Theory and Applications of Categories, Vol. 28, No. 27, 2013, pp. 933–980.

A DOUBLE CATEGORICAL MODEL OF WEAK 2-CATEGORIES

SIMONA PAOLI, DORETTE PRONK

Abstract. We introduce the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories. We show that this model is suitably equivalent to bicategories and give an explicit description of the functors involved in this biequivalence. As an application we show that groupoidal weakly globular double categories model homotopy 2-types.

1. Introduction

Category theory has seen the development of several types of higher dimensional structures, which find applications to diverse areas such as homotopy theory, mathematical physics, algebraic geometry, and computer science. The structures for higher dimensional categories include three main classes; namely, strict n-categories, n-fold categories and weak n-categories.

The notion of strict n-categories is the simplest of the higher structures and is obtained by iterated enrichment: a strict n-category is a category enriched in strict (n−1)-categories with respect to the cartesian monoidal structure. Although widely studied and wellunderstood, strict n-categories are often too simple to capture the complexity needed for applications. For instance, strict n-groupoids are not enough to model n-types of topological spaces.

A much wider type of higher dimensional categorical structure is formed by the n-fold categories, obtained by iterated internalization: an n-fold category is a category internal to (n − 1)-fold categories. The notion of a double category, and the general idea of considering internal categories inside another category was introduced by Charles Ehresmann in 1963 [14]. The foundational properties of n-fold categories (at that time called multiple categories) were studied in a series of papers by Andrée and Charles Ehresmann [1, 13, 12, 11]. A homotopical treatment of n-fold categories, in the context of Quillen model structures, was given in [15] and [16]. The case n = 2, where they are called double categories, has been studied extensively. In their full generality, double categories give rise to surprisingly complex phenomena, such as the non-composability of certain tiling diagrams of double cells [9] and [8].

Another very important class of higher dimensional structures is that of weak ncategories. This comprises several different models (see for instance [21] and [22]); in the case n = 2, these are all suitably equivalent to the classical notion of bicategory [2].

Received by the editors 2013-09-10 and, in revised form, 2013-09-29.

Transmitted by Robert Paré. Published on 2013-10-06.

2010 Mathematics Subject Classification: 18D05, 18G55, 18G30.

Key words and phrases: double categories, strict 2-categories, bicategories, Tamsamani weak 2categories, pseudo-functors, strictification.

\begin{matrix}c \\ ◯ \end{matrix} Simona Paoli, Dorette Pronk, 2013. Permission to copy for private use granted.

The fundamental question we are addressing in this work concerns the relations between these three classes of higher dimensional structures. Some of these relations are well known: it is easily seen that there are full and faithful embeddings of strict n-categories into n-fold categories; likewise, most definitions of weak n-categories contain strict ncategories as a special case. However, the relation between n-fold categories and weak n-categories is less understood.

A hint in this direction comes from homotopy theory. Blanc and the first author showed in [3] and [4] that a suitable subcategory of n-fold groupoids, called weakly globular n-fold groupoids, model n-types of topological spaces. Likewise, in the pathconnected case, the first author showed [24] that weakly globular cat^{n−1}-groups model path-connected n-types.

These results motivate us to search for a suitable subcategory of n-fold categories which models weak n-categories.

In this paper we present a solution to this problem for the case n = 2. This low dimensional case presents a special interest in this context because it links with the vast existing literature on double categories, see for instance, [14], [6], [17], [30] and [15]. The extension of this work to general n is the subject of a subsequent paper [26].

The elements of our double categorical model of weak 2-categories are called weakly globular double categories. This model is based on the idea of weak globularity. In a strict 2-category, the 0-cells have a discrete structure, that is, they form a set. This is called the globularity condition, as it determines the globular shape of the 2-cells. In a weakly globular double category we relax the globularity condition into weak globularity: the 0-cells have a categorical structure of their own which is not discrete but it is equivalent to a discrete one. The double categorical context gives the possibility to encode this extra structure precisely. We refer the reader to Section 2 for a longer discussion of the intuition behind the notion of weakly globular double category.

Our main result is that weakly globular double categories, with the appropriate classes of morphisms and 2-cells, are suitably equivalent to bicategories: more precisely, there is a biequivalence of 2-categories:

Bic : WGDbl_{ps,v} ≃ Bicat_{icon} : Dbl.  (1)

The proof of this result goes via a comparison with another existing model of weak 2-categories due to Tamsamani [33], which was proved by Lack and the first author to be suitably equivalent to bicategories [20]. Thus, we obtain the biequivalence (1) from biequivalences

WGDbl_{ps,v} ≃ (Ta_{2})_{ps} ≃ Bicat_{icon}

The comparison functors between weakly globular double categories and the Tamsamani model factors through a subcategory of pseudo-functors Ps[Δ^{op}, Cat]. We refer the reader to Section 2 for an overview of our methodology.

The biequivalence (1) gives a new way of rigidifying a bicategory into a strict double structure, which differs from the classical strictification of bicategories into strict 2categories. An explicit description of both functors involved in this biequivalence reveals some interesting details about how weakly globular double categories model the weak higher structure. Specifically, we see that the weakly globular double category associated to a bicategory is obtained by local rigidifications of the bicategory which are glued together in a suitable sense by vertical isomorphisms, whereas the bicategory associated to a weakly globular double category is obtained as a kind of quotient.

We end this paper by highlighting some features of our model which one would expect from a model of weak 2-categories. Namely, we show that suitably groupoidal weakly globular double categories model homotopy 2-types and we show that for weakly globular double categories, pinwheel-tilings do form composable pasting diagrams. A complete proof establishing that every tiling diagram in a weakly globular double category is composable is forthcoming [27].

Acknowledgements. The first author is supported by a Marie Curie International Reintegration Grant no. 256341. She would also like to thank the Department of Mathematics and Statistics of Dalhousie University for their hospitality during a visit in June 2012. The second author is supported by an NSERC Discovery grant. She would also like to thank the Department of Mathematics of the University of Leicester for their hospitality during a visit in August 2012. Both authors thank the Department of Computing of Macquarie University and CoAct for their hospitality during July 2013.

2. Overview of the main results

This section provides a summary of the main ideas and results of this paper. This paper presents a new notion of a weak 2-category via a subcategory of strict double categories, which we call weakly globular double categories.

While several different models of weak 2-categories exist in the literature (see for instance [21]), the prototype is the classical notion of bicategory [2]. The main result of this paper (Theorem 7.10) asserts that there is a biequivalence between the 2-category of weakly globular double categories with pseudo-functors and vertical transformations and the 2-category of bicategories with homomorphisms and icons:

Bic : WGDbl_{ps,v} ≃ Bicat_{icon} : Dbl.  (2)

Weakly globular double categories also form an intermediate structure between strict 2-categories and (strict) double categories in the following way:

2-Cat ↪ WGDbl ↪ DblCat,

where the first inclusion considers a 2-category as a double category with only identities as vertical arrows, i.e., the category of objects and vertical morphisms is discrete. This discreteness condition is also called the globularity condition, as it determines the globular shape of the 2-cells in a strict 2-category.

The first step in the process to obtain the notion of weakly globular double category (Definition 6.7) is to replace the globularity condition by weak globularity: the category of objects and vertical arrows in a weakly globular double category 𝕏 is no longer discrete but merely equivalent to a discrete category, and therefore a posetal groupoid (which can also be viewed as a set with an equivalence relation). The set of connected components of this posetal groupoid is the set of objects of the corresponding bicategory Bic𝕏.

The second step in obtaining the correct definition of a weakly globular double category is to realize that arrows in Bic𝕏 correspond to horizontal arrows in 𝕏. So we need to provide extra structure allowing one for instance to compose in Bic𝕏 pairs of horizontal arrows in 𝕏 whose target and source may not be identical, but which are in the same vertical connected component:

[figure: •; g; •; f; •]  (3)

The general version of this extra condition asserts that for each n ≥ 2, the equivalence of categories γ : 𝕏_{0} → 𝕏_{0}^{d} between the posetal groupoid 𝕏_{0} and its discretization induces equivalences of categories

𝕏_{1}×_{𝕏_{0}} ⋯^{n}×_{𝕏_{0}}𝕏_{1} ≃ 𝕏_{1}×_{𝕏_{0}^{d}} ⋯^{n}×_{𝕏_{0}^{d}}𝕏_{1} .  (4)

This implies in particular that the maps f and g in the configuration (3) can be lifted to a composable pair of horizontal arrows in 𝕏 (see Section 8.1 for details).

The methodology to establish the comparison result (2) makes use of another model of weak 2-categories, due to Tamsamani, which we review in Section 4. This is a simplicial model based on the functor category [Δ^{op}, Cat] of simplicial objects in Cat and of the notion of Segal maps, i.e, the maps 𝕏_{n} → 𝕏_{1}×_{𝕏_{0}} ⋯^{n}×_{𝕏_{0}}𝕏_{1} that are part of the simplicial structure. The details are described in Section 4.1.

Notice that this setting is natural in our context as both strict 2-categories and double categories can be described in this simplicial language: a double category is the same as an object 𝕏 ∈ [Δ^{op}, Cat] such that the Segal maps are isomorphisms, and if we further require 𝕏_{0} to be discrete we obtain a strict 2-category.

It was shown in [20] that there is a biequivalence of 2-categories

N : (Ta_{2})_{ps} ≃ Bicat_{icon} : G  (5)

where (Ta_{2})_{ps} is the 2-category of Tamsamani weak 2-categories with pseudo-functors. In this paper we show (Theorem 7.6) that there is a biequivalence of 2-categories

D: WGDbl_{ps,v} ≃ (Ta_{2})_{ps} : Q .  (6)

Hence the main result (2) follows by composition of (5) and (6); that is, Bic = ND and Dbl = QG.

The functors D and Q are called discretization and rigidification respectively. The functor D replaces the posetal groupoid of vertical arrows in the horizontal nerve N_{h}𝕏 of a weakly globular double category 𝕏 by its equivalent discrete category. This produces an object of [Δ^{op}, Cat] which is discrete at level zero, so we recover the globularity condition. This however comes at the expense of the strictness of the Segal maps, which from being isomorphisms in N_{h}𝕏 now become mere equivalences of categories. We therefore obtain an object D𝕏 ∈ Ta_{2}, which is suitably equivalent to 𝕏 (see Section 7.3 for further details).

The functor Q associates to a Tamsamani object X ∈ Ta_{2}, in which composition is only weakly associative and unital, a weakly globular double category QX in which all compositions are strictly associative and unital, but in which we no longer have the globularity condition. Thus, we think of QX as a rigidification of X. We note that this is however completely different from the classical strictification of a bicategory into a strict 2-category [23].

Unlike the functor D, the construction of the rigidification functor Q is rather elaborate and requires an intermediate passage, via the category Ps[Δ^{op}, Cat] of pseudo-functors from Δ^{op} to Cat.

We show in Section 5 how to associate an object of Ps[Δ^{op}, Cat] to a Tamsamani weak 2-category in a functorial way, using a standard categorical technique known as transport of structure along an adjunction. For X ∈ Ta_{2} ∈ [Δ^{op}, Cat] this allows us to replace each category X_{n} for n ≥ 2 with the equivalent category X_{1}×_{X_{0}} ⋯^{n}×_{X_{0}}X_{1} and equip the resulting object with a pseudo simplicial structure, so we have a functor

S : Ta_{2} → Ps[Δ^{op}, Cat] .

The pseudo-functors in the image of S are of a particular type. We write P̃s[Δ^{op}, Cat] for the subcategory of pseudo-functors H: Δ^{op} → Cat, called Segalic, such that H_{0} is a discrete category and H_{n} ≅ H_{1}×_{H_{0}} ⋯^{n}×_{H_{0}}H_{1} for each n ≥ 2. We see that the functor S in fact lands in the subcategory of Segalic pseudo-functors,

S : Ta_{2} → P̃s[Δ^{op}, Cat] .

On the other hand, pseudo-functors can be strictified to strict functors (see [28]); that is, there is a strictification functor

St : Ps[Δ^{op}, Cat] → [Δ^{op}, Cat]

together with, for each X ∈ Ps[Δ^{op}, Cat], a pseudo-natural transformation St X → X, which is a levelwise equivalence of categories.

We show (see Sections 6 and 7) that the strictification functor St, when restricted to the subcategory P̃s[Δ^{op}, Cat] of Segalic pseudo-functors, lands in the subcategory N_{h}(WGDbl) of [Δ^{op}, Cat] whose objects are horizontal nerves of weakly globular double categories:

St : P̃s[Δ^{op}, Cat] → N_{h}(WGDbl) .

Since, on the other hand, N_{h}(WGDbl) is isomorphic to WGDbl itself, we finally obtain by composition the rigidification functor

Q : (Ta_{2})_{ps} ⟶^{S} P̃s[Δ^{op}, Cat] ⟶^{St} N_{h}(WGDbl) ⟶^{P} WGDbl_{ps,v}

together with a pseudo-functor N_{h}QX → X for every X ∈ (Ta_{2})_{ps} which is a levelwise equivalence of categories.

It is instructive to give an explicit description of the functors Bic and Dbl in the biequivalence (2). The associated double category DblB of a bicategory B has an interesting description in terms of marked paths. The horizontal arrows in this double category consist of paths in the original bicategory with two marked objects. One can think of this arrow as corresponding to a particular chosen composite of the path between the two marked objects. This is then a local rigidification of the bicategory in the sense that it only rigidifies or strictifies the composition with other arrows along the path (which may extend beyond the marked objects). The vertical arrow structure allows us to glue all these local rigidifications together into a strict double category. For details on how this works, see Section 8.2.

Section 8.1 contains the description of Bic𝕏 for a weakly globular double category 𝕏, which can be viewed as a fundamental bicategory of 𝕏. Its objects correspond to vertical isomorphism classes of the objects in 𝕏. However, for the arrows and double cells we do not take equivalences classes and the second condition in the definition of weakly globular double categories, involving the equivalences (4), plays a central role in the definition of horizontal composition of the arrows and 2-cells of Bic𝕏. This will also be called the induced Segal maps condition.

This condition plays a crucial role in Section 9, where we show that the pathological double categorical pasting diagrams known as pinwheels can be factored and then composed when working in a weakly globular double category. This is in some sense an expected feature, as each weakly globular double category models a bicategory.

Another expected feature concerns the modelling of 2-types for those weakly globular double categories satisfying suitable invertibility conditions. We establish this in Section 10, where an appropriate subcategory of groupoidal weakly globular double categories is shown to correspond to Tamsamani weak 2-groupoids, and therefore to model 2-types of topological spaces.

3. Double categories

Since we will introduce weakly globular double categories as a special kind of double categories, we use this section to review the definition of a double category and some of the basic concepts related to double categories, such as morphisms and transformations. In particular, we consider two types of morphisms between double categories, strict functors and pseudo-functors. We inherit these notions from the category of simplicial sets via the adjoint to the nerve functor.

Strict functors correspond to strict transformations of simplicial sets and are what a category theorist would expect them to be, but the notion of pseudo-functor, which is inherited from pseudo-natural transformations between the nerve functors, turns out to be a bit weaker than what one usually assumes for double categories (as in [10], for instance). We introduce horizontal and vertical transformations and note that for pseudo-functors it only makes sense to consider vertical transformations.

3.1. Double categories. A double category 𝕏 is an internal category in Cat, i.e., a diagram of the form

𝕏 = ( \xymatrix{𝕏_{1} ×_{𝕏_{0}} 𝕏_{1} \ar[r]|{m} & 𝕏_{1} \ar[r]^{d_{0}} \ar[r]_{d_{1}} & 𝕏_{0} \ar[l]|{s}} ) .  (7)

The elements of 𝕏_{00}, i.e., the objects of the category 𝕏_{0}, are the objects of the double category. The elements of 𝕏_{01}, i.e., the arrows of the category 𝕏_{0}, are the vertical arrows of the double category. Their domains, codomains, identities, and composition in the double category 𝕏 are as in the category 𝕏_{0}. For objects A, B ∈ 𝕏_{00}, we write

𝕏_{v}(A, B) = 𝕏_{0}(A, B)

for the set of vertical arrows from A to B. We denote a vertical identity arrow by 1_{A} and write · for vertical composition. The elements of 𝕏_{10}, i.e., the objects of the category 𝕏_{1}, are the horizontal arrows of the double category, and their domain, codomain, identities and composition are determined by d_{0}, d_{1}, s, and m in (7). For objects A, B ∈ 𝕏_{00}, we write

𝕏_{h}(A, B) = {f ∈ 𝕏_{10}| d_{0}(f) = A and d_{1}(f) = B}.

We will also use ∘ for the composition of a horizontal arrows, g ∘ f = m(g, f). In order to make a notational distinction between horizontal and vertical arrows, we denote the vertical arrows by [figure: •]and the horizontal arrows by [figure] . The elements of 𝕏_{11}, i.e., the arrows of the category 𝕏_{1}, are the double cells of the double category. An element α ∈ 𝕏_{11} has a vertical domain and codomain in 𝕏_{10} (since 𝕏_{1} is a category), which are horizontal arrows, say h and k respectively. The cell α also has a horizontal domain, d_{0}(α), and a horizontal codomain, d_{1}(α). The arrows d_{0}(α) and d_{1}(α) are vertical arrows. Furthermore, the horizontal and vertical domains and codomains of these arrows match up in such a way that all this data fits together in a diagram

[figure: h; •; d_{0}(α); α; •; d_{1}(α); k; .]

These double cells can be composed vertically by composition in 𝕏_{1} (again written as ·) and horizontally by using m, and written as α_{1} ∘ α_{2} = m(α_{1}, α_{2}). The identities in 𝕏_{1} give us vertical identity cells, denoted by

[figure: A; •; 1_{A}; 1_{f}; f; B; •; 1_{B}; or; A; 1_{f}; f; B; A; f; B; A; f; B.]

The image of s gives us horizontal identity cells, denoted by

[figure: A; •; v; _{id_{v}}; ^{Id^{A}}; A; •; v; or; A; •; v; _{id_{v}}; A; •; v; B; _{Id_{B}}; B; B; B,]

where Id_{A} = s(A) and id_{v} = s(v). Composition of squares satisfies horizontal and vertical associativity laws and the middle four interchange law. Further, id_{1_{A}} = 1_{Id_{A}} and we will denote this cell by ι_{A},

[figure: A; •; 1_{A}; ^{Id^{A}}; ι_{A}; A; •; 1_{A}; A; _{Id_{A}}; A.]

For any double category 𝕏, the horizontal nerve N_{h}𝕏 is defined to be the functor

N_{h}𝕏: Δ^{op} → Cat such that (N_{h}𝕏)_{0} = 𝕏_{0}, (N_{h}𝕏)_{1} = 𝕏_{1} and (N_{h}𝕏)_{k} = 𝕏_{1}×_{𝕏_{0}} ⋯^{k} ×_{𝕏_{0}}𝕏_{1}

for k ≥ 2. So N_{h}𝕏 is given by the diagram

[figure: ···; 𝕏_{1} ×_{𝕏_{0}} 𝕏_{1}; m; π_{1}; π_{2}; 𝕏_{1}; d_{0}; d_{1}; 𝕏_{0}; s]

3.2. Pseudo-functors and strict functors. As maps between double categories we consider those functors that correspond to natural transformations between their horizontal nerves.

3.3. Definition.

1. A strict functor F: 𝕏 → 𝕐 between double categories is given by a (strict) natural

transformation

F: N_{h}𝕏 ⇒ N_{h}𝕐: Δ^{op} → Cat.

2. A pseudo-functor F: 𝕏 → 𝕐 between double categories is given by a pseudo-natural transformation F: N_{h}𝕏 ⇒ N_{h}𝕐: Δ^{op} → Cat.

So strict functors send objects to objects, horizontal arrows to horizontal arrows, vertical arrows to vertical arrows, and double cells to double cells, and preserve domains, codomains, identities and horizontal and vertical composition strictly. Pseudo-functors will preserve all this structure only up to appropriate vertical isomorphisms. Note that pseudo-functors between double categories as defined here are weak in a different way from what is described in [10], for instance. A pseudo-functor (F, φ, σ, μ): 𝕏 → 𝕐 in our definition consists of functors F_{0} : 𝕏_{0} → 𝕐_{0}, F_{1} : 𝕏_{1} → 𝕐_{1} and F_{k}: 𝕏_{1}×_{𝕏_{0}} ⋯^{k} ×_{𝕏_{0}}𝕏_{1} → 𝕐_{1}×_{𝕐_{0}} ⋯^{k} ×_{𝕐_{0}}𝕐_{1} for k ≥ 2, together with invertible natural transformations,

[figure: 𝕏_{1}; d_{i}; F_{1}; φ_{i}; 𝕐_{1}; d_{i}; 𝕏_{0}; s; F_{0}; σ; 𝕐_{0}; s; 𝕏_{0}; F_{0}; 𝕐_{0}; 𝕏_{1}; F_{1}; 𝕐_{1}; 𝕏_{1} ×_{𝕏_{0}} 𝕏_{1}; m; F_{2}; μ; 𝕐_{1} ×_{𝕐_{0}} 𝕐_{1}; m; 𝕏_{1} ×_{𝕏_{0}} 𝕏_{1}; π_{i}; F_{2}; θ_{i}; 𝕐_{1} ×_{𝕐_{0}} 𝕐_{1}; π_{i}; 𝕏_{1}; F_{1}; 𝕐_{1}; 𝕏_{1}; F_{1}; 𝕐_{1} ,]

(where i = 1, 2), and analogously for F_{k} with k ≥ 2. These satisfy the usual naturality and coherence conditions, that can be derived from their (pseudo) simplicial description.

Note that vertical composition and domains and codomains are preserved strictly, but horizontal domains and codomains are only preserved up to a vertical isomorphism. (Compare this to the notion of pseudo-morphism in [10] which requires that domains and codomains are preserved strictly, but horizontal composition and units are only preserved up to coherent isomorphisms. So the pseudo-morphisms considered there form a strict subclass of the ones considered in this paper.)

3.4. Vertical and horizontal transformations. Since double categories have two types of arrows, there are two possible choices for types of transformations between maps of double functors: vertical and horizontal transformations. Vertical transformations correspond to modifications between natural transformations of functors from Δ^{op} into Cat, so these are the ones that are relevant to our study of the rigidification functor from Tamsamani weak 2-categories to weakly globular double categories.

3.5. Definition. A vertical transformation γ: F ⇒ G: 𝕏 ⇉ 𝕐 between strict double functors has components vertical arrows \xymatrix{γ_{A}: F A \ar[r]|{•} & GA} indexed by the objects of 𝕏

and for each horizontal arrow \xymatrix{A \ar[r]^{h} & B} in 𝕏, a double cell

[figure: F A; F h; •; γ_{A}; γ_{h}; F B; •; γ_{B}; GA; Gh; GB,]

such that γ is strictly functorial in the horizontal direction, i.e., γ_{h_{2}∘h_{1}} = γ_{h_{2}} ∘ γ_{h_{1}}, and natural in the vertical direction, i.e.,

[figure: F A; F h; •; F v; F ζ; F B; •; F w; F A; •; γ_{A}; F h; γ_{h}; F B; •; Gw; F C; •; γ_{C}; γ_{k}; F k; F D; •; γ_{D}; ≡; GA; •; Gv; Gζ; Gh; GB; •; Gw; GC; Gk; GD; GC; Gk; GD,]  (8)

for any double cell ζ in 𝕏.

To give a vertical transformation between pseudo-functors of double categories, we need to require that the data above fits together with the structure cells of the pseudotransformations. We spell out part of the details and will leave the rest for the reader.

3.6. Definition. A vertical transformation γ: F ⇒ G: 𝕏 ⇉ 𝕐 between pseudo-functors has components vertical arrows \xymatrix{γ_{A}: F A \ar[r]|{•} & GA} indexed by the objects of 𝕏 and for each

horizontal arrow \xymatrix{A \ar[r]^{h} & B} in 𝕏, a double cell

[figure: d_{0}F h; F h; •; d_{0}γ_{h}; γ_{h}; d_{1}F h; •; d_{1}γ_{h}; d_{0}Gh; Gh; d_{1}Gh,]

such that the following squares of vertical arrows commute:

\xymatrix{F_{0}A \ar[r]|{•} \ar[d]|{•}_{γ_{A}} & d_{0}F h \ar[d]|{•}^{d_{0}γ_{h}} & F_{0}B \ar[r]|{•} \ar[d]|{•}_{γ_{B}} & d_{1}F_{1}h \ar[d]|{•}^{d_{1}γ_{h}} \\ G_{0}A \ar[r]|{•} & d_{0}G_{1}h & G_{0}B \ar[r]|{•} & d_{1}G_{1}h}

where the unlabelled arrows are the structure isomorphisms corresponding to F and G.

We require that γ is natural in the vertical direction, in the sense that the following square of vertical arrows commutes for a vertical arrow \xymatrix{A \ar[r]|{•}^{v} & B} in 𝕏,

\xymatrix{F_{0}A \ar[r]|{•}^{F_{1}v} \ar[d]|{•}_{γ_{A}} & F_{0}B \ar[d]|{•}^{γ_{B}} \\ G_{0}A \ar[r]|{•}^{G_{1}v} & G_{0}B}

and furthermore, for any double cell ζ in 𝕏,

[figure: d_{0}F_{1}h; F h; •; d_{0}F_{1}ζ; F_{1}ζ; d_{1}F_{1}h; •; d_{1}F_{1}ζ; d_{0}F_{1}h; •; d_{0}γ_{h}; F h; γ_{h}; d_{1}F_{1}h; •; d_{1}γ_{h}; d_{0}F_{1}k; •; d_{0}γ_{k}; γ_{k}; F k; d_{1}F_{1}k; •; d_{1}γ_{k}; ≡; d_{0}G_{1}h; •; d_{0}G_{1}ζ; G_{1}ζ; Gh; d_{1}G_{1}h; •; d_{1}G_{1}ζ; d_{0}G_{1}k; Gk; d_{1}G_{1}k; d_{0}G_{1}k; Gk; d_{1}G_{1}k.]

In the horizontal direction we require pseudo-functoriality, which means that

[figure: F_{1}(gf); •; μ_{g,f}^{F}; •; •; π_{2}F_{2}(g,f); (θ_{2})_{g,f}; •; •; π_{1}F_{2}(g,f); (θ_{1})_{g,f}; •; F_{1}f; •; d_{0}γ_{f}; γ_{f}; •; •; F_{1}g; γ_{g}; •; d_{1}γ_{g}; =; F_{1}(gf); •; γ_{gf}; •; •; (θ_{2})_{g,f}^{−1}; G_{1}f; •; •; G_{1}g; (θ_{1})_{g,f}^{−1}; •; G_{1}(gf); .; •; π_{2}G_{2}(g,f); (μ_{g,f}^{G})^{−1}; π_{2}G_{2}(g,f); •; G_{1}(gf)]

For pseudo-functors, it only makes sense to consider vertical transformations. (Note that since the pseudo-aspect of these transformations is completely determined by their domain and codomain pseudo-functors, we will speak of vertical transformations without the adjective strict or pseudo.) For further details on pseudo-functors and vertical transformations between them, see the sequel [25] to this paper where we will spell out some of the coherence and naturality conditions and use them in the study of the weakly globular double category of fractions.

For strict double functors there is also the dual notion of horizontal transformation. Although this notion of 2-cell is not mentioned in the equivalence with the 2-category of bicategories, horizontal transformations do play an important role in the category of weakly globular double categories. We will illustrate this in a sequel to this paper [25] in which we define and study the construction of a weakly globular double category of fractions. The fact that horizontal transformations play a role next to the vertical transformations should not surprise us since vertical transformations correspond to a very special class of 2-cells in the category of bicategories, namely the icons.

The definition of a horizontal transformation is dual to that of a vertical transformation in that all mentions of vertical and horizontal have been exchanged.

3.7. Definition. A horizontal transformation a: G ⇒ K: 𝔻 ⇉ 𝔼 between functors of double categories has components horizontal arrows \xymatrix{GX \ar[r]^{a_{X}} & KX} indexed by the objects of 𝔻 and for each vertical arrow \xymatrix{X \ar[r]|{•}^{v} & Y} in 𝔻, a double cell

[figure: GX; a_{X}; •; Gv; a_{v}; KX; •; Kv; GY; a_{Y}; KY,]

such that a is strictly functorial in the vertical direction, i.e., a_{v_{1}·v_{2}} = a_{v_{1}} · a_{v_{2}} and natural in the horizontal direction, i.e., the composition of

[figure: GX; •; Gv; Gζ; Gf; GX′; •; Gv′; a_{X′}; a_{v′}; KX′; •; Kv′; GY; Gg; GY′; a_{Y′}; KY′]

is equal to the composition of

[figure: GX; •; Gv; a_{v}; a_{X}; KX; •; Kv; Kf; Kζ; KX′; •; Kv′; GY; a_{Y}; KY; Kg; KY′]

for any double cell ζ in 𝔻.

We write DblCat_{st,v}, respectively DblCat_{st,h}, for the 2-categories of weakly globular double categories, strict functors, and vertical transformations, respectively horizontal transformations. We will write DblCat_{ps,v} for the 2-category of double categories, pseudofunctors, and vertical transformations of pseudo-functors.

There is an inclusion functor H: Cat → DblCat_{st,h} that sends a category C to a double category with the category C as horizontal arrows and only identity arrows (on objects of C) as vertical arrows and vertical identity cells as squares. Adjoint to this functor there is h: DblCat_{st,h} → Cat sending a double category to its category of horizontal arrows. Analogously, there is a functor V : Cat → DblCat_{st,v} that sends a category C to a double category with the arrows of C in the vertical direction (and a discrete horizontal category) with adjoint v: DblCat_{st,v} → Cat.

4. Tamsamani weak 2-categories

We recall some background on the Tamsamani model of weak 2-categories, see also [29],

[33] and [20]. This is a simplicial model based on the concept of Segal maps. The latter can be used to describe strict 2-categories as simplicial objects in Cat which are discrete at level zero and such that the Segal maps are isomorphisms. The Segal map isomorphisms give the associativity and unit laws in a strict 2-category, as part of the simplicial identities.

The idea of a Tamsamani weak 2-category is to relax the structure by requiring that the Segal maps are no longer isomorphisms but merely equivalences of categories. The associativity and unit laws do not hold strictly but only up to coherent isomorphism, so one obtains an associated bicategory. Conversely, there is a 2-nerve construction that associates to a bicategory a Tamsamani weak 2-categories, and an appropriate equivalence between the two notions, as recalled below.

4.1. Tamsamani weak 2-categories. We first recall the notion of Segal map. Let C be a category with pullbacks and let X ∈ [Δ^{op}, C], the category of simplicial objects in C, that is, strict functors from Δ^{op} to C. For each k ≥ 2, we denote by

X_{1}×_{X_{0}} ⋯^{k}×_{X_{0}}X_{1}

the limit of the diagram

X_{1} ⟶^{∂_{1}} X_{0} ⟵^{∂_{0}} ⋯ ⟶^{∂_{1}} X_{0} ⟵^{∂_{0}} X_{1} .

For each 1 ≤ j ≤ k, let ν_{j} : X_{k} → X_{1} be induced by the map [1] → [k] in Δ sending 0 to j − 1 and 1 to j. Then the following diagram commutes:

[figure: X_{k}; X_{1}; X_{1}; X_{1}; X_{0}; X_{0}; X_{0}; X_{0}; X_{0}; ⋯; ⋯; ν_{1}; ν_{2}; ν_{k}; ∂_{0}; ∂_{1}; ∂_{0}; ∂_{1}; ∂_{0}; ∂_{1}]

By definition of limit, there is a unique map

η_{k} : X_{k} → X_{1}×_{X_{0}} ⋯^{k}×_{X_{0}}X_{1}

such that pr_{j} η_{k} = ν_{j}, where pr_{j} is the j^{th} projection.

The maps η_{k} are called Segal maps and they play an important role in Tamsamani’s model of weak 2-categories. They can also be used to characterize nerves of internal categories: a simplicial object in C is the nerve of an internal category in C if and only if the Segal maps are isomorphisms for all k ≥ 2.

4.2. Definition. [8] The category Ta_{2} of Tamsamani weak 2-categories is the full subcategory of [Δ^{op}, Cat] whose objects X are such that X_{0} is discrete and, for all k ≥ 2, the Segal map η_{k} : X_{k} → X_{1}×_{X_{0}} ⋯^{k}×_{X_{0}}X_{1} is a categorical equivalence.

Let π_{0} : Cat → Set associate to a category the set of isomorphism classes of its objects. The functor π_{0} induces a functor π_{0}^{∗} : [Δ^{op}, Cat] → [Δ^{op}, Set], (π_{0}^{∗}X)_{n} = π_{0}X_{n}. If X ∈ Ta_{2}, then π_{0}^{∗}X is the nerve of a category. In fact, since π_{0} sends categorical equivalences to isomorphisms and preserves fibered products over discrete objects, for all k ≥ 2, we have

π_{0}X_{k} ≅ π_{0}(X_{1}×_{X_{0}} ⋯^{k}×_{X_{0}}X_{1}) ≅ π_{0}X_{1}×_{π_{0}X_{0}} ⋯^{k}×_{π_{0}X_{0}}π_{0}X_{1}.

We write Π_{0}X for the category whose nerve is π_{0}^{∗}X. This defines a functor

Π_{0} : Ta_{2} → Cat .

Given X ∈ Ta_{2} and a, b ∈ X_{0} let X_{(a,b)} be the full subcategory of X_{1} whose objects z are such that d_{0}z = a and d_{1}z = b. By considering the functor (d_{0}, d_{1}) : X_{1} → X_{0} × X_{0}, since X_{0} is discrete, we obtain a coproduct decomposition X_{1} = ∐_{a,b∈X_{0}} X_{(a,b)}.

4.3. Definition. A morphisms F : X → Y in Ta_{2} is a 2-equivalence if, for all a, b ∈ X_{0}, F_{(a,b)} : X_{(a,b)} → Y_{(F a,F b)} and Π_{0}F are categorical equivalences.

4.4. Remark. Notice that if a morphism in Ta_{2} is a levelwise equivalence of categories, it is in particular a 2-equivalence.

4.5. Bicategories and Tamsamani weak 2-categories. In this section we recall some results from [20]. We consider the 2-category Bicat_{icon} whose objects are bicategories, whose morphisms are normal homomorphisms and whose 2-cells are icons; the latter are oplax natural transformations with identity components. The fully faithful inclusion J : Δ → Bicat_{icon} gives rise to a 2-nerve functor

N : Bicat_{icon} → [Δ^{op}, Cat] ,
(NB)_{n} = Bicat_{icon}([n], B) .

It is shown in [20] that N is fully faithful and that the 2-nerve of a bicategory is in fact a Tamsamani weak 2-category. Given a bicategory B, (NB)_{0} is the discrete category with objects the objects of B. An object of (NB)_{1} is a morphism of B while a morphism in (NB)_{1} is a 2-cell in B. A complete characterization of 2-functors X : Δ^{op} → Cat which are 2-nerves of bicategories is given in [20, Theorem 7.1].

The 2-nerve functor N has a left 2-adjoint G, which was defined in [33]. Given a Tamsamani weak 2-category X, the objects of GX are the element of X_{0}, the 1- and 2-cells are the objects and morphisms of X_{1} and the vertical composition of 2-cells is the composition in X_{1}.

Since the Segal map η_{2} : X_{2} → X_{1} ×_{X_{0}} X_{1} is an equivalence, we can choose a functor M : X_{1} ×_{X_{0}} X_{1} → X_{1} and an isomorphism σ : d_{1} ≅ Mη_{2} as follows:

[figure: X_{2}; d_{1}; η_{2}; σ; X_{1} ×_{X_{0}} X_{1}; M; X_{1}]

This gives the composition of 1-cells and the horizontal composition of 2-cells.

The identity isomorphisms are σs_{0}, σs_{1} (where s_{0}, s_{1} : X_{1} → X_{2} are the degeneracy maps). For the associativity isomorphisms, one needs to consider the following pasting diagrams, where we denote X_{1}^{k} = X_{1}×_{X_{0}} ⋯^{k}×_{X_{0}}X_{1}, for k = 2, 3.

[figure: X_{3}; ^{\binom{d_{0}}{d_{2}d_{2}}}; d_{2}; X_{2} ×_{X_{0}} X_{2}; d_{1}×1; η_{2}×1; σ×1; X_{1}^{3}; M×1; X_{2}; η_{2}; d_{1}; σ; X_{1}^{2}; M; X_{1}; X_{3}; ^{\binom{d_{0}d_{1}}{d_{3}}}; d_{1}; X_{2} ×_{X_{0}} X_{2}; 1×d_{1}; 1×η_{2}; 1×σ; X_{1}^{3}; 1×M; X_{2}; η_{2}; d_{1}; σ; X_{1}^{2}; M; X_{1}]

Since the left hand composites of the diagrams are equal and the top composites are both equal to the equivalence η_{3} : X_{3} → X_{1}^{3}, there is a unique invertible cell M(M × 1) ≅ M(1 × M) which pasted onto the left diagram gives the right diagram. The proof of the coherence laws uses the fact that η_{4} is an equivalence, see [20] for details.

The relation between the functors N and G is summarized in the following theorem.

4.6. Theorem. [6, Theorem 7.2]. The 2-nerve 2-functor N : Bicat_{icon} → Ta_{2}, seen as landing in the 2-category Ta_{2} of Tamsamani weak 2-categories, has a left 2-adjoint given by G. Since N is fully faithful, the counit GN → 1 is invertible. Each component u : X → NGX of the unit is a pointwise equivalence, and u_{0} and u_{1} are identities.

A morphism f : X → Y in Ta_{2} is a 2-equivalence if and only if Gf is a biequivalence of bicategories. It is not hard to see that inverting these maps in Bicat_{icon} and in Ta_{2} gives equivalent categories. Another approach consists in enlarging the class of morphisms in Ta_{2} to include pseudo-natural transformations. One then obtains:

4.7. Theorem. [6, Theorem 7.3]. The 2-nerve 2-functor N : Bicat_{icon} → (Ta_{2})_{ps} is a biequivalence of 2-categories with pseudo-inverse G.

5. From Tamsamani weak 2-categories to pseudo-functors

In this section we associate functorially to a Tamsamani weak 2-category a pseudo-functor from Δ^{op} to Cat. The idea of this construction is, given H ∈ Ta_{2}, to replace H_{n} with its equivalent category H_{1}×_{H_{0}} ⋯^{n}×_{H_{0}}H_{1} when n ≥ 2. The resulting structure is no longer simplicial but pseudo-simplicial, that is an object of Ps[Δ^{op}, Cat]. The proof of this is based on a general fact (Lemma 5.3) which is essentially known, and which is an instance of transport of structure along an adjunction (Theorem 5.2).

The pseudo-functors H constructed from a Tamsamani weak 2-category are such that H_{0} is discrete and H_{n} ≅ H_{1}×_{H_{0}} ⋯^{n}×_{H_{0}}H_{1} for n ≥ 2. We call such pseudo-functors Segalic. In the next section, we will see that the strictification of a Segalic pseudo-functor yields the horizontal nerve of a weakly globular double category.

5.1. Transport of structure along an adjunction. We now recall a general categorical property, known as transport of structure along an adjunction, with one of its applications.

5.2. Theorem. [19, Theorem 6.1] Given an equivalence η, ε : f ⊣ f^{∗} : A → B in the complete and locally small 2-category A, and an algebra (A, a) for the monad T = (T, i, m) on A, the equivalence enriches to an equivalence

η, ε : (f, f^{=}) ⊢ (f^{∗}, f^{∗}^{=}) : (A, a) → (B, b, b̂, \overline{b})

in Ps-T -alg, where b̂ = η, \overline{b} = f^{∗}a · T ε · T a · T^{2}f, f^{=} = ε^{−1}a · T f, and f^{∗}^{=} = f^{∗}a · T ε.

Let η′, ε′ : f′ ⊢ f^{′∗} : A′ → B′ be another equivalence in A and let (B′, b′, b̂′, \overline{b′}) be the corresponding pseudo-T -algebra as in Theorem 5.2. Suppose g : (A, a) → (A′, a′) is a morphism in A and γ is an invertible 2-cell in A

[figure: B; B′; A; A′; f^{∗}; h; f^{′∗}; g; γ]

Let \overline{γ} be the invertible 2-cell given by the following pasting:

[figure: T B; T B′; B; B′; T A; T A′; A; A′; T h; b; b′; h; T g; g; T f^{∗}; f^{∗}; T f^{′∗}; f^{′∗}; (T γ)^{−1}; γ; f^{′∗}^{=}; f^{∗}^{=}]

Then it is not difficult to show that (h, \overline{γ}) : (B, b, b̂, \overline{b}) → (B′, b′, b̂′, \overline{b′}) is a pseudo-T -algebra morphism.

The following fact is essentially known and, as sketched in the proof below, it is an instance of Theorem 5.2.

5.3. Lemma. Let C be a small 2-category, F, F′ : C → Cat be 2-functors, α : F → F′ a 2-natural transformation. Suppose that, for all objects C of C, the following conditions hold:

i) G(C), G′(C) are objects of Cat and there are adjoint equivalences of categories μ_{C} ⊢ η_{C}, μ′_{C} ⊢ η′_{C},

μ_{C} : G(C) ⇄ F (C) : η_{C} μ′_{C} : G′(C) ⇄ F′(C) : η′_{C},

ii) there are functors β_{C} : G(C) → G′(C),

iii) there is an invertible 2-cell

γ_{C} : β_{C}η_{C} ⇒ η′_{C}α_{C}.

Then

a) There exists a pseudo-functor G : C → Cat given on objects by G(C), and pseudonatural transformations η : F → G, μ : G → F with η(C) = η_{C}, μ(C) = μ_{C}; these are part of an adjoint equivalence μ ⊢ η in the 2-category Ps[C, Cat].

b) There is a pseudo-natural transformation β : G → G′ with β(C) = β_{C} and an invertible 2-cell in Ps[C, Cat], γ : βη ⇒ ηα with γ(C) = γ_{C}.

Proof. Recall [28] that the functor 2-category [C, Cat] is 2-monadic over [|C|, Cat], where |C| is the set of objects in C. Let T be the 2-monad; then the pseudo-T -algebras are precisely the pseudo-functors from C to Cat. Let

U : Ps-T -alg ≡ Ps[C, Cat] → [|C|, Cat]

be the forgetful functor.

Then the adjoint equivalences μ_{C} ⊢ η_{C} amount precisely to an adjoint equivalence in [|C|, Cat], μ_{0} ⊢ η_{0}, μ_{0} : G_{0} ⇄ UF : η_{0} where G_{0}(C) = G(C) for all C ∈ |C|. By Theorem 5.2, this equivalence enriches to an adjoint equivalence μ ⊢ η in Ps[C, Cat]

μ : G ⇄ F : η

between F and a pseudo-functor G; it is UG = G_{0}, Uη = η_{0}, Uμ = μ_{0}; hence on objects G is given by G(C), and η(C) = Uη(C) = η_{C}, μ(C) = Uμ(C) = μ_{C}.

Let ν_{C} : id_{G(C)} ⇒ η_{C}μ_{C} and ε_{C} : μ_{C}η_{C} ⇒ id_{F (C)} be the unit and counit of the adjunction μ_{C} ⊢ η_{C}. From Theorem 5.2, given a morphism f : C → D in C, it is

G(f) = η_{D}F (f)μ_{C}

and we have natural isomorphisms:

η_{f} : G(f)η_{C} = η_{D}F (f)μ_{C}η_{C} ⟹^{η_{D}F (f)ε_{C}} η_{D}F (f)

μ_{f} : F (f)μ_{C} ⟹^{ν_{F (f)}μ_{C}} μ_{D}η_{D}F (f)μ_{C} = μ_{D}G(f).

Also, the natural isomorphism

β_{f} : G′(f)β_{C} ⇒ β_{D}G(f)

is the result of the following pasting 5.4. From Tamsamani weak 2-categories to pseudofunctors. We now use the results of the previous section to associate to a Tamsamani weak 2category a pseudo-functor.

[figure: G(C); G′(C); G(D); G′(D); F (C); F′(C); F (D); F′(D); β_{C}; G(f); G′(f); β_{D}; α_{C}; F (f); F′(f); α′_{D}; γ_{C}; γ_{D}^{−1}; η′_{f}; η_{f}]

5.5. Proposition. There is a functor

S : Ta_{2} → Ps[Δ^{op}, Cat].

which associates to an object X of Ta_{2} a pseudo-functor SX ∈ Ps[Δ^{op}, Cat] with

(SX)_{n} = \begin{cases}X_{1}×_{X_{0}} ⋯^{n}×_{X_{0}}X_{1} & n ≥ 2, \\ X_{1} & n = 1, \\ X_{0} & n = 0. \end{cases}

To a morphism F : X → X′, S associates a pseudo-natural transformation β(F) : SX → SX′ with

β(F)_{n} = \begin{cases}(F_{1}, …, F_{1}) & n ≥ 2, \\ F_{1} & n = 1, \\ F_{0} & n = 0. \end{cases}

Further, there is a pseudo-natural transformation α : SX → X which is a levelwise categorical equivalence.

Proof. We apply Lemma 5.3 to the case where C = Δ^{op}, considered as a 2-category with identity 2-cells; let X : Δ^{op} → Cat be an object of Ta_{2}. By definition, for each n ≥ 2, there is an equivalence of categories X_{n} ≃ X_{1}×_{X_{0}} ⋯^{n}×_{X_{0}}X_{1}. We can always choose this equivalence to be an adjoint equivalence; thus let η_{n} : X_{n} → X_{1}×_{X_{0}} ⋯^{n}×_{X_{0}}X_{1} be the Segal map and μ_{n} its left adjoint. By Lemma 5.3, we deduce that there is a pseudo-functor SX ∈ Ps[Δ^{op}, Cat] with

(SX)_{n} = \begin{cases}X_{1}×_{X_{0}} ⋯^{n}×_{X_{0}}X_{1} & n ≥ 2, \\ X_{1} & n = 1, \\ X_{0} & n = 0. \end{cases}

Suppose F : X → X′ is a morphism in Ta_{2} and let β_{n} : (SX)_{n} → (SX′)_{n} be

β_{n} = \begin{cases}(F_{1}, …, F_{1}) & n ≥ 2, \\ F_{1} & n = 1, \\ F_{0} & n = 0. \end{cases}

It is immediate to check from the definition of Segal map that the following diagram commutes for all n ≥ 0,

\xymatrix{X_{n} \ar[r]^{F_{n}} \ar[d]_{η_{n}} & X′_{n} \ar[d]^{η′_{n}} \\ SX_{n} \ar[r]_{β_{n}} & SX′_{n}.}

Thus the condition in the hypothesis of Lemma 5.3 is satisfied, with γ_{n} the identity 2-cell. It follows from Lemma 5.3 that there is a pseudo-natural transformation β : SX → SX′ with β(F)_{n} = β_{n}.

Suppose that X ⟶^{F} X′ ⟶^{F′} X′′ is a pair of composable morphisms in Ta_{2}; then, for each n ≥ 2,

β(F′F)_{n} = ((F′F)_{1}, …, (F′F)_{1}) = (F′_{1}, …, F′_{1})(F_{1}, …, F_{1}) = β(F′)_{n}β(F)_{n}.

Therefore, β(F′F) = β(F′)β(F).

Finally, the existence of a pseudo-natural transformation α : SX → X follows immediately by Lemma 5.3, taking α_{i} = id for i = 0, 1 and α_{n} = μ_{n} for n ≥ 2.

We see from Proposition 5.5 that the pseudo-functors arising from Tamsamani weak 2-categories have a special form. We call them Segalic pseudo-functors, as in the following definition.

5.6. Definition. The category P̃s[Δ^{op}, Cat] of Segalic pseudo-functors from Δ^{op} to Cat is the full subcategory of Ps[Δ^{op}, Cat] whose objects H are such that H_{n} ≅ H_{1}×_{H_{0}} ⋯^{n}×_{H_{0}}H_{1} for each n ≥ 2 and H_{0} is discrete.

Then Proposition 5.5 immediately implies

5.7. Corollary. There is a functor

S : Ta_{2} → P̃s[Δ^{op}, Cat],

and a pseudo-natural transformation α : SX → X which is a levelwise categorical equivalence.

6. Strictification of Segalic pseudo-functors

In this section we apply the strictification method of Power [28] to Segalic pseudo-functors (Definition 5.6). We show in Theorem 6.5 that the resulting strict functor from Δ^{op} to Cat is the horizontal nerve of a double category satisfying additional properties. We identify these double categories as our central notion of weakly globular double categories (Definition 6.7), and hence conclude in Corollary 6.9 that the strictification of a Segalic pseudo-functor yields the horizontal nerve of a weakly globular double category. In the next section we will use these results to build a rigidification functor from Tamsamani weak 2-categories to weakly globular double categories.

We start by recalling the construction of the strictification functor from [28],

St : Ps[Δ^{op}, Cat] → [Δ^{op}, Cat].

6.1. Strictification of pseudo-functors. As explained in [28, 4.2], the functor 2-category [Δ^{op}, Cat] is 2-monadic over [ |Δ^{op}|, Cat], where |Δ^{op}| is the set of objects of Δ^{op}.

Let U : [Δ^{op}, Cat] → [ |Δ^{op}|, Cat] be the forgetful functor, (UX)_{n} = X_{n} for all [n] ∈ Δ^{op}, X ∈ [Δ^{op}, Cat]. Then its left adjoint F is given on objects by

(F H)_{n} = ∐_{[m]∈|Δop|} Δ^{op}([m], [n]) × H_{m},

for H ∈ [ |Δ^{op}|, Cat], [n] ∈ |Δ^{op}|. If T is the monad corresponding to the adjunction F ⊣ U then

(T H)_{n} = ∐_{[m]∈|Δop|} Δ^{op}([m], [n]) × H_{m},

for H ∈ [ |Δ^{op}|, Cat], and [n] ∈ |Δ^{op}|.

A pseudo-T -algebra is given by H ∈ [ |Δ^{op}|, Cat], functors

h_{n} : ∏_{[m]∈|Δop|} Δ^{op}([m], [n]) × H_{m} → H_{n}

and additional data as described in [28, 4.2]. This amounts precisely to a pseudo-functor from Δ^{op} to Cat and the 2-category Ps-T -alg of pseudo-T -algebras corresponds to the 2-category Ps[Δ^{op}, Cat] of pseudo-functors, pseudo-natural transformations and modifications.

The general strictification result proved in [28, 3.4], when applied to this case, yields that every pseudo-functor from Δ^{op} to Cat is equivalent, in Ps[Δ^{op}, Cat], to a 2-functor.

The construction given in [28] is as follows. Given a pseudo-T -algebra as above, factor h : T H → H as T H ⟶^{r} L ⟶^{g} H with r_{n} bijective on objects and g_{n} fully faithful for each [n] ∈ Δ^{op}. Then it is shown in [28] that it is possible to give a strict T -algebra structure T L → L such that (g, T g) is an equivalence of pseudo-T -algebras.

6.2. Remark. Since (g, T g) is an equivalence of pseudo-T -algebras, g_{n} is an equivalence of categories for every [n] ∈ Δ^{op}. In fact, by definition there is a map (g′, T g′) with invertible 2-cells α : (g, T g)(g′, T g′) ⟹ id and β : (g′, T g′)(g, T g) ⟹ id. A 2-cell in Ps-T -alg amounts to a 2-cell in [ |Δ^{op}|, Cat] satisfying the condition of [28, 2.6]. Since the 2-cells in [ |Δ^{op}|, Cat] are modifications, this implies that for each [n] ∈ Δ^{op} there are natural transformations id ≅ g_{n}g′_{n}, and g′_{n}g_{n} ≅ id; that is, g_{n} is an equivalence of categories for each n.

6.3. A special case of strictification. We now apply the strictification technique of Section 6.1 to the class of Segalic pseudofunctors.

We first need a preliminary lemma that establishes a more explicit description of T UH and of the structure map h : T UH → H for a Segalic pseudo-functor H.

6.4. Lemma. Let U : Ps[Δ^{op}, Cat] → [|Δ^{op}|, Cat] be the forgetful functor, (UH)_{n} = H_{n} for all n ≥ 0, and let T be the monad on [|Δ^{op}|, Cat] as in Section 6.1. Then, if H ∈ P̃s[Δ^{op}, Cat],

a) The pseudo-T -algebra corresponding to H has structure map h : T UH → UH given as follows: for each k ≥ 0,

(T UH)_{k} = ∐_{[n]∈Δ}Δ([k], [n]) × H_{n} = ∐_{[n]∈Δ} ∐_{Δ([k],[n])}H_{n}.

For n ≥ 0 and f ∈ Δ([k], [n]), let i_{n} : ∐_{Δ([k],[n])}H_{n} → (T UH)_{k} and j_{f} : H_{n} →

∐_{Δ([k],[n])}H_{n} be the coproduct injections. Then h_{k}i_{n}j_{f} = H(f).

b) There are functors ∂′_{0}, ∂′_{1} : (T UH)_{1} ⇉ (T UH)_{0} making the following diagram com-

mute:

\xymatrix{(T UH)_{1} \ar[r]^{h_{1}} \ar[d]^{∂′_{1}} \ar[d]_{∂′_{0}} & H_{1} \ar[d]^{∂_{1}} \ar[d]_{∂_{0}} \\ (T UH)_{0} \ar[r]^{h_{0}} & H_{0}}  (9)

that is, ∂_{i}h_{1} = h_{0}∂′_{i}, for i = 0, 1.

c) For each k ≥ 2, (T UH)_{k} ≅ (T UH)_{1}×_{(T UH)_{0}} ⋯^{k}×_{(T UH)_{0}}(T UH)_{1}.

d) For each k ≥ 2, the morphism h_{k} : (T UH)_{k} → H_{k} is h_{k} = (h_{1}, …, h_{1}).

Proof.

a) From the general correspondence between pseudo-T -algebras and pseudo-functors, the pseudo-T -algebra corresponding to H has structure map h : T UH → UH as stated. Recalling that, if X is a set and C is a category, X × C ≅ ∐_{X}C, we have

(T UH)_{k} = ∐_{[n]∈Δop}Δ^{op}([n], [k]) × H_{n} = ∐_{[n]∈Δ}Δ([k], [n]) × H_{n} ≅ ∐_{[n]∈Δ} ∐_{Δ([k],[n])}H_{n}.

b) Let δ_{i} : [0] → [1], δ_{i}(0) = 0, δ_{i}(1) = i, for i = 0, 1. For f ∈ Δ([1], [n]) let

j_{f} : H_{n} → ∐_{Δ([1],[n])}H_{n} and i_{n}: ∐_{Δ([1],[n])}H_{n} → ∐_{[n]∈Δ} ∐_{Δ([1],[n])}H_{n} be the coproduct

injections. Let ∂′_{i} : (T UH)_{1} → (T UH)_{0} be the functors determined by

∂′_{i}i_{n}j_{f} = i_{n}j_{fδ_{i}}.  (10)

From a), we have

h_{0}∂′_{i}i_{n}j_{f} = h_{0}i_{n}j_{fδ_{i}} = H(fδ_{i}) and ∂_{i}h_{1}i_{n}j_{f} = H(δ_{i})H(f) .

Since H ∈ Ps[Δ^{op}, Cat] and H_{0} is discrete, it follows that H(fδ_{i}) = H(δ_{i})H(f), so that, from above, h_{0}∂′_{i}i_{n}j_{f} = ∂_{i}h_{1}i_{n}j_{f} for each [n] ∈ Δ, f ∈ Δ([1], [n]). We conclude that h_{0}∂′_{i} = ∂_{i}h_{1}.

c) For each k ≥ 2, [k] is the colimit in Δ of the diagram

\xymatrix{[1] &  & [1] &  &  &  & [1] \\  & [0] \ar@{-->}[ul]^{0} \ar@{-->}[ur]_{1} &  & [0] \ar@{-->}[ul]^{0} & ⋯ & [0] \ar@{-->}[ur]_{1}}

that is, [k] = [1]∐_{[0]}⋯^{k}∐_{[0]}[1]. In fact, it is easy to check by direct computation that

there is a pushout in Δ:

\xymatrix{[0] \ar[r]^{0} \ar[d]_{1} & [k − 1] \ar[d]^{p} \\ [1] \ar[r]_{q} & [k]}

where q(i) = i, for i = 0, 1, and p(t) = t + 1, for t = 0, …, k − 1. In particular,

[2] = [1]∐_{[0]}[1]. Inductively, if [k − 1] = [1]∐_{[0]}…^{k−1}∐_{[0]}[1] then, from the above pushout,

[k] = [k − 1]∐_{[0]}[1] = [1]∐_{[0]}…^{k}∐_{[0]}[1], as claimed. It follows that there is a bijection, for

k ≥ 2

Δ([k], [n]) ≅ Δ([1], [n])×_{Δ([0],[n])}⋯^{k}×_{Δ([0],[n])}Δ([1], [n]).

From the proof of b), the functors ∂_{i} : (T UH)_{1} → (T UH)_{0} are determined by the functors (\overline{δ}_{i}, id) : Δ([1], [n]) × H_{n} → Δ([0], [n]) × H_{n} where \overline{δ}_{i}(f) = fδ_{i} for f ∈ Δ([1], [n]). Hence, from above, we obtain

(T UH)_{1}×_{(T UH)_{0}} ⋯^{k}×_{(T UH)_{0}}(T UH)_{1} =

= ∐_{[n]∈Δ}(Δ([1], [n]) × H_{n})×_{∐_{[n]∈Δ}(Δ([0],[n])×H_{n})}⋯^{k}×_{∐_{[n]∈Δ}(Δ([0],[n])×H_{n})} ∐_{[n]∈Δ}(Δ([1], [n]) × H_{n})

≅ ∐_{[n]∈Δ}(Δ([1], [n]) × H_{n})×_{(Δ([0],[n])×H_{n})}⋯^{k}×_{(Δ([0],[n])×H_{n})}(Δ([1], [n]) × H_{n})

= ∐_{[n]∈Δ}(Δ([1], [n])×_{Δ([0],[n])}⋯^{k}×_{Δ([0],[n])}Δ([1], [n])) × (H_{n}×_{H_{n}} ⋯^{k}×_{H_{n}}H_{n})

= ∐_{[n]∈Δ}Δ([k], [n]) × H_{n} = (T UH)_{k}.

d) From a), h_{k}i_{n}j_{f} = H(f) for f ∈ Δ([k], [n]), n > 0. Let f correspond to (δ_{1}, …, δ_{k}) in the isomorphism Δ([k], [n]) ≅ Δ([1, [n])×_{Δ([0,[n])}⋯^{k}×_{Δ([0,[n])}Δ([1, [n]). Then j_{f} =

(j_{δ_{1}}, …, j_{δ_{k}}). Since H_{k} ≅ H_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}H_{1}, H(f) corresponds to (H(δ_{1}), …, H(δ_{k}))

with p_{i}H(f) = H(δ_{i}). Then, for all f ∈ Δ([k], [n]), n > 0 we have

\begin{matrix}h_{k}i_{n}j_{f} = H(f) = (H(δ_{1}), …, H(δ_{k})) = (h_{1}i_{n}j_{δ_{1}}, …, h_{1}i_{n}j_{δ_{k}}) = \\ = (h_{1}, …, h_{1})i_{n}(j_{δ_{1}}, …, j_{δ_{k}}) = (h_{1}, …, h_{1})i_{n}j_{f}. \end{matrix}

We conclude that h_{k} = (h_{1}, …, h_{1})

6.5. Theorem. Let H ∈ P̃s[Δ^{op}, Cat] and L = StH be the strictification of H as in Section 6.1. Then

a) There is a morphism g : L → H in Ps[Δ^{op}, Cat] such that, for each k ≥ 0, g_{k} is an equivalence of categories.

b) L_{k} ≅ L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1} for all k ≥ 2.

c) The functor g_{0} : L_{0} → H_{0} induces equivalences of categories

L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1} ≃ L_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}L_{1} for all k ≥ 2.

Proof.

a) This follows directly from [28] (see Remark 6.2).

b) Let h : T UH → UH be as in Lemma 6.4. As described in Section 6.1, factor h = gr so that, for each i ≥ 0, h_{i} factors as (T UH)_{i} ⟶^{r_{i}} L_{i} ⟶^{g_{i}} H_{i} with r_{i} bijective on objects and g_{i} fully faithful. As explained in [28], the g_{i} are in fact equivalences.

Since the bijective on objects and fully faithful functors form a factorization system in Cat, the commutativity of (9) implies that there are functors d_{0}, d_{1} : L_{1} → L_{0} such that the following commutes:

\xymatrix{(T UH)_{1} \ar[r]^{r_{1}} \ar[d]_{∂′_{0}} \ar[d]^{∂′_{1}} & L_{1} \ar[r]^{g_{1}} \ar[d]_{d_{0}} \ar[d]^{d_{1}} & H_{1} \ar[d]_{∂_{0}} \ar[d]^{∂_{1}} \\ (T UH)_{0} \ar[r]_{r_{0}} & L_{0} \ar[r]_{g_{0}} & H_{0}}

that is, ∂_{i}r_{1} = r_{0}∂′_{i}, ∂_{i}g_{1} = g_{0}d_{i} for i = 0, 1. By Lemma 6.4 d) , h_{k} factors as

\begin{matrix}(T UH)_{k} = (T UH)_{1}×_{(T UH)_{0}} ⋯^{k}×_{(T UH)_{0}}(T UH)_{1} ⟶^{(r_{1},...,r_{1})} L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1} \\ ⟶^{(g_{1},...,g_{1})} H_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}H_{1} ≅ H_{k}. \end{matrix}  (11)

Since r_{0}, r_{1} are bijective on objects, so is (r_{1}, …, r_{1}). Since g_{0}, g_{1} are fully faithful, so is (g_{1}, …, g_{1}). Hence (11) is the factorization of h_{k} and we conclude that L_{k} ≅

L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1}.

c) Since H_{1} ≃ L_{1} and H_{0} is discrete, H_{k} ≅ H_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}H_{1} ≃ L_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}L_{1}. On

the other hand, H_{k} ≃ L_{k} ≅ L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1}. In conclusion, L_{1}×_{H_{0}} ⋯^{k}×_{H_{0}}L_{1}

≃ L_{1}×_{L_{0}} ⋯^{k}×_{L_{0}}L_{1}.

6.6. Weakly globular double categories. We see from Theorem 6.5 b) that L is a simplicial object in Cat whose Segal maps are isomorphisms: it is therefore the horizontal nerve of a double category. Further, parts a) and c) of Theorem 6.5 show that this double category satisfies additional conditions. We identify such double categories as our central notion of weakly globular double category, as in the following definition:

6.7. Definition. A weakly globular double category 𝕏 is a double category which satisfies the following two conditions

• (the weak globularity condition) there is an equivalence of categories γ : 𝕏_{0} → 𝕏_{0}^{d}, where 𝕏_{0}^{d} is the discrete category of the path components of 𝕏_{0}; • (the induced Segal maps condition) γ induces an equivalence of categories, for all

n ≥ 2,

𝕏_{1}×_{𝕏_{0}} ⋯^{n}×_{𝕏_{0}}𝕏_{1} ≃ 𝕏_{1}×_{𝕏_{0}^{d}} ⋯^{n}×_{𝕏_{0}^{d}}𝕏_{1} .  (12)

Analogously to the notation introduced above we write WGDbl_{ps,v} for the 2-category of weakly globular double categories, pseudo-functors, and vertical transformations between them. We also write WGDbl_{st,v} and WGDbl_{st,h} for the 2-categories with strict functors and vertical, respectively horizontal, transformations.

6.8. Remark. For each n > 0, the equivalence (12) in the induced Segal maps condition gives us for each ‘stair-case-path’ of alternating horizontal and vertical arrows,

[figure: f_{1}; •; v_{1}; •; v_{2}; f_{3}; ···; •; v_{n−2}; •; v_{n−1}; f_{n}; f_{2}; f_{n−1}]

a corresponding horizontal path with vertically invertible double cells,

[figure: f′_{1}; •; u_{0}; φ_{1}; •; u_{1}; φ_{1}; f′_{2}; •; u_{2}; f′_{3}; φ_{3}; •; u_{3}; ···; •; u_{n−2}; f′_{n−1}; φ_{n−1}; •; u_{n−1}; f′_{n}; φ_{n}; •; u_{n}; f_{1}; •; v_{1}; •; v_{2}; f_{3}; ···; •; v_{n−2}; •; v_{n−1}; f_{n}; f_{2}; f_{n−1}]

If we denote by N_{h}(WGDbl) the subcategory of [Δ^{op}, Cat] of horizontal nerves of weakly globular double categories, we finally obtain:

6.9. Corollary. The strictification functor St : Ps[Δ^{op}, Cat] → [Δ^{op}, Cat] when restricted to Segalic pseudo-functors gives rise to a functor

St : P̃s[Δ^{op}, Cat] → N_{h}(WGDbl) .

Further, for any H ∈ P̃s[Δ^{op}, Cat] there is a morphism g : St H → H in Ps[Δ^{op}, Cat] such that, for each k ≥ 0, g_{k} is an equivalence of categories.

Proof. It follows immediately from Theorem 6.5 and Definition 6.7.

We end this section with a remark which will be used in Section 8.2 in giving the explicit description of the weakly globular double category associated to a bicategory.

6.10. Remark. Recall that the factorization of any functor F : C → D as the composite

C ⟶^{S} E ⟶^{T} D with S bijective on objects and T fully faithful is constructed as follows.

Consider the pullbacks of sets

\xymatrix{E_{1} \ar[r]^{(d̃_{0},d̃_{1})} \ar[d]_{T_{1}} & C_{0} × C_{0} \ar[d]^{F_{0}×F_{0}} \\ D_{1} \ar[r]_{(d_{0},d_{1})} & D_{0} × D_{0}}

where d_{0}, d_{1} are the source and target map in the category D. It is easy to see that there is a category E with objects E_{0} = C_{0} and E_{1} and source and target maps d̃_{0} and d̃_{1} as in the pullback diagram above. Further, there are functors S : C → E, T : E → D with S_{0} = id, T_{0} = F_{0}, and S_{1} determined by F_{1} : C_{1} → D_{1} and (d′_{0}, d′_{1}) : C_{1} → C_{0} × C_{0}. Hence, in the notation of Theorem 6.5, we have

\begin{aligned}L_{00} & = (T UH)_{00} = ∐_{[n]∈Δ}Δ([0], [n]) × H_{n0} \\ L_{10} & = (T UH)_{10} = ∐_{[n]∈Δ}Δ([1], [n]) × H_{n0} \end{aligned}

while L_{01} and L_{11} are given by the following pullbacks:

\xymatrix{L_{11} \ar[r] \ar[d]_{g_{11}} & (T UH)_{10} × (T UH)_{10} \ar[d]^{h_{10}×h_{10}} \\ H_{11} \ar[r]_{(d_{0},d_{1})} & H_{10} × H_{10}}

\xymatrix{L_{01} \ar[r] \ar[d]_{g_{01}} & (T UH)_{00} × (T UH)_{00} \ar[d]^{h_{00}×h_{00}} \\ H_{00} \ar[r]_{(id,id)} & H_{00} × H_{00}}

7. Weakly globular double categories, Tamsamani weak 2-categories, and bicategories

In this section we prove our main result (Theorem 7.10) that there is a biequivalence between the 2-category of weakly globular double categories with pseudo-functors and the 2-category of bicategories with icons.

We obtain this result first by comparing weakly globular double categories to Tamsamani weak 2-categories. Using the results of Sections 5 and 6 we build a rigidification functor from the latter to weakly globular double categories. In section 7.3 we build a discretization functor in the opposite direction. We show in Theorem 7.6 that these two functors give a biequivalence between the 2-categories of weakly globular double categories and Tamsamani weak 2-categories. Together with the comparison result Theorem 4.7 from Tamsamani weak 2-categories to bicategories, this yields our main result in Theorem 7.10.

7.1. Rigidifying Tamsamani weak 2-categories. We now use the results of Sections 5 and 6 to build a rigidification functor from Tamsamani weak 2-categories to weakly globular double categories.

7.2. Theorem. There is a functor

Q: Ta_{2} → WGDbl_{st,v}

such that, for all X ∈ Ta_{2}, there is a pseudo-morphism α_{X}: N_{h}QX → X, natural in X, which is a levelwise categorical equivalence.

Proof. Let P be the left adjoint to the horizontal nerve N_{h}, P N_{h} = id. Let Q be the composite

Q: (Ta_{2})_{ps} ⟶^{S} P̃s[Δ^{op}, Cat] ⟶^{St} N_{h}(WGDbl) ⟶^{P} WGDbl_{ps,v}

where S: (Ta_{2})_{ps} → P̃s[Δ^{op}, Cat] is as in Corollary 5.7 and St: Ps[Δ^{op}, Cat] → N_{h}(WGDbl) is as in Corollary 6.9. Now take X ∈ (Ta_{2})_{ps}. By Corollaries 5.7 and 6.9 there are pseudonatural transformations SX → X and St SX → SX which are levelwise equivalences of categories. Since St SX ≅ N_{h}P St SX = N_{h}QX, we obtain by composition a pseudonatural transformation N_{h}QX → X, which is a levelwise equivalence of categories.

7.3. Discretization. In the next proposition we construct a functor in the opposite direction, from weakly globular double categories to Tamsamani weak 2-categories. The idea is to replace the category of objects and vertical morphisms in a weakly globular double category by its equivalent discrete category. This recovers the globularity condition, but at the expense of the strictness of the Segal maps: from being isomorphisms they become equivalences.

7.4. Proposition. There is a functor

D : WGDbl_{st,v} → Ta_{2}

such that, for every 𝕏 ∈ WGDbl there is a pseudo-morphism η_{𝕏}: D𝕏 → N_{h}𝕏, natural in 𝕏, which is a levelwise categorical equivalence.

Proof. Let 𝕏 ∈ WGDbl. By definition, there is a categorical equivalence γ: 𝕏_{0} → 𝕏_{0}^{d}. Write γ′: 𝕏_{0}^{d} → 𝕏_{0} for its pseudo-inverse. Then γγ′ = id, since 𝕏_{0}^{d} is discrete. Let

(D𝕏)_{0} = 𝕏_{0}^{d},
(D𝕏)_{1} = 𝕏_{1},
(D𝕏)_{k} = 𝕏_{1}×_{𝕏_{0}} ⋯^{k} ×_{𝕏_{0}}𝕏_{1}, for k ≥ 2.

Let ∂_{i}, σ_{i} be the face and degeneracy operators of N_{h}𝕏. Define d_{i}: (D𝕏)_{1} → (D𝕏)_{0} and s_{0} : (D𝕏)_{0} → (D𝕏)_{1} by d_{i} = γ∂_{i}, for i = 0, 1 and s_{0} = σ_{0}γ′. All other face and degeneracy operators in D𝕏 are as in N_{h}𝕏. Notice that, since γγ′ = id, D𝕏 ∈ [Δ^{op}, Cat].

By construction, (D𝕏)_{0} is discrete. To show that D𝕏 ∈ Ta_{2} we need to show that all Segal maps are categorical equivalences. Since 𝕏 is weakly globular, by definition we have for n ≥ 2,

(D𝕏)_{n} = 𝕏_{1}×_{𝕏_{0}} ⋯^{n}×_{𝕏_{0}}𝕏_{1}
≃ 𝕏_{1}×_{𝕏_{0}^{d}} ⋯^{n}×_{𝕏_{0}^{d}}𝕏_{1}
= (D𝕏)_{1}×_{(D𝕏)_{0}} ⋯^{n}×_{(D𝕏)_{0}}(D𝕏)_{1}.

This shows that D𝕏 ∈ Ta_{2}.

Let η_{0} = γ′, η_{k} = id for k > 0. This defines a pseudo-natural transformation η: D𝕏 → N_{h}𝕏 which is a levelwise categorical equivalence.

7.5. Remark. From the pseudo-inverses (μ_{𝕏})_{k} to (η_{𝕏})_{k} for each k, using Lemma 5.3 we construct the pseudo-inverse μ_{𝕏} : N_{h}𝕏 → D𝕏 to the pseudo-functor η_{𝕏} : D𝕏 → N_{h}𝕏 in Ps[Δ^{op}, Cat]. Likewise, for each Y ∈ Ta_{2} the pseudo-functor N_{h}QY → Y of Theorem 7.2 has a pseudo-inverse Y → N_{h}QY.

Notice that the functors Q : Ta_{2} → WGDbl_{st,v} and D : WGDbl_{st,v} → Ta_{2} of Theorem 7.2 and Proposition 7.4 extend to functors Q : (Ta_{2})_{ps} → (WGDbl)_{ps,v} and D : (WGDbl)_{ps} → (Ta_{2})_{ps,v} (we shall denote them with the same letters).

7.6. Theorem. There is a biequivalence of 2-categories:

(WGDbl)_{ps,v} ≃ (Ta_{2})_{ps} .

Proof. Since the horizontal nerve functor N_{h} : (WGDbl)_{ps} → Ps[Δ^{op}, Cat] is fully faithful, there is an isomorphism

(WGDbl)_{ps,v}(𝕏, 𝕐) ≅ Ps[Δ^{op}, Cat](N_{h}𝕏, N_{h}𝕐) .  (13)

We claim that there is an equivalence of categories

F: Ps[Δ^{op}, Cat](N_{h}𝕏, N_{h}𝕐) ≃ (Ta_{2})_{ps}(D𝕏, D𝕐) : G  (14)

This is constructed as follows. Let η_{𝕏}: D𝕏 → N_{h}𝕏 and μ_{𝕏}: N_{h}𝕏 → D𝕏 be as in Remark 7.5.

Define

F f = μ_{𝕐}fη_{𝕏}, Gg = η_{𝕐}gμ_{𝕏} .

Then F Gg = μ_{𝕐}(η_{𝕐}gμ_{𝕏})η_{𝕏} ≅ g and GF f = η_{𝕐}(μ_{𝕐}fη_{𝕏})μ_{𝕏} ≅ f, showing that (14) is an equivalence of categories as claimed.

From (13) and (14) we deduce

(WGDbl)_{ps,v}(𝕏, 𝕐) ≃ (Ta_{2})_{ps}(D𝕏, D𝕐) ,

that is, the functor D is locally an equivalence of categories.

On the other hand, D is also biessentially surjective on objects. In fact by Theorem 7.2 and by Proposition 7.4, for every X ∈ (Ta_{2})_{ps} there is a composite morphism DQX → N_{h}QX → X in Ps[Δ^{op}, Cat] which is levelwise a categorical equivalence, hence an equivalence in (Ta_{2})_{ps}. In conclusion, D is a biequivalence.

7.7. Remark. The biequivalence of the previous theorem is not an adjoint equivalence. The functor D cannot be a right adjoint, since it doesn’t preserve general limits, as it is π_{0} at level 0. On the other hand, Q clearly does not preserve products, so it cannot be a right adjoint either.

7.8. The main result. Theorems 4.7 and 7.6 imply our main result, which is a biequivalence between the 2categories of weakly globular double categories with pseudo-functors and vertical transformations, and the 2-category of bicategories with homomorphisms and icons. We give a more descriptive name to each of the functors realizing this biequivalence in the following definition:

7.9. Definition. The fundamental bicategory functor Bic: WGDbl_{ps} → Bicat_{icon} is the composite GD. The associated double category functor Dbl: Bicat_{icon} → WGDbl_{ps} is the composite QN. This last functor will also be called the double category of marked paths functor, for reasons that will be come clear in Section 8.2 below.

7.10. Theorem. There is a biequivalence of 2-categories

Bic : WGDbl_{ps,v} ≃ Bicat_{icon} : Dbl.

Proof. This follows immediately from theorems 4.7 and 7.6.

7.11. Remark. Note that we have Dbl ∘ Bic ≃ Id_{WGDbl} and Bic ∘ Dbl ≃ Id_{Bicat}, but these functors are not adjoint, since D and Q are not adjoint as shown in Remark 7.7, but G and N form a biadjoint biequivalence as proved in [20].

8. Bicategories and Double Categories

In the previous section we established our main result, namely the existence of a biequivalence of 2-categories,

Bic : WGDbl_{ps,v} ≃ Bicat_{icon} : Dbl.

Both 2-functors, Bic and Dbl, were obtained as compositions of other functors. Whenever we want to translate bicategorical concepts into concepts of weakly globular double categories, or vice versa, we will need to have an explicit description of the correspondence. In other words, we will need an explicit description of Bic𝕏 for a weakly globular double category 𝕏 and of Dbl(B) for a bicategory B. Studying the explicit description of Bic(𝕏) will also give us a clearer understanding of the role of the induced Segal maps condition characterizing weakly globular double categories. Conversely, the explicit description of Dbl(B) will give us a better understanding of how rigidification of a bicategory is different from the strictification of a bicategory into a strict 2-category.

8.1. The fundamental bicategory. Let 𝕏 be a weakly globular double category. The objects of Bic𝕏 are obtained as the connected components π_{0}𝕏_{0} of the vertical arrow category 𝕏_{0}. When A is an object of 𝕏, i.e., an element of 𝕏_{00}, we write Ā for the corresponding object in Bic𝕏. Note that Ā = B̄ if and only if there is a (unique) vertical arrow v: \xymatrix{A \ar[r]|{•} & B} in 𝕏 (since the vertical arrow category 𝕏_{0} is posetal and groupoidal).

For any two objects Ā and B̄ in 𝕏, the set of arrows, Bic𝕏(Ā, B̄) is obtained as a disjoint union of horizontal hom-sets in 𝕏,

Bic𝕏(Ā, B̄) = ∐ ; \begin{matrix}Ā′ = Ā \\ B̄′ = B̄ \end{matrix} 𝕏_{h}(A′, B′).

Note that we do not put an equivalence relation on the horizontal arrows of 𝕏 to obtain the arrows of the fundamental bicategory; we will therefore use the same symbol to denote a horizontal arrow in 𝕏 and the corresponding arrow in Bic(𝕏).

For any two arrows \xymatrix{Ā \ar[r]^{f} \ar[r]_{g} & B̄} in Bic𝕏 represented by horizontal arrows \xymatrix{A_{1} \ar[r]^{f} & B_{1}}

and \xymatrix{A_{2} \ar[r]^{g} & B_{2}} in 𝕏, the 2-cells from f to g correspond to double cells of the form

[figure: A_{1}; •; v; f; α; B_{1}; •; w; A_{2}; g; B_{2} .]

Since v and w are unique, we will denote the corresponding 2-cell in Bic𝕏 by α: f ⇒ g.

Let f: A_{1} → B_{1} and g: B_{2} → C_{2} be horizontal arrows such that there is an invertible vertical arrow \xymatrix{v: B_{2} \ar[r]|{•} & B_{1}}. Then the induced Segal maps condition gives rise to a diagram

\xymatrix{A_{3} \ar[dd]|{•}_{x} \ar[r]_{φ_{f_{3},f}}^{f_{3}} & B_{3} \ar[d]|{•}^{y} \ar[r]_{φ_{g_{3},g}}^{g_{3}} & C_{3} \ar[d]|{•}^{z} \\  & B_{2} \ar[d]|{•}^{v} \ar[r]_{g} & C_{2} \\ A_{1} \ar[r]_{f} & B_{1}}

as in Remark 6.8. Then composition of f: Ā_{1} → B̄_{1} and g: B̄_{2} → C̄_{2} (where B̄_{1} = B̄_{2}) in Bic𝕏 is defined to be the horizontal composite g_{3} ∘ f_{3} : Ā_{3} → C̄_{3}.

The horizontal composition of 2-cells is defined as follows: Let

\xymatrix{Ā \ar[r]^{f} \ar[r]|{⇓α}_{g} & B̄ \ar[r]^{h} \ar[r]|{⇓β}_{k} & C̄}

be a diagram of arrows and cells in the fundamental bicategory, represented by double cells in 𝕏,

[figure: A_{2}; •; u_{21}; f; α; B_{2}; •; v_{21}; and; B_{4}; •; v_{43}; h; β; C_{4}; •; z; A_{1}; g; B_{1}; B_{3}; k; C_{3}.]

Let the composite of g and k in Bic𝕏 be the arrow k_{5} ∘ g_{5} with a corresponding diagram (as in Remark 6.8),

\xymatrix{A_{5} \ar[dd]|{•}_{u_{51}} \ar[r]_{φ_{g_{5},g}}^{g_{5}} & B_{5} \ar[d]|{•}_{v_{53}} \ar[r]_{φ_{k_{5},k}}^{k_{5}} & C_{5} \ar[d]|{•}^{w_{53}} \\  & B_{3} \ar[d]|{•}^{v_{31}} \ar[r]_{k} & C_{3} \\ A_{1} \ar[r]_{g} & B_{1}}

and let the composite of f and h be the arrow h_{6} ∘ f_{6} as in the diagram

\xymatrix{A_{6} \ar[dd]|{•}_{u_{62}} \ar[r]_{φ_{f_{6},f}}^{f_{6}} & B_{6} \ar[d]|{•}_{v_{64}} \ar[r]_{φ_{h_{6},h}}^{h_{6}} & C_{6} \ar[d]|{•}^{w_{64}} \\  & B_{4} \ar[d]|{•}^{v_{42}} \ar[r]_{h} & C_{4} \\ A_{2} \ar[r]_{f} & B_{2}}

Then the composition of α and β is represented by the following pasting of double cells:

[figure: A_{6}; •; u_{62}; φ_{f_{6},f}; f_{6}; B_{6}; •; v_{64}; φ_{h_{6},h}; h_{6}; C_{6}; •; w_{64}; B_{4}; •; v_{43}; β; h; C_{4}; •; w_{43}; B_{3}; k; •; v_{32}; φ_{k_{5},k}^{−1}; C_{3}; •; v_{35}; A_{2}; •; u_{21}; α; f; B_{2}; •; v_{21}; A_{1}; •; u_{15}; φ_{g_{5},g}^{−1}; g; B_{1}; •; v_{15}; A_{5}; g_{5}; B_{5}; k_{5}; C_{5}.]

(Here, u_{ij} = u_{ji}^{−1} and u_{jk} · u_{ij} = u_{ik}, and analogous for v and w, since the vertical category is groupoidal posetal. Furthermore, the same holds for the cells, because they are components of a vertical transformation.)

The units for the composition are obtained from the functor μ_{0} : 𝕏_{0}^{d} → 𝕏_{0} which is part of the equivalence of categories, 𝕏_{0}^{d} ≃ 𝕏_{0}. For an object Ā in Bic𝕏, 1Ā is the horizontal arrow Id_{μ_{0}(Ā)}.

There are associativity and unit isomorphisms for this composition that satisfy the usual coherence conditions by the results in [33] and [20].

8.2. The double category of marked paths. Let B be a bicategory. Before we begin the construction of Dbl(B), we first choose a

composite φ_{f_{1},...,f_{n}} for each finite path \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{1} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} of (composable) arrows in B. If the path is empty, we take φ_{A_{0}} = 1_{A_{0}}. For each path of such paths,

\xymatrix{( \ar[r]^{f_{i_{1}}} & ⋯ \ar[r]^{f_{in_{1}}} & )( \ar[r]^{f_{i_{n_{1}+1}}} & ⋯ \ar[r]^{f_{in_{2}}} & ) ⋯ ( \ar[r]^{f_{i_{n_{m−1}+1}}} & ⋯ \ar[r]^{f_{in_{m}}} & )}

the associativity and unit cells give rise to unique invertible comparison 2-cells, which we denote by

[figure: φ_{φ_{f_{i_{1}} ,...,f_{in_{1}}} ,φ_{f_{i_{n_{1}+1}} ,...,f_{in_{2}}} ,...,φ_{f_{i_{n_{m−1}+1}} ,...,f_{in_{m}}}}; φ_{f_{i_{1}} ,...,f_{in_{m}}}; Φ_{f_{i_{1}} ···f_{in_{1}} ,f_{i_{n_{1}+1}} ···f_{in_{2}} ,...,f_{i_{n_{m−1}+1}} ···f_{in_{m}}}; .]

(The uniqueness of these cells follows from the associativity and unit coherence axioms.) With these chosen composites and cells, we will walk through the constructions corresponding to the functors in the composition Dbl = QN = P St SN. The 2-nerve NB: Δ^{op} → Cat has the following components:

• (NB)_{0} is the discrete category with objects B_{0}.

• (NB)_{1} has objects B_{1}, i.e., the arrows of B, and arrows the 2-cells of B.

• (NB)_{2} has objects diagrams of the form

[figure: f; α≅; h; g]

in B, and arrows cylinders between such diagrams, i.e., an arrow from (f, g, h, α) to (f′, g′, h′, α′) is a triple (φ, γ, θ) of 2-cells, φ: f ⇒ f′, γ: g ⇒ g′, and θ: h ⇒ h′,

such that θ · α = α′ · (γ ∘ φ),
[figure: f; α≅; h; g; θ⇓; =; f; φ⇓; γ⇓; g; h′; f′; α′≅; h′; g′]

The pseudo-functor SNBΔ^{op} → Cat has then

• (SNB)_{0} is the discrete category on the objects of B;

• (SNB)_{1} has as objects the arrows of B and as arrows the 2-cells of B;

• (SNB)_{2} has as objects paths of length 2 in B, [figure: f_{1}; f_{2}]and as arrows horizontal

paths of 2-cells of length 2 in B, [figure: f_{1}; g_{1}; α_{1}; f_{2}; g_{2}; α_{2}] .

• (SNB)_{n} has as objects paths of length n in B, [figure: f_{1}; f_{2}; f_{n}]and as arrows

horizontal paths of 2-cells of length n in B, [figure: f_{1}; g_{1}; α_{1}; f_{2}; g_{2}; α_{2}; f_{n}; g_{n}; α_{n}] .

By the construction described in Remark 6.10 and taking the pseudo-inverse of the horizontal nerve, we obtain the following description of the double category Dbl(B).

The objects of Dbl(B) are given as pairs of an arrow ψ: [0] → [n] in Δ with a path,

\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}}, of length n in B, for all n. Since the arrow ψ is determined by its image i_{0} = ψ(0) ∈ [n], we will denote this object in Dbl(B) by

( \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} ; i_{0})

and think of A_{i_{0}} as a marked object along the path. So we will also use the notation

\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{i_{0}}} & [A_{i_{0}}] \ar[r]^{f_{i_{0}+1}} & ⋯ \ar[r]^{f_{n}} & A_{n}.}

There is a unique vertical arrow from \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{i_{0}}} & [A_{i_{0}}] \ar[r]^{f_{i_{0}+1}} & ⋯ \ar[r]^{f_{n}} & A_{n}} to

\xymatrix{B_{0} \ar[r]^{g_{1}} & B_{2} \ar[r]^{g_{2}} & ⋯ \ar[r]^{g_{j_{0}}} & [B_{j_{0}}] \ar[r]^{g_{j_{0}+1}} & ⋯ \ar[r]^{g_{m}} & A_{m}} if and only if A_{i_{0}} = B_{j_{0}}. In diagrams we

will include this vertical arrow in the following way

[figure: A_{0}; f_{1}; A_{2}; f_{2}; ⋯; f_{i_{0}}; [A_{i_{0}}]; f_{i_{0}+1}; ⋯; f_{n}; A_{n}; B_{0}; g_{1}; B_{2}; g_{2}; ⋯; g_{j_{0}}; [B_{j_{0}}]; g_{j_{0}+1}; ⋯; g_{m}; A_{n}]

Horizontal arrows in Dbl(B) are given as pairs of an arrow ψ: [1] → [n] in Δ with a path of length n in B, for all n. Analogous to what we did for objects we denote horizontal arrows by

( \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} ; i_{0}, i_{1}) with i_{0} ≤ i_{1},

or by

\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{i_{0}}} & [A_{i_{0}}] \ar[r]^{f_{i_{0}+1}} & ⋯ \ar[r]^{f_{i_{1}}} & [A_{i_{1}}] \ar[r]^{f_{i_{1}+1}} & ⋯ \ar[r]^{f_{n}} & A_{n}.}

The domain of (\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}}; i_{0}, i_{1}) is (\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}}; i_{0}) and the codomain

is ( \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} ; i_{1}). For a horizontal identity arrow,

( \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} ; i_{0}, i_{0})

we will also use the notation

[figure: A_{0}; f_{1}; A_{2}; f_{2}; f_{i_{0}}; [[A_{i_{0}}]]; f_{i_{0}+1}; ⋯; f_{n}; A_{n}]

or

[figure: A_{0}; f_{1}; A_{2}; f_{2}; f_{i_{0}}; [A_{i_{0}}]; [A_{i_{0}}]; f_{i_{0}+1}; ⋯; f_{n}; A_{n}]

when this makes it easier to fit such an arrow into a diagram representing a double cell as shown below.

A double cell consists of two horizontal arrows

( \xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}} ; i_{0}, i_{1}) and ( \xymatrix{B_{0} \ar[r]^{g_{1}} & B_{2} \ar[r]^{g_{2}} & ⋯ \ar[r]^{g_{m}} & B_{m}} ; j_{0}, j_{1})

(for the vertical domain and codomain respectively), such that A_{i_{0}} = B_{j_{0}} and A_{i_{1}} = B_{j_{1}} (such that there are unique vertical arrows between the domains of these arrows and between the codomains of these arrows), together with a 2-cell in B between the chosen composites,

[figure: φ_{f_{i_{0}+1},...,f_{i_{1}}}; φ_{g_{j_{0}+1},...,g_{j_{1}}}; ⇓α; .]

We combine all this information together in the following diagram

[figure: A_{0}; f_{1}; ⋯; f_{i_{0}}; [A_{i_{0}}]; φ_{f_{i_{0}+1},...,f_{i_{1}}}; α; f_{i_{0}+1}; ⋯; f_{i_{1}}; [A_{i_{1}}]; f_{i_{1}+1}; ⋯; f_{n}; A_{n}; B_{0}; g_{1}; ⋯; g_{j_{0}}; [B_{j_{0}}]; φ_{g_{j_{0}+1},...,g_{j_{1}}}; g_{j_{0}+1}; ⋯; g_{j_{1}}; [B_{j_{1}}]; g_{j_{1}+1}; ⋯; g_{m}; B_{m}]

So this represents a double cell in Dbl(B).

Two horizontal arrows,

(\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}}; i_{0}, i_{1}) and (\xymatrix{B_{0} \ar[r]^{g_{1}} & B_{2} \ar[r]^{g_{2}} & ⋯ \ar[r]^{g_{n}} & A_{n}}; j_{0}, j_{1}),

are composable if and only if the two paths are the same, i.e., m = n, A_{i} = B_{i} and f_{i} = g_{i} for all i = 1, …, n, and furthermore, i_{1} = j_{0}. In that case, the horizontal composition of

these arrows is given by (\xymatrix{A_{0} \ar[r]^{f_{1}} & A_{2} \ar[r]^{f_{2}} & ⋯ \ar[r]^{f_{n}} & A_{n}}; i_{0}, j_{1}).

The horizontal composition of double cells

[figure: A_{0}; f_{1}; ⋯; f_{i_{0}}; [A_{i_{0}}]; φ_{f_{i_{0}+1},...,f_{i_{1}}}; α; f_{i_{0}+1}; ⋯; f_{i_{1}}; [A_{i_{1}}]; f_{i_{1}+1}; ⋯; f_{n}; A_{n}; B_{0}; g_{1}; ⋯; g_{j_{0}}; [B_{j_{0}}]; φ_{g_{j_{0}+1},...,g_{j_{1}}}; g_{j_{0}+1}; ⋯; g_{j_{1}}; [B_{j_{1}}]; g_{j_{1}+1}; ⋯; g_{m}; B_{m}]

and

[figure: A_{0}; f_{1}; ⋯; f_{i_{1}}; [A_{i_{1}}]; φ_{f_{i_{1}+1},...,f_{i_{2}}}; β; f_{i_{1}+1}; ⋯; f_{i_{2}}; [A_{i_{2}}]; f_{i_{2}+1}; ⋯; f_{n}; A_{n}; B_{0}; g_{1}; ⋯; g_{j_{1}}; [B_{j_{1}}]; φ_{g_{j_{1}+1},...,g_{j_{2}}}; g_{j_{1}+1}; ⋯; g_{j_{2}}; [B_{j_{2}}]; g_{j_{2}+1}; ⋯; g_{m}; B_{m}]

is defined to be

[figure: A_{0}; f_{1}; ⋯; f_{i_{0}}; [A_{i_{0}}]; φ_{f_{i_{0}+1},...,f_{i_{2}}}; α⊗β; f_{i_{0}+1}; ⋯; f_{i_{2}}; [A_{i_{2}}]; f_{i_{2}+1}; ⋯; f_{n}; A_{n}; B_{0}; g_{1}; ⋯; g_{j_{0}}; [B_{j_{0}}]; φ_{g_{j_{0}+1},...,g_{j_{2}}}; g_{j_{0}+1}; ⋯; g_{j_{2}}; [B_{j_{2}}]; g_{j_{2}+1}; ⋯; g_{m}; B_{m}]

where α ⊗ β is the 2-cell in B given by the following pasting diagram

[figure: φ_{f_{i_{0}+1},...,f_{i_{2}}}; φ_{g_{j_{0}+1},...,g_{j_{2}}}; Φ_{f_{i_{0}+1}···f_{i_{1}} ,f_{i_{1}+1}···f_{i_{2}}}; Φ_{g_{j_{0}+1}···g_{j_{1}} ,g_{j_{1}+1}···g_{j_{2}}}; φ_{f_{i_{0}+1},...,f_{i_{1}}}; φ_{g_{j_{0}+1},...,g_{j_{1}}}; α; φ_{f_{i_{1}+1},...,f_{i_{2}}}; φ_{g_{j_{1}+1},...,g_{j_{2}}}; β]

8.3. Remarks.

1. Note that both the category of horizontal arrows and the category of vertical arrows of Dbl(B) are posetal.

2. We call Dbl(B) the double category of marked paths in B.

3. For a 2-category C there is a double category HC with the arrows of C in the horizontal direction and only identity arrows in the vertical direction, and the double cells correspond to the 2-cells in C. The double category Dbl(C) is not isomorphic to this double category, HC, but it is 2-equivalent to it. And the same statement applies to a category C: HC ≅ D̸bl(C), but HC ≃_{2} Dbl(C).

9. Pasting diagrams in weakly globular double categories

It was observed by Dawson and Paré [9] that not every rectangular tiling diagram of double cells is composable. However, when it is composable, their general associativity result states that any two ways of composing it by successive binary compositions will give the same result.

Since the tilings in our weakly globular double categories are related to pasting diagrams in bicategories via the functors Bic and Dbl and all pasting diagrams in a bicategory are composable, we expect that all rectangular tilings of double cells in a weakly globular double category are composable.

Whereas Dawson and Paré proved in [9] that every rectangular tiling in a double category can be composed under certain factorization conditions we will show in an upcoming paper [27] that the existence of certain vertically invertible double cells is sufficient to prove this result. Although the complete proof has several technical parts, the main idea can be illustrated with a single pinwheel tiling. So we want to include this here as an illustration of how one uses the special properties of a weakly globular double category.

9.1. Proposition. Every pinwheel diagram,

[figure: f_{1}; •; v_{1}; α; •; v_{3}; f_{2}; β; •; v_{7}; f_{3}; •; v_{4}; ε; •; v_{5}; f_{4}; γ; •; v_{8}; f_{5}; •; v_{2}; δ; f_{6}; •; v_{6}; f_{7}; f_{8}; ,]  (15)

of double cells in a weakly globular double category is composable.

Proof. We begin by applying the induced Segal maps condition as in Remark 6.8 with n = 3 to the configuration

[figure: f_{1}; •; v_{3}; f_{3}; f_{4}]

in the diagram (15). This gives us vertically invertible double cells as in the following diagram,

[figure: f_{1}; •; u_{1}; φ_{1}; •; v_{3}; •; u_{2}; φ_{3}; f_{3}; •; u_{3}; f_{4}; φ_{4}; •; u_{4}; f′_{1}; f′_{3}; f′_{4}; .]

These cells and their inverses can be inserted in the pinwheel diagram so that its pasting, if it exists, is equal to the pasting of the following diagram:

[figure: f_{1}; •; u_{1}; φ_{1}; •; v_{3}; f_{2}; β; •; v_{7}; •; u_{2}; φ_{3}; f_{3}; •; u_{3}; φ_{4}; f_{4}; •; u_{4}; f′_{1}; •; u_{1}^{−1}; φ_{1}^{−1}; f′_{3}; φ_{3}^{−1}; •; u_{2}^{−1}; •; u_{3}^{−1}; f′_{4}; φ_{4}^{−1}; •; u_{4}^{−1}; f_{3}; •; v_{3}^{−1}; 1_{f_{3}}; f_{4}; 1_{f_{4}}; f_{1}; •; v_{1}; α; •; v_{3}; f_{3}; •; v_{4}; ε; •; v_{5}; f_{4}; γ; •; v_{8}; f_{5}; •; v_{2}; δ; f_{6}; •; v_{6}; f_{7}; f_{8}; .]

The effect of inserting these cells is the creation of a new horizontal line - a vertical factorization of the whole tiling that we did not have before. The tiling above this line is clearly composable: first compose φ_{3} and φ_{4} horizontally, and then compose the result vertically with β and finally compose φ_{1} horizontally with this composite, ((φ_{4}∘φ_{3})·β)∘φ_{1}.

The pasting diagram below the horizontal line is also composable as indicated in the following diagram

[figure: •; u_{1}^{−1}; φ_{1}^{−1}; f′_{1}; •; u_{2}^{−1}; u_{2}^{−1}; •; φ_{3}^{−1}; f′_{3}; •; u_{3}^{−1}; •; φ_{4}^{−1}; f′_{4}; •; u_{4}^{−1}; •; v_{3}^{−1}; •; v_{3}^{−1}; 1_{f_{3}}; f_{3}; f_{4}; 1_{f_{4}}; •; v_{1}; α; f_{1}; •; v_{3}; •; v_{3}; •; v_{4}; •; v_{4}; f_{3}; ε; •; v_{5}; •; v_{5}; f_{4}; γ; •; v_{8}; f_{5}; f_{6}; •; v_{6}; f_{5}; δ; •; v_{2}; f_{6}; •; v_{6}; f_{7}; f_{8}]

So the pinwheel diagram (15) is indeed composable.

Note that the opposite pinwheel,

[figure: •; v_{1}; α; f_{1}; •; v_{2}; β; f_{2}; •; v_{3}; f_{3}; •; v_{4}; γ; •; v_{5}; ε; f_{4}; •; v_{7}; •; v_{6}; f_{5}; δ; f_{6}; •; v_{8}; f_{7}; ]; f_{8}]

can be shown to be composable by the mirror image of this factorization.

10. Homotopical application

In this section we give a homotopical application of weakly globular double categories. We introduce a subcategory of the latter, whose objects we call groupoidal weakly globular double categories.

We show that this category provides an algebraic model of 2-types. This generalizes a result of [4], where weakly globular double groupoids are proved to model 2-types.

10.1. 2-equivalences in weakly globular double categories. We are going to introduce a notion of equivalence in (WGDbl)_{ps} modeled over the comparison with Tamsamani weak 2-categories. We need the following preliminary lemma. Let π_{0} : Cat → Set associate to a category the set of isomorphism classes of its objects.

10.2. Lemma.

i) There is a functor

Π_{0} : (WGDbl)_{ps} → Cat

such that, for all 𝕏 ∈ (WGDbl)_{ps} and n ≥ 0, (NΠ_{0}𝕏)_{n} = (π_{0}^{∗}N_{h}𝕏)_{n} where N : Cat → [Δ^{op}, Set] is the nerve functor.

ii) If 𝕏 ∈ (WGDbl)_{ps} and a, b ∈ 𝕏_{0}^{d}, let 𝕏_{(a,b)} be the full subcategory of 𝕏_{1}, whose objects z are such that γ∂_{0}(z) = a, γ∂_{1}(z) = b, where ∂_{i} : 𝕏_{1} → 𝕏_{0} are the face operators

and γ : 𝕏_{0} → 𝕏_{0}^{d}. Then 𝕏_{1} ≅ ∐_{a,b∈𝕏_{0}^{d}}𝕏_{(a,b)}.

iii) A morphism F : 𝕏 → 𝕐 in (WGDbl)_{ps} induces functors F_{(a,b)} : 𝕏_{(a,b)} → 𝕐_{(F a,F b)} for all a, b ∈ 𝕏_{0}^{d}.

Proof.

i) By definition, since 𝕏 is weakly globular, there is a categorical equivalence

𝕏_{1}×_{𝕏_{0}} ⋯^{n}×_{𝕏_{0}}𝕏_{1} ≃ 𝕏_{1}×_{𝕏_{0}^{d}} ⋯^{n}×_{𝕏_{0}^{d}}𝕏_{1}

for all n ≥ 2. Since π_{0} sends categorical equivalences to isomorphisms and preserves fiber products over discrete objects, this implies

π_{0}(𝕏_{1}×_{𝕏_{0}^{d}} ⋯^{n}×_{𝕏_{0}^{d}}𝕏_{1}) ≅ π_{0}𝕏_{1}×_{π_{0}𝕏_{0}^{d}} ⋯^{n}×_{π_{0}𝕏_{0}^{d}}π_{0}𝕏_{1}.

This shows that π_{0}N_{h}𝕏 is the nerve of a category, which we denote by Π_{0}𝕏. The functor π_{0} induces a functor π_{0}^{∗} : Ps[Δ^{op}, Cat] → [Δ^{op}, Set]. In particular, if F is a morphism in (WGDbl)_{ps}, π_{0}^{∗}F is a morphism between nerves of categories. This defines Π_{0} on morphisms.

ii) This follows immediately by considering the functor γ(∂_{0}, ∂_{1}) : 𝕏_{1} → 𝕏_{0}^{d} × 𝕏_{0}^{d}, since 𝕏_{0}^{d} is discrete.

iii) Since F is a pseudo-natural transformation, there is a pseudo-commutative diagram

\xymatrix{𝕏_{1} \ar[r]^{(∂_{0},∂_{1})} \ar[d] & 𝕏_{0} × 𝕏_{0} \ar[d] \\ 𝕐_{1} \ar[r]_{(∂′_{0},∂′_{1})} & 𝕐_{0} × 𝕐_{0}}

and therefore, since 𝕏_{0}^{d} and 𝕐_{0}^{d} are discrete, a commutative diagram

\xymatrix{𝕏_{1} \ar[r]^{γ_{𝕏}(∂_{0},∂_{1})} \ar[d] & 𝕏_{0}^{d} × 𝕏_{0}^{d} \ar[d] \\ 𝕐_{1} \ar[r]_{γ_{𝕐}(∂′_{0},∂′_{1})} & 𝕐_{0}^{d} × 𝕐_{0}^{d}}

This determines the functor F_{(a,b)} : 𝕏_{(a,b)} → 𝕐_{(F a,F b)} for all a, b ∈ 𝕏_{0}^{d}.

10.3. Definition. We say that a morphism F : 𝕏 → 𝕐 in (WGDbl)_{ps} is a 2-equivalence if

i) For all a, b ∈ 𝕏_{0}^{d}, F_{(a,b)} : 𝕏_{(a,b)} → 𝕐_{(F a,F b)} is an equivalence of categories.

ii) Π_{0}F is an equivalence of categories, where Π_{0} is as in Lemma 10.2

10.4. Proposition. The functors Q and D preserve 2-equivalences.

Proof. The fact that D preserves 2-equivalences is immediate from the definitions. Let f : X → Y be a 2-equivalence in (Ta_{2})_{ps}. By Theorem 7.2, there is a pseudo-commutative diagram in Ps[Δ^{op}, Cat]

\xymatrix{N_{h}QX \ar[rr]^{α_{X}} \ar[dd]_{N_{h}Qf} &  & X \ar[dd]^{f} \\  & ∼ \\ N_{h}QY \ar[rr]_{α_{Y}} &  & Y}  (16)

in which α_{X} and α_{Y} are levelwise categorical equivalences. Applying the functor π_{0}^{∗} : Ps[Δ^{op}, Cat] → [Δ^{op}, Set] we obtain a commutative diagram in [Δ^{op}, Set]

\xymatrix{π_{0}^{∗}N_{h}QX \ar[r]^{π_{0}^{∗}α_{X}} \ar[d]_{π_{0}^{∗}N_{h}Qf} & π_{0}^{∗}X \ar[d]^{π_{0}^{∗}f} \\ π_{0}^{∗}N_{h}QY \ar[r]_{π_{0}^{∗}α_{Y}} & π_{0}^{∗}Y.}

Recalling that π_{0}^{∗}N_{h}QX = NΠ_{0}QX, π_{0}^{∗}X = NΠ_{0}X and similarly for the other terms, and applying the functor P : [Δ^{op}, Set] → Cat which is left adjoint to the nerve, we obtain the commutative diagram in Cat

\xymatrix{Π_{0}QX \ar[r]^{P π_{0}^{∗}α_{X}} \ar[d]_{Π_{0}Qf} & Π_{0}X \ar[d]^{Π_{0}f} \\ Π_{0}QY \ar[r]_{P π_{0}^{∗}α_{Y}} & Π_{0}Y.}  (17)

Since α_{X} and α_{Y} are levelwise categorical equivalences, π_{0}^{∗}α_{X}, and π_{0}^{∗}α_{Y} are isomorphisms, hence P π_{0}^{∗}α_{X} and P π_{0}^{∗}α_{Y} are isomorphisms. Since f is a 2-equivalence in (Ta_{2})_{ps}, by definition Π_{0}F is an equivalence of categories. Hence the commutativity of (17) implies that Π_{0}Qf is an equivalence of categories. Also, for each a, b ∈ (N_{h}QX)_{0∗}^{d} ≅ X_{0∗}, by (16) we obtain a pseudo-commutative diagram in Cat

\xymatrix{(QX)_{(a,b)} \ar[rr]^{(α_{X})_{(a,b)}} \ar[dd]_{(Qf)_{(a,b)}} &  & X_{(a,b)} \ar[dd]^{f_{(a,b)}} \\  & ∼ \\ (QY)_{(fa,fb)} \ar[rr]_{(α_{Y} )_{(fa,fb)}} &  & Y_{(fa,fb)}.}  (18)

But (α_{X})_{(a,b)} and (α_{Y})_{(fa,fb)} are categorical equivalences because (α_{X})_{1} and (α_{Y})_{1} are categorical equivalences. Also f_{(a,b)} is a categorical equivalence because f is a 2-equivalence in (Ta_{2})_{ps}. Hence by (18), (Qf)_{(a,b)} is a categorical equivalence. In conclusion, Qf is a 2-equivalence in (WGDbl)_{ps}.

10.5. Tamsamani weak 2-groupoids. By imposing suitable invertibility conditions to Tamsamani weak 2-categories one obtains the notion of Tamsamani weak 2-groupoid, which was shown in [33] to give an algebraic model of 2-types of topological spaces. We now recall this background.

10.6. Definition. The category GTa_{2} of Tamsamani weak 2-groupoids is the full subcategory of Ta_{2} whose objects X are such that, for all a, b ∈ X_{0}, X_{(a,b)} and Π_{0}X are groupoids.

10.7. Theorem. [33] There are functors

B : GTa_{2} ⇄ 2-Types : F

inducing an equivalence of categories

B : GTa_{2}/∼^{2} ⇄ Ho(2-Types) : F

In particular, for each X ∈ GTa_{2}, there is a 2-equivalence, γ_{X} : X → F BX, natural in X and, for each 2-type Y, there is a weak homotopy equivalence Γ_{Y} : BF Y → Y, natural in Y.

10.8. Remark. Given X ∈ Ta_{2}, BX is homotopy equivalent to the nerve of the Grothendieck construction on X ∈ [Δ^{op}, Cat], see [7], [32]. The latter can also be used to define the classifying space for objects of Ps[Δ^{op}, Cat], so that in fact we have a functor B : (GTa_{2})_{ps} → 2-Types. In particular if f ∈ (GTa_{2})_{ps} is such that Bf_{k} is a weak equivalence for all k, then Bf is a weak equivalence (see [7] or [32]).

10.9. Groupoidal weakly globular double categories. We now introduce the subcategory of groupoidal weakly globular double categories, which we are going to show form an algebraic model of 2-types.

10.10. Definition. The category GWGDbl of groupoidal weakly globular double categories is the full subcategory of WGDbl whose objects 𝕏 are such that

i) For all a, b ∈ 𝕏_{0}^{d}, 𝕏_{(a,b)} is a groupoid.

ii) Π_{0}𝕏 is a groupoid, where Π_{0} : WGDbl → Cat is as in Lemma 10.2.

A morphism in GWGDbl is a 2-equivalence if and only if it is a 2-equivalence in WGDbl. The category (GWGDbl)_{ps} is the full subcategory of (WGDbl)_{ps} whose objects are in GWGDbl.

10.11. Remark. It follows immediately from the definitions that the functors

D : (WGDbl)_{ps} ⇆ (Ta_{2})_{ps} : Q

restrict to functors

D : (GWGDbl)_{ps} ⇆ (GTa_{2})_{ps} : Q

The notion of groupoidal weakly globular double category is more general than the one of weakly globular double groupoid introduced in [4]. In particular, objects of GWGDbl are not necessarily double groupoids, as the horizontal categories 𝕏_{∗0} and 𝕏_{∗1} are not required to be groupoids if 𝕏 ∈ GWGDbl. This notion is similar to the one of Tamsamani weak 2-groupoids, where inverses of horizontal morphisms exist only in a weak sense. Indeed, the next proposition establishes a comparison with Tamsamani’s model.

10.12. Modelling 2-types. We will not prove that groupoidal weakly globular double categories do indeed model homotopy 2-types, and find that they provide an algebraic Postnikov decomposition for their classifying space.

10.13. Theorem. The functors

BD : (GWGDbl)_{ps} ⇆ 2-Types : QF

induce an equivalence of categories

(GWGDbl)_{ps}/∼^{2} ≅ Ho(2-Types) .

Proof. Given 𝕏 ∈ GWGDbl, there is a commutative diagram in Ps[Δ^{op}, Cat]

\xymatrix{N_{h}QD𝕏 \ar[r]^{N_{h}Qγ_{D𝕏}} \ar[d]_{α_{D𝕏}} & N_{h}QF BD𝕏 \ar[d]^{α_{F BDX}} \\ D𝕏 \ar[r] & F BD𝕏}

in which the vertical maps are levelwise equivalences and the horizontal maps are 2equivalences.

On the other hand, by Remark 7.5, there is a map μ_{𝕏} : N_{h}𝕏 → D𝕏 which is a levelwise equivalence. Since levelwise equivalences are 2-equivalences, we obtain a zig-zag of 2-equivalences between N_{h}𝕏 and N_{h}QF BD𝕏; hence 𝕏 ≅ QF BD𝕏 in (GWGDbl)_{ps}/∼^{2}.

Conversely, let Y ∈ 2-type. By Theorem 7.2 and Proposition 7.4, we have a map in Ps[Δ^{op}, Gpd]

DQF Y → F Y

which is a levelwise equivalence of categories. It follows (see Remark 10.8) that there is a weak homotopy equivalence BDQF Y → BF Y. On the other hand, by Theorem 10.7 there is a weak equivalence BF Y ≃ Y. Hence we have a zig-zag of weak homotopy equivalences between BDQF Y and Y, so that BDQF Y ≅ Y in Ho(2-Types).

10.14. Remark. For every object 𝕏 of GWGDbl it is possible, as in the case of the weakly globular double groupoids of [4], to give an algebraic description of the homotopy groups and of the Postnikov decomposition of the classifying space BD𝕏. This is done as follows. Let BΠ_{0}𝕏 be the classifying space of the groupoid Π_{0}𝕏; for each x ∈ 𝕏_{0}^{d} let id_{x} ∈ 𝕏_{10} be id_{x} = σ_{0}γ′(x), where γ′ : 𝕏_{0}^{d} → 𝕏_{0} is the inverse of γ : 𝕏_{0} → 𝕏_{0}^{d} and σ_{0} : 𝕏_{00} → 𝕏_{10} is the degeneracy operator. Then it follows from [33] that

π_{i}(BD𝕏, x) = π_{i}(BΠ_{0}𝕏, x) i = 0, 1
π_{2}(BD𝕏, x) = Hom_{𝕏_{1}}(id_{x}, id_{x}).

Further, there is a morphism 𝕏 → cΠ_{0}𝕏, where cΠ_{0}𝕏 is the double category which, as a category object in Cat in the vertical direction, is discrete with object of objects Π_{0}𝕏. From above, we see that the morphism 𝕏 → cΠ_{0}𝕏 induces isomorphisms of homotopy groups π_{i}(BD𝕏) ≅ π_{i}(BcΠ_{0}𝕏) i = 0, 1. Hence this gives algebraically the Postnikov decomposition of BD𝕏.

References

[1] Andrée Bastiani, Charles Ehresmann, Multiple functors I. Limits relative to double categories, Cahiers Topologie Géom. Différentielle 15 (1974), no. 3, pp. 215–292.

[2] J. Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer L.N.M. 47 (1967), pp. 1–77.

[3] David Blanc, Simona Paoli, n-Fold groupoids, n-Types and n-Track categories (2012) arXiv: 1204.5101.

[4] David Blanc, Simona Paoli, Two-track categories, Journal of K-theory, 8 no.1 (2011), pp. 59–106.

[5] Ronald Brown, Ghafar Mosa, Double categories, 2-categories, thin structures and connections, Theory and Applications of Categories 5 (1999), pp. 163–175.

[6] Ronald Brown, Christopher B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Différentielle 17 (1976), no. 4, pp. 343–362.

[7] P. Carrasco, A. M. Cegarra, A. R. Garzon, Nerves and classifying spaces for bicategories, Alg. and Geom. Topology, 10 (2010), pp. 219–274.

[8] Robert Dawson, A forbidden-suborder characterization of binarily-composable diagrams in double categories, Theory and Applications of Categories, 1 (1995), pp. 146– 155.

[9] R. Dawson and R. Paré, General associativity and general composition for double categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques 34 (1993), pp. 57–79.

[10] Robert Dawson, Robert Paré, Dorette Pronk, The span construction, Theory and Applications of Categories 24 (2010), no. 13, pp. 302–377.

[11] Andrée Ehresmann, Charles Ehresmann, Multiple functors IV. Monoidal closed structures on Cat_{n}, Cahiers de Topologie et Géométrie Différentielle Catégoriques 20 (1979), no. 1, pp. 59–104.

[12] Andrée Ehresmann, Charles Ehresmann, Multiple functors III. The cartesian closed category Cat_{n}, Cahiers de Topologie et Géométrie Différentielle Catégoriques 19 (1978), no. 4, pp. 387–443.

[13] Andrée Ehresmann, Charles Ehresmann, Multiple functors II. The monoidal closed category of multiple categories, Cahiers Topologie Géom. Différentielle 19 (1978), no. 3, pp. 295–334.

[14] Charles Ehresmann, Catégories doubles et catégories structurées, C. R. Acad. Sci. Paris 256 (1963) pp. 1198–1201.

[15] Thomas M. Fiore, Simona Paoli, Dorette Pronk, Model structures on the category of small double categories, Algebr. Geom. Topol. 8 (2008), no. 4, pp. 1855–1959.

[16] Thomas M. Fiore, Simona Paoli, A Thomason model structure on the category of small n-fold categories, Algebr. Geom. Topol. 10 (2010), pp. 1933–2008.

[17] Marco Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Categ. Structures 5 (1997), no. 4, pp. 363–413.

[18] M. Grandis, R. Paré, Limits in double categories, Cahiers Topologie Géom. Différentielle Catég. 40 (1999), no. 3, pp. 162–220.

[19] G.M. Kelly, Stephen Lack, Monoidal functors generated by adjunctions, with applications to transport of structure, Galois theory, Hopf algebras, and semiabelian categories, pp. 319–340, Fields Inst. Commun. 43, Amer. Math. Soc., Providence, RI, 2004.

[20] Stephen Lack, Simona Paoli, 2-nerves for bicategories, K-Theory 38 (2008), no. 2, pp. 153–175.

[21] Tom Leinster, A survey of definitions of n-category. Theory Appl. Categ. 10 (2002), pp. 1–70.

[22] Tom Leinster, Higher operads, higher categories. London Mathematical Society Lecture Note Series, 298. Cambridge University Press, Cambridge, 2004.

[23] S. MacLane and R. Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), pp. 59–80.

[24] S. Paoli, Weakly globular cat^{n}-groups and Tamsamani’s model, Adv. in Math. 222, (2009), pp. 621-727.

[25] S. Paoli, D.A. Pronk, The weakly globular double category of fractions of a category, in preparation.

[26] S. Paoli, D.A. Pronk, Weakly globular n-fold categories and Tamsamani’s model, in preparation.

[27] S. Paoli, D.A. Pronk, General pasting diagrams in weakly globular double categories, in preparation.

[28] A.J. Power, A general coherence result, J.Pure and Applied Algebra 57 (1989), no 2, pp. 165–173.

[29] Carlos Simpson, Homotopy theory of higher categories. New Mathematical Monographs, 19. Cambridge University Press, Cambridge, 2012.

[30] Michael Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008), no. 18, pp. 650–738.

[31] C.B. Spencer, An abstract setting for homotopy pushouts and pullbacks, Cahiers Tp. Géo. Diff. 18 (1977), pp. 409–430.

[32] R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Phil. Soc., 85 (1979), pp. 91–109.

[33] Zouhair Tamsamani, Sur des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux, K-theory, 16 (1999), pp. 51–99.

Department of Mathematics, University of Leicester, LE17RH, UK and Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada

Email: sp424@le.ac.uk, pronk@mathstat.dal.ca

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