated variants of these that have additional directions for morphisms. For

instance, a 2-uple Segal space Xcontains a space of objects, and between any

two objects, a space of vertical and horizontal arrows. Thus a 2-uple Segal space

has two categorical directions instead of merely one as is the case for an ordi-

nary Segal space. Compatible arrows of Xthat have the same direction, can be

composed in a coherently associative fashion. Moreover, Xcontains a space of

2-cells. A 2-cell may be pictured as a square

a b

c d

in X. Here the arrows that point horizontally are horizontal arrows of Xand

similarly for the vertical ones. That is, 2-cells have a source and target verti-

cal arrow and a source and target horizontal arrow. Again, Xhas a coherently

associative composition for these 2-cells, which is compatible with the composi-

tion of 1-cells. In general, d-uple Segal spaces contain dcategorical directions,

which may interact coherently using higher cells.

These d-uple Segal spaces are rich structures that play a useful role in ∞-

category theory. For instance, by ‘truncating’ all but one of the categorical

directions, they can be used to model (∞,d)-categories. This is the model for

(∞,d)-categories due to Barwick [Bar05], which we will brieﬂy discuss in Section 5.

It has been (directly) compared to other models for (∞,d)-categories by Bergner

and Rezk in [BR13] and [BR20], and Loubaton, Ozornova and Rovelli in [Lou22]

and [OR22]. Consequently, d-uple Segal spaces may act as a useful interme-

diate step towards constructing (∞,d)-categories. For instance, in order to

construct (∞,d)-categories of iterated spans or to construct the Morita (∞,d)-

categories it is more convenient to deﬁne their encompassing d-uple Segal spaces

ﬁrst (see [Hau18] and [Hau17]). We also believe that d-uple Segal spaces can

be used to study phenomena of (∞,d)-categories. For instance, the universal

property of the (∞,2)-category of spans [EH23] should be the shadow of a uni-

versal property of 2-uple Segal space of spans that is analogous to one in the

strict case [DPP10]. We hope to study this in future work.

This work grew out of the author’s study of double ∞-categories, which can

be viewed as 2-uple Segal spaces that satisfy a completeness assumption (see

Remark 2.9). More speciﬁcally, out of the interest in a particular class of dou-

ble ∞-categories: so-called ∞-equipments, which are generalizations of proar-

row equipments to the ∞-categorical context. These ∞-equipments oﬀer a con-

text for synthetic or formal category theory. There exist suitable ∞-equipments

for equivariant, indexed, internal, ﬁbered, enriched and ordinary ∞-category

theory. We commenced the study of these ∞-equipments in [Rui23].

Content of the paper. We will commence the paper by setting up the necessary

preliminaries, including revisiting the deﬁnition of iterated Segal spaces, in

Section 2.

4 JACO RUIT

Subsequently, in Section 3, we introduce the novel notion of d-dimensional

pasting shapes whose nerves classify arrangements of cells in a d-uple Segal

space. For d= 3, these include arrangements of rectangular cuboids, whose

faces may be subdivided into smaller rectangles. The faces of these rectangles

may in turn be subdivided into smaller edges. We will also discuss some funda-

mental properties that pasting shapes may have. In particular, we single out an

important class of pasting shapes: the so-called (locally) composable ones. The

composable pasting shapes are certain well-behaved pasting shapes. For d= 2,

the aforementioned pinwheel of Dawson and Par´

e is an example of a pasting

shape that is not composable.

After setting up the theory of pasting shapes, we can formulate the main

result of this paper, the pasting theorem for d-uple Segal spaces:

Theorem A (The pasting theorem).Suppose that I1,...,Inis a covering of a d-

dimensional locally composable pasting shape I. Then Ican be written as a union

[

i=1

Ii,

and this colimit description is preserved when passing to the ∞-category of d-uple

Segal spaces via the nerve functor for pasting shapes.

The precise statement appears as Theorem 3.49 in the paper. This theorem can

be used to show Corollary 3.52, which states that the spine inclusion associated

with a composable pasting shape is an equivalence. The composable pasting

shapes thus classify arrangements of cells that admit a composite that is unique

up to contractible choice (see Corollary 3.55).

The technical heart of the paper is in Section 4. Here, we proceed by induc-

tion on the dimension of pasting shapes, to give a proof of Theorem A. After

this demonstration, we conclude the paper in Section 5 by giving an idea of

how our pasting theorem for d-uple Segal spaces may yield a pasting theorem

for (∞,d)-categories.

Conventions. We will use the language of ∞-categories throughout this article.

For deﬁniteness, we work with the model of quasi-categories for ∞-categories

as developed by Joyal and Lurie [Lur09a]. The readers that prefer to use the lan-

guage of model categories, may interpret the main results of this paper within

the appropriate model categories associated with the ∞-categories in question.

We will use the following customary notation:

•The (large) ∞-categories of spaces [Lur09a, Deﬁnition 1.2.16.1] and ∞-

categories [Lur09a, Deﬁnition 3.0.0.1] are denoted by Sand Cat∞, re-

spectively.

•If Cis an ∞-category, we write

MapC(−,−) : Cop ×C→S

for its associated mapping space functor [Lur09a, Subsection 5.1.3].

•Every 1-category Cwill be viewed as an ∞-category, suppressing the

notation of the nerve [Lur09a, Subsection 1.1.2].

A PASTING THEOREM FOR ITERATED SEGAL SPACES 5

Acknowledgements. I want to thank my PhD-supervisor, Lennart Meier, for

the helpful conversations during the writing of this paper and his useful com-

ments on the draft versions. Furthermore, I would like to express my gratitude

to the anonymous referee for the useful suggestions that greatly improved the

presentation of this article.

During the writing of this paper, the author was funded by the Dutch Re-

search Council (NWO) through the grant “The interplay of orientations and

symmetry”, grant no. OCENW.KLEIN.364.

2. Preliminaries

A clear deﬁnition of the (strict) d-uple categories introduced by Ehresmann

[Ehr63] can be given in terms of Grothendieck’s notion of categorical objects

[Gro95]. Following [Hau17] and [Hau18], we will introduce an ∞-categorical

variant on d-uple categories via the same principle.

2.1. Categorical objects. To this end, we ﬁrst need the notion of categorical ob-

jects in an ∞-category. Throughout this section, we will ﬁx an ∞-category Cand

assume that it has all pullbacks. The following deﬁnition is due to [Lur09b]:

Deﬁnition 2.1. Acategorical object Xin Cis a simplicial object X:∆op →Csuch

that the so-called Segal map

X([n]) →X({0≤1})×X({1})·· · ×X({n−1})X({n−1≤n})

is an equivalence for all n. The full subcategory of Fun(∆op,C) spanned by the

categorical objects in Cis denoted by Cat(C).

Example 2.2. We have the following examples:

•A categorical object in the (2,1)-category of categories is a pseudo dou-

ble category. These were deﬁned by Grandis and Par´

e [GP99].

•If Cis given by the ∞-category of spaces S, then Cat(S) is the ∞-category

that underlies the model category of Segal spaces, constructed by Rezk in

[Rez01, Theorem 7.1]. Henceforth, we will refer to categorical objects

in Sas Segal spaces.

•A categorical object in Cat∞is called a double ∞-category. These were

ﬁrst studied by Haugseng in [Hau13].

Note that the subcategory Cat(C)⊂Fun(∆op,C) is closed under limits, so that

limits of categorical objects may be computed pointwise. In particular, we de-

duce that Cat(C) again admits all pullbacks. Thus, we may iterate Deﬁnition 2.1:

Deﬁnition 2.3. We deﬁne the ∞-category of d-uple categorical objects in Cby

Catd(C) := Cat(···Cat(Cat(C))···).

By adjunction, the ∞-category of d-uple categorical objects in Ccan be de-

scribed as a full subcategory of the ∞-category

Fun(∆op,×d,C)

of d-uple simplicial objects in C. In the case that Cis presentable, then Catd(C) is

a (left) reﬂective subcategory of Fun(∆op,×d,C). I.e. the inclusion admits a left

adjoint (see also [Lur09a, Remark 5.2.7.9]) in this case. We will demonstrate

how this can be established.

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arXiv:2210.04549v2[math.CT]23May2024APASTINGTHEOREMFORITERATEDSEGALSPACESJACORUITAbstract.WeintroduceanovelnotionofpastingshapesforiteratedSegalspaceswhichclassifyparticulararrangementsofcomposingcellsind-upleSegalspaces.Usingthisformalism,wethencontinuetoproveapastingtheo-remfortheseiteratedSegalsp...

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