aa r X i v : . [ m a t h . C T ] A ug WEAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS
SIMONA PAOLI
Abstract.
Weakly globular double categories are a model of weak 2-categoriesbased on the notion of weak globularity, and they are known to be suitably equiv-alent to Tamsamani 2-categories. Fair 2-categories, introduced by J. Kock, modelweak 2-categories with strictly associative compositions and weak unit laws. In thispaper we establish a direct comparison between weakly globular double categoriesand fair 2-categories and prove they are equivalent after localisation with respectto the 2-equivalences. This comparison sheds new light on weakly globular doublecategories as encoding a strictly associative, though not strictly unital, composition,as well as the category of weak units via the weak globularity condition. Introduction
Higher category theory is a rapidly developing field with applications to disparateareas, from homotopy theory, mathematical physics, algebraic geometry to, more re-cently, logic and computer science.Higher categories comprise not only objects and morphisms (like in a category)but also higher morphims, which compose and have identities. A key point in highercategory theory is the behaviour of these compositions. In a category, compositionof morphisms is associative and unital. Higher categories in which these rules forcompositions hold for morphisms in all dimensions are called strict higher categories:they are not difficult to formalize, but they are of limited use in applications. A strikingexample is the case of strict n -groupoids, which are strict n -categories with invertiblehigher morphisms. These are algebraic models for the building blocks of topologicalspaces (the n -types) only when n = 0 , ,
2, see [11] for a counterexample.To model n -types for all n (that is, to satisfy the ’homotopy hypothesis’), a morecomplex class of higher structures is needed, the weak n -categories. In a weak n -category, compositions are associative and unital only up to an invertible cell in thenext dimension, in a coherent way.There are several different models of weak n -categories: a survey was given in [7],and several new approaches appeared later on. Of particular relevance for this workare the Segal-type models [8], based on multi-simplicial structures. These comprise theTamsamani model Ta n , originally introduced by Tamsamani [13] and further studiedby Simpson [12] as well as two new models introduced by the author in [8]: the weaklyglobular Tamsamani n -categories Ta nwg and the weakly globular n -fold categories Cat nwg .These models use a new paradigm to encode weakness in a higher category, the notionof weak globularity. The sets of higher morphisms in dimensions 0 , · · · , n in Ta n arereplaced in Ta nwg and Cat nwg by homotopically discrete structures which are only equiv-alent of sets. This allows to obtain the model
Cat nwg of weak n -categories based on the Date : August 2020.2010
Mathematics Subject Classification. simple structure of n -fold categories. The three Segal-type models Ta n , Ta nwg , Cat nwg areproved in [8] to be equivalent up to homotopy .A model of higher categories with associative compositions and weak units wasproposed by Joachim Kock [4] and called fair n -categories Fair n . This model is similarin spirit to Ta n , but with the simplicial category ∆ replaced by the ’fat delta’ category∆ of coloured finite non-empty semi-ordinals. To date it is not yet known if this modelsatisfies the homotopy hypothesis, except for the special case of 1-connected 3-types[2]. It was conjectured earlier on by Simpson [11] that there should exist a modelof higher categories with associative compositions and weak units that satisfies thehomotopy hypothesis and is suitably equivalent to the fully weak models.In this paper we concentrate on the case n = 2. We construct a pair of functorsbetween Cat and
Fair and show they induce an equivalence of categories after lo-calization with respect to the 2-equivalences. This equivalence is not surprising, sinceboth models are known to be equivalent to bicategories [9], [4]. The significance andnovelty of our result lies in the method of proof: we establish a direct comparison be-tween Cat and
Fair , which does not use their equivalence to bicategories. This directcomparison is very non-trivial, and makes use of several novel ideas and constructions,which we believe will lead to higher dimensional generalizations.The passage from weakly globular double categories to fair 2-categories uses a prop-erty of Cat that was not observed so far, namely that it is possible to extract fromit a strictly associative (though not strictly unital) composition. It also gives a newmeaning to the weak globularity condition of weakly globular double categories asencoding the category of weak units.The functor in the other direction, from fair 2-categories to weakly globular doublecategories, factors thorough the category of Segalic pseudo-functors
SegPs [∆ op , Cat ]from ∆ op to Cat , already introduced in [8]; it also uses novel properties of
Fair and ofthe ’fat delta’ category ∆ that we establish in this work.Finally, the categories Fair , Cat and
SegPs [∆ op , Cat ] are not sufficient to provethe final comparison result. To establish the zig-zags of 2-equivalences giving rise tothe equivalence of categories after localization between
Cat and
Fair , we need toenlarge the context by introducing two new players: the category of Segalic pseudo-functors SegPs [∆ op , Cat ] from from the opposite of the ’fat delta’ category to
Cat andthe category
Fair of weakly globular fair 2-categories.We envisage that the new ideas and techniques of this work will provide a basis forhigher dimensional generalizations. These will be tackled in future work.
Organization of the paper
Sections 1 to 5 cover the necessary background: 2-categorical techniques (Section 2), weakly globular double categories (Section 3), fair2-categories (Sections 4 and 5). These sections are expository except for Section 4.2where we highlight some properties of the ’fat delta’ category ∆ which, as far as weare aware, have not appeared in the literature.The comparison between
Cat and
Fair is made of two parts. In Section 6 weexplain the passage from weakly globular double categories to fair 2-categories. Weconstruct in Theorem 6.4 the functor F : Cat → Fair EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 3 and in Proposition 6.9 a natural transformation in [∆ op , Cat ] S ( X ) : F ( X ) → ˜ π ∗ X (with ˜ π ∗ X as in Definition 6.6) which is a levelwise equivalence of categories.In Section 8 we treat the other direction, from fair 2-categories to weakly globulardouble categories. We define the functor R : Fair → Cat (Definition 8.4) and we show in Theorem 8.6 our main result that the functors F and R induce an equivalence of categories after localization with respect to the 2-equivalences. We prove this result by constructing a 2-equivalence R F X → X in Cat for each X ∈ Cat and a zig-zag of 2-equivalences in
Fair between Y and F R Y for each Y ∈ Fair . The construction of this zig-zag requires new notions andresults developed in Section 7: the category of Segalic pseudo-functors SegPs [∆ op , Cat ],the category
Fair of weakly globular fair 2-categories and Theorem 7.7 relating thetwo.
Acknowledgements
This material is based upon work supported by the NationalScience Foundation under Grant No. DMS-1440140 while the author was in residenceat the Mathematical Sciences Research Institute in Berkeley, California, during the’Higher Categories and Categorification’ program in Spring 2020. I thank the organiz-ers for their invitation to this program. I also thank the University of Leicester for itsfinancial support during my study leave.2.
Techniques from -category theory In this Section we recall two techniques from 2-category theory. The first is thestrictification of pseudo-functors: this plays an important role in the theory of weaklyglobular double categories, as recalled in Section 3, and it will also be used in Section 7whose results are crucial to the proof of our main Theorem 8.6. The second techniqueis the transport of structure along an adjunction, which will be used in Proposition8.3, leading to the functor R : Fair → Cat (Definition 8.4).2.1.
Strictification of pseudo-functors.
Let C be a small category. The 2-categoryof functors [ C op , Cat ] is 2-monadic over [ ob ( C op ) , Cat ] where ob ( C op ) is the set of objectsof C op . Let U : [ C op , Cat ] → [ ob ( C op ) , Cat ]be the forgetful functor given by (
U X ) k = X k for each k ∈ C op and X ∈ [ C op , Cat ]. Itsleft adjoint F is given on objects by( F Y ) k = ` r ∈ ob ( C op ) C op ( r, k ) × Y r for Y ∈ [ ob ( C op ) , Cat ], k ∈ C op . If T is the monad corresponding to the adjunction F ⊣ U , then ( T Y ) k = ( U F Y ) k = ` r ∈ ob ( C op ) C op ( r, k ) × Y r . A pseudo T -algebra is given by Y ∈ [ ob ( C op ) , Cat ], functors h k : ` r ∈ ob ( C op ) C op ( r, k ) × Y r → Y k SIMONA PAOLI and additional data given by the axioms of pseudo T -algebra (see for instance [10]).This amounts precisely to functors from C op to Cat and the 2-category
Ps- T -alg ofpseudo T -algebras corresponds to the 2-category Ps [ C op , Cat ] of pseudo-functors, pseudo-natural transformations and modifications. Note that there is a commuting diagram[ C op , Cat ] (cid:31) (cid:127) / / U (cid:15) (cid:15) Ps [ C op , Cat ] U v v ♥♥♥♥♥♥♥♥♥♥♥♥ [ ob ( C op ) , Cat ]Recalling that, if X is a set, X × C ∼ = ` X C , we see that the pseudo T -algebra corre-sponding to H ∈ Ps [ C op , Cat ] has structure map h : T U H → U H as follows:(
T U H ) k = ` r ∈C C ( k, r ) × H r = ` r ∈C ` C ( k,r ) H r . If f ∈ C ( k, r ), let i r = ` C ( k,r ) H r → ` r ∈C ` C ( k,r ) H r = ( T U H ) r ,j f : H r → ` C ( k,r ) H r be the coproduct inclusions, then h k i r j f = H ( f ) . (1)The structure map T U H → U H carries a canonical enrichment to a pseudo-naturaltransformation
F U H → H . The strictification of pseudo-algebras result proved in[10] yields that every pseudo-functor from C op to Cat is equivalent, in Ps [ C op , Cat ], toa 2-functor, that is, an object of [ C op , Cat ].Given a pseudo T -algebra as above, [10] consider the factorization of h : T U H → U H as T U H v −→ L g −→ U H with v k bijective on objects and g k fully faithful, for each k ∈ C op . It is shown in [10]that g is a pseudo-natural transformation and it is possible to give a strict T -algebrastructure T L → L such that ( g, T g ) is an equivalence of pseudo T -algebras. It isimmediate to see that, for each k ∈ C op , g k is an equivalence of categories.Further, it is shown in [5] that St : Ps [ C op , Cat ] → [ C op , Cat ] as described above isleft adjoint to the inclusion J : [ C op , Cat ] → Ps [ C op , Cat ]and that the components of the units are equivalences in Ps [ C op , Cat ]. Remark 2.1.
It is straightforward from [10] that if H ∈ [ C op , Cat ] the pseudo-naturaltransformation g : St H → H is a natural transformation.In this work we use the strictification of pseudo-functors in Section 7 in the casewhere C = ∆ and in Section 8 in the case where C = ∆. As we recall in Section 3, thistechnique also plays a crucial role in the theory of weakly globular double categories. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 5
Transport of structure.
The following 2-categorical technique will be used inSection 8. Its proof relies on [3, Theorem 6.1].
Lemma 2.2. [9]
Let C be a small -category, F, F ′ : C →
Cat be -functors, and α : F → F ′ a -natural transformation. Suppose that, for all objects C of C , thefollowing conditions hold: (1) 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 , (2) there are functors β C : G ( C ) → G ′ ( C ) , (3) there is an invertible -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 ) , andpseudo-natural 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 -category Ps [ C , Cat ] . b) There is a pseudo-natural transformation β : G → G ′ with β ( C ) = β C and aninvertible -cell in Ps [ C , Cat ] , γ : βη ⇒ ηα with γ ( C ) = γ C . Weakly globular double categories and Segalic pseudo-functors
We recall the theory of weakly globular double categories and of Segalic pseuso-functors, originally introduced in [9] and further developed in [8].3.1.
Weakly globular double categories.
We first need the notion of Segal mapsand of induced Segal maps.
Definition 3.1.
Let X ∈ [∆ op , C ] be a simplicial object in a category C with pullbacks.For each 1 ≤ j ≤ k and k ≥
2, let ν j : X k → X be induced by the map [1] → [ k ] in ∆sending 0 to j − j . Then the following diagram commutes: X kν u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ν (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ν k ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ X d (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) d (cid:30) (cid:30) ❃❃❃❃❃❃❃ X d (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) d (cid:31) (cid:31) ❃❃❃❃❃❃❃ . . . X d } } ④④④④④④④④ d (cid:30) (cid:30) ❃❃❃❃❃❃❃ X X X . . . X X (2) If X × X k · · ·× X X denotes the limit of the lower part of the diagram (2), the k -thSegal map of X is the unique map µ k : X k → X × X k · · ·× X X such that pr j µ k = ν j where pr j is the j th projection. SIMONA PAOLI
Let C be the category of internal categories in C and internal functors [1] and let N : Cat
C → [∆ op , C ] be the nerve functor. The Segal maps characterize the essentialimage of N . Namely, an object X ∈ [∆ op , C ] is in the essential image of N if and onlyif its Segal maps X k → X × X k · · ·× X X are isomorphisms for all k ≥ Remark 3.2.
When C = Set , N : Cat → [∆ op , Set ] is the nerve of small categories.This functor is fully faithful, so we can identify
Cat with the essential image of N . Wewill make this identification throughout this work. Definition 3.3.
Let X ∈ [∆ op , C ] and suppose that there is a map γ : X → Y in C such that the limit of the diagram X γd (cid:1) (cid:1) ☎☎☎☎☎☎☎ γd (cid:29) (cid:29) ✿✿✿✿✿✿✿ X γd (cid:1) (cid:1) ☎☎☎☎☎☎☎ γd ❅❅❅❅❅❅❅❅ · · · k · · · X γd ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ γd (cid:29) (cid:29) ✿✿✿✿✿✿✿ Y Y Y · · · · · ·
Y Y exists; denote the latter by X × Y k · · ·× Y X . Then the following diagram commutes,where ν j is as in Definition 3.1, and k ≥ X kν t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ν (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦ ν k ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ X γd (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) γd (cid:31) (cid:31) ❃❃❃❃❃❃❃ X γd (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) γd ❅❅❅❅❅❅❅ . . . X γd } } ③③③③③③③③ γd (cid:31) (cid:31) ❃❃❃❃❃❃❃ Y Y Y . . . Y Y
The k -th induced Segal map of X is the unique mapˆ µ k : X k → X × Y k · · ·× Y X such that pr j ˆ µ k = ν j where pr j is the j th projection. If Y = X and γ is the identity,the induced Segal map coincides with the Segal map of Definition 3.1. Definition 3.4.
A homotopically discrete category is an equivalnce relation, that is agroupoid with no non-trivial loops. We denote by
Cat hd the category of homotopicallydiscrete categories.Let p : Cat → Set be the isomorphism classes of objects functor. As discussedfor instance in [8, Lemma 4.1.4] p preserves pullbacks over discrete objects and sendsequivalences of categories to isomorphisms. Definition 3.5. If X ∈ Cat hd , we denote by X d the discrete category on the set pX .There is a map γ : X → X d , called discretization, which is an equivalence of categories.The category of weakly globular double categories was originally introduced in [9]and further studied in [8]: Definition 3.6.
The category
Cat of weakly globular double categories is the fullsubcategory of [∆ op , Cat ] whose objects X are such that EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 7 a) X ∈ Cat hd .b) For each k ≥ µ k : X k → X × X k · · ·× X X are isomorphisms.c) For each k ≥ µ k : X k → X × X d k · · ·× X d X which are induced by the discretization map γ : X → X d are equivalences ofcategories.Note that because of condition b), Cat is a full subcategory of the category
Cat of double categories, that is of internal categories in Cat . Remark 3.7.
Let p (1) : Cat → [∆ op , Set ] be given by ( p (1) X ) k = pX k for all k ≥ p (1) X is the nerve of a category. In fact, since p sends equivalences of categoriesto isomorphisms and preserves pullbacks over discrete objects, for each X ∈ Cat and k ≥ p (1) X ) k = p ( X k ) ∼ = p ( X × X d k · · ·× X d X ) ∼ = ∼ = p ( X ) × p ( X d ) k · · ·× p ( X d ) p ( X ) ∼ = p ( X ) × p ( X ) k · · ·× p ( X ) p ( X ) . Thus, using the notational convention of remark 3.2 we can write p (1) : Cat → Cat . In what follows, given X ∈ Cat and a, b ∈ X d we denote by X ( a, b ) the fiber at( a, b ) of the map given by the composite X ∂ ,∂ ) −−−−→ X × X γ × γ −−→ X d × X d . where γ : X → X d is the discretization map as in Definition 3.5. The category X ( a, b ) plays the role of ’hom-category’. Definition 3.8.
A morphism F : X → Y in Cat is a 2-equivalence if:i) For all a, b ∈ X d the morphism F ( a,b ) : X ( a, b ) → Y ( F a, F b ) is an equivalenceof categories.ii) p (1) F is an equivalence of categories. Remark 3.9.
The following properties were shown in [8]:a) If F is a levelwise equivalence of categories it is in particular a 2-equivalence.When F = Id, the two notions coincide.b) Condition ii) in Definition 3.8 can be relaxed to requiring that pp (1) X is sur-jective.c) 2-Equivalences in Cat have the 2-out-3 property.
SIMONA PAOLI
Definition 3.10.
Given X ∈ Cat let D X ∈ [∆ op , Cat ] be( D X ) n = (cid:26) X d , n = 0 X n , n > . The face operators ∂ ′ , ∂ ′ : X ⇒ X d are given by ∂ ′ i = γ∂ i where ∂ i : X ⇒ X , i = 0 , σ ′ : X d → X is σ ′ = σ γ ′ where γ ′ : X d → X is apseudo-inverse of γ , γγ ′ = Id. All the other face and degeneracy operators of D X areas in X . Remark 3.11. D X can be obtained by transport of structure along the equivalencesof categories ( D X ) k ≃ X k given by γ ′ for k = 0 and id for k >
0. Thus by Lemma 2.2there is a pseudo-natural transformation D X → X in Ps [∆ op , Cat ] which is a levelwiseequivalence of categories.3.2.
Weakly globular Tamsamani -categories and Segalic pseudo-functors. We recall from [8] the definitions of the categories Ta of weakly globular Tamsamani2-categories and SegPs [∆ op , Cat ] of Segalic pseudo-functors, as well as the constructionof the functor
T r : Ta → SegPs [∆ op , Cat ] . These play an important role in Section 6 in building the functor F : Cat → Fair . Definition 3.12.
The category Ta of weakly globular Tamsamani 2-categories is thefull subcategory of [∆ op , Cat ] whose objects X are such thati) X ∈ Cat hd .ii) The induced Segal maps ˆ µ k : X k → X k × X d k · · ·× X d X k are equivalences ofcategories for all k ≥ Remark 3.13. a) From the definitions,
Cat is the full subcategory of Ta whose objects X aresuch that all the Segal maps are isomorphisms.b) There is a functor p (1) : Ta → Cat given by ( p (1) X ) k = pX k , k ≥
0. Theproof that the essential image of p (1) : Ta → [∆ op , Set ] consists of nerves ofcategories is as in the case of
Cat .The category Ta of Tamsamani 2-categories was originally introduced in [13] but cannow be seen as a subcategory of Ta as follows: Definition 3.14.
The full subcategory of Ta whose objects X are such that X isdiscrete is the category Ta of Tamsamani 2-categories.Note that for Tamsamani 2-categories the induced Segal maps and the Segal mapscoincide. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 9
Let H ∈ Ps [∆ op , Cat ] be such that H is discrete. There is a commuting diagram in Cat for each k ≥ H kν u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ν (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ν k ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ H d (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) d (cid:30) (cid:30) ❃❃❃❃❃❃❃ H d (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) d (cid:31) (cid:31) ❃❃❃❃❃❃❃ . . . H d } } ④④④④④④④④ d (cid:30) (cid:30) ❃❃❃❃❃❃❃ H H H . . . H H where ν j is induced by the map [1] → [ k ] sending 0 to j − j . Hence there is aSegal map H k → H × H k · · ·× H H . Definition 3.15.
The category
SegPs [∆ op , Cat ] is the full subcategory of Ps [∆ op , Cat ]whose objects H are such thati) H is discrete.ii) All Segal maps are isomorphisms for all k ≥ H k ∼ = H × H k · · ·× H H . In [8] we constructed a functor
T r : Ta → SegPs [∆ op , Cat ] (3)by applying transport of structure to X ∈ Ta ⊂ [∆ op , Cat ] along the equivalence ofcategories γ : X → X d , Id : X → X , ˆ µ k : X k → X × X d k · · ·× X d X for k ≥
2. Thusby construction (
T r X ) k = X d , k = 0 X , k = 1 X × X d k · · ·× X d X , k > . and there is a pseudo-natural transformation t ( X ) : T r X → X which is a levelwiseequivalence of categories.Segalic pseudo-functors and weakly globular double categories are related by thefollowing important result, which we will use later on in this work: Theorem 3.16. [8]
The strictification functor St : Ps [∆ op , Cat ] → [∆ op , Cat ] restrictsto a functor St : SegPs [∆ op , Cat ] → Cat . Since Ta ⊂ Ta , by composition we obtain a functor ’rigidification’ Q : Ta T r −−→ SegPs [∆ op , Cat ] St −→ Cat . In [8] we also built a functor ’discretization’ in the opposite direction
Disc : Cat → Ta and we showed that Q and Disc induce an equivalence of categories after localizationwith respect to the 2-equivalences. Combining this with the result of [6] relating Ta to bicategories, we obtained in [9] a 2-categorical equivalence between weakly globulardouble categories and bicategories, showing that Cat is a suitable model of weak2-categories. 4.
The fat delta
We recall from [4] the category ’fat delta’, denoted ∆, and we prove some newproperties it satisfies.4.1.
Definition of the fat delta.
A coloured category is a category C with a specificsubcategory W comprising all objects. The arrows in W are called coloured arrows (orequimorphisms, or equivalences). Let CCat be the category of coloured categories andcoloured-preserving functors. A coloured graph is a graph in which some of the edgeshave been singled out as colours. The free coloured category on a coloured graph isdefined by taking the free category on the whole graph and taking W to be the freecategory on the coloured part of the graph (including vertices). Thus in a free colouredcategory the composite of two arrows is an equimorphism if and only if both arrowsare equimorphisms.Ordinals can be seen as free categories on linearly ordered graphs. Similarly, a (finite)coloured ordinal is defined as a free coloured category on a (finite) linearly orderedgraph. Intuitively these consist of string of arrows, some of which are coloured.Let T ⊂ CCat be the subcategory of finite non-empty coloured ordinals. We canrepresent k ∈ T as dots arranged in a column:Ordinary arrows are not shown and equimorphisms are shown as links. Morphismsare as usual ordinals for the dots (strands are not allowed to cross), but a link canbe set and not broken. The functor π : T → ∆ is given by ’taking equiconnectedcomponents’, that is contracting all the links.Recall that a semi-category is just like a category except that the identity arrows arenot required. A semi-functor is a map compatible with the the composition law. Thecategory ∆ mono is obtained from ∆ by removing the face maps. The ∆ mono -diagramssatisfying the Segal condition are semi-categories. A semi-ordinal is the semi-categoryassociated to a finite total strict order relation. A coloured semi-category is a semi-category with a sub-semicategory comprising all objects; a morphism between colouredsemi-categories is a semi-functor required to preserve colours. The coloured (finite)semi-ordinals are the free coloured semi-categories on (finite) linearly ordered colouredgraphs. Definition 4.1. [4] The fat delta ∆ is the category of all coloured finite non-emptysemi-ordinals.It is shown in [4] that ∆ = T mono . There are functors∆ mono ֒ → ∆ ֒ → ∆ EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 11 showing that the fat delta is intermediate between ∆ mono and ∆. The functor π : T → ∆ gives rise to a functor π : ∆ = T mono → ∆ mono ֒ → ∆ (4)4.2. Further properties the fat delta.
In the rest of this Section we describe prop-erties about the category ∆ that will be used in Section 8 and which, as far as we areaware, have not appeared in the literature.
Remark 4.2.
Let π : ∆ → ∆ be as in (4). Denote by 0( k ) the object of ∆ with π (0( k )) = [0] ∈ ∆ formed by k links. We observe that 0( k ) is given by the colimit in∆ of a diagram of the form0(1) 0(1) .... δ \ \ ✽✽✽✽✽✽✽ δ B B ✝✝✝✝✝✝✝ [0] δ \ \ ✽✽✽✽✽✽✽ @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) [0] ^ ^ ❃❃❃❃❃❃❃❃ B B ✝✝✝✝✝✝✝ where the maps δ , δ are given byWe write 0( k ) = 0(1) ` [0] k . . . ` [0] k ∈ ∆ with π ( k ) = k >
2. Suppose that k has links at j , . . . , j t with 0 ≤ j < · · · < j t ≤ k of length n , . . . , n t respectively. Then k is the colimit in ∆ of the diagram [1] [1] 0( n ) · · · [1] [1] 0( n ) · · · [0] δ Y Y ✸✸✸✸✸✸ δ E E ☛☛☛☛☛☛ . . . [0] B B ✆✆✆✆✆✆✆ [0] \ \ ✾✾✾✾✾✾✾ E E ✡✡✡✡✡✡✡ [0] Y Y ✸✸✸✸✸✸ E E ☛☛☛☛☛☛ . . . [0] B B ✆✆✆✆✆✆✆ [0] \ \ ✾✾✾✾✾✾✾ that is k = [ j ] × [0] n ) × [0] [ j ] × [0] n ) · · · [ j t ] × [0] n t ) × [0] [ k − j t ]which, from Remark 4.2, is the same as a colimit of the form[1] [1] 0(1) 0(1)[0] δ Y Y ✸✸✸✸✸✸ δ E E ☛☛☛☛☛☛ . . . [0] W W ✴✴✴✴✴✴✴ C C ✞✞✞✞✞✞ [0] [ [ ✼✼✼✼✼✼ C C ✞✞✞✞✞✞ [0] [ [ ✼✼✼✼✼✼ Denote by the object of ∆ given by either [1] ∈ ∆ or 0(1) = ∈ ∆. Then k is acolimit in ∆ of a diagram of the form [0] δ [ [ ✽✽✽✽✽✽✽ δ C C ✝✝✝✝✝✝✝ [0] δ [ [ ✽✽✽✽✽✽✽ . . . [0] δ C C ✝✝✝✝✝✝✝ that is k = ` [0] s . . . ` [0] where s = k + n , + · · · , + n t . It follows that there is a bijection∆( k, r ) ∼ = ∆( , r ) × ∆([0] ,r ) s · · ·× ∆([0] ,r ) ∆( , r ) (5)This bijection will play an important role in the proof of Lemma 7.3. Lemma 4.3.
Let f : m → n be a map in ∆ and let πf = εη , η : m → r , ε : r → n bethe epi-mono factorization of the map πf in ∆ . Then there are maps in ∆ m η −→ r ε −→ n with f = εη , π ( ε ) = ε , π ( η ) = η .Proof. Let i s , . . . , i , be elements of n (with π ( i s ) < · · · < π ( i )) not in the image of f and let j , . . . , j t (with π ( j ) < · · · < π ( j t )) be the elements of m in the image of f such that ( πf )( πj ) = ( πf )( πj + 1).Let r = m − t = n − s and let r be obtained from r by inserting all the links identifiedby η ( j ). Then f factors as m η −→ r ε −→ n with η = η j · · · η j t and ε = ε i · · · ε i s . By construction πη = η , πε = ε and πf = εη . (cid:3) Remark 4.4.
For each n ∈ ∆ and n ∈ ∆ such that π ( n ) = n let ν n : n → n be themap in ∆ which sends jwhere the link on the right contracts to j under π . For instance, the map ν : 4 → ε : r → n is a map in ∆ mono , it is also a map in ∆. Thus given ε : r → n such that π ( ε ) = ε there is a commuting diagram r ε / / ν r (cid:15) (cid:15) n ν n (cid:15) (cid:15) r ε / / n (6) EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 13
Lemma 4.5.
Let m η −→ r ε −→ n be maps in ∆ as in Lemma 4.3. Then there arefactorizations in ∆ r ν r / / ! ! ❇❇❇❇❇❇❇❇ rr ( m ) > > ⑤⑤⑤⑤⑤⑤⑤⑤ (7) n ν n / / ! ! ❈❈❈❈❈❈❈❈ nn ( m ) = = ④④④④④④④④ (8) and a map r ( m ) → n ( m ) making the following diagrams commute r ( m ) (cid:15) (cid:15) } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ m η / / r (9) r ε / / (cid:15) (cid:15) n (cid:15) (cid:15) r ( m ) / / (cid:15) (cid:15) n ( m ) (cid:15) (cid:15) r ε / / n (10) Further, if m ′ η ′ −→ r ′ ε ′ −→ n ′ are maps in ∆ such that π ( η ′ ) = π ( η ) and π ( ε ′ ) = π ( ε ) thenthere are factorizations in ∆ r ν r ′ / / ❆❆❆❆❆❆❆❆❆ r ′ r ( m ) > > ⑤⑤⑤⑤⑤⑤⑤⑤⑤ (11) n ν n ′ / / ❇❇❇❇❇❇❇❇❇ n ′ n ( m ) = = ⑤⑤⑤⑤⑤⑤⑤⑤⑤ (12) making the following diagram commute m (cid:15) (cid:15) ν m | | ②②②②②②②②② ν m ′ " " ❋❋❋❋❋❋❋❋❋ m η (cid:15) (cid:15) m ′ η ′ (cid:15) (cid:15) r ( m ) a a ❇❇❇❇❇❇❇❇ = = ④④④④④④④④ } } ⑤⑤⑤⑤⑤⑤⑤⑤ ! ! ❈❈❈❈❈❈❈❈ r r ′ r ν r b b ❊❊❊❊❊❊❊❊❊❊ O O ν r ′ < < ①①①①①①①①①① (13) Proof.
Let j , . . . , j t (with 1 ≤ j < · · · < j t ≤ m ) be such that η ( j ) = η ( j + 1) where η = π ( η ) : m → r . Then r ( m ) is obtained from r by inserting one link in each η ( j ).The map r → r ( m ) sends η ( j ) while the map r ( m ) → r sends this link in r ( m ) to the link in r that contracts to η ( j ) = η ( j + 1) under the functor π : ∆ → ∆ mono sending r to r :By construction (7) commutes. The map r ( m ) → m sends this link to the link in m which is sent by η to the above link in r . By construction, (9) commutes.Since ε is a map in ∆ mono , it is also a map in ∆. Let n ( m ) be given by the pushoutin ∆ r ε / / (cid:15) (cid:15) n (cid:15) (cid:15) r ( m ) / / n ( m ) (14)From the commuting diagram 6 in ∆ (see Remark 4.4) r ε / / ν r (cid:15) (cid:15) n ν n (cid:15) (cid:15) r ε / / n since (14) is a pushout, we obtain a map n ( m ) → n making the following diagramcommute r / / (cid:15) (cid:15) n (cid:15) (cid:15) (cid:17) (cid:17) r ( m ) / / ●●●●●●●●● n ( m ) ! ! ❈❈❈❈❈❈❈❈ r / / n Therefore (8) and (10) commute.The map r ( m ) → r ′ sends the above link in r ( m ) to the link in r ′ that contracts to η ( j ) = η ( j + 1) under π : r ′ → r . EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 15
By construction (11) commutes. From the commuting diagram r ε / / ν r ′ (cid:15) (cid:15) n ν n ′ (cid:15) (cid:15) r ′ ε ′ / / n ′ since (14) is a pushout, we obtain the commuting diagram r ε / / (cid:15) (cid:15) n (cid:15) (cid:15) (cid:17) (cid:17) r ( m ) / / ❋❋❋❋❋❋❋❋❋ n ( m ) ! ! ❇❇❇❇❇❇❇❇ r ′ / / n ′ so that (12) commutes.The maps m → r ( m ) → m send j where the map r ( m ) → m is as in (9). Similarly for the map m → r ( m ) → m ′ . Thus(13) commutes. (cid:3) Lemma 4.6.
Let n f −→ m g −→ s be maps in ∆ op . There are maps n f −→ m g −→ s in ∆ op with πf = f and πg = g .Proof. Let f ′ : n ′ → m ′ be a map in ∆ op with πf ′ = f . Let g = ηε be the mono-epifactorization of the map g in ∆ op , with ε : m → r and η : r → s . Since ε is a map in∆ op mono , it is also a map in ∆ op . Hence there is a map in ∆ op given by the composite m ′ → m → r , which is sent to ε by π . Let η ′ : r → s be a map in ∆ op with πη ′ = η ,where r is constructed by inserting a link in r for each j, j + 1 such that η ( j ) = η ( j + 1).The map η ′ sends this link to jj + 1and there is a map r → r given by j with πr = r . Let m be the pullback in ∆ op m / / ε (cid:15) (cid:15) m ′ (cid:15) (cid:15) r / / r and n be the pullback in ∆ op n f / / (cid:15) (cid:15) m (cid:15) (cid:15) n ′ f ′ / / m ′ In summary, we have the following diagram, where the dotted arrows are arrows in∆ op and the rest are arrows in ∆ op : n f / / ❴❴❴ m ε / / r η / / ❴❴❴ sn ′ O O f ′ / / m ′ O O ? ? ⑦⑦⑦⑦⑦⑦⑦ r O O η ′ @ @ ✁✁✁✁✁✁✁✁ n O O f / / m O O ε ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧ Let g : m → s = s be the composite m ε / / r / / s then πf = f and πg = ηε = g . (cid:3) Corollary 4.7.
Given n f −→ m g −→ s h −→ r in ∆ op , there are maps n f −→ m g −→ s h −→ r in ∆ op with π ( f ) = f , π ( g ) = g , π ( h ) = h .Proof. By Lemma 4.5 there is n ′ f ′ −→ m ′ g ′ −→ s ′ with π ( f ′ ) = f , π ( g ′ ) = g . Let s ε −→ v η −→ r be the mono-epi factorization in ∆ op of h . Since ε is a map in ∆ op mono , it is also a mapin ∆ op . So there is map in ∆ op given by the composite s ′ → s ε −→ v , which is sent to ε by π . Reasoning as in the proof of Lemma 4.6, there is v η −→ r with πη = η . Let s bethe pullback in ∆ op s / / (cid:15) (cid:15) v (cid:15) (cid:15) s ′ / / s ε / / v EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 17
By pulling back further we obtain n f / / ❴❴❴ m g / / ❴❴❴ sn ′ O O f ′ / / m ′ O O g ′ / / s ′ O O n O O f / / m O O g / / s O O with πf = f , πg = g while h : s → r = r is given by the composite s → v η −→ r . Byconstruction, π ( h ) = h . (cid:3) Fair -categories We recall from [4] the category
Fair of fair 2-categories. We start from a moregeneral notion. Let S be a coloured category with a notion of discrete object. Given X : ∆ op → S , denote O = X • , A = X , U = X and think of these as objects, arrows, weak identity arrows respectively. Definition 5.1.
A fair S -category is a colour-preserving functor X : ∆ op → S , pre-serving discrete objects and pullbacks over discrete objects. Remark 5.2.
1) The object O is discrete. Taking • as the only discrete object in ∆ op , the firstaxiom says that X : ∆ op → S preserves discrete objects.2) The second axiom is the Segal condition. Let m + • n be the pushout in ∆ op of m ← • → n . Then X m + • n / / (cid:15) (cid:15) X n (cid:15) (cid:15) X m / / O should be a pullback square. As a consequence, the restriction to either copyof ∆ opmono ⊂ ∆ op is a ∆ opmono -diagram satisfying the Segal condition. Hence A and U are each semicategories.3) It is shown in [4] that in a fair S -category X = ( O , A , U ) the two maps U ⇒ O coincide.When S = Cat , fair S -categories are called fair 2-categories. There is a truncationfunctor p (1) : Fair → Cat given by ( p (1) X ) n = p ( X n ) where p : Cat → Set is the isomorphism classes of objects functor. Given a, b ∈ X ,let X ( a, b ) be the fiber at ( a, b ) of the map X ∂ ,∂ ) −−−−→ X × X . Definition 5.3.
A morphism f : X → Y in Fair is a 2-equivalence if(i) For all a, b ∈ X , f ( a,b ) : X ( a, b ) → Y ( f a, f b ) is an equivalence of categories.(ii) p (1) f is an equivalence of categories. Remark 5.4.
As observed in [4, § X it is enough to givethe following data:a) A discrete category of objects O = X , a category of arrows A = X and acategory of weak units U = X together with a commuting diagram O A t o o s o o U O O O O u ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ b) Semi-category structures on U / / / / O and A / / / / O such that U / / / / (cid:15) (cid:15) OA / / / / O is a semi-functor.c) The maps U / / / / O as well as the composition maps U × O A → A ← A× O U , U × O U → U are equivalences of categories.The Segal condition (stating that X preserves discrete objects and pullbacks overdiscrete objects) implies that the rest of the diagram can be constructed from a) andb). The condition that X preserves colours is equivalent to c). In fact, the maps in c)are all images of vertical map in ∆ and generate the category of vertical arrows in∆. By the Segal condition and the fact that equivalences of categories are stableunder pullbacks over discrete objects, requiring the five maps in c) are equivalencesof categories is equivalent to requiring that every vertical map in ∆ is sent to anequivalence of categories. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 19
We also observe that to give a morphism f : X → Y in Fair is equivalent to givesemi-functors A / / / / (cid:15) (cid:15) O (cid:15) (cid:15) A ′ / / / / O ′ and U / / / / (cid:15) (cid:15) O (cid:15) (cid:15) U ′ / / / / O ′ making the following diagram commute:( A ⇒ O ) / / ( A ′ ⇒ O ′ )( U ⇒ O ) / / O O ( U ′ ⇒ O ′ ) O O Lemma 5.5.
Let F : X → Y be a morphism in Fair which is levelwise equivalenceof categories (i.e. F n is an equivalence of categories for all n ∈ ∆ op ). Then F is a -equivalence.Proof. Since X , Y are discrete, F is an isomorphism. Thus Y = ` a ′ ,b ′ ∈ Y Y ( a ′ , b ′ ) ∼ = ` F a,F ba,b ∈ X Y ( F a, F b ) . Since X = ` a,b ∈ X X ( a, b ) and F is an equivalence of categories, it follows that X ( a, b ) → Y ( F a, F b ) is an equivalence of categories for all a, b ∈ X . Also, since F n is an equiva-lence of categories for all n ∈ ∆ opmono , pF n = ( p (1) F ) n is a bijection. Therefore p (1) F isan isomorphism. By definition, it follows that F is a 2-equivalence. (cid:3) From weakly globular double categories to fair -categories In this Section we construct the first half of the comparison between weakly globulardouble categories and fair 2-categories, namely we build in Theorem 6.4 a functor F : Cat → Fair . To achieve this, we prove in Proposition 6.3 that the essentialimage of the functor
T r : Cat → SegPs [∆ op , Cat ] which we recalled in Section 3consists of Segalic pseudo-functors such that their restriction to ∆ op mono is a functor.We call these pseudo-functors strong Segalic pseudo-functors (Definition 6.1). Thisproperty allows to construct from X ∈ Cat a semi-category structure internal to
Cat with object of objects X d and object of arrows X . The corresponding fair 2-category F X has X d as category of objects, X as category of arrows and X as category ofweak units. The rest of the axioms of fair 2-category for F X are checked using theproperties of weakly globular double categories for X .6.1. Strong Segalic pseudo-functors.
The inclusion functor i : ∆ op mono → ∆ op induces a functor i ∗ : Ps [∆ op , Cat ] → Ps [∆ op mono , Cat ] . Since
SegPs [∆ op , Cat ] ⊂ Ps [∆ op , Cat ] there is also a functor i ∗ : SegPs [∆ op , Cat ] → Ps [∆ op mono , Cat ] . Definition 6.1.
A Segalic pseudo-functor X ∈ SegPs [∆ op , Cat ] is called strong if i ∗ X is a functor from ∆ op mono to Cat . A morphism of strong Segalic pseudo-functorsis a pseudo-natural transformation F in SegPs [∆ op , Cat ] such that i ∗ F is a naturaltransformation in [∆ op mono , Cat ]. We denote by
SSegPs [∆ op , Cat ] the category of strongSegalic pseudo-functors, so that i ∗ : SSegPs [∆ op , Cat ] → [∆ op mono , Cat ] . Remark 6.2. a) We recall that an object Z of [∆ op mono , Cat ] is a semi-simplicial object in
Cat ;that is, a sequence of objects Z i ∈ Cat ( i ≥
0) together with face operators ∂ i : Z n → Z n − ( i = 0 , . . . , n ) satisfying the semi-simplicial identities ∂ i ∂ j = ∂ j − ∂ i if i < j .b) Recall that a semi-category internal to Cat consists of a semi-simplicial ob-ject Z ∈ [∆ op mono , Cat ] such that the Segal maps Z k → Z × Z k · · ·× Z Z areisomorphisms for all k ≥ F of Theorem 6.4. Proposition 6.3.
The restriction to
Cat ⊂ Ta of the functor T r : Ta → SegPs [∆ op , Cat ] in (3) is a functor T r : Cat → SSegPs [∆ op , Cat ] . Proof.
By definition of strong Segalic pseudo-functor, given X ∈ Cat we need toshow that i ∗ T r X ∈ [∆ op mono , Cat ] and that i ∗ T r F is a natural transformation for eachmorphism F in Cat .Let ∂ i : X n → X n − be the face operators of X . By Remark 6.2 we need show that ∂ ′ i = T r ∂ i : ( T r X ) n → ( T r X ) n − satisfy the semi-simplicial identities ∂ ′ i ∂ ′ j = ∂ ′ j − ∂ ′ i if i < j . By construction of T r [8, Theorem 10.1.1]( T r X ) n = X d , n = 0; X , n = 1; X × X d n · · ·× X d X , n ≥ . and T r X is built from X by transport of structure from the equivalences of categories γ : X → X d = ( T r X ) Id : X → X = ( T r X ) ˆ µ k : X k = X × X k · · ·× X X → X × X d k · · ·× X d X = ( T r X ) k k ≥ EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 21 where ˆ µ k is the k th induced Segal map of X . Since ˆ µ k is injective on objects, itspseudo-inverse ν k satisfies ν k ˆ µ k = id for k ≥ . By [8, Lemma 4.3.2] the face maps ∂ ′ i : ( T r X ) k → ( T r X ) k − are given as follows:i) For k = 1, i = 0 , ∂ ′ i = γ∂ i : X → X d . ii) For k = 2, i = 0 , , ∂ ′ i = ∂ i ν : X × X d X → X . iii) For k > i = 0 , . . . , k∂ ′ i = ˆ µ k − ∂ i ν k : X × X d k · · ·× X d X → X × X d k − · · ·× X d X . We now verify the semi-simplicial identities when i < j :a) (
T r X ) ∂ ′ j −→ ( T r X ) ∂ ′ i −→ ( T r X ) ∂ ′ i ∂ ′ j = γ∂ i ∂ j ν = γ∂ j − ∂ i ν = ∂ ′ j − ∂ ′ i . b) ( T r X ) ∂ ′ j −→ ( T r X ) ∂ ′ i −→ ( T r X ) ∂ ′ i ∂ ′ j = ∂ i ν ˆ µ ∂ j ν = ∂ i ∂ j ν = ∂ j − ∂ i ν = ∂ j − ν ˆ µ ∂ i ν = ∂ ′ j − ∂ ′ i . c) For k > T r X ) k +1 ∂ ′ j −→ ( T r X ) k ∂ ′ i −→ ( T r X ) k − ∂ ′ i ∂ ′ j = ˆ µ k − ∂ i ν k ˆ µ k ∂ j ν k +1 = ˆ µ k − ∂ i ∂ j ν k +1 == ˆ µ k − ∂ j − ∂ i ν k +1 = ˆ µ k − ∂ j − ν k ˆ µ k ∂ i ν k +1 = ∂ ′ j − ∂ ′ i . Thus
T r X satisfies the semi-simplicial identities, hence T r X ∈ [∆ op mono , Cat ].Let F : X → Y be a morphism in Cat . By [8, Lemma 4.3.2]
T r F is given by( T r F ) k = F d , k = 0Id , k = 1ˆ µ k ( F , k . . ., F ) ν k , k ≥ . Using the functoriality of F , the definition of ∂ ′ i , the fact that F d γ = γF and ν j ˆ µ k = Idwe see that the following diagrams commute for all k ≥ X ∂ ′ i / / F (cid:15) (cid:15) X d F d (cid:15) (cid:15) Y ∂ ′ i / / Y d X × X d X ∂ ′ i / / ν (cid:15) (cid:15) X (cid:15) (cid:15) X × X X ∂ i / / ( F ,F ) (cid:15) (cid:15) X F (cid:15) (cid:15) Y × Y Y ∂ i / / ˆ µ (cid:15) (cid:15) Y (cid:15) (cid:15) Y × Y d Y ∂ ′ i = ∂ i ν / / Y X × X d k · · ·× X d X µ k − ∂ i ν k / / ν k (cid:15) (cid:15) X × X d k − · · ·× X d X ν k − (cid:15) (cid:15) X × X k · · ·× X X ∂ i / / ( F , k ...,F ) (cid:15) (cid:15) X × X k − · · ·× X X F , k − ... ,F ) (cid:15) (cid:15) Y × Y k · · ·× Y Y ∂ i / / ˆ µ k (cid:15) (cid:15) Y × Y k − · · ·× Y Y µ k − (cid:15) (cid:15) Y × Y d k · · ·× Y d Y µ k − ∂ i ν k / / Y × Y d k − · · ·× Y d Y In conclusion, for each k ≥ i = 0 , . . . , k + 1 the following diagram commutes( T r n X ) k +1 ∂ ′ i / / ( T r n F ) k +1 (cid:15) (cid:15) ( T r n X ) k ( T r n F ) k (cid:15) (cid:15) ( T r n Y ) k +1 ∂ ′ i / / ( T r n Y ) k This shows that
T r n F is a natural transformation of functors in [∆ op mono , Cat ]. (cid:3) The functor F . We now prove the first main theorem of this paper, namely theexistence of a functor F from weakly globular double categories to fair 2-categoriesthat preserves 2-equivalences. In Proposition 6.9 we also compare X ∈ Cat and F X ∈ Fair by suitably modifying X to an object ˜ π ∗ X ∈ [∆ op , Cat ] and by establishingan equivalence S ( X ) : F ( X ) → ˜ π ∗ X . These results will be used in Section 8 toestablish the equivalence after localization of Cat and
Fair (Theorem 8.6). Theorem 6.4.
There is a functor F : Cat → Fair such that ( F X ) = X d , p (1) X = p (1) F X and, for each a, b ∈ X d , X ( a, b ) ∼ = F X ( a, b ) .In particular, F sends -equivalences in Cat to -equivalences in Fair .Proof. Let X ∈ Cat . Define( F X ) = X d , ( F X ) = X , ( F X ) = X together with the commuting diagram X d X γ∂ o o γ∂ o o X γ O O γ O O σ > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 23 where ∂ , ∂ : X → X (resp. σ : X → X ) are the face (resp. degeneracy) operatorsin X .By Proposition 6.3, i ∗ T r X ∈ [∆ op mono , Cat ] with( i ∗ T r X ) k = X d , k = 0 X , k = 1 X × X d k · · ·× X d X , k > . Thus by Remark 6.2 b), i ∗ T r X is a semi-category object internal to Cat . Whenrestricted to X (cid:31) (cid:127) σ / / X , this becomes a semi-category structure internal to Cat X × X d X / / X γ / / γ / / X d . Note that since, when restricted to X , all face operators in X become identities, allthe maps X × X d X → X are equal to ν and, for each k ≥
2, all the maps X × X d k +1 · · ·× X d X → X × X d k · · ·× X d X are equal to ˆ µ k ν k +1 . We also have a semi-functor · · · X × X d X × X d X / / / / / / / / ( σ ,σ ,σ ) (cid:15) (cid:15) X × X d X / / / / / / ( σ ,σ ) (cid:15) (cid:15) X / / / / σ (cid:15) (cid:15) X d (cid:15) (cid:15) · · · X × X d X × X d X / / / / / / / / X × X d X / / / / / / X / / / / X d Since X ∈ Cat hd , γ : X → X d is an equivalence of categories. By Remark 6.2 toprove that F X ∈ Fair it remains to show that the composition maps X × X d X → X , X × X d X → X , X × X d X → X are equivalence of categories. As noted above, all maps X × X d X → X are equal to ν , so in particular the composition map is an equivalence of categories.Consider the commuting diagram X × X d X σ , Id) / / X × X d X cν / / X X = X × X X σ , Id) / / ˆ µ O O X × X X c / / ˆ µ O O X O O Note that the bottom morphism is c ( σ , Id) = Id : X → X . Since ˆ µ is an equivalenceof categories, it follows that such is cν ( σ , Id) : X × X d X → X . The case for themap cν (Id , σ ) : X × X d X → X is completely similar. In conclusion, F X ∈ Fair .If f : X → Y is a morphism in Cat , by Proposition 6.3 i ∗ T r f is a naturaltransformation in [∆ op mono , Cat ]. Thus there is a semi-functor A ( f ) : A ( X ) → A ( Y ): · · · X × X d X × X d X / / / / / / / / ( f ,f ,f ) (cid:15) (cid:15) X × X d X / / / / / / ( f ,f ) (cid:15) (cid:15) X / / / / f (cid:15) (cid:15) X d f d (cid:15) (cid:15) · · · Y × Y d Y × Y d Y / / / / / / / / Y × Y d Y / / / / / / Y / / / / Y d which restricts to a semi-functor U ( X ) → U ( Y ): · · · X × X d X × X d X / / / / / / / / (cid:15) (cid:15) X × X d X / / / / / / (cid:15) (cid:15) X / / / / (cid:15) (cid:15) X d (cid:15) (cid:15) · · · Y × Y d Y × Y d Y / / / / / / / / Y × Y d Y / / / / / / Y / / / / Y d making the following diagram in [∆ op mono , Cat ] commute: A ( X ) A ( f ) / / A ( Y ) U ( X ) O O U ( f ) / / U ( Y ) O O By Remark 6.2 it follows that F f is a morphism in Fair . By construction, ( p (1) F X ) k = p (1) X k for all k ≥
0, so that p (1) F X = p (1) X and, for all a, b ∈ X d , ( F X )( a, b ) = X ( a, b ). It follows that a 2-equivalence in Cat is sent by F to a 2-equivalence in Fair . (cid:3) We next want to relate F X and X . For this purpose, we first note that X ∈ Cat ⊂ [∆ op , Cat ] gives rise to an object of [∆ op , Cat ] closely related to X , as illustrated in thefollowing definitions and lemma. Definition 6.5.
Let π ∗ : [∆ op , Cat ] → [∆ op , Cat ] be induced by π : ∆ op → ∆ op . Thatis, for each X ∈ [∆ op , Cat ], ( π ∗ X ) n = X π ( n ) = X n , where π ( n ) = n . Definition 6.6.
Let ˜ π ∗ : Cat → [∆ op , Cat ] be given by(˜ π ∗ X ) n = (cid:26) ( π ∗ X ) n , if n = 0 X d , if n = 0 . where the maps (˜ π ∗ X ) = X ⇒ (˜ π ∗ X ) = X d are γ∂ i i = 0 , π ∗ X ) = X ⇒ (˜ π ∗ X ) = X d are both equal to γ . All other maps (˜ π ∗ X ) n → (˜ π ∗ X ) m are as in( π ∗ X ) n → ( π ∗ X ) m . Remark 6.7.
It is immediate that ˜ π ∗ X can be obtained from π ∗ X by transport ofstructure along the equivalences of categories (˜ π ∗ X ) k ≃ ( π ∗ X ) k given by γ ′ : X d → X for k = 0 and id for k >
0. Therefore by Lemma 2.2 there is a pseudo-natural trans-formation ˜ π ∗ X → π ∗ X in Ps [∆ op , Cat ] which is a levelwise equivalence of categories.
Lemma 6.8.
Let F : Cat → Fair be as in Theorem 6.4. a) For each n ∈ ∆ op and X ∈ Cat , there is an equivalence of categories X n → ( F X ) n (where n = π ( n )) which is injective on objects whenever n = 0 . b) For each X ∈ Cat and n ∈ ∆ op there is an injective equivalence of categories z n : (˜ π ∗ X ) n → ( F X ) n . (15) Proof. a) When n = π ( n ) = n ∈ ∆ op , this is immediate from the definition of F andthe fact that X ∈ Cat . EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 25 If n ∈ ∆ op with n = 0, π ( n ) = 0 and n has one link of lenght n , it is ( F X ) n = X × X d n · · ·× X d X . Thus the induced Segal map condition for X gives an injectiveequivalence of categories(˜ π ∗ X ) n = X π ( n ) = X = X × X n · · ·× X X → X × X d n · · ·× X d X = ( F X ) n . If n ∈ ∆ op has two links of length n , n and π ( n ) = 1, it is( F X ) n = X × X d X n · · · X × X d X × X d X n · · · X × X d X so the induced Segal map condition for X gives an injective equivalence of categories(˜ π ∗ X ) n = X π ( n ) = X = X × X X n · · · X × X X × X X n · · · X × X X →→ X × X d X n · · · X × X d X × X d X n · · · X × X d X = ( F X ) n . As in Remark 4.2, let n ∈ ∆ with n = π ( n ) ≥ j , . . . , j t with 0 ≤ j < · · · < j t ≤ n of length n , . . . , n t with s = n + n + · · · + n t , n = j + · · · + j t . Byconstruction of F ,( F X ) n = X × X d j · · ·× X d X × X d n · · ·× X d X × X d j · · ·× X d X × X d · · · Since F X ∈ Fair , there is equivalence of categories ν s : ( F X ) n → X × X d j · · ·× X d X × X d n · · ·× X d X × X d j · · · ≃≃ X × X s − ( n + ··· + n t ) · · · × X X = X × X n · · ·× X X = X n with inverse ˆ µ s such that ν s ˆ µ s = Id.b) This follows from a) and the fact that ( F X ) = X d = (˜ π ∗ X ) . (cid:3) The following proposition, together with Theorem 6.4, will be crucially used in theproof of the main result Theorem 8.6.
Proposition 6.9.
Let F : Cat → Fair be as in Theorem 6.4 and ˜ π ∗ as in Definition6.6. There is a natural transformation S ( X ) : F ( X ) → ˜ π ∗ X in [∆ op , Cat ] which is alevelwise equivalence of categories.Proof. For each n ∈ ∆ op , ( S ( X )) n is given by the pseudo-inverse to the equivalence ofcategories z n in (15).Let f : n → m be a map in ∆ op . We claim that the map ( F X ) n → ( F X ) m is givenby the composite ( F X ) n ( S X ) n −−−−→ (˜ π ∗ X ) n ˜ π ∗ f −−→ (˜ π ∗ X ) m z m −→ ( F X ) m . (16)In fact, by the construction of F in the proof of Theorem 6.4, the structure of F X asa diagram in [∆ op , Cat ] is determined by the following maps:a) The maps giving the semi-category structures( F X ) = X × X d X → X = ( F X ) ⇒ X d = ( F X )( F X ) = X × X d X → X = ( F X ) ⇒ X d = ( F X ) which are given by the composites X × X d X ν −→ X × X X c −→ X X × X d X ν −→ X × X X = X b) The maps ( F X ) = X × X d X → X = ( F X )( F X ) = X × X d X → X = ( F X )which are given as composites X × X d X ν −→ X × X X = X X × X d X ν −→ X × X X = X . We see that the maps in a) and b) satisfy (16). Since all other maps ( F X ) n → ( F X ) m are determined by these, they also satisfy (16). This proves the claim,so that F X ( f ) = z m (˜ π ∗ f )( S X ) n . Since by Proposition 6.9 z n is an injective equivalence, ( S X ) n z n = Id for all n ∈ ∆ op . Therefore the following diagram commutes( F X ) n F X ( f ) / / ( S X ) n (cid:15) (cid:15) ( F X ) m ( S X ) m (cid:15) (cid:15) (˜ π ∗ X ) n ˜ π ∗ f / / (˜ π ∗ X ) m since ( S X ) m ( F X )( f ) = ( S X ) m z m (˜ π ∗ f )( S X ) n = (˜ π ∗ f )( S X ) n . This shows that S X : F X → ˜ π ∗ X is a natural transformation in [∆ op , Cat ]. (cid:3) Weakly globular fair -categories In this Section we introduce the category of Segalic pseudo-functors from ∆ op to Cat and the category
Fair of weakly globular fair 2-categories, and we show in Theorem7.7 that they are related by the strictification of pseudo-functors. We also introducein Lemma 7.8 a functor D : Fair → Fair which preserves 2-equivalences. Theseconstructions and results will be used only in the next Section in the proof of our lastand main result Theorem 8.6. This Section may therefore be skipped at first reading. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 27
Segalic pseudo-functors from ∆ op to Cat . Let H ∈ Ps [∆ op , Cat ] be such that H is discrete. Let k ∈ ∆ k = ` [0] k + n , + ··· , + n t . . . ` [0] be as in Remark 4.2. Let ν i : H k → H ( i = 1 , . . . , s ) be given as follows. If = [1], ν i is induced by the map [1] → k sending 0 to i − i , where we re-labeled the s vertices of k . For example if k ∈ ∆ (with π ( k ) = 4, n = 1, n = 1) is •••••• then we re-label the 4 + 2 = 6 vertices as • • • • • • Below we draw pictures of the maps ν i = [1] → k for i = 1 , . . . , • • • • •• • • • •• • • • •• • • • •• • • tttttt • • ✞✞✞✞✞✞✞✞✞ • • ✎✎✎✎✎✎✎✎✎✎✎✎ • • ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ •• • • tttttt • • ✞✞✞✞✞✞✞✞✞ • • ✎✎✎✎✎✎✎✎✎✎✎✎ • • ✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓✓ • ν ν ν ν ν If = 0(1) = , there are maps H k → H induced by the maps → k as follows. Foreach block of n j links (1 ≤ j ≤ t ) each map sends to a consecutive link, as follows... ... ... • • •• • •• • • · · · • • • ✉✉✉✉✉✉ • • ✟✟✟✟✟✟✟✟✟ •• • • tttttt • • ✞✞✞✞✞✞✞✞✞ • so there are n + · · · + n t maps → k . Since H is discrete, the following diagramcommutes H k v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:15) (cid:15) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ H } } ⑤⑤⑤⑤⑤⑤⑤⑤ ! ! ❇❇❇❇❇❇❇❇ H } } ⑤⑤⑤⑤⑤⑤⑤⑤ ●●●●●●●● · · · H } } ⑤⑤⑤⑤⑤⑤⑤⑤ ! ! ❇❇❇❇❇❇❇❇ H H H · · · H H Therefore there are Segal maps η k = H k → H × H s · · ·× H H (17)where k = π ( k ) and s = k + n + · · · + n t . Definition 7.1.
The category
SegPs [∆ op , Cat ] of Segalic pseudo-functors from ∆ op to Cat is the full subcategory of Ps [∆ op , Cat ] whose objects H are such thati) H is discreteii) For each k ∈ ∆ op with π ( k ) = k ≥
2, the Segal maps (17) are isomorphisms.iii) The maps H ⇒ H , H × H H → H ← H × H H , H × H H → H which are images of (18)are equivalences of categories. Remark 7.2.
From the definitions, if X ∈ Fair , then X ∈ SegPs [∆ op , Cat ]. In fact theinclusion [∆ op , Cat ] ⊂ Ps [∆ op , Cat ] restricts to the inclusion
Fair ⊂ SegPs [∆ op , Cat ].Recall [10] that the functor 2-category [∆ op , Cat ] is 2-monadic over [ ob (∆ op ) , Cat ].Let U : [∆ op , Cat ] → [ ob (∆ op ) , Cat ] be the forgetful functor; then its left adjoint is(
F Y ) k = ` r ∈ ob (∆ op ) ∆ op ( r, k ) × Y r for Y ∈ [ ob (∆ op ) , Cat ], k ∈ ∆ op .Let T the monad corresponding to the adjunction F ⊣ U . Then the pseudo T -algebracorresponding to H ∈ Ps [∆ op , Cat ] has structure map h : T U H → H as follows:( T U H ) k = ` r ∈ ∆ ∆( k, r ) × H r = ` r ∈ ∆ ` ∆( k,r ) H r . If f ∈ ∆( k, r ), let i r : ` ∆( k,r ) H r → ` r ∈ ∆ ` ∆( k,r ) H r = ( T U H ) r j f : H r → ` ∆( k,r ) H r EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 29 be the coproduct inclusions, then h k i r j f = H ( f ) . Lemma 7.3.
Let
U, T, F be as above. Let H ∈ SegPs [∆ op , Cat ] . Then a) There are functors ∂ ′ , ∂ ′ : ( T U H ) ⇒ ( T U H ) making the following diagramcommute: ( T U H ) h / / ∂ ′ (cid:15) (cid:15) ∂ ′ (cid:15) (cid:15) H ∂ (cid:15) (cid:15) ∂ (cid:15) (cid:15) ( T U H ) h / / H (19) that is, ∂ i h = h ∂ ′ i i = 0 , . b) For each k ∈ ∆ as in Remark 4.2 with k = π ( k ) ≥ , and s = k + n + · · · + n t ( T U H ) k ∼ = ( T U H ) × ( T UH ) s · · ·× ( T UH ) ( T U H ) . c) For each k ∈ ∆ with k ≥ the morphism h k : ( T U H ) k → H k is given by h k = ( h , . . . , h ) .Proof. a) Let δ i : [0] → , i = 0 , ∂ ′ i : ( T U H ) → ( T U H ) i = 0 , ∂ ′ i i r j f = i r j fδ i where f ∈ ∆( k, r ). Then h ∂ ′ i i r j f = h i r j fδ i = H ( f δ i ) ∂ i h i r j f = H ( δ i ) H ( f ) . Since H ∈ Ps [∆ op , Cat ] and H is discrete, it follows that H ( f δ i ) = H ( δ i ) H ( f )so that, from above, h ∂ ′ i i r j f = ∂ i h i r j f . We conclude that h ∂ ′ i = ∂ i h .b) From the proof of a), the functors ∂ ′ i : ( T U H ) → ( T U H ) for i = 0 , δ i , Id) : ∆( , r ) × H r → ∆(0 , r ) × H r where δ i ( g ) = gδ i for g ∈ ∆( , r ) and( T U H ) k = ` r ∈ ∆ ∆( , r ) × H r ( T U H ) = ` r ∈ ∆ ∆(0 , r ) × H r Using the bijection (5) in Remark 4.2 it follows that(
T U H ) × ( T UH ) s · · ·× ( T UH ) ( T U H ) == ` r ∈ ∆ { ∆( , r ) × ∆(0 ,r ) s · · ·× ∆(0 ,r ) ∆( , r ) } × H r == ∆ r ∈ ∆ ( k, r ) × H r = ( T U H ) k . This proves b).c) From above, for each f ∈ ∆( k, r ), h k i r j f = H ( f ). Let f correspond to δ , . . . , δ s in the bijection (5) in Section 4.1. Then j f = ( jδ , . . . , jδ s ). Since by hypothesis H k ∼ = H × H s · · ·× H H H ( f ) corresponds to ( H ( δ ) , . . . , H ( δ s )) with p i H ( f ) = H ( δ i ). Then for all f we have h k i r j f = ( H ( δ ) , . . . , H ( δ s )) == ( h i r j δ , . . . , h i r j δ s ) = ( h , . . . , h ) i r j f . It follows that h k = ( h , . . . , h ). (cid:3) Weakly globular fair -categories. We now introduce the category
Fair ofweakly globular fair 2-categories. This is a weakly globular version of
Fair . Wereplace the discrete object X with a homotopically discrete X ∈ Cat hd while retainingthe strict Segal condition. This gives semi-category structures (internal to Cat ) to X ⇒ X and to X ⇒ X . The set underlying the discrete category X d plays therole of ’set of objects’. By analogy with the category Cat , we require induce Segalmaps conditions. In Lemma 7.8 we show that this allows to obtain semi-categorystructures on X ⇒ X d and X ⇒ X d and thus build a functor D : Fair → Fair which discretizes X to X d .The main property of the category Fair is that it arises as strictification of Segalicpseudo-functors from ∆ op to Cat (see Theorem 7.7) in a way formally analogous to how
Cat arises as strictification of Segalic pseudo-functors from ∆ op to Cat (see Theorem3.16).
Definition 7.4.
The category
Fair of weakly globular fair 2-categories is the fullsubcategory of [∆ op , Cat ] whose objects X are such thata) X ∈ Cat hd .b) All Segal maps are isomorphisms: for all k ∈ ∆ op with π ( k ) ≥ k , as inRemark 4.2 and s = k + n + · · · + n t , X k ∼ = X × X s · · ·× X X . c) The induced Segal maps (where k and s are as in b)) X × X s · · ·× X X → X × X d s · · ·× X d X are equivalences of categories.d) X preserves colours. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 31
It is clear that
Fair ⊂ Fair . As in the case of
Fair the Segal condition for X ∈ Fair means that the restriction to either copy of ∆ mono ⊂ ∆ is a ∆ opmono -diagramthat satisfies the Segal condition. Hence X ⇒ X and X ⇒ X are each semi-category objects in Cat , that is they carry associative composition operations over X .The Segal condition means that the rest of the diagram can be constructed from thetriangle X X o o o o X ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ O O O O provided that X → X is a semi-functor (that is, compatible with compositions in X and X ).The preservation of colours is equivalent to requiring that the maps X ⇒ X , X × X X → X ← X × X X , X × X X → X . which are in the images of the map (18) (see Definition 7.1) are equivalences of cate-gories.There is a functor p (1) : Fair → Cat given by ( p (1) X ) k = pX k for all k ∈ ∆ opmono and X ∈ Fair . In fact, by the induced Segal map condition and the fact that p preservespullbacks over discrete objects, for all k ≥ p (1) X ) k = p ( X × X k · · ·× X X ) ∼ = p ( X × X d k · · ·× X d X ) == pX × X d k · · ·× X d pX . So all Segal maps of p (1) X are isomorphisms, hence, using the notation of Remark 3.2, p (1) X ∈ Cat .Given X ∈ Fair and a, b ∈ X d , we denote by X ( a, b ) ⊂ X the fiber at ( a, b ) of themap X ∂ ,∂ ) −−−−→ X × X γ × γ −−→ X d × X d . Definition 7.5.
A morphism F : X → Y in Fair is a 2-equivalence if(i) For all a, b ∈ X d , F ( a, b ) : X ( a, b ) → Y ( F a, F b ) is an equivalence of categories.(ii) p (1) F is an equivalence of categories. Remark 7.6.
Let X ∈ Cat and π ∗ X ∈ [∆ op , Cat ] be as in Definition 6.5. We notethat π ∗ X ∈ Fair . In fact ( π ∗ X ) = X ⇒ X = ( π ∗ X ) has a semi-category structuresince X ∈ Cat . The semi-category structure on ( π ∗ X ) = X ⇒ X = ( π ∗ X ) is thediscrete one. By the weak globularity condition, X ∈ Cat hd . The remaining conditionsin the definition of Fair follow from the induced Segal map condition for X ∈ Cat .Since
Fair ⊂ Fair and, if X ∈ Cat , π ∗ X ∈ Fair , we can consider
Fair as acommon generalization of both
Fair and Cat . We now reconcile the notions of weakly globular fair 2-category and of Segalicpseudo-functor through the strictification functor (the latter is as in Section 2.1). Theproof of the following theorem is formally analogous to the one of Theorem 3.16.
Theorem 7.7.
The strictification functor St : Ps [∆ op , Cat ] → [∆ op , Cat ] restricts to a functor St : SegPs [∆ op , Cat ] → Fair
Further, for each H ∈ SegPs [∆ op , Cat ] there is a pseudo-natural transformation St H → H whose components are equivalences of categories.Proof. From [10], to construct the strictification L = St H of the pseudo-functor H weneed to factorize h : T U H → U H as h = gv in such a way that, for each k ∈ ∆ op , h k factorizes as ( T U H ) k v k −→ L k g k −→ ( U H ) k = H k with v k bijective on objects and g k fully and faithful. As explained on [10], g k is in factan equivalence of categories.Since the bijective on objects and the fully faithful functors form a factorizationsystem in Cat , the commutativity of (19) in Lemma 7.3 implies that there are functors d i : L ⇒ L i = 0 , T U H ) / / ∂ ′ (cid:15) (cid:15) ∂ ′ (cid:15) (cid:15) L / / d (cid:15) (cid:15) d (cid:15) (cid:15) H ∂ (cid:15) (cid:15) ∂ (cid:15) (cid:15) ( T U H ) / / L / / H By Lemma 7.3 h k factorizes as( T U H ) k ∼ = ( T U H ) × ( T UH ) s · · ·× ( T UH ) ( T U H ) ( v ...v ) −−−−→ L × L s · · ·× L L g ... g ) −−−−→ H × H s · · ·× H H . Since v and v are bijective on objects, so is ( v . . . v ). Since g , g are fully faithful,so is ( g ... g ). Therefore the above is the required factorization of h k and we concludethat L k ∼ = L × L s · · ·× L L . By [10], g : L → H is a pseudo-natural transformation with g k an equivalence ofcategories for all k ∈ ∆ op . In particular, H ≃ L . Since H is discrete, we have H k ∼ = H × H s · · ·× H H ≃ L × H s · · ·× H L . On the other hand, from above H k ≃ L k ∼ = L × L s · · ·× L L . In conclusion we obtain an equivalence of categories L × L s · · ·× L L ≃ L × H s · · ·× H L ∼ = L × L d s · · ·× L d L . EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 33
Since L → H is a pseudo-natural transformation, from the above the following dia-grams pseudo-commute: L / / / / (cid:15) (cid:15) L (cid:15) (cid:15) H / / / / H L × L L / / (cid:15) (cid:15) L (cid:15) (cid:15) H × H H / / HL × L L / / (cid:15) (cid:15) L (cid:15) (cid:15) L × L L o o (cid:15) (cid:15) H × H H / / H H × H L o o Since H ∈ SegPs [∆ op , Cat ], the bottom maps are equivalences of categories. Thevertical maps are also equivalences of categories since L → H is a levelwise equivalence.It follows that the top maps are also equivalences of categories. This completes theproof that L ∈ Fair . (cid:3) Lemma 7.8.
There is a functor D : Fair → Fair sending -equivalences in Fair to -equivalences in Fair . Further, if X ∈ Fair , DX = X .Proof. For each k ∈ ∆ opmono , define( DX ) k = X d , k = 0 X , k = 1 X × X d k · · ·× X d X , k > . We also define the composite maps (for k ≥ X ∂ i −→ X γ −→ X d X × X d X ν −→ X × X X ∂ i −→ X X × X d k +1 · · ·× X d X ν k +1 −−→ X × X k +1 · · ·× X X ∂ i −→→ X × X k · · ·× X X µ k −→ X × X d k · · ·× X d X (20)Since X ∈ Fair , X ⇒ X has a semi-category structure and ˆ µ k is an injectiveequivalence of categories. Thus, reasoning as in the proof of Theorem 6.4, the maps(20) define a semi-category structure on X ⇒ X d . Let ( DX ) = X and define the composite maps (for k ≥ X ∂ i −→ X γ −→ X d X × X d X ν −→ X × X X ∂ i −→ XX × X d k +1 · · ·× X d X ν k +1 −−→ X × X k +1 · · ·× X X ∂ i −→→ X × X k · · ·× X X ˆ µ k −→ X × X d k · · ·× X d X (21)Since X ∈ Fair , X ⇒ X has a semi-category structure and ˆ µ k is an injectiveequivalence. Thus, reasoning as in the proof of Theorem 6.4, the maps (21) define asemi-category structure on X ⇒ X d . Since X ∈ Fair , there is a semi-functor X × X X / / (cid:15) (cid:15) X (cid:15) (cid:15) / / / / X X × X X / / X / / / / X It follows that the following diagram commutes: X × X d X / / ν (cid:15) (cid:15) X / / / / X / / X d X × X X / / (cid:15) (cid:15) X (cid:15) (cid:15) / / / / X / / X d X × X X / / ˆ µ (cid:15) (cid:15) X / / / / X / / X d X × X d X / / X / / / / X / / X d That is, there is a semi-functor X / / / / (cid:15) (cid:15) X d (cid:15) (cid:15) X / / / / X d By Remark 5.4 to show that DX ∈ Fair it remains to show that the maps X / / / / X d as well as the composition maps X × X d X / / XX × X d X / / X X × X d X o o EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 35 are equivalences of categories. This follows from the fact that X ∈ Fair and from thecommutativity of the diagram X × X d X / / (cid:15) (cid:15) XX × X X : : tttttttttttttt X × X d X / / (cid:15) (cid:15) X X × X d X (cid:15) (cid:15) o o X × X X : : tttttttttttttt X × X X d d ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ From the definitions it is straightforward that for each X ∈ Fair p (1) X = p (1) DX .Further, for each a, b ∈ X d X ( a, b ) = DX ( a, b ) . It follows that D sends 2-equivalence to 2-equivalences. It is straightforward from thedefinition that if X ∈ Fair , DX = X . (cid:3) From fair -categories to weakly globular double categories In this Section we construct the functor R : Fair → Cat . This functor factorsthrough the category of strong Segalic pseudo-functors
SSegPs [∆ op , Cat ] and the func-tor from fair 2-categories to the latter is built in Proposition 8.3.We then prove our main result, Theorem 8.6, stating that the functor R : Fair → Cat and the functor F : Cat → Fair of Section 6 induce an equivalence afterlocalization with respect to the 2-equivalences. The proof of this theorem uses thenotions and results of Section 7.8.1. The functor R . The following lemma and proposition describe some new prop-erties of the category
Fair . They will be used in the proof of Proposition 8.3 in buildingstrong Segalic pseudo-functors from fair 2-categories.
Lemma 8.1.
Let π : ∆ → ∆ be as in (4) . For each k ∈ ∆ and X ∈ Fair there is anequivalence of categories α k : X π ( k ) ⇆ X k : β k (22) such that β k α k = Id .Proof. We distinguish various cases:i) Let k ∈ ∆ mono , then π ( k ) = k and α k = Id.ii) Let k = , so X k = U , X π ( k ) = O . Since X ∈ Fair there is an equivalence ofcategories β k : U → O , with pseudo-inverse α k . Since O is discrete, β k α k = Id. iii) Let k be a link of lenght s , so X k = U × O s · · ·× O U for s ∈ ∆ mono , s ≥ X π ( k ) = O× O s · · ·× O O = O . Since X ∈ Fair there is an equivalence ofcategories β k = ( β , · · · β ), with pseudo-inverse α k = ( α , · · · α ). Therefore β k α k = Id.iv) Let k = so that X k = A× O U and X π ( k ) = X = A× O O . Since pullbacksin Cat over discrete objects are equivalent to pseudo-pullbacks, there is anequivalence of categories β k = (Id , β ) with pseudo-inverse α k = (Id , α ). Byii), β k α k = Id.v) Let k = , so X k = U × O A . This case is completely similar to iv).vi) By the Segal condition, for every other k ∈ ∆, X k is an iterated pullback over O of either A× O U , U × O A , U × O s · · ·× O U for s ∈ ∆ mono , s ≥ A× O r · · ·× O A for r ∈ ∆ mono , r ≥
2, while X π ( k ) is the same iterated pullback over O of A× O O = A , O× O A = A , O× O s · · ·× O O = O and A× O r · · ·× O A . Themap β k is thus given by (Id , β ), ( β , Id), ( β , · · · β ), Id respectively on eachcomponent, with pseudo-inverse (Id , α ), ( α , Id),( α , · · · α ), Id respectively oneach component. By iii), iv) and v), β k α k = Id. (cid:3) Proposition 8.2.
Let f : n → m and f ′ : n ′ → m ′ be maps in ∆ op with πf = πf ′ andlet α k , β k for each k ∈ ∆ be as in Lemma 8.1. Then, if X ∈ Fair , β m X ( f ) α n = β m ′ X ( f ′ ) α n ′ . Proof.
Let m η −→ r ε −→ n , εη = f and m ′ η ′ −→ r ′ ε ′ −→ n ′ , ε ′ η ′ = f ′ be maps in ∆ as inLemma 4.3. By Lemma 4.5 there are commuting diagrams X n X ( ε ) / / X r X n ( m ) t / / s n O O X r ( m ) s r O O X n X ( ε ) γ n A A ✂✂✂✂✂✂✂✂✂✂✂ X n ′ X ( ε ′ ) γ n ′ ^ ^ ❁❁❁❁❁❁❁❁❁❁❁ X r γ r A A ✄✄✄✄✄✄✄✄✄✄✄ X r ′ γ r ′ ] ] ❁❁❁❁❁❁❁❁❁❁❁ (23) EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 37 and X m X r ( m ) w m O O X r γ r < < ②②②②②②②② X ( η ) (cid:15) (cid:15) X r ′ γ r ′ b b ❋❋❋❋❋❋❋❋ X ( η ′ ) (cid:15) (cid:15) X m γ m E E ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ X m ′ γ m ′ Y Y ✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹✹ (24)Let v n : X n → X n ( m ) be a pseudo-inverse to s n with s n v n = Id. Let δ n : X n ( m ) → X n , δ n ′ : X n ( m ) → X n ′ be the pseudo-inverses to γ n , γ n ′ , γ r , γ r ′ respectively, with γ n δ n = Id, γ n ′ δ n ′ = Id, γ r δ r = Id, γ r ′ δ r ′ = Id. Let α n = δ n v n , α n ′ = δ n ′ v n . Then γ n α n = γ n δ n v n = v n γ n ′ α n ′ = γ n ′ δ n ′ v n = v n . (25)Since X ( f ) = X ( η ) X ( ε ), from (23), (24) and (25) we obtain β m X ( f ) α n = w m γ m X ( η ) X ( ε ) α n = w m γ r X ( ε ) α n = w m tγ n α n = w m tv n . Similarly, since X ( f ′ ) = X ( η ′ ) X ( ε ′ ), from (23), (24) and (25) we obtain β m ′ X ( f ′ ) α n ′ = w m γ m ′ X ( η ′ ) X ( ε ′ ) α n ′ = w m γ r ′ X ( ε ′ ) α n ′ = w m tγ n ′ α n ′ = w m tv n . In conclusion β m X ( f ) α n = β m ′ X ( f ′ ) α n ′ = w m tv n . (cid:3) Proposition 8.3.
There is a functor T : Fair → SSegPs [∆ op , Cat ] such that, for each X ∈ Fair , ( T X ) = X , ( T X ) = X and ( T X ) r = X × X r · · ·× X X for r ≥ .Proof. Since X ∈ Fair , X ∈ [∆ op , Cat ]. Note that if k ∈ ∆ op , π ( k ) ∈ ∆ op mono ⊂ ∆ op .By Lemma 2.2 we can therefore use transport of structure on X along the equivalencesof categories (22) to build a pseudo-functor t X ∈ Ps [∆ op , Cat ] with( t X ) k = X π ( k ) for all k ∈ ∆ op . Given f : n → m in ∆ op , the map t f : X π ( n ) → X π ( m ) is given by thecomposite X π ( n ) α n −→ X n X ( f ) −−−→ X m β m −→ X π ( m ) . Given morphisms n f −→ m g −→ s in ∆ op , the 2-cell X π ( n ) X π ( m ) X π ( s ) t f ⇓ t ( gf ) t g is obtained by the following pasting diagram, where ϕ m : α m β m ⇒ Id is the counit inthe adjoint equivalence ( α m , β m ): X π ( n ) X π ( m ) X π ( s ) X n X m X m X st fα n t gα m X ( f ) X ( gf ) β m Id ww(cid:127) X ( g ) β s (26)Given Id n : n → n in ∆ op , the 2-cell X π ( n ) t (Id π ( n ) ) / / α n (cid:15) (cid:15) X π ( n ) X n Id / / X nβ n O O (27)is the identity since β n α n = Id by Lemma 8.1. Note that, where restricted to ∆ op mono , t X is a functor satisfying the strict Segal condition, and if f : n → m is a map in∆ op mono , since f = π ( f ), it is t f = X ( f ).We now construct from t X ∈ Ps [∆ op , Cat ] a pseudo-functor T X ∈ SSegPs [∆ op , Cat ].For each n ∈ ∆ op , ( T X ) n = ( t X ) n . Given f : n → m in ∆ op , choose f : n → m in∆ op such that πf = f and let T X ( f ) = t X ( f ) . We claim that this is well defined. In fact, given f ′ : n ′ → m ′ in ∆ op with πf = πf ′ ,by Proposition 8.2 it is t X ( f ) = β m X ( f ) α n = β m ′ X ( f ′ ) α n ′ = t X ( f ′ ) . Given maps n f −→ m g −→ s in ∆ op , by Lemma 4.3 there are maps n f −→ m g −→ s in ∆ op with π ( f ) = f and π ( g ) = g , π ( gf ) = gf so that T X ( gf ) = t ( X ( g f )). The 2-cell T X ( g ) T X ( f ) ⇒ T X ( gf )is thus as in (26) and similarly for the 2-cell (27).Given maps n f −→ m g −→ s h −→ r in ∆ op , by Lemma 4.3 there are maps n f −→ m g −→ s h −→ r in ∆ op with π ( f ) = f , π ( g ) = g , π ( h ) = h . By construction T X ( hgf ) = t X ( h g f ) T X ( hg ) T X ( f ) = t X ( h g ) t X ( f ) T X ( h ) T X ( gf ) = t X ( h ) t ( X ( g f ) EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 39
Thus, since t is a pseudo-functor, we have the following coherence diagrams T X n T X r T X m T X sT X ( hgf ) T X ( gf ) ⇑ T X ( f ) T X ( g ) ⇑ T X ( h ) ≡ T X n T X r T X m T X sT X ( hgf ) ⇑ T X ( f ) T X ( hg ) T X ( g ) ⇑ T X ( h ) Since the 2-cell (27) is the identity, there are no further coherence axioms to check. Inconclusion, T X ∈ Ps [∆ op , Cat ]. By construction, T X ∈ SSegPs [∆ op , Cat ]. (cid:3) We can now define our second main comparison functor, from fair 2-categories toweakly globular double categories.
Definition 8.4.
Let R : Fair → Cat be the composite
Fair T −→ SSegPs [∆ op , Cat ] St −→ Cat , where T is as in Proposition 8.3 and St is the restriction to SSegPs [∆ op , Cat ] of thefunctor St : SegPs [∆ op , Cat ] → Cat of Theorem 3.16.8.2.
The main result.
We finally reach our main result, Theorem 8.6, establishingthat the functors F : Cat → Fair and R : Fair → Cat induce an equivalenceof categories after localization with respect to the 2-equivalences. We will use thefollowing general fact:
Remark 8.5.
Recall (see Section 2.1) the adjunction St ⊣ JSt : Ps [ C , Cat ] ⇄ [ C , Cat ]where St is the strictification functor and J is the inclusion. Let X ∈ [ C , Cat ] andsuppose there is a pseudo-natural transformation t : Z → J X in Ps [ C , Cat ] such that t c is an equivalence of categories for all c ∈ C . Then by the adjunction St ⊣ J thiscorresponds to a natural transformation w : St Z → X in [ C , Cat ] making the followingdiagram commute Z η / / t $ $ ■■■■■■■■■■■■■■■ J St Z Jw (cid:15) (cid:15) J X
Since η c and t c are equivalences of categories for all c ∈ C , such is w c .The method of proof of Theorem 8.6 is as follows: we show that, for each X ∈ Cat and Y ∈ Fair , there are zig-zags of levelwise equivalences (and thus in particular 2-equivalences) between X and R F X in Cat and between Y and F R Y in Fair . Themain steps are: a) Given X ∈ Cat , there is a levelwise equivalence pseudo-natural transforma-tion in Ps [∆ op , Cat ] given by the composite t F X → F X S ( X ) −−−→ ˜ π ∗ X. This gives rise to a levelwise equivalence pseudo-natural transformation T F X → X in Ps [∆ op , Cat ]. By Remark 8.5, this corresponds to a levelwise equivalencenatural transformation in [∆ op , Cat ] R F X = St T F X → X. This is in particular a 2-equivalence in
Fair between X and R F X .b) Given Y ∈ Fair , there is a levelwise equivalence pseudo-natural transformationin Ps [∆ op , Cat ] given by the composite F R Y S ( R Y ) −−−−−→ ˜ π ∗ R Y → π ∗ R Y = π ∗ St T Y → π ∗ T Y = t Y → Y .
By Remark 8.5 this corresponds to a levelwise equivalence natural transforma-tion in [∆ op , Cat ] St F R Y → Y .
Since F R Y ∈ Fair , it is also F R Y ∈ SegPs [∆ op , Cat ], so that by Theorem7.7
St F R Y ∈ Fair and (see Remark 2.1) there is a levelwise equivalencenatural transformation in [∆ op , Cat ] St F R Y → F R Y . c) From b) there is a zig-zag of 2-equivalences in
Fair F R Y ← St F R Y → Y with F R Y and Y in Fair . Applying the functor D of Lemma 7.8 we obtaina zig-zag of 2-equivalences in Fair F R Y = DF R Y ← DSt F R Y → DY = Y .
Theorem 8.6.
The functors F : Cat ⇄ Fair : R induce an equivalence of categories after localization with respect to the -equivalences Cat / ∼ ≃ Fair / ∼ . Proof.
Let X ∈ Cat . By Proposition 8.3 there is a natural transformation in[∆ op , Cat ] S ( X ) : F X → ˜ π ∗ X which is a levelwise equivalence of categories. Since, by the proof of Proposition 8.3, t is obtained by transport of structure, by Lemma 2.2 there is a pseudo-natural trans-formation in Ps [∆ op , Cat ] t F X → F X which is a levelwise equivalence of categories. Composing it with natural transforma-tion S ( X ) we obtain a pseudo-natural transformation in Ps [∆ op , Cat ] t F X → ˜ π ∗ X (28)which is a levelwise equivalence of categories. EAKLY GLOBULAR DOUBLE CATEGORIES AND WEAK UNITS 41
The same construction used in the proof of Proposition 8.3 to build T F X ∈ Ps [∆ op , Cat ] from t F X , when applied to ˜ π ∗ X ∈ [∆ op , Cat ], yields D X ∈ [∆ op , Cat ](where D X is as in Definition 3.10). Thus from (28) we obtain a pseudo-naturaltransformation in Ps [∆ op , Cat ] T F X → D X (29)which is a levelwise equivalence of categories.By Remark 3.11 there is a pseudo-natural transformation D X → X in Ps [∆ op , Cat ]which is a levelwise equivalence of categories. Composing it with (29) we obtain apseudo-natural transformation T F X → X = J X (30)which is a levelwise equivalence of categories. Applying Remark 8.5 to (30) we obtaina natural transformation in [∆ op , Cat ] R F X = StT F X → X (31)which is a levelwise equivalence of categories. Thus (31) is in particular a 2-equivalencein Cat (see Remark 3.9), so that R F X ∼ = X in Cat / ∼ .The functor π : ∆ op → ∆ op induces a functor π ∗ : Ps [∆ op , Cat ] → Ps [∆ op , Cat ]which restricts to the functor π ∗ : [∆ op , Cat ] → [∆ op , Cat ] of Definition 6.5.Let Y ∈ Fair . We claim that π ∗ T Y = t Y . (32)In fact, for each n ∈ ∆ op , by construction of T and t ,( π ∗ T Y ) n = ( T Y ) π ( n ) = Y π ( n ) = ( t Y ) n . Given f : n → m in ∆ op ,( π ∗ T Y )( f ) = ( T Y )( πf ) = ( t Y )( f ) . Given maps n f −→ m g −→ s in ∆ op the 2-cell( π ∗ T Y )( g f ) = ( T Y )( π ( g f )) = ( t Y )( g f ) ⇒⇒ ( t Y )( g )( t Y )( f ) = ( T Y )( πg ) ◦ ( T Y )( πf ) == ( π ∗ T Y )( g )( π ∗ T Y )( f )is as for t Y . This proves (32).Since, by the proof of Proposition 8.3, t is obtained by transport of structure, byLemma 2.2 there is a pseudo-natural transformation in Ps [∆ op , Cat ] π ∗ T Y = t Y → Y (33)which is a levelwise equivalence of categories. By the properties of the strictificationfunctor (see Section 2.1), there is a pseudo-natural transformation in Ps [∆ op , Cat ] R Y = St T Y → T Y which is a levelwise equivalence of categories. This induces a pseudo-natural transfor-mation in Ps [∆ op , Cat ] π ∗ R Y → π ∗ T Y (34) which is a levelwise equivalence of categories. By Remark 6.7 there is a pseudo-naturaltransformation in Ps [∆ op , Cat ] ˜ π ∗ R Y → π ∗ R Y (35)which is a levelwise equivalence of categories. On the other hand, by Proposition 8.3,there is a pseudo-natural transformation F R Y → ˜ π ∗ R Y (36)which is a levelwise equivalence of categories.Combining (33), (34), (35), (36) we obtain a pseudo-natural transformation in Ps [∆ op , Cat ] F R Y = F St T Y → Y = J X (37)which is a levelwise equivalence of categories.Applying Remark 8.5 to (37) we obtain a natural transformation in [∆ op , Cat ] St F R Y → Y (38)which is a levelwise equivalence of categories.By Remark 7.2, since F St T Y ∈ Fair then F R Y ∈ SegPs [∆ op , Cat ] so, by Theo-rem 7.7,
St F R Y ∈ Fair . Also, by Remark 2.1 there is a natural transformation in[∆ op , Cat ] St F R Y → F R Y (39)which is a levelwise equivalence of categories.By Lemma 5.5, both (38) and (39) are in particular 2-equivalences. Applying thefunctor D of Lemma 7.8 we obtain a zig-zag of 2-equivalences in Fair DF R Y = F R Y ← DSt F R Y → DY = Y .
It follows that Y ∼ = F R Y in Fair / ∼ . (cid:3) As an immediate Corollary, we obtain an equivalence of
Fair and Ta after localiza-tion with respect to the 2-equivalences. Corollary 8.7.
There is an equivalence of categories
Fair / ∼ ≃ Ta / ∼ . Proof.
By Theorem 8.6 there is an equivalence of categories
Fair / ∼ ≃ Cat / ∼ whileby [8, theorem 12.2.6] there is an equivalence of categories Cat / ∼ ≃ Ta / ∼ . Hencethe result. (cid:3)
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