Bi-initial objects and bi-representations are not so different
aa r X i v : . [ m a t h . C T ] S e p BI-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SODIFFERENT tslil clingman AND LYNE MOSER
Abstract.
We introduce a functor V : DblCat h,nps → h,nps extracting from a dou-ble category a 2-category whose objects and morphisms are the vertical morphisms andsquares. We give a characterisation of bi-representations of a normal pseudo-functor F : C op → Cat in terms of double bi-initial objects in the double category el ( F ) ofelements of F , or equivalently as bi-initial objects of a special form in the 2-category V el ( F ) of morphisms of F . Although not true in general, in the special case where the2-category C has tensors by the category = { → } and F preserves those tensors, weshow that a bi-representation of F is then precisely a bi-initial object in the 2-category el ( F ) of elements of F . We give applications of this theory to bi-adjunctions and weightedbi-limits. Introduction
In ordinary category theory, properties of a categorical object are often formulated asquestions of representability of a presheaf . By a presheaf, we mean a functor F : C op → Set ,where C is a category and Set is the category of sets and functions. A representation ofa presheaf comprises the data of an object I ∈ C together with a natural isomorphism C ( − , I ) ∼ = = ⇒ F . In particular, this gives isomorphisms of sets C ( C, I ) ∼ = F C , for each object C ∈ C . A classical theorem, which we shall refer to as the “Representation Theorem”,establishes that a presheaf F has a representation precisely when its category of elements has an initial object; see for example [12, Proposition III.2.2] or [14, Proposition 2.4.8] .This category of elements of F is defined as the slice category {∗} ↓ F , where {∗} denotesthe singleton set.Two main examples of properties that can be rephrased in terms of representability arethe existence of limits and adjoints for a functor. Indeed, asking that a functor F : I → C admits a limit amounts to asking whether the presheaf [ I , C ](∆( − ) , F ) : C op → Set has arepresentation. Therefore, by the Representation Theorem, this is equivalent to requiringthe presence of a terminal object in the slice category ∆ ↓ F of cones over F . Similarly, theexistence of a right adjoint to a functor L : C → D may equivalently be reformulated as theexistence of a representation of the presheaf C ( L − , D ) : C op → Set , for each object D ∈ D , Mathematics Subject Classification.
Key words and phrases.
Bi-representations, (double) bi-initial objects, bi-adjunctions, weighted bi-limits,pseudo-commas. Riehl defines the category of elements by hand along with a projection functor to C , rather than C op ;therefore, representations correspond to terminal objects in this setting. or equivalently of the existence of a terminal object in the slice category L ↓ D , for eachobject D ∈ D .In passing from ordinary categories to 2-categories, we may seek to elevate discussionsof representations of ordinary presheaves to their 2-dimensional counter-parts. By a 2-dimensional presheaf, we mean a normal pseudo-functor F : C op → Cat , where C is a2-category and Cat is the 2-category of categories, functors, and natural transformations.The data of a 2-dimensional representation is once more an object I ∈ C , but when it comesto comparing the categories C ( C, I ) and
F C we may retain the idea that this comparisonis mediated by an isomorphism of categories, or we may require only the presence of anequivalence of categories. The former choice leads to the notion of a 2 -representation , whilethe latter leads to the more general notion of a bi-representation .Recall that an object I in a category C is initial if we have an isomorphism of sets C ( I, C ) ∼ = {∗} for all objects C ∈ C . If we wish to formulate the 2-dimensional definitionanalogously for an object I in a 2-category C , as before we now have the option of retainingthe idea that the universal property should be governed by an isomorphism of categories C ( I, C ) ∼ = , for all objects C ∈ C , where is the terminal category, or instead asking thatthe universal property is governed by an equivalence of categories. The former requirementleads to the notion of a 2 -initial object , while the latter leads to the more general notionof a bi-initial object .Now that the players are ready the game is afoot. The question underpinning the mostgeneral 2-dimensional version of the Representation Theorem is this: Question.
Can bi-representations of a normal pseudo-functor F be characterised as certain bi-initial objects in some -category of elements el ( F ) of F would be the correct setting for an affirmative answer.The 2-category el ( F ) is defined as the pseudo-slice ↓ F , where by pseudo-slicewe mean a relaxation of the slice 2-category where the triangles of morphisms commute upto a general 2-isomorphism, rather than an identity.Although we hate to disappoint the reader, to see that this is not the case we willturn our interest to a specific kind of bi-representation. Generalising ordinary limits, bi-representations of the 2-presheaf [ I , C ](∆( − ) , F ), known as bi-limits , were first introducedby Street in [17, 18] and further studied by Kelly in [9]. The comparatively strongerspecial case – 2-representations of the above 2-presheaf, known as 2 -limits – had previouslybeen introduced, independently, by Auderset [1] and Borceux-Kelly [2], and was furtherdeveloped by Street [16], Kelly [8,9] and Lack in [10]. As el ([ I , C ](∆( − ) , F )) is the oppositeof the pseudo-slice 2-category ∆ ↓ F of cones over F , the question now becomes whetherbi-limits may be characterised as bi-terminal objects in the pseudo-slice 2-category of cones.Unfortunately, as the authors show in [3], such a characterisation is not possible ingeneral. The failure stems from the fact that the data of a bi-limit is not wholly capturedby a bi-terminal object in the pseudo-slice 2-category of cones (see [3, § I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 3
We have thus eliminated our first guess from the possible affirmative answers to ourquestion above. To cast further doubt on any positive resolution, correct characterisationsof 2-limits as some form of 2-dimensional terminal object that are present in the literatureare all phrased in the language of double categories ; such results are explored by Gran-dis [4], Grandis-Par´e [5, 6], and Verity [19]. These results may all be seen to share thefollowing approach: the slice 2-category of cones does not capture enough data to success-fully characterise 2-limits, and so instead more data must be necessarily added in the formof a slice double category of cones. Indeed, in [6] Grandis and Par´e write:“On the other hand, there seems to be no natural way of expressing the2-dimensional universal property of weighted (strict or pseudo) limits byterminality in a 2-category.”The state of the art thus seems to suggest that our question admits no positive answerin general. However, our main contribution in response to Grandis and Par´e above, is asuccessful and purely 2-categorical characterisation of 2-limits as certain 2-terminal objectsin a “shifted” slice 2-category of cones. In fact, we obtain this result as an application ofa purely 2-categorical formulation of a generalisation of the Representation Theorem inthe case of bi-representations. To do this, we extend the results of Grandis, Par´e, andVerity to general bi-representations of normal pseudo-functors, and obtain in this fashion adouble-categorical characterisation of bi-representations in terms of “bi-type” double-initialobjects. From this work and some new methods we are able to extract our results.Let us explore these results in greater detail. Recall that a double category has two sortsof morphisms between objects – the horizontal and vertical morphisms – and 2-dimensionalmorphisms called squares . So as to distinguish double categories from 2-categories, we willalways be careful to name the former by double-struck letters A , B , C , . . . whereas thelatter will always appear as named by bold letters A , B , C , . . .A 2-category A can always be seen as a horizontal double category H A with only trivialvertical morphisms. This construction extends functorially to an assignment on normalpseudo-functors F H F . Associated to each such normal pseudo-functor is a doublecategory of elements el ( F ) given by the pseudo-slice double category ↓↓ H F , where thesepseudo-slices are double-categorical analogues of pseudo-slice 2-categories. Furthermore,we introduce a new notion of double bi-initial objects I in a double category A ; objects I ∈ A for which the projection I ↓↓ A → A is given by an equivalence in the 2-category ofdouble categories, double functors, and horizontal natural transformations. Note that, inthe case of double-initial objects as defined by Grandis and Par´e in [5, § Theorem A.
Let C be a -category, and F : C op → Cat be a normal pseudo-functor. Thefollowing statements are equivalent.(i) The normal pseudo-functor F has a bi-representation ( I, ρ ) .(ii) There is an object I ∈ C together with an object i ∈ F I such that ( I, i ) is doublebi-initial in el ( F ) . t. clingman AND L. MOSER By applying this result to the 2-presheaf [ I , C ](∆( − ) , F ), where F : I → C is a normalpseudo-functor, we derive a generalisation in Corollary 7.22 of results by Grandis, Par´e,and Verity, characterising bi-limits as double bi-terminal objects in the pseudo-slice doublecategory ∆ ↓↓ F . This follows from the fact that the double category el ([ I , C ](∆( − ) , F ))is isomorphic to the horizontal opposite of the pseudo-slice double category ∆ ↓↓ F .We now aim to extract a fully 2-categorical statement from Theorem A above. Forthis, it is enough to characterise double bi-initial objects in a double category A as certain bi-initial objects in some H A of objects, horizontal morphisms, andsquares with trivial vertical boundaries of A . However, the general vertical structure of thedouble category A is not captured by this operation, and therefore the 2-category H A alonedoes not suffice for our purposes. To remedy this issue, we introduce a functor V whichextracts from a double category A a 2-category V A whose objects and morphisms are thevertical morphisms and squares of A , respectively. This captures precisely the additionaldata that was lacking in H A in our application, and allows us to prove the below result. Infact, there is natural embedding of H A into V A and so, with some care, we may leverage V A alone to characterise double bi-initial objects in A . The below appears as Theorem 5.13in the paper. Theorem B.
Let A be a double category, and I ∈ A be an object. The following statementsare equivalent.(i) The object I is double bi-initial in A .(ii) The object I is bi-initial in H A and the vertical identity e I is bi-initial in V A .(iii) The vertical identity e I is bi-initial in V A . As a direct application of this result to the double category of elements el ( F ) of anormal pseudo-functor F : C op → Cat , we obtain our fully 2-categorical characterisationof bi-representations. Note that the underlying horizontal 2-category of el ( F ) is preciselythe 2-category of elements el ( F ) of F , but new here is V el ( F ) which we refer to as the2 -category of morphisms of F , denoted by mor ( F ). Indeed, while the objects in el ( F )are pairs ( C, x ) of an object C ∈ C and an object x ∈ F C , the objects of mor ( F ) arepairs ( C, α ) of an object C ∈ C and a morphism α : x → y in F C , which justifies theterminology. The following result extends Theorem A and appears as the second part ofour main theorem, Theorem 6.8.
Theorem C.
Let C be a -category, and F : C op → Cat be a normal pseudo-functor. Thefollowing statements are equivalent.(i) The normal pseudo-functor F has a bi-representation ( I, ρ ) .(ii) There is an object I ∈ C together with an object i ∈ F I such that ( I, i ) is bi-initialin el ( F ) and ( I, id i ) is bi-initial in mor ( F ) .(iii) There is an object I ∈ C together with an object i ∈ F I such that ( I, id i ) is bi-initialin mor ( F ) . The equivalence of (i) and (iii) in the above theorem gives a satisfying answer to ouroriginal question. In particular, to respond Grandis and Par´e, we specialise the above
I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 5 theorem to the case of bi-limits to see that bi-limits are equivalently certain types of bi-terminal objects in a 2-category whose objects are given by the morphisms of cones –known as modifications – as we will see in Corollary 7.22. Thus the counter-examples of [3]for bi-limits show the presence of mor ( F ) in (ii) is necessary in general.Although the correct characterisation of bi-limits in a 2-category C above depends ontaking morphisms of cones as objects, in the presence of tensors in C these can be simplyseen as cones whose summit is a tensor by the category = { → } . In the case of 2-limits,Kelly observed in [9, §
3] that the presence of tensors by causes the 1-dimensional aspectof the universal property of a 2-limit to imply the 2-dimensional aspect. As a consequence,we showed in [3, Proposition 2.13] that a 2-limit is precisely a 2-terminal object in the slice2-category of cones under such a hypothesis.This result is part of a far more general framework and we shall approach this in parts.A double categorical analogue of tensors by is given by the notion of tabulators ofvertical morphisms; these are defined by Grandis and Par´e in [5, § Theorem D.
Let A be a double category with tabulators, and I ∈ A be an object. Thenthe following statements are equivalent.(i) The object I is double bi-initial in A .(ii) The object I is bi-initial in H A . We now aim to simplify the characterisation of bi-representations given in Theorem Cwhen the 2-category C has tensors by , which can be seen as tabulators in the dou-ble category H C op . For this simplification, we further need the normal pseudo-functor F : C op → Cat to preserve these tensors so that overall the double category of elements el ( F ) admits tabulators. Although we were not able to use the 2-category of elements el ( F ) to give an answer to our question in general, in this special case we may apply Theo-rem D to recover the following verbatim translation of the Representation Theorem to the2-categorical setting, which appears as Theorem 6.14. Theorem E.
Let C be a -category with tensors by , and F : C op → Cat be a normalpseudo-functor which preserves these tensors. The following statements are equivalent.(i) The normal pseudo-functor F has a bi-representation ( I, ρ ) .(ii) There is an object I ∈ C together with an object i ∈ F I such that ( I, i ) is bi-initialin el ( F ) . This applies to the case of bi-limits, and we formulate in Corollary 7.25 a more generalversion of [3, Proposition 2.13]: a bi-limit is precisely a bi-terminal object in the pseudo-slice 2-category of cones when the ambient 2-category admits tensors by . This applicationprovides the promised proof of [3, Proposition 5.5]. In fact these notions are somehow dual, but our applications all involve the horizontal double categoryassociated to the opposite of a 2-category and so tabulators there coincide with tensors by . t. clingman AND L. MOSER While we have only mentioned the case of bi-limits so far, in this paper the differenttheorems characterising bi-representations are first specialised to the case of weighted bi-limits , which were introduced by Street [16] and Kelly [9]. The cone of a weighted limit is ofa special shape, determined by the weight – a normal pseudo-functor W taking values in Cat – and a bi-limit can be seen as a weighted limit with conical weight W = ∆ , i.e., a constantweight at the terminal category. More still, when the weight is conical the pseudo-slice ofcones is isomorphic to the opposite of the pseudo-slice of weighted cones . Since weighted bi-limits can also be seen as bi-representations of a normal pseudo-functor of a special kind,we also obtain characterisations Theorems 7.19 and 7.21 of weighted bi-limits in termsof double bi-initial and bi-initial objects. From these we extract the characterisations ofbi-limits in terms of double bi-terminal and bi-terminal objects mentioned above.Another application of the Representation Theorem is to the existence of a right adjointto a given functor. Going one dimension up, we can define an analogous notion of bi-adjunction between 2-categories C and D . A bi-adjunction comprises the data of a pairof normal pseudo-functors L : C → D and R : D → C together with a pseudo-naturalequivalence D ( L − , − ) ≃ = ⇒ C ( − , R − ). In order to apply our main results to the existenceof a right bi-adjoint to a given normal pseudo-functor, there is first the delicate matter ofreformulating such a question in terms of bi-representations.Theorem 7.3 states that a normal pseudo-functor L : C → D has a right bi-adjoint ifand only if there is a bi-representation of the normal pseudo-functor D ( L − , D ) for eachobject D ∈ D . This shows that the pseudo-naturality of D ( L − , − ) ≃ = ⇒ C ( − , R − ) in oneof the variables is superfluous data, and may always be recovered from merely object-wise information – in analogy with the corresponding result for ordinary adjunctions andrepresentations. Although this result about bi-adjunctions is known or expected, we wereunable to find even a statement of this theorem in the literature. Capitalising on this gapwe provide a proof in Section 7.1 using some cool 2-dimensional Yoneda tricks rather thana direct construction.This formulation of the existence of a right bi-adjoint is then amenable to our theoremsabout bi-representations above and we prove in Theorem 7.11 that L has a right bi-adjointif and only if there is a double bi-initial object in the pseudo-slice double category L ↓↓ D for each object D ∈ D . As before we derive a purely 2-categorical statement by applyingthe functors H and V to L ↓↓ D . The resulting 2-categories are isomorphic to the pseudo-slice 2-category L ↓ D and a “shifted” pseudo-slice 2-category Ar ∗ L ↓ D , whose objects are2-morphisms between LC and D . Finally bi-adjunctions also benefit from the presence oftensors and we prove in Theorem 7.15 that, if the 2-category C has tensors by whichare preserved by L , then L has a right bi-adjoint if and only if there is a bi-initial objectin the pseudo-slice 2-category L ↓ D for each object D ∈ D . This special case givesa straightforward 2-categorical version of the characterisation of the existence of a rightadjoint to an ordinary functor.As we saw throughout the introduction, there are also 2-type versions of the bi-typenotions of considered here. All of the theorems given in this paper may also be proven in Note that tensors by are weighted colimits and that left bi-adjoints preserve those (LBAPWBC). I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 7 this stronger setting; the proofs are predictably less involved as there are less coherenceconditions to check here. For example, Theorem A in this stronger setting can be formu-lated as follows: there is a 2-representation of a normal pseudo-functor F : C op → Cat ifand only if there is a double-initial object in the double category of elements of F , definedhere as the strict slice double category ↓↓ H F . Similarly, Theorem C for the 2-type casewould concern 2-initial objects and stricter versions of el ( F ) and mor ( F ).1.1. Outline.
The paper is organised as follows. Sections 2 and 3 present the setting of 2-categories and double categories in which we will be working and the functors relating thesetwo settings. Several notions of equivalences between two 2-categories, and respectivelybetween two double categories, are also studied in these sections. Then Sections 4 and 5introduce pseudo-comma objects and bi-initiality, as well as many results comparing the2-categorical notions to their double categorical analogues. Finally, Section 6 contains themain results characterising bi-representations in terms of bi-initiality, and Section 7 studiestheir applications to bi-adjunctions and weighted bi-limits.We now give a more detailed outline of each section. We first introduce in Section 2 the2-categories nps of 2-categories, normal pseudo-functors, and pseudo-natural transfor-mations, and
DblCat h,nps of double categories, normal pseudo-double functors, and hori-zontal pseudo-natural transformations. A reader already familiar with these notions couldskip ahead to the next section. In Section 3 we first recall the definitions of the horizontalembedding functor H of 2-categories into double categories, and its right adjoint H whichextracts from a double category its underlying horizontal 2-category. We further introducethe functor V , which extracts from a double category a 2-category whose objects and mor-phisms are the vertical morphisms and squares, respectively. We then define the notionsof pseudo-equivalences , as equivalences in the 2-category nps , and horizontal pseudo-equivalences , as equivalences in the 2-category DblCat h,nps . We further introduce anothernotion of equivalences of double categories, that of double pseudo-equivalence , as the nor-mal pseudo-double functors whose images under H and V are part of a pseudo-equivalence.We then compare this last notion to that of horizontal pseudo-equivalences.In Section 4, we recall the definitions of pseudo-comma 2-categories, and pseudo-commadouble categories, and describe the data explicitly in the case of pseudo-slices. Additionallywe show that the functors H and V preserve these pseudo-type comma objects. Then,in Section 5, an object in a 2-category (double category) is defined to be (double) bi-initial when the projection from the pseudo-slice under this object is a (double) pseudo-equivalence. Although this might seem too weak a requirement, we prove the surprisingresult that an object is double bi-initial as defined above if and only if the projection issplit surjective on objects, and fully faithful on horizontal morphisms, vertical morphisms,and squares; and similarly so in the 2-categorical case. Note that this results reproducesthe usual notion of bi-initial objects for 2-categories. The main result of this sectioncharacterises a double bi-initial object in a double category through the bi-initiality of itsimages under both H and V , or equivalently only under V . In the case where the ambientdouble category admits tabulators, this we improve our main result to the following: anobject is double bi-initial if and only if its image under H is bi-initial. t. clingman AND L. MOSER In Section 6, we introduce bi-representations of a normal pseudo-functor F : C op → Cat and prove our main theorem characterising bi-representations as double bi-initial objects inthe double category of elements el ( F ), or equivalently as objects which are simultaneouslybi-initial in the 2-category of elements el ( F ) and the 2-category of morphisms mor ( F ).Moreover, as we show, the 2-category of morphisms mor ( F ) itself captures enough datato successfully characterise bi-representations. In particular, when C admits tensors by and F preserves them, we prove that the double category of elements has tabulators. Inthis case we recover a 2-categorical analogue of the ordinary Representation Theorem: abi-representation of F is precisely a bi-initial object in the 2-category of elements el ( F ).Finally, Section 7 is devoted to applying our main results to the case of bi-adjunctionsand weighted bi-limits. While weighted bi-limits are defined as bi-representations, it is notthe case for bi-adjunctions. Therefore, we first need to prove that bi-adjunctions are equiv-alently object-wise bi-representations. We also formulate the obtained characterisationsfor weighted bi-limits in the special case of (conical) bi-limits, in order to make contactwith the counter-examples of [3].1.2. Acknowledgements.
This work began when both authors were at the MathematicalSciences Research Institute in Berkeley, California, during the Spring 2020 semester. Thefirst-named author benefited from support by the National Science Foundation under GrantNo. DMS-1440140, while at residence in MSRI. The second-named author was supportedby the Swiss National Science Foundation under the project P1ELP2 188039. The first-named author was additionally supported by the National Science Foundation grant DMS-1652600, as well as the JHU Catalyst Grant.2.
Background on -categories and double categories To state and prove our main result, Theorem 6.8 below, we will make use of the lan-guages of 2-categories and of double categories. In particular we will employ the notionsof normal pseudo-functor, pseudo-natural transformation, modification, as well as hori-zontal double categorical counterparts to these notions – double functors and horizontalnatural transformations which exhibit pseudo-type behaviour in the horizontal direction.To cement terminology and familiarise ourselves with these notions we will briefly recallthe 2-categorical and double categorical concepts at issue in Sections 2.1 and 2.2 below.Readers comfortable with these definitions should skip ahead to Section 3.2.1. 2 -categories.
Recall that a 2-category is a category enriched in categories. It com-prises the data of objects and hom-categories between each pair of objects, together with a horizontal composition operation. The objects of the hom-categories are called morphisms ,the morphisms therein are called 2 -morphisms , and the composition operation therein iscalled vertical composition of 2-morphisms.Morphisms between 2-categories which preserve all the 2-categorical structure strictlyare called 2-functors. However, in this paper, we consider the more general notion ofmorphisms of 2-categories, namely normal pseudo-functors . I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 9
Definition 2.1.
Given 2-categories A and B , a pseudo-functor ( F, φ ) : A → B comprisesthe data of(i) an assignment on objects A ∈ A F A ∈ B ,(ii) functors F : A ( A, A ′ ) → B ( F A, F A ′ ) for each pair of objects A, A ′ ∈ A ,(iii) 2-isomorphisms φ A : id F A ∼ = = ⇒ F id A in B for each object A ∈ A , called unitors ,(iv) 2-isomorphisms φ a,a ′ : ( F a ′ )( F a ) ⇒ F ( a ′ a ) in B for each pair of composable mor-phisms a : A → A ′ and a ′ : A ′ → A ′′ in A , called compositors ,such that these data satisfy naturality, associativity, and unitality conditions. For detailssee, for example, [7, Definition 4.1.2].If, for all A ∈ A , the unitor φ A is given by the identity 2-morphism id id F A , we say thatthe pseudo-functor (
F, φ ) is normal . Notation . We denote by nps the category of 2-categories and normal pseudo-functors.This category nps can be upgraded into a 2-category in which the 2-morphisms aregiven by the following.
Definition 2.3.
Given pseudo-functors
F, G : A → B , a pseudo-natural transforma-tion α : F ⇒ G comprises the data of(i) morphisms α A : F A → GA in B for each object A ∈ A ,(ii) 2-isomorphisms α a : ( Ga ) α A ∼ = = ⇒ α A ′ ( F a ) in B for each morphism a : A → A ′ in A as depicted below. F A GAF A ′ GA ′ α A α A ′ F a Gaα a ∼ = such that the 2-morphisms α a above are natural with respect to 2-morphisms in A , andcompatible with the compositors and unitors of F and G . For details see, for example, [7,Definition 4.2.1].If, for all morphisms a : A → A ′ in A , the 2-isomorphism component α a is an identity,i.e., ( Ga ) α A = α A ′ ( F a ), then we say that α is 2 -natural . Remark . As a consequence of the compatibility with the unitors above, a pseudo-natural transformation α : F ⇒ G for which F and G are both normal automaticallysatisfies α id A = id α A for all A ∈ A .As every pseudo-functors is isomorphic to a normal one by [11, Proposition 5.2], wechoose to simplify our arguments by forgoing the extra data and coherence associated tothe former class by working only with normal pseudo-functors. Notation . We denote by nps the 2-category of 2-categories, normal pseudo-functors,and pseudo-natural transformations.
The category nps is cartesian closed. Its internal homs have as objects and mor-phisms normal pseudo-functors and pseudo-natural transformations, while the 2-morphismsare the modifications defined next.
Definition 2.6.
Given pseudo-natural transformations α, β : F ⇒ G , a modification Γ : α β comprises the data of 2-morphisms Γ A : α A ⇒ β A in B for each object A ∈ A which are compatible with the 2-isomorphism components of α and β . For details see, forexample, [7, Definition 4.4.1]. Definition 2.7.
Let A and B be 2-categories. The internal hom Ps ( A , B ) in nps is the 2-category whose objects are normal pseudo-functors from A to B , morphisms arepseudo-natural transformations, and 2-morphisms are modifications. Proposition 2.8.
There are isomorphisms of sets nps ( A × B , C ) ∼ = nps ( A , Ps ( B , C )) natural in A , B , and C in nps . Moreover, these isomorphisms of sets extend naturallyto isomorphisms of -categories Ps ( A × B , C ) ∼ = Ps ( A , Ps ( B , C )) . Proof.
One can take A to be the 2-category free on an object, free on a morphism,and Σ free on a 2-morphism, respectively, to see that the objects, morphisms, and 2-morphisms of Ps ( B , C ) are precisely the ones described in Definition 2.7. Note that, forexample, a pseudo-natural transformation between normal pseudo-functors B → C asdefined in Definition 2.3 is equivalently a normal pseudo-functor × B → C . The firstisomorphism of sets follows then from the facts that any 2-category can be obtained as acolimit of , , and Σ , and that the product preserves these colimits.The second isomorphism of 2-categories follows from the Yoneda lemma applied to thefollowing sequence of isomorphisms of sets nps ( D , Ps ( A × B , C )) ∼ = nps ( D × A × B , C ) ∼ = nps ( D × A , Ps ( B , C )) ∼ = nps ( D , Ps ( A , Ps ( B , C ))) , which holds naturally in D ∈ nps . (cid:4) Double categories.
In addition to the 2-dimensional concepts above, we will makemuch use of the possibly less familiar notions of double categories and their morphisms.To prepare for this, invite the reader to join us in recalling some of the early definitions.
Definition 2.9. A double category A comprises the data of(i) objects A, A ′ , B, B ′ , . . . ,(ii) horizontal morphisms a : A → A ′ ,(iii) vertical morphisms u : A B ,(iv) squares α with both horizontal and vertical sources and targets, written inline as α : ( u ab u ′ ) or drawn as I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 11
AB A ′ B ′ u u ′ abα ,(v) horizontal and vertical identity morphisms for each object A , written id A : A = A and e A : A A respectively,(vi) a horizontal identity square for each vertical morphism u and a vertical identitysquare for each horizontal morphism f , written respectively as AB AB u u id A id B id u AA A ′ A ′ e A e A ′ aa e a (vii) a horizontal composition operation on horizontal morphisms and squares along ashared vertical boundary,(viii) a vertical composition operation on vertical morphisms and squares along a sharedhorizontal boundary,such that the composition operations are all appropriately associative and unital, and suchthat horizontal and vertical composition of squares obeys the interchange law. We directthe reader to [4, Definition 3.1.1] for details. Remark . Note that a 2-category A can be seen as a horizontal double category H A ,with only trivial vertical morphisms; see Definition 3.1. Dually, we can also see a 2-category A as a vertical double category V A .The horizontal embedding is preferred in this document, as it corresponds to the inclusionof 2-categories seen as internal categories to Cat whose category of objects is discrete, intogeneral internal categories to
Cat , which are precisely the double categories. This inclusionitself agrees with the inclusion
Cat = IntCat ( Set ) IntCat ( Cat ) =
DblCat arising from
Set Cat , when categories are seen as 2-categories with only trivial 2-morphisms.Much as in the case of 2-categories above, we will be interested not in (strict) doublefunctors, which preserve the double categorical structure strictly, but in certain pseudo-type ones. As we choose here to see 2-categories as horizontal double categories, in orderto extend this assignment on objects to a functor from nps , we need to require thatour pseudo-double functors are pseudo in the horizontal direction.
Definition 2.11.
Given double categories A and B , a (horizontally) pseudo-doublefunctor ( F, φ ) : A → B comprises the data of(i) assignments of objects A , horizontal morphisms a : A → A ′ , vertical morphisms u : A B , and squares α : ( u ab u ′ ) in A to objects F A , horizontal morphisms
F a : F A → F A ′ , vertical morphisms F u : F A F B , and squares
F α : (
F u
F aF b
F u ′ )in B respectively, (ii) for each A ∈ A , a vertically invertible unitor square of B of the form F AF A F AF A id FA F id A φ A ∼ = ,(iii) for each pair of composable horizontal morphisms a : A → A ′ and a ′ : A ′ → A ′′ in A , a vertically invertible compositor square of B of the form F A F A ′ F A F A ′′ F A ′′ F a F a ′ F ( a ′ a ) φ a,a ′ ∼ = ,such that(1) vertical compositions of vertical morphisms and squares, as well as vertical identi-ties, are preserved strictly,(2) the unitor squares are natural with respect to vertical morphisms of A ,(3) the compositor squares are natural with respect to vertical composition by squaresof A ,(4) the compositor squares are associative and unital with respect to the unitor squaresIf, for all A ∈ A , the unitor square φ A is given by the vertical identity square e id F A , wesay that the pseudo-double functor (
F, φ ) is normal . Notation . We denote by
DblCat h,nps the category of double categories and normalpseudo-double functors.We direct the reader to [4, Definition 3.5.1] for a full elaboration of these conditionsfor lax-type double functors – though we have interchanged the vertical and horizontaldirections by comparison.As in the 2-categorical case, the category
DblCat h,nps may be upgraded to a 2-categorywhose 2-morphisms are given by the following.
Definition 2.13.
Given pseudo-double functors
F, G : A → B , a horizontal pseudo-natural transformation α : F ⇒ G comprises the data of(i) horizontal morphisms α A : F A → GA for each A ∈ A ,(ii) squares α u : ( F u α A α B Gu ) for each vertical morphism u : A B of A ,(iii) vertically invertible squares F A GA GA ′ F A F A ′ GA ′ α A GaF a α A ′ α a ∼ = I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 13 for each horizontal morphism a : A → A ′ ,such that the squares α u are coherent with respect to vertical composition and identities,and together with the squares α a and the compositors and unitors of F and G satisfyhorizontal conditions of naturality and unitality. For a full expansion of these conditions,see [4, Definition 3.8.2] – though note again that our horizontal and vertical directions havebeen interchanged.If, for all horizontal morphisms a , the square component α a is an identity, we call α a horizontal natural transformation . Remark . As a consequence of the axioms, a horizontal pseudo-natural transformation α : F ⇒ G for which F and G are normal is such that the vertically invertible square α id A is given by the vertical identity square e α A for all A ∈ A . Notation . We denote by
DblCat h,nps the 2-category of double categories, normalpseudo-double functors, and horizontal pseudo-natural transformations.The category
DblCat h,nps is also cartesian closed. Its internal homs have as objectsand horizontal morphisms, the normal pseudo-double functors and the horizontal pseudo-natural transformations. The vertical morphisms and squares are the vertical natural trans-formations and modifications . We will not have use of the details of such notions, and referthe curious reader to [4, Definitions 3.8.1 and 3.8.3] for these. As ever, we caution thereader that our horizontal and vertical directions are interchanged.
Definition 2.16.
Let A and B be double categories. The internal hom P s( A , B ) in DblCat h,nps is the double category whose objects are normal pseudo-double functors from A to B , horizontal morphisms are horizontal pseudo-natural transformations, vertical mor-phisms are vertical (strict-)natural transformations, and squares are modifications. Proposition 2.17.
There are isomorphisms of sets
DblCat h,nps ( A × B , C ) ∼ = DblCat h,nps ( A , P s( B , C )) natural in A , B , and C in DblCat h,nps . Moreover, these isomorphisms of sets extend natu-rally to isomorphisms of double categories P s( A × B , C ) ∼ = P s( A , P s( B , C )) . Proof.
One can take A to be the double category free on an object, H free on a horizontalmorphism, V free on a vertical morphism, and H × V free on a square, respectively,to see that the objects, horizontal morphisms, vertical morphisms, and squares of P s( B , C )are precisely the ones described in Definition 2.16. Note that, for example, a horizontalpseudo-natural transformation between normal pseudo-double functors B → C as definedin Definition 2.13 is equivalently a normal pseudo-double functor H × B → C . The firstisomorphism of sets follows then from the fact that any double category can be obtainedas a colimit of , H , V , and H × V , and that the product preserves these colimits. The second isomorphism of double categories follows from the Yoneda lemma applied tothe following sequence of isomorphisms of sets
DblCat h,nps ( D , P s( A × B , C )) ∼ = DblCat h,nps ( D × A × B , C ) ∼ = DblCat h,nps ( D × A , P s( B , C )) ∼ = DblCat h,nps ( D , P s( A , P s( B , C ))) , which holds naturally in D ∈ DblCat h,nps . (cid:4) The functor V and pseudo-equivalences A 2-category A can be seen as a horizontal double category H A with only trivial verticalmorphisms. This construction has a right adjoint, which extracts from a double category A its underlying horizontal 2-category H A of objects, horizontal morphisms, and squareswith trivial vertical boundaries. Another 2-category V A that can be extracted from adouble category A has as objects the vertical morphisms of A , as morphisms the squares of A , and 2-morphisms as described in Definition 3.4. These 2-categories H A and V A allowone to retrieve most of the structure of the double category A , except for composition ofvertical morphisms.After introducing the functors H , H , and V in Section 3.1, we study in Section 3.2the relation between equivalences in the 2-categories nps and DblCat h,nps , which wecall pseudo-equivalences of 2-categories and horizontal pseudo-equivalences of double cat-egories, respectively. We further introduce another notion of equivalences between doublecategories, that of double pseudo-equivalences which correspond to the normal pseudo-double functors whose images under H and V are pseudo-equivalences of 2-categories. Weshow that a normal pseudo-double functor mediates a double pseudo-equivalence if it is apart of a horizontal pseudo-equivalence, but that the converse does not hold in general.These notions find use in Section 5.2 where the double functors considered are of a spe-cial form. We will show there that, for these special double functors, the two notions areequivalent.3.1. The functors H , H, and V . Let us first introduce the horizontal full embeddingfunctor from nps to DblCat h,nps . Definition 3.1.
We define a functor H : nps → DblCat h,nps which sends a 2-category A to the horizontal double category H A with the same objects as A , horizontal morphismsgiven by the morphisms of A , only trivial vertical morphisms, and squares α : (e A ab e A ′ )given by the 2-morphisms α : a ⇒ b of A .Given a normal pseudo-functor F : A → B , the induced normal pseudo-double functor H F : H A → H B acts as F does on the corresponding data, and respects vertical identi-ties. The compositor vertically invertible squares of H F are the ones corresponding to thecompositor 2-isomorphisms of F .The functor H has a right adjoint, given by the following functor. Definition 3.2.
The functor H : DblCat h,nps → nps sends a double category A to its underlying horizontal -category H A with the same objects as A , morphisms given I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 15 by the horizontal morphisms of A , and 2-morphisms α : a ⇒ b given by the squares in A of the form AA A ′ A ′ abα .Given a normal pseudo-double functor F : A → B , the induced normal pseudo-functor H F : H A → H B acts as F does on the corresponding data, and the data of its compositor2-isomorphisms are given by the compositor squares of F . Proposition 3.3.
The functors H : nps → DblCat h,nps and H : DblCat h,nps → nps form an adjunction H ⊣ H such that the unit η : id nps ⇒ H H is an identity.Proof. Let C be a 2-category, and A be a double category. By specialising Definition 2.11to the case where the source is a double category with only trivial vertical morphisms, wesee that normal pseudo-double functors H C → A correspond precisely to normal pseudo-functors C → H A , i.e., we have an isomorphism of sets DblCat h,nps ( H C , A ) ∼ = nps ( C , H A ) , natural in C and A . Moreover, a straightforward computation shows that H H C = C . (cid:4) We want to extract another 2-category from a double category which contains the dataof all vertical morphisms and squares, and this is done through the following functor
DblCat h,nps → nps . In [13, Definition 2.10], the second-named author, Sarazola, andVerdugo give a similar definition but in a setting where the morphisms of 2-categories anddouble categories are strict . Under the inclusion of the appropriate subcategories into ourweaker setting, our functor may be seen to restrict to theirs.Although this functor appeared chronologically prior in the work of [13], the originalmotivation to isolate and treat its definition was our Theorem 5.13 below. Indeed, as weshall see in Remark 5.14, from this context the below definition naturally emerges. Definition 3.4.
Let V denote the free double category on a vertical morphism. We definethe functor V : DblCat h,nps → nps to be the composite DblCat h,nps P s( V , − ) −−−−−→ DblCat h,nps H −→ nps . In particular, it sends a double category A to the 2-category V A whose(i) objects are the vertical morphisms of A ,(ii) morphisms α : u → u ′ are squares in A of the form AB A ′ B ′ u u ′ abα , (iii) 2-morphisms ( σ , σ ) : α ⇒ α ′ are pairs of squares σ : (e A aa ′ e A ′ ) and σ : (e B bb ′ e B ′ )satisfying the following pasting equality. AA A ′ A ′ B B ′ aa ′ σ u u ′ b ′ α ′ B = B B ′ B ′ bb ′ σ A A ′ u u ′ aα Remark . The functor V : nps → DblCat h,nps is also a right adjoint since it is thecomposite of two right adjoints H and P s( V , − ), and its left adjoint is given by H ( − ) × V .As we saw in Section 2, the categories nps and DblCat h,nps can be extended to 2-categories by adding pseudo-natural transformations and horizontal pseudo-natural trans-formations, respectively. The functors H , H , and V defined above then extend to 2-functors. Proposition 3.6.
The functors H , H , and V extend to -functors H : nps → DblCat h,nps and H , V : DblCat h,nps → nps . Proof.
Comparing Definition 2.3 and Definition 2.13, the data of a pseudo-natural trans-formation α : F ⇒ G in nps evidently gives the data of a horizontal pseudo-naturaltransformation H α : H F ⇒ H G in DblCat h,nps . One can verify that this assignment extends H to a 2-functor.Given a horizontal pseudo-natural transformation α : F ⇒ G in DblCat h,nps , by forget-ting the square components of α associated to vertical morphisms, we get a pseudo-naturaltransformation H α : H F ⇒ H G in nps . One can verify that this assignment extends H to a 2-functor.Since P s( A , − ) is also a 2-functor for any double category A , we conclude that V is a2-functor as a composite of two 2-functors. (cid:4) Remark . Although we will have no use for the fact, it is the case that the adjunctions H ⊣ H and H ( − ) × V ⊣ V extend to 2-adjunctions.3.2. Notions of equivalences.
We now introduce the notion of a pseudo-equivalence . Itis defined as an equivalence in the 2-category nps . More precisely:
Definition 3.8.
Let A and B be 2-categories. A pseudo-equivalence between A and B comprises the data of normal pseudo-functors F : A → B and G : B → A and pseudo-natural isomorphisms η : id A ∼ = = ⇒ GF and ε : F G ∼ = = ⇒ id B . Remark . Note that, a bi-equivalence between 2-categories is defined instead to com-prise the data of normal pseudo-functors F : A → B and G : B → A and pseudo-natural I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 17 equivalences η : id A ≃ = ⇒ GF and ε : F G ≃ = ⇒ id B . Therefore, a pseudo-equivalence is a specialcase of a bi-equivalence. A pseudo-equivalence can be characterised as a split essentially surjective on objects, splitessentially full on morphisms, and fully faithful on 2-morphisms normal pseudo-functor.This is given more precisely by the following Whitehead Theorem.
Theorem 3.10 (Whitehead Theorem for pseudo-equivalences) . Let A and B be -catego-ries. A normal pseudo-functor F : A → B is part of a pseudo-equivalence if and only if(i) for every object B ∈ B , there is a chosen object A ∈ A together with a chosenisomorphism F A ∼ = −→ B in B ,(ii) for every morphism b : F A → F A ′ in B , there is a chosen morphism a : A → A ′ in A together with a chosen -isomorphism F a ∼ = = ⇒ b ,(iii) for every -morphism β : F a ⇒ F b in B , there is a unique -morphism α : a ⇒ b in A such that F α = β .Proof. By the Whitehead Theorem for bi-equivalences (see [7, Theorem 7.4.1]), we havethat a normal pseudo-functor F is part of a bi-equivalence, as defined in Remark 3.9,if and only if F is split surjective on objects up to equivalence, split essentially full onmorphisms, and fully faithful on squares. By requiring further that the data η and ε ofthe bi-equivalence are pseudo-natural isomorphisms, this changes the split surjectivity onobjects from “up to equivalence” to “up to isomorphism”. Therefore, we have that F is partof a pseudo-equivalence if and only if F is split surjective on objects up to isomorphism,split essentially full on morphisms, and fully faithful on 2-morphisms. (cid:4) Remark . It follows from the Whitehead Theorem for (1-)categories that (ii) and (iii)are equivalent to F inducing an equivalence on hom-categories. Therefore, we have that anormal pseudo-functor F : A → B is part of a pseudo-equivalence if and only if F is splitessentially surjective on morphisms and F induces an equivalence on hom-categories A ( A, A ′ ) F −→ B ( F A, F A ′ ) . In the double categorical setting, we similarly define a horizontal pseudo-equivalence asan equivalence in the 2-category
DblCat h,nps . More precisely:
Definition 3.12.
Let A and B be double categories. A horizontal pseudo-equivalence between A and B comprises the data of normal pseudo-double functors F : A → B and G : B → A and horizontal pseudo-natural isomorphisms η : id A ∼ = = ⇒ GF and ε : F G ∼ = = ⇒ id B .As we show in Corollary 3.15, horizontal pseudo-equivalences are in particular hor-izontally split essentially surjective on objects, split essentially full on horizontal mor-phisms, split essentially surjective on vertical morphisms, and fully faithful on squares.Unlike pseudo-equivalences (see Theorem 3.10), they are however not characterized bythese properties, as we show in Example 3.16. We therefore introduce a notion of doublepseudo-equivalences as the normal pseudo-double functors satisfying these properties. We could therefore have called pseudo-equivalences instead ‘1+ bi2 -equivalences’ because their definitionis half-way point between 2-type and bi-type equivalences, and = 1 + bi2 . Definition 3.13.
Let A and B be double categories. A normal pseudo-double functor F : A → B mediates a double pseudo-equivalence if(i) for every object B ∈ B , there is a chosen object A ∈ A together with a chosenhorizontal isomorphism F A ∼ = −→ B in B ,(ii) for every horizontal morphism b : F A → F A ′ in B , there is a chosen horizontalmorphism a : A → A ′ together with a chosen vertically invertible square in B of theform F A F A ′ F A F A ′ F ab ∼ = ,(iii) for every vertical morphism v : B D in B , there is a chosen vertical morphism u : A C in A together with a chosen horizontally invertible square in B of theform F A BF C D
F u v ∼ = ∼ = ∼ = ,(iv) for every square β in B of the form F A F A ′ F C F C ′ F u F u ′ F aF cβ ,there is a unique square α : ( u ac u ′ ) in A such that F α = β .These double pseudo-equivalences can actually be characterised as the normal pseudo-double functors whose images under H and V are pseudo-equivalences. Proposition 3.14.
Let A and B be double categories. A normal pseudo-double functor F : A → B mediates a double pseudo-equivalence if and only if the normal pseudo-functors H F : H A → H B and V F : V A → V B are pseudo-equivalences of -categories.Proof. This result is proven in [13] for double bi-equivalences (see [13, Definition 3.5]).Although the context there involves strict double functors and 2-functors, the proof goesthrough unchanged for normal pseudo-double functors and normal pseudo-functors – thecompositors play no role. First note that, by definition, we have that F is a double pseudo-equivalence if and only if F is a double bi-equivalence which is split surjective on objectsand on vertical morphisms up to isomorphism (instead of up to equivalence). Then, byapplying [13, Proposition 3.10] with this further restriction, we can see that F is sucha double bi-equivalence if and only if H F and V F are bi-equivalences which are split I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 19 surjective on objects up to isomorphism. But this holds if and only if H F and V F arepseudo-equivalences, as noted in the proof of Theorem 3.10. (cid:4) As a corollary, we obtain that a horizontal pseudo-equivalence is in particular a doublepseudo-equivalence.
Corollary 3.15.
Let A and B be double categories. If a normal pseudo-double functor F : A → B is part of a horizontal pseudo-equivalence, then it mediates a double pseudo-equivalence.Proof. Let (
F, G, η, ε ) be a horizontal pseudo-equivalence between double categories. By ap-plying the 2-functors H : DblCat h,nps → nps and V : DblCat h,nps → nps to the data( F, G, η, ε ), we obtain pseudo-equivalences ( H F, H G, H η, H ε ) and ( V F, V G, V η, V ε ) be-tween the induced 2-categories. This shows that H F and V F are both pseudo-equivalences,and therefore F is a double pseudo-equivalence by Proposition 3.14. (cid:4) However, the converse does not generally hold, as we see now.
Example 3.16.
We give an example of a double functor which mediates a double pseudo-equivalence but is not a horizontal pseudo-equivalence.Let A be the double category generated by the following data:(i) four objects: A , B , B ′ , and C ,(ii) one horizontal isomorphism: f : B ∼ = −→ B ′ ,(iii) three vertical morphisms: u : A B , v : B ′ C , and w : A C ,(iv) no non-trivial squares.And let B be the double category free on two composable vertical morphisms x : 0 1and y : 1 2. In pictures, we have A AB B ′ CC ∼ = fu vw A = 210 xy B =Let F : A → B be the double functor sending(i) the objects A B, B ′ C f, f -1 id ,(iii) the vertical morphisms u x ; v y ; and w yx .Then F is surjective on objects and vertical morphisms, and fully faithful on horizontalmorphisms and squares. In particular, it is a double pseudo-equivalence.In order to obtain a contradiction, suppose that F is a horizontal pseudo-equivalence.Then there is a normal pseudo-double functor G : B → A and horizontal pseudo-naturalisomorphisms η : id A ∼ = = ⇒ GF and ε : F G ∼ = = ⇒ id B . In particular, to give the data of ε , we need to give horizontal isomorphisms in B ε : F G (0) ∼ = −→ , ε : F G (1) ∼ = −→ , and ε : F G (2) ∼ = −→ . Since B has only trivial horizontal morphisms, we must have F G (0) = 0,
F G (1) = 1, and
F G (2) = 2. Therefore, we have G (0) = A , G (1) ∈ { B, B ′ } , and G (2) = C . To define G onvertical morphisms, we need to set what the images in A of the vertical morphisms x and y are: G (0) = AG (1) Gx G (1) G (2) = C Gy By setting G (1) = B , there is no vertical morphism in A as depicted above right, and, bysetting G (1) = B ′ , there is no vertical morphism in A as depicted above left. Therefore,such an inverse G cannot exist. Remark . A result in the vein of [4, Theorem 4.4.5] applies to our weaker setting:when the double categories considered are “horizontally invariant” (see [4, Theorem andDefinition 4.1.7]), then one can show that a normal pseudo-double functor between themis part of a horizontal pseudo-equivalence if and only if it mediates a double pseudo-equivalence. 4.
Pseudo-comma -dimensional categories We now introduce pseudo-comma double categories and pseudo-comma 2-categories, andshow that they are related through the functors H , V : DblCat h,nps → nps . We treatthese objects in general so that we may later variously specialise the theory to pseudo-slicesboth over and under objects. We will use these results for the purposes of comparing doublebi-initial objects in a double category with bi-initial objects in the induced 2-categoriesobtained by applying H and V in Section 5, as well as for computing the double categoriesof elements in the case of bi-adjunctions and weighted bi-limits in Section 7.Let us first define the pseudo-comma double category of a cospan of normal pseudo-double functors. With an eye to our applications of this theory in Sections 5 and 7, wethen give a more explicit description of the data in a pseudo-slice double category: apseudo-comma where one of the double categories involved is terminal. Definition 4.1.
Let G : C → A and F : B → A be normal pseudo-double functors. The pseudo-comma double category G ↓↓ F of G and F is defined as the following pullbackin DblCat h,nps , G ↓↓ F C × B A × AP s( H , A ) Π (
G, F ) ( s, t ) I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 21 where H is the free double category on a horizontal morphism and P s( − , − ) is the internalhom described in Definition 2.16. Remark . Note that Π : G ↓↓ F → C × B is a strict double functor. Remark . We give an explicit description of the pseudo-comma double category in thecase where C = is the terminal category and G = I : → A is an object in A . This isthe double category I ↓↓ F , called pseudo-slice double category , whose(i) objects are pairs ( B, f ) of an object B ∈ B and a horizontal morphism f : I → F B in A ,(ii) horizontal morphisms ( b, ψ ) : ( B, f ) → ( B ′ , f ′ ) comprise the data of a horizontalmorphism b : B → B ′ in B and a vertically invertible square ψ in A of the form II F B F B ′ F B ′ f ′ f F bψ ∼ = ,(iii) vertical morphisms ( u, γ ) : ( B, f ) (
C, g ) comprise the data of a vertical morphism u : B C in B and a square γ in A of the form II F CF B
F ufgγ ,(iv) squares (
B, f )( C, g ) ( C ′ , g ′ )( B ′ , f ′ ) ( u, γ ) ( u ′ , γ ′ )( b, ψ )( c, ϕ ) β comprise the data of a square β : ( u bc u ′ ) in B such that the following pastingequality holds in A . II F B F B ′ F B ′ f ′ f F bψ ∼ = I F C F C ′ F u F u ′ g F cγ F β I = II F C F C ′ F C ′ F B ′ F u ′ f ′ g ′ g F cϕ ∼ = γ ′ The double functor Π : I ↓↓ F → × B ∼ = B is the projection onto the B -component.If B = A and F = id A , we write I ↓↓ A := I ↓↓ id A .We now define the pseudo-comma 2-category of a cospan of normal pseudo-functors, andalso give an explicit description of the special case of a pseudo-slice 2-category. Definition 4.4.
Let G : C → A and F : B → A be normal pseudo-functors. The pseudo-comma -category G ↓ F is defined as the following pullback in nps . G ↓ F C × B A × APs ( , A ) π ( G, F ) ( s, t ) where is the free 2-category on a morphism and Ps ( − , − ) is the internal hom describedin Definition 2.7. Remark . Note that π : G ↓ F → C × B is a strict Remark . We give an explicit description of the pseudo-comma 2-category in the casewhere C = is the terminal category and G = I : → A is an object in A . This is the2-category I ↓ F , called a pseudo-slice -category , whose(i) objects are pairs ( B, f ) of an object B ∈ B and a morphism f : I → F B in A ,(ii) morphisms ( b, ψ ) : ( B, f ) → ( B ′ , f ′ ) comprise the data of a morphism b : B → B ′ in B and a 2-isomorphism ψ in A of the form I F BF B ′ ff ′ F bψ ∼ = ,(iii) 2-morphisms β : ( b, ψ ) ⇒ ( c, ϕ ) comprise the data of a 2-morphism β : b ⇒ c in B such that the following pasting equality holds in A . I F BF B ′ F b F cff ′ F βψ ∼ = = I F BF B ′ F cf ′ f ϕ ∼ = The 2-functor π : I ↓ F → × B ∼ = B is the projection onto the B -component.If B = A and F = id A , we write I ↓ A := I ↓ id A with I ∈ A . Remark . Given the explications of Remarks 4.3 and 4.6, we wish to draw the reader’sattention to an important disparity between the double categories I ↓↓ H F and H ( I ↓ F ),for a 2-functor F : B → A and an object I ∈ A . While the latter double category has onlytrivial vertical morphisms, the former has all 2-morphisms of A of the form γ : f ⇒ g for I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 23 f, g : I → F B as vertical morphisms – a far richer stock of information. This is symptomaticof a broader truth: the double category P s( H , H A ) has all 2-morphisms of A as verticalmorphisms, while H Ps ( , A ) is the double category associated to its underlying horizontal2-category Ps ( , A ) = H P s( H , H A ) and therefore has only trivial vertical morphisms.While H thus does not preserve pseudo-comma objects, the main result of this sectionsays that the functors H and V do preserve pseudo-comma objects, in the sense that the2-category obtained by applying H or V to the pseudo-comma double category associatedto a cospan is isomorphic to the pseudo-comma 2-category of the image under H or V ofthe original cospan. This is the content of the following proposition and the rest of thesection will be devoted to its proof. Proposition 4.8.
Let G : C → A and F : B → A be normal pseudo-double functors. Thenthere are canonical isomorphisms of -categories as in the following commutative squares. H ( G ↓↓ F ) H G ↓ H F H ( C × B ) H C × H B ∼ = H Π π ∼ = V ( G ↓↓ F ) V G ↓ V F V ( C × B ) V C × V B ∼ = V Π π ∼ = To prove this we first show that the functors H and V behave well with respect to theinternal homs. Lemma 4.9.
For every -category C and every double category A , there is an isomorphismof -categories Ps ( C , H A ) ∼ = H P s( H C , A ) natural in C and A .Proof. Recall from Proposition 3.3 that H ⊣ H form an adjunction between nps and DblCat h,nps . Moreover, note that the functor H preserves products. Hence we have thefollowing isomorphisms nps ( B , Ps ( C , H A )) ∼ = nps ( B × C , H A ) (internal hom) ∼ = DblCat h,nps ( H ( B × C ) , A ) ( H ⊣ H ) ∼ = DblCat h,nps ( H B × H C , A ) ( H preserves products) ∼ = DblCat h,nps ( H B , P s( H C , A )) (internal hom) ∼ = nps ( B , H P s( H C , A )) , ( H ⊣ H )natural in B ∈ nps . By the Yoneda lemma, we have that Ps ( C , H A ) ∼ = H P s( H C , A ) . (cid:4) Corollary 4.10.
For every -category C and every double category A , there is an isomor-phism of -categories Ps ( C , V A ) ∼ = V P s( H C , A ) natural in C and A . Proof.
We have the following isomorphisms Ps ( C , V A ) = Ps ( C , H P s( V , A )) (definition of V ) ∼ = H P s( H C , P s( V , A )) (Lemma 4.9) ∼ = H P s( H C × V , A ) (internal hom) ∼ = H P s( V × H C , A ) (symmetry of × ) ∼ = H P s( V , P s( H C , A )) (internal hom)= V P s( H C , A ) . (definition of V )natural in C and A . (cid:4) The proof of Proposition 4.8 now follows from these results and the fact that H and V are right adjoints, and therefore preserve limits. Proof (Proposition 4.8) . Let us consider the following diagram. H G ↓ H F H C × H B H ( C × B ) H ( G ↓↓ F ) H A × H A Ps ( , H A ) H P s( H , A ) H ( A × A ) ∼ = ∼ = ∼ = H ( s, t ) π H Π H ( G × F ) ( s, t ) ∼ = H G × H F (1)(2) (3) (4)(5) First note that H G ↓ H F is a pullback of the commutative square (3), and, since H preservespullbacks, H ( G ↓↓ F ) is a pullback of the outer commutative square. The commutativesquare (4) is obtained in two steps. First, apply Lemma 4.9 to the 2-categories ⊔ and , respectively, and to the double category A , and use the naturality of these isomorphismswith respect to the 2-functor ⊔ → given by the inclusion at the two endpoints.Second, apply the isomorphisms Ps ( ⊔ , H A ) ∼ = H A × H A and H P s( ⊔ , H A ) ∼ = H ( A × A ) . Note that the bottom isomorphism H A × H A ∼ = H ( A × A ) of the square (4) is the canonicalone coming from the fact that H preserves products. Similarly, we have a canonical isomor-phism H B × H C ∼ = H ( B × C ) and the diagram (5) commutes. By the universal propertyof pullbacks, we get an isomorphism H ( F ↓↓ G ) ∼ = H F ↓ H G such that the diagrams (1)and (2) commute.The argument is similar for the case of V since this functor also preserves pullbacks andproducts, and Corollary 4.10 holds. (cid:4) I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 25
5. 2 -dimensional initiality
In Section 5.1 we introduce the new notion of a double bi-initial object in a doublecategory, which we aim to compare with that of a bi-initial object in a 2-category inSection 5.2.Double bi-initial objects are defined by requiring that the projection double functor fromthe pseudo-slice double category under this object mediates a double pseudo-equivalence,and similarly so for bi-initial objects in 2-categories. We choose here to use doublepseudo-equivalences, as they admit a characterisation in terms of pseudo-equivalences of 2-categories, and therefore allow for an easy comparison with bi-initial objects in 2-categories.As we will see, in the definition of a double bi-initial object, we could have equivalentlyrequired the projection double functor to be a horizontal pseudo-equivalence – the morenatural version of equivalences in
DblCat h,nps . In fact, the associated projection will beshown to actually satisfy the stronger condition of being split surjective on objects, andfully faithful on horizontal morphisms, vertical morphisms, and squares.We then show that a similar result holds for bi-initial objects in 2-categories: the as-sociated projection 2-functor is a pseudo-equivalence if and only if it is split surjectiveon objects, and fully faithful on morphisms and 2-morphisms. The main result then saysthat an object of a double category A is double bi-initial if and only if its images in the2-category H A and V A are bi-initial. In fact, we improve upon this by showing that anobject I is bi-initial in H A if its vertical identity e I is bi-initial in V A . That is, we showthat double bi-initial objects in A may be successfully detected purely 2-categorically asbi-initial objects of a suitable form in V A .Finally we show that in the presence of double limits of vertical morphisms in A , called tabulators , the reverse implication also holds: the object e I is bi-initial in V A when I isbi-initial in H A . Taken together these results show that, in the presence of tabulators, thecharacterisation of double bi-initial objects is now as good as one could hope for: a doublebi-initial object in a double category A is precisely a bi-initial object in the underlyinghorizontal 2-category H A .Both 2-categories and double categories have several duals, but of interest is the oppo-site C op of a 2-category C and the horizontal opposite A op of a double category A . Theseoperations agree with one another under applications of the functors H , H , and V . Inparticular, later we will have interest in (double) bi-terminal objects, which are simply(double) bi-initial objects in the (horizontal) opposite. Correspondingly, all the results ofthis section dualise to the setting of (double) bi-terminal objects.5.1. Double bi-initial objects.
Let us first give the definition of a double bi-initial ob-ject. Recall Remark 4.3, where we described explicitly pseudo-slice double categories.
Definition 5.1.
Let A be a double category. An object I in A is double bi-initial if theprojection double functor Π : I ↓↓ A → A mediates a double pseudo-equivalence.Expanding definitions, this holds if:(i) for every object A ∈ A , there is a chosen horizontal morphism f : I → A in A , (ii) for every tuple of horizontal morphisms f : I → A , f ′ : I → A ′ , and a : A → A ′ in A , there is a chosen vertically invertible square ψ in A II A A ′ A ′ f ′ f aψ ∼ = ,(iii) for every vertical morphism u : A B in A , there is a chosen square γ in A II AB ufgγ ,(iv) for every tuple of squares γ , γ ′ , and α in A II AB ufgγ
II A ′ B ′ u ′ f ′ g ′ γ ′ AB u A ′ B ′ u ′ abα and for every pair of vertically invertible squares in A II A A ′ A ′ f ′ f aψ ∼ = II B B ′ B ′ g ′ g bϕ ∼ = ,the following pasting equality holds in A . II A A ′ A ′ f ′ f aψ ∼ = I B B ′ u u ′ g bγ α I = I B ′ A ′ u ′ f ′ g ′ γ ′ I B B ′ g bϕ ∼ = Remark . Note that split essential surjectivity on objects of Π : I ↓↓ A → A only givesthat, for every object A ∈ A , there is a chosen horizontal morphism g : I → B in A togetherwith a chosen isomorphism h : B ∼ = −→ A . But, by choosing instead f = hg : I → A , we seethat Π is actually split surjective on objects. Similarly so for the split fullness on horizontalmorphisms and the split surjectivity on vertical morphisms. I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 27
More is true: when I is double bi-initial in a double category A , then the projectiondouble functor Π : I ↓↓ A → A is actually fully faithful on vertical morphisms. Lemma 5.3.
Let I be a double bi-initial object in a double category A . Then, for every pairof horizontal morphisms f : I → A and g : I → B , and every vertical morphism u : A B in A , there is a unique square in A of the form II AB ufgγ .In other words, the double functor
Π : I ↓↓ A → A is fully faithful on vertical morphisms.Proof. Let f : I → A and g : I → B be two horizontal morphisms in A , and let u : A B be a vertical morphism in A . By Definition 5.1 (iii), there is a chosen square γ in A II AB ufgγ .By Definition 5.1 (ii) applied to ( f , f, id A ) and ( g, g, id B ), there are vertically invertiblesquares ψ and ϕ in A as depicted below. II AA ffψ ∼ = II BB ggϕ ∼ = Then we set γ to be the following composite II AB ufgγ I = I AA ffψ ∼ = I B ugγ
I B gϕ ∼ = which proves the existence.Now suppose γ ′ is another such square in A II AB ufgγ ′ .By applying Definition 5.1 (iv) to the squares ( γ, γ ′ , id u ) and to the vertically invertiblesquares (e f , e g ), we directly get that γ = γ ′ from the pasting equality. (cid:4) Corollary 5.4.
Let I be a double bi-initial object in a double category A . Then the pro-jection double functor Π : I ↓↓ A → A is faithful on horizontal morphisms.Proof. Given horizontal morphisms f : I → A , f ′ : I → A ′ , and a : A → A ′ in A , wecan apply Lemma 5.3 to the horizontal morphisms f ′ : I → A ′ and f a : I → A ′ , and thevertical identity e A ′ . This gives that the choice of the vertically invertible square ψ inDefinition 5.1 (ii) is unique and therefore Π is faithful on horizontal morphisms. (cid:4) We summarise the results of Lemma 5.3 and Corollary 5.4 in the following proposition.In particular, given these results, it is straightforward to construct the rest of the data of ahorizontal pseudo-equivalence for Π, and so by Corollary 3.15 the second statement belowfollows.
Proposition 5.5.
Let A be a double category and I ∈ A be an object. Then the projectiondouble functor Π : I ↓↓ A → A mediates a double pseudo-equivalence if and only if it is splitsurjective on objects, and fully faithful on horizontal morphisms, vertical morphisms, andsquares. In particular, this holds if and only if Π is a horizontal pseudo-equivalence. (cid:4) Remark . The previous proposition implies that the conditions of a bi-initial object I ina double category A can be improved as follows. First, the choice of ψ in Definition 5.1 (ii)is unique. Then Definition 5.1 (iii) can be replaced by the following: for every verticalmorphism u : A B and every pair of horizontal morphisms f : I → A and g : I → B ,there is a unique square γ : (e I fg u ) with this boundary. And finally, note that the verticallyinvertible squares ψ and ϕ in Definition 5.1 (iv) are now unique and therefore need not bepart of the given data.5.2. Double bi-initial objects vs bi-initial objects.
We now want to compare thenotion of double bi-initial objects in a double category with the notion of bi-initial objectsin related 2-categories. We first recall the definition of a bi-initial object, which uses thenotion of a pseudo-slice 2-category as elaborated in Remark 4.6.
Definition 5.7.
Let A be a 2-category. An object I ∈ A is bi-initial if the projection2-functor π : I ↓ A → A mediates a pseudo-equivalence.Expanding definitions, this holds if:(i) for every object A ∈ A , there is a chosen morphism f : I → A in A ,(ii) for every tuple of morphisms f : I → A , f ′ : I → A ′ , and a : A → A ′ in A , there isa chosen 2-isomorphism ψ in A I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 29
I AA ′ ff ′ aψ ∼ = ,(iii) for every 2-morphism α : a ⇒ b in A , and every pair of 2-isomorphisms ψ : f ∼ = = ⇒ af ′ and ϕ : f ∼ = = ⇒ bf ′ the following pasting equality holds in A . I AA ′ a bf ′ f αψ ∼ = = I AA ′ bff ′ ϕ ∼ = Lemma 5.8.
Let I be a bi-initial object in a -category A . Then the projection -functor π : I ↓ A → A is faithful on morphisms.Proof. Let f : I → A , f ′ : I → A ′ , and a : A → A ′ be morphisms in A . Suppose that ψ, ψ ′ : af ′ ⇒ f are two 2-isomorphisms as in Definition 5.7 (ii). By applying Defini-tion 5.7 (iii) to the tuple (id a , ψ, ψ ′ ), we directly get that ψ = ψ ′ . (cid:4) We may recast the previous lemma as the following result.
Proposition 5.9.
Let A be a -category, and I ∈ A be an object. Then the projection -functor π : I ↓ A → A mediates a pseudo-equivalence if and only if it is split surjectiveon objects, and fully faithful on morphisms, and -morphisms. (cid:4) Remark . The previous proposition implies that the conditions of a bi-initial object I in a 2-category A can be improved as follows. First, the choice of ψ in Definition 5.7 (ii)is unique. Second, note that the 2-isomorphisms ψ and ϕ in Definition 5.7 (iii) are nowuniquely determined by f , f ′ , a , and b , and therefore need not be part of the given data.It can now easily be checked that this notion of bi-initial object, as given in Remark 5.10,coincides with the usual definition. Proposition 5.11.
Let A be a -category, and I ∈ A be an object. Then the object I isbi-initial in A if and only if, for every object A ∈ A , the unique functor A ( I, A ) ≃ −→ ispart of an equivalence of categories.Proof. Suppose I ∈ A is bi-initial and let A ∈ A . By Definition 5.7 (i), there is a chosenmorphism f : I → A in A : this gives split surjectivity of A ( I, A ) → . Now, given twomorphisms f : I → A and f ′ : I → A in A , by applying Definition 5.7 (ii) to the tuple( f, f ′ , id A ), there is a unique 2-isomorphism ψ : f ′ ∼ = = ⇒ f : this gives fully faithfulness onmorphisms of A ( I, A ) → . This shows that A ( I, A ) → is part of an equivalence.Conversely, suppose now that A ( I, A ) ≃ −→ is part of an equivalence, for all A ∈ A . Weprove that Definition 5.7 (i-iii) hold for I . Given A ∈ A , by split surjectivity on objects of A ( I, A ) ≃ −→ , there is a chosen morphism f : I → A , which proves (i). Now, givenmorphisms f : I → A , f ′ : I → A ′ , and a : A → A ′ in A , consider the morphisms f a and f ′ in A ( I, A ′ ). By fully faithfulness on morphisms of A ( I, A ′ ) ≃ −→ , there is a unique2-isomorphism ψ : f ′ ∼ = = ⇒ af , which proves (ii). Condition (iii) follows from faithfulness onmorphisms of A ( I, A ′ ) ≃ −→ as the two pastings must therefore be equal. (cid:4) Since pseudo-slices are special cases of pseudo-commas, Proposition 4.8 may be spe-cialised in this context to give the following result.
Corollary 5.12.
Let A be a double category, and I ∈ A be an object. Then there arecanonical isomorphisms of -categories as in the following commutative triangles. H ( I ↓↓ A ) I ↓ H A H A H Π ∼ = π V ( I ↓↓ A ) e I ↓ V A V A V Π ∼ = π Proof.
This directly follows from Proposition 4.8, by taking B = , C = A , F = I : → A and G = id A : A → A . Note that V I = e I : V ( ) = → V A . (cid:4) With this result and the fact that double pseudo-equivalences are exactly the doublefunctors whose images under H and V are pseudo-equivalences, we may give a 2-categoricalcharacterisation of double bi-initial objects by leveraging this fact as follows. Theorem 5.13.
Let A be a double category, and I ∈ A be an object. The followingstatements are equivalent.(i) The object I ∈ A is double bi-initial.(ii) The corresponding objects I ∈ H A and e I ∈ V A are bi-initial.(iii) The corresponding object e I ∈ V A is bi-initial.Proof. We first prove that (i) and (ii) are equivalent. By definition, an object I ∈ A is double bi-initial if and only if the projection double functor Π : I ↓↓ A → A mediatesa double pseudo-equivalence. By Proposition 3.14, this is equivalent to saying that theinduced 2-functors H Π and V Π are pseudo-equivalences and, by Corollary 5.12, this holdsif and only if the projection 2-functors π : I ↓ H A → H A and π : e I ↓ V A → V A are pseudo-equivalences. By definition of a bi-initial object, this holds if and only if the objects I ∈ H A and e I ∈ V A are bi-initial.We now prove that (ii) and (iii) are equivalent. It is clear that (ii) implies (iii), andso it remains to prove that if e I is bi-initial in V A , then I is bi-initial in H A . We proveDefinition 5.7 (i-iii) for I ∈ H A with the unicity of choices of Remark 5.10. Let A ∈ H A be an object. Then the vertical identity e A is an object in V A . By Definition 5.7 (i) fore I ∈ V A , there is a chosen morphism γ : e I → e A in V A , i.e., a square in A of the form I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 31
I AI A fgγ .In particular, the horizontal morphism f : I → A gives a choice of morphism in H A asdesired, which proves (i). Now let f : I → A , f ′ : I → A ′ , and a : A → A ′ be morphismsin H A , i.e., horizontal morphisms in A . Then the vertical identity squares e f , e f ′ , ande a are morphisms e f : e I → e A , e f ′ : e I → e A ′ , and e a : e A → e A ′ in V A . By applyingDefinition 5.7 (ii) and Remark 5.10 for e I ∈ V A to the tuple (e f , e f ′ , e a ), there is a unique2-isomorphism ( ψ, ϕ ) : e f ′ ∼ = = ⇒ e a e f in V A . In particular, we have ψ = ϕ by the pastingequality for 2-morphisms in V A , and this gives a unique 2-isomorphism ψ : f ′ ∼ = = ⇒ af in H A as required. This proves (ii). Finally, let f : I → A and f ′ : I → A ′ be morphisms in H A ,and let α : a ⇒ b be a 2-morphism in H A between morphisms a, b : A → A ′ . Then thesquares II AA ff e f II A ′ A ′ f ′ f ′ e f ′ AA A ′ A ′ abα are morphisms e f : e I → e A , e f ′ : e I → e A ′ , and α : e A → e A ′ in V A . By applying Def-inition 5.7 (ii) and Remark 5.10 for e I ∈ V A to the tuple (e f , e f ′ , α ), there is a unique2-isomorphism ( ψ, ϕ ) : e f ′ ∼ = = ⇒ α e f in V A , i.e., two vertically invertible squares ψ and ϕ satisfying the following pasting equality in A . II A A ′ A ′ f ′ f aψ ∼ = I A A ′ f b e f α I = I A ′ A ′ f ′ f ′ e f ′ I A A ′ f bϕ ∼ = This corresponds to the pasting equality of Definition 5.7 (iii), and therefore shows (iii). (cid:4)
Remark . This theorem served as the initial motivation for the definition of the functor V whose role is so central in this paper.Observe that double bi-initial objects in a double category A have two aspects to their weak universal properties: one concerning objects and one concerning vertical morphisms.The former is entirely horizontal in nature and so is completely captured by the underlyinghorizontal 2-category H A . The latter, despite concerning vertical morphisms, does not infact need the full strength of vertical composition in A to be expressed. Indeed, except for vertical composition by squares with trivial boundary of the form of Definition 3.4 (iii), thisaspect of the weak universal property is also somehow horizontal. That is to say, the un-derlying horizontal H P s( V , A ) is precisely the setting in which to capture thislast data as it has vertical morphisms as objects and understands horizontal compositionsof general squares. But this 2-category H P s( V , A ) is exactly our V A !5.3. Double bi-initial objects and tabulators.
We have seen that double bi-initialobjects may be detected through purely 2-categorical means. In this section we show thata substantial simplification of the 2-categorical criteria is possible when the double categoryin question has tabulators . These correspond to double limits of vertical morphisms andwere introduced by Grandis and Par´e in [5, § Definition 5.15.
Let A be a double category, and u : A B be a vertical morphism in A .A tabulator of u is a double limit of the double functor u : V → A , where V is thedouble category free on a vertical morphism. In other words, it is a pair ( ⊤ u, τ u ) of anobject ⊤ u ∈ A together with a square τ u : (e ⊤ u pq u ) in A satisfying the following universalproperties.(i) For every square γ : (e I fg u ) in A , there is a unique horizontal morphism t : I → ⊤ u in A such that the following pasting equality holds. II AB fg uγ I = I ⊤ u ⊤ u AB tt pq u e t τ u (ii) For every tuple of squares γ : (e I fg u ), γ ′ : (e I ′ f ′ g ′ u ), θ : ( v f ′ f e A ) and θ : ( v g ′ g e B )in A satisfying the following pasting equality, I ′ AII AB f ′ fg uv θ γ I ′ = AI ′ I BB f ′ g ′ g uv γ ′ θ there is a unique square θ : ( v t ′ t e ⊤ u ), where t : I → ⊤ u and t ′ : I ′ → ⊤ u are theunique horizontal morphisms given by (i) applied to γ and γ ′ respectively, suchthat θ satisfies the following pasting equalities. I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 33 I ′ I AA f ′ fv θ I ′ = I ⊤ u ⊤ u AA t ′ t ppv θ e p I ′ I BB g ′ gv θ I ′ = I ⊤ u ⊤ u BB t ′ t qqv θ e q We say that A has chosen tabulators if there is a chosen tabulator for each verticalmorphism u in A . Theorem 5.16.
Let A be a double category with chosen tabulators, and I ∈ A be an object.Then the following statements are equivalent.(i) The object I is double bi-initial in A .(ii) The object I is bi-initial in H A .Proof. If I is double bi-initial in A , then, by Theorem 5.13, I is bi-initial in H A .Now suppose that I is bi-initial in H A . We prove that I satisfies Definition 5.1 (i-iv).First note that Definition 5.7 (i) and (ii) applied to I ∈ H A correspond to Definition 5.1 (i)and (ii) applied to I ∈ A . Therefore, it remains to show Definition 5.1 (iii) and (iv).Let u : A B be a vertical morphism and let ( ⊤ u, τ u ) be the chosen tabulator of u .By Definition 5.7 (i) applied to the object ⊤ u , there is a chosen horizontal morphism t : I → ⊤ u . By the first universal property of tabulators, we get a square in A II AB fg uγ as desired. This proves Definition 5.1 (iii).Now suppose that we have squares in A II AB ufgγ
II A ′ B ′ u ′ f ′ g ′ γ ′ AB u A ′ B ′ u ′ abα and suppose ( ⊤ u, τ u ) and ( ⊤ u ′ , τ u ′ ) are tabulators for u and u ′ respectively. By the firstuniversal property of tabulators, the squares γ and γ ′ uniquely correspond to horizontalmorphisms t : I → ⊤ u and t ′ : I → ⊤ u ′ respectively. Moreover, the square α uniquelycorresponds to a horizontal morphism ⊤ α : ⊤ u → ⊤ u ′ . By applying Definition 5.7 (ii) to t : I → ⊤ u , t ′ : I → ⊤ u ′ , and ⊤ α : ⊤ u → ⊤ u ′ , we get a square θ in A of the form II ⊤ u ⊤ u ′ ⊤ u ′ t ′ t ⊤ αθ ∼ = . By the second universal property of tabulators, this uniquely correspond to squares ψ := θ and ϕ := θ satisfying the following pasting. II A A ′ A ′ f ′ f aψ ∼ = I B B ′ u u ′ g bγ α I = I B ′ A ′ u ′ f ′ g ′ γ ′ I B B ′ g bϕ ∼ = Note that ψ and ϕ are the unique squares for the tuples ( f, f ′ , a ) and ( g, g ′ , b ) respectively;see Remark 5.10. This shows Definition 5.1 (iv). (cid:4) Corollary 5.17.
Let A be a double category with chosen tabulators, and I ∈ A be anobject. Then the object I is bi-initial in H A if and only if the corresponding object e I isbi-initial in V A .Proof. By Theorem 5.16, I is bi-initial in H A if and only if I is double bi-initial in A . ByTheorem 5.13, this holds if and only if e I is bi-initial in V A . (cid:4) Bi-representations of normal pseudo-functors
In Section 6.1 we state and prove our main result characterising bi-representations ofnormal pseudo-functors F : C op → Cat as various sorts of bi-initial objects, where
Cat isthe 2-category of categories, functors, and natural transformations. We give two flavoursof such a theorem, one stated in the language of double categories, and the other statedcompletely in terms of 2-categories. The former predictably relies on the double categoryof elements of F construction, but in the latter case, we will define from the data of anormal pseudo-functor F not only the 2-category of elements of F , but also a 2-categoryof morphisms of F . Moreover, by specialising Theorem 5.13, we see that the latter 2-category subsumes the former for this purpose: bi-representations of a normal pseudo-functor F : C op → Cat are precisely bi-initial objects of a particular form in the 2-categoryof morphisms of F .We then show in Section 6.2 that, when the 2-category C has tensors by and thenormal pseudo-functor F : C op → Cat preserves them, the expected characterisation actu-ally holds: bi-representations of F are now precisely bi-initial objects in the 2-category ofelements of F .6.1. The general case.
Let us begin by defining the central objects at issue.
Definition 6.1.
Let C be a 2-category, and F : C op → Cat be a normal pseudo-functor.A bi-representation of F is a pair ( I, ρ ) of an object I ∈ C and a pseudo-natural adjointequivalence ρ − : C ( − , I ) ≃ = ⇒ F , i.e., an adjoint equivalence in the 2-category Ps ( C op , Cat ). I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 35
Remark . Recall that an equivalence in a 2-category can always be promoted to an adjoint equivalence (see, e.g. [15, Lemma 2.1.11]). Therefore, by requiring the pseudo-natural equivalence in Definition 6.1 to be adjoint, we do not lose any generality whilesimultaneously making the data easier to handle in the forthcoming proofs.To the data of such a normal pseudo-functor C op → Cat we will associate a doublecategory of elements . This double category will play an analogous role to the classicalcategory of elements in detecting representations.
Definition 6.3.
Let C be a 2-category, and F : C op → Cat be a normal pseudo-functor.The double category of elements el ( F ) of F is defined to be the pseudo-slice doublecategory ↓↓ H F induced by the cospan −→ H Cat H F ←−− H C op . More explicitly, it is the double category whose(i) objects are pairs (
C, x ) of an object C ∈ C and a functor x : → F C , i.e., anobject x ∈ F C ,(ii) horizontal morphisms ( c, ψ ) : ( C ′ , x ′ ) → ( C, x ) comprise the data of a morphism c : C → C ′ of C and a natural isomorphism ψ of the form F C ′ F C x ′ x F cψ ∼ = ,i.e., an isomorphism ψ : x ∼ = −→ ( F c ) x ′ in F C ,(iii) vertical morphisms α : ( C, x ) (
C, y ) are natural transformations α : x ⇒ y offunctors x, y : → F C , i.e., morphisms α : x → y in F C ,(iv) squares γ : ( α ′ ( c,ψ )( d,ϕ ) α ) comprise the data of a 2-morphism γ : c ⇒ d of C , asdisplayed below-left, which satisfies the below-right pasting equality, CC ′ c dγ F C ′ F C
F c F dxx ′ F γy ′ ψ ∼ = α ′ = F C ′ F C
F dy ′ yx α ϕ ∼ = i.e., the following diagram in F C is commutative. x ( F c ) x ′ ( F c ) y ′ ( F d ) y ′ y αψ ∼ = ( F c ) α ′ ( F γ ) y ′ ϕ ∼ = Much like in the 1-dimensional case, from a normal pseudo-functor F : C op → Cat weare able to construct the 2-category of elements el ( F ) of F , but new here is the 2-categoryof morphisms of F . As we shall see, the joint properties of these 2-categories may beleveraged to successfully characterise bi-representations. Definition 6.4.
Let C be a 2-category, and F : C op → Cat be a normal pseudo-functor.We define the following two 2-categories associated to F . • The 2 -category of elements el ( F ) of F is defined to be H el ( F ). • The 2 -category of morphisms mor ( F ) of F is defined to be V el ( F ).The ardently 2-categorical reader may be dismayed by the foray into the realm of doublecategories to give the above definition. In the coming discussion we will find that we are ableto comfortably re-seat these 2-categories as the result of purely 2-categorical considerations.Observe that exponentiation by the category = { → } gives rise to the classicalfunctor Ar := ( − ) : Cat → Cat , the category of arrows functors, where
Cat is the categoryof categories and functors.
Definition 6.5.
We define the functor Ar ∗ : nps → nps as follows. It sends a2-category C to the 2-category Ar ∗ C with the same objects as C and hom-categoriesAr ∗ C ( C, C ′ ) := Ar( C ( C, C ′ )) for each pair of objects C, C ′ ∈ C . That is, a morphism inAr ∗ C is a 2-morphism of C and a 2-morphism in Ar ∗ C is a commutative square of verticalcomposites of 2-morphisms in C .Given a normal pseudo-functor F : C → D , we define the normal pseudo-functor Ar ∗ F to act as F on objects and as Ar F on hom-categories. The compositors of Ar ∗ F are givencomponent-wise by the compositors of F . Remark . The functor Ar ∗ is a shadow of our double categorical approach of the previoussections. Indeed, we have the equality of functors V H = Ar ∗ to complement H H = id nps .Recall that el ( F ) was defined as the pseudo-slice double category ↓↓ H F . This, cou-pled with the fact that H and V preserve slices allows us to give the following, purely 2-categorical formulations of the 2-categories of elements and morphisms of a normal pseudo-functor. Remark . If C is a 2-category and F : C op → Cat is a normal pseudo-functor then, byCorollary 5.12 and Remark 6.6, the 2-categories el ( F ) and mor ( F ) are isomorphic to thepseudo-slice 2-categories induced by the cospans −→ Cat F ←− C op and −→ Ar ∗ Cat Ar ∗ F ←−−− Ar ∗ C op , respectively. In particular, el ( F ) is the pseudo-type version of the usual 2-category ofelements of F . I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 37
We are now in a position to give the central result of this paper, a 2-dimensional analogueto the classical relationship between representations and initial objects. The equivalencebetween (i) and (iii) below gives the promised 2-categorical account of the theorem, andwhile it may be derived directly, we will find that our work of the previous sections allowsfor a more efficient approach via (ii).
Theorem 6.8.
Let C be a -category, and ( F, φ ) : C op → Cat be a normal pseudo-functor.The following statements are equivalent.(i) The normal pseudo-functor F has a chosen bi-representation ( I, ρ ) .(ii) There is a chosen object I ∈ C together with a chosen object i ∈ F I such that ( I, i ) is double bi-initial in el ( F ) .(iii) There is a chosen object I ∈ C together with a chosen object i ∈ F I such that ( I, i ) is bi-initial in el ( F ) and ( I, id i ) is bi-initial in mor ( F ) .(iv) There is a chosen object I ∈ C together with a chosen object i ∈ F I such that ( I, id i ) is bi-initial in mor ( F ) . First note that the equivalence of conditions (ii), (iii), and (iv) follows directly fromTheorem 5.13. The rest of this section will be devoted to the proof of the equivalencebetween conditions (i) and (ii). For this, we first introduce the following “unique filler”lemma to make efficient the proof of the forwards implication.
Lemma 6.9.
Under the assumptions of Theorem 6.8 (i), for every pair of horizontal mor-phisms ( f, ψ ) : ( I, i ) → ( C, x ) and ( g, ϕ ) : ( I, i ) → ( C, y ) , and every vertical morphism α : ( C, x ) (
C, y ) in el ( F ) , there is a unique square in el ( F ) of the below form. ( I, i ) (
C, x )( I, i ) (
C, y ) ( f, ψ )( g, ϕ ) αγ .Proof. Let (
I, ρ ) be a bi-representation of a normal pseudo-functor (
F, φ ) : C op → Cat ,( f, ψ ) : ( I, i ) → ( C, x ) and ( g, ϕ ) : (
I, i ) → ( C, y ) be horizontal morphisms in el ( F ), and α : ( C, x ) (
C, y ) be a vertical morphism in el ( F ). Define υ to be the unique morphismof F C fitting in the following diagram,(
F f ) i x y ( F g ) iρ C ( f ) ρ C ( g ) ψ -1 α ϕ ( ρ f ) id I ∼ = ( ρ g ) id I ∼ = υ where ρ f : ( F f ) ρ I ∼ = = ⇒ ρ C C ( I, f ) and ρ g : ( F g ) ρ I ∼ = = ⇒ ρ C C ( I, g ) are the 2-isomorphism com-ponents of ρ at f and g , respectively. By Definition 6.3 (iv), a square γ : (id i ( f,ψ )( g,ϕ ) α )in el ( F ) is the data of a 2-morphism γ : f ⇒ g of C such that( F f ) i ( F γ ) i −−−→ ( F g ) i = ( F f ) i ψ -1 −→ x α −→ y ϕ −→ ( F g ) i. Therefore, we may deduce that this equation holds if and only if ( ρ g ) id I ( F γ ) i = υ ( ρ f ) id I , bydefinition of υ . The left-hand composite of this equality appears as the result of evaluatingthe below-left pasting at id I , and this pasting is equal to the below-right pasting by pseudo-naturality of ρ . C ( I, I ) F I C ( I, C ) F C ρ I g ∗ ρ C F g F fρ g ∼ = F γ C ( I, I ) F I C ( I, C ) F C ρ I ρ C f ∗ F fg ∗ γ ∗ ρ f ∼ = =We deduce therefore that( ρ g ) id I ( F γ ) i = υ ( ρ f ) id I iff ( ρ C ( γ ))( ρ f ) id I = υ ( ρ f ) id I iff ρ C ( γ ) = υ. All in all then, γ : (id i ( f,ψ )( g,ϕ ) α ) is a square in el ( F ) if and only if ρ C ( γ ) = υ . Since ρ C : C ( I, C ) → F C is an equivalence and is therefore fully faithful on morphisms, there isa unique such γ . (cid:4) With this lemma established, the proof of the forward implication (i) ⇒ (ii) of Theo-rem 6.8 is readily given. Proof (Theorem 6.8, (i) ⇒ (ii)) . Suppose (i), that is, we have a specified bi-representation(
I, ρ ) of F . From this data we will select an object i ∈ F I and demonstrate that (
I, i ) isdouble bi-initial in el ( F ). To begin, let us define i ∈ F I as i := ρ I (id I ). We address eachof conditions (i-iv) of Definition 5.1 in turn.Let ( C, x ) be an object of el ( F ). Since ρ C : C ( C, I ) → F C is an equivalence and x ∈ F C , there is a chosen morphism f : C → I in C together with a chosen isomorphism ψ : x ∼ = −→ ρ C ( f ) in F C . By post-composing with the inverse of ( ρ f ) id I : ( F f ) i ∼ = −→ ρ C ( f ),arising from the 2-isomorphism component ρ f : ( F f ) ρ I ∼ = = ⇒ ρ C C ( f, I ) of ρ at f , we finda horizontal morphism ( f, ( ρ f ) -1 id I ψ ) : ( I, i ) → ( C, x ) in el ( F ), and so we have establishedDefinition 5.1 (i).The rest of conditions (ii-iv) each follow from applications of Lemma 6.9 above, whichwe elaborate below. First, Definition 5.1 (ii) grants us the existence of a boundary of el ( F )of the form depicted below, and charges us with finding a unique, vertically invertible filler.( I, i )( I, i ) ( C ′ , x ′ ) ( C, x )( C, x ) ( f, ψ )( f ′ , ψ ′ ) ( c, ϕ ) By composing the bottom horizontal morphisms we see that Lemma 6.9 supplies us with aunique filler for this square. That this filler is vertically invertible follows from consideringthe vertical opposite of the above square combined with further applications of Lemma 6.9.Next, Definition 5.1 (iii) grants us a vertical morphism α : ( C, x ) (
C, y ) of el ( F )and demands the existence of a square from ( I, i ) (
I, i ) to α . By our construction I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 39 of Definition 5.1 (i) above, we may give horizontal morphisms ( f, ψ ) : (
I, i ) → ( C, x ) and( g, ϕ ) : (
I, i ) → ( C, y ), and thus produce the below boundary in el ( F ). An application ofLemma 6.9 shows (iii). ( I, i ) (
C, x )( I, i ) (
C, y ) ( f, ψ )( g, ϕ ) α Finally, we must show that Definition 5.1 (iv) holds. That is, we must demonstratethat there is an equality of squares filling a fixed boundary. Fortunately we may applyLemma 6.9 to this boundary and so conclude the proof. (cid:4)
We conclude this section by proving the reverse implication.
Proof (Theorem 6.8, (ii) ⇒ (i)) . Suppose (ii), that is, we have a chosen double bi-initialobject (
I, i ) in el ( F ). From this data we will construct equivalences ρ C : C ( C, I ) → F C for each C ∈ C and then show that they assemble into a pseudo-natural transformation. Bya standard result of 2-categories, any equivalence is canonically rectifiable into an adjointequivalence and so we do not trouble ourselves with additional work after giving ρ .For a fixed C ∈ C , let us define the functor ρ C : C ( C, I ) → F C on objects f : C → I as ρ C ( f ) := ( F f ) i and on morphisms γ : f ⇒ g as ρ C ( γ ) := ( F γ ) i . As F respects verticalcomposition of 2-morphisms strictly, it is clear that ρ C is a functor by construction.With the functors ρ C defined, we now show that each of these functors is an equivalence.To that end, let us fix C ∈ C and x ∈ F C . Observe that (
C, x ) is an object of el ( F )so that, since ( I, i ) is double bi-initial in el ( F ), there is a chosen horizontal morphism( f, ψ ) : ( I, i ) → ( C, x ) in el ( F ). This is precisely the data of an object f ∈ C ( C, I )and an isomorphism ψ : ρ C ( f ) = ( F f ) i ∼ = −→ x , which shows that ρ C is split essentiallysurjective. To see that each ρ C is fully faithful on morphisms, let f, g : C → I be objectsin C ( C, I ) and α : ρ C ( f ) → ρ C ( g ) be a morphism between their images in F C . Thisdata is equivalently a pair of horizontal morphisms ( f, id ρ C ( f ) ) : ( I, i ) → ( C, ρ C ( f )) and( g, id ρ C ( g ) ) : ( I, i ) → ( C, ρ C ( g )), since ρ C ( f ) = ( F f ) i and ρ C ( g ) = ( F g ) i by definition,together with a vertical morphism α : ( C, ρ C ( f )) ( C, ρ C ( g )) in el ( F ). Since ( I, i ) isdouble bi-initial in el ( F ), by Lemma 5.3, there is a unique square in el ( F ) of the form( I, i ) (
C, ρ C ( f ))( I, i ) (
C, ρ C ( g )) ( f, id ρ C ( f ) )( g, id ρ C ( g ) ) αγ ,that is, a unique 2-morphism γ : f ⇒ g such that ρ C ( γ ) = ( F γ ) i = α . This shows fullyfaithfulness of ρ C . Now that we have a collection of object-wise equivalences ρ C we seek to construct thedata of the pseudo-naturality comparison natural isomorphisms ρ c : ( F c ) ρ C ′ ∼ = = ⇒ ρ C C ( c, I )depicted below, for each morphism c : C → C ′ in C . C ( C ′ , I ) F C ′ C ( C, I ) F C ρ C ′ ρ C C ( c, I ) F cρ c ∼ = For f ∈ C ( C ′ , I ) observe that ( F c ) ρ C ′ ( f ) = ( F c )( F f ) i and ρ C C ( c, I )( f ) = F ( f c ) i , so thatwe can set ρ c to be ( φ c, − ) i , the compositor of F at ( c, − ) evaluated at i . This satisfies allof the required properties of pseudo-naturality. (cid:4) In fact, we have additionally proven that bi-representations (
I, ρ ) of a normal pseudo-functor are determined up to isomorphism by their values on id I . This conclusion may beseen as a special case of a suitable 2-dimensional Yoneda lemma. Corollary 6.10.
Let C be a -category, and F : C op → Cat be a normal pseudo-functor.Suppose that ( I, ρ ) is a bi-representation of F . Then there is a canonical bi-representation ( I, ρ ) of F given by ρ C = ( F − )( ρ I (id I )) : C ( C, I ) → F C, for every C ∈ C . Moreover, we have that ρ ∼ = ρ .Proof. The construction is given by tracing the proofs above of Theorem 6.8 through(i) ⇒ (ii) and then (ii) ⇒ (i). Finally, the isomorphism ρ ∼ = ρ is given by the 2-isomorphismcomponents ( ρ f ) id I : ( F f ) ρ I (id I ) ∼ = −→ ρ C ( f ) of ρ itself evaluated at id I , for every f : C → I in C . (cid:4) Remark . In particular, when F is a strict 2-functor, without loss of generality abi-representation of F may be taken to be a 2-natural adjoint equivalence. Indeed, thebi-representation constructed in Corollary 6.10 is 2-natural.6.2. The case in presence of tensors by . Finally we explore a substantial improve-ment of Theorem 6.8 which is possible when the 2-category C has tensors , defined below,which are preserved by F . Definition 6.12.
Let C be a 2-category, C ∈ C be an object, and A be a category.A power of C by A is a weighted 2-limit of the 2-functor C : → C by the weight A : → Cat . In other words, it is a pair ( A ⋔ C, λ ) of an object A ⋔ C ∈ C and a functor λ : A → C ( A ⋔ C, C ) such that, for every object C ′ ∈ C , pre-composition by λ induces anisomorphism of categories λ ∗ ◦ C ( − , C ) : C ( C ′ , A ⋔ C ) ∼ = −→ Cat ( A , C ( C ′ , C )) . We say that
C has chosen powers by A if there is a chosen power of C by A for eachobject C ∈ C . I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 41
Dually, a tensor of C by A is a power of C by A in the opposite 2-category C op . In otherwords, it is a pair ( C ⊗A , ζ ) of an object C ⊗A ∈ C and a functor ζ : A → C ( C, C ⊗A ) suchthat, for every object C ′ ∈ C , pre-composition by ζ induces an isomorphism of categories ζ ∗ ◦ C ( C, − ) : C ( C ⊗ A , C ′ ) ∼ = −→ Cat ( A , C ( C, C ′ )) . We say that
C has chosen tensors by A if there is a chosen tensor of C by A for eachobject C ∈ C . Remark . Powers may be viewed as a lower-dimensional shadow of the double categor-ical notion of tabulators seen in Definition 5.15. Indeed, a power of an object C ∈ C by thecategory = { → } in a 2-category C is precisely a tabulator of the vertical identity e C in its associated horizontal double category H C ; see [4, Exercise 5.6.2 (c)]. In particular,tabulators in H C op correspond to tensors by in C .From the universal property of powers, it is straightforward to see that the 2-category Cat has chosen powers by any category A given by A ⋔ C := Cat ( A , C ). Given a 2-category C with chosen tensors by a category A , we then say that a normal pseudo-functor F : C op → Cat preserves powers by A if, for every object C ∈ C , we have anisomorphism of categories F ( C ⊗ A ) ∼ = Cat ( A , F C ) which is natural with respect to thedefining cones. Theorem 6.14.
Let C be a -category with chosen tensors by , and F : C op → Cat bea normal pseudo-functor which preserves powers by . Then the following statements areequivalent.(i) The normal pseudo-functor F has a chosen bi-representation ( I, ρ ) .(ii) There is a chosen object I ∈ C together with a chosen object i ∈ F I such that ( I, i ) is double bi-initial in el ( F ) .(iii) There is a chosen object I ∈ C together with a chosen object i ∈ F I such that ( I, i ) is bi-initial in el ( F ) .In particular, ( I, i ) is bi-initial in el ( F ) if and only if ( I, id i ) is bi-initial in mor ( F ) . In order to prove this result we will make use of the following lemma.
Lemma 6.15.
Let C be a -category with chosen tensors by , and F : C op → Cat be anormal pseudo-functor which preserves powers by . Then the double category el ( F ) haschosen tabulators.Proof. Let α : ( C, x ) (
C, y ) be a vertical morphism in el ( F ) and let ( C ⊗ , ζ ) be a tensorof C by . Recall that ζ is a functor ζ : → C ( C, C ⊗ ), and therefore it corresponds toa 2-morphism C C ⊗ ζ ζ ζ . Moreover, the morphism α : x → y in F C is equivalently a functor α : → F C andtherefore it corresponds to an object α ∈ F ( C ⊗ ) as Cat ( , F C ) ∼ = F ( C ⊗ ). We set ⊤ u := ( C ⊗ , α ) ∈ el ( F ) and τ u to be the following square in el ( F ).( C ⊗ , α )( C ⊗ , α ) ( C, x )( C, y ) ( ζ , id x )( ζ , id y ) αζ We show that it satisfies the universal properties of tabulators of Definition 5.15. Let( C ′ , x ′ )( C ′ , x ′ ) ( C, x )( C, y ) ( c, ψ )( d, ϕ ) αγ be a square in el ( F ). By the universal property of tensors, the 2-morphism γ : c ⇒ d corresponds to a morphism γ : C ⊗ → C ′ . Moreover, note that the pair ( ψ, ϕ ) gives anisomorphism in Cat ( , F C ) from α to ( F γ ) x ′ , and since F preserves powers by , then( ψ, ϕ ) corresponds to an isomorphism ( ψ, ϕ ) : α ∼ = ( F γ ) x ′ in F ( C ⊗ ). We get the requiredhorizontal morphism ( γ, ( ψ, ϕ )) : ( C ′ , x ′ ) → ( C ⊗ , α ) for Definition 5.15 (i).Similarly, Definition 5.15 (ii) follows from the fact that 2-morphisms in C of the form C ′ C ⊗ γγ ′ θ uniquely correspond to 2-morphisms θ , θ in C between morphisms C → C ′ such that γθ = θ γ ′ , by the universal property of tensors. (cid:4) Proof (Theorem 6.14) . First note that (i) and (ii) are equivalent by Theorem 6.8.To see that (ii) and (iii) are equivalent consider the following. By Lemma 6.15, thedouble category el ( F ) admits tabulators. Thus, by Theorem 5.16, an object ( I, i ) is doublebi-initial in el ( F ) if and only if ( I, i ) is bi-initial in el ( F ).Finally, that ( I, i ) is bi-initial in el ( F ) if and only if ( I, id i ) is bi-initial in mor ( F ) followsfrom Corollary 5.17. (cid:4) Applications to bi-adjunctions and weighted bi-limits
Now that we have satisfied ourselves with the characterisation of Theorem 6.8 of bi-representations of normal pseudo-functor we focus now on two formal applications. InSection 7.1, we will leverage some 2-dimensional arguments to give a characterisation ofbi-adjunctions in terms of bi-terminal objects in pseudo-slices. Then, in Section 7.2 we willconnect to the counter-examples given in [3] by proving a correct characterisation of bi-limits in terms of bi-terminal objects in pseudo-slices, specialising the supporting theorem
I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 43 in the same section about weighted bi-limits. In both sections we will additionally giveimprovements on these results by specialising Theorem 6.14 when the 2-categories at issuehave chosen tensors by , which in the case of weighted bi-limits subsumes a known specialcase.7.1. Bi-adjunctions.
We begin by introducing the notion of a bi-adjunction.
Definition 7.1.
Let C and D be 2-categories. A bi-adjunction between C and D com-prises the data of normal pseudo-functors L : C → D and R : D → C , and adjoint equiva-lences of categories Φ C,D : C ( C, RD ) ≃ −→ D ( LC, D )pseudo-natural in C ∈ C op and D ∈ D . Remark . We wish to draw the reader’s attention to the following relationship betweenbi-adjunctions and bi-representations. If L and R are embroiled in a bi-adjunction, thenin particular for each object D ∈ D we may observe that we have a bi-representationΦ − ,D : C ( − , RD ) ≃ = ⇒ D ( L − , D )of the normal pseudo-functor D ( L − , D ). In this sense, bi-adjunctions are “locally” bi-representations.The major goal of this section is to provide a converse to the above observation. Thatis, we will concentrate our efforts on establishing that being “locally bi-represented” is, infact, enough to determine a right bi-adjoint in the sense of Definition 7.1. Such a result is ofcourse expected by analogy to the ordinary categorical version. Giving such a formulationof bi-adjunctions in terms of bi-representations allows us to apply Theorem 6.8 and therebygive a characterisation of bi-adjunctions in terms of bi-terminal objects in pseudo-slices. Theorem 7.3.
Let C and D be -categories, and ( L, δ ) : C → D be a normal pseudo-functor. The following statements are equivalent.(i) The normal pseudo-functor L : C → D has a chosen right bi-adjoint R : D → C .(ii) For all objects D ∈ D , there is a chosen bi-representation ( RD, Ψ D ) of D ( L − , D ) ,where Ψ D : C ( − , RD ) ≃ = ⇒ D ( L − , D ) in Ps ( C op , Cat ) is a pseudo-natural adjointequivalence. In order to prove this theorem we will make use of some purely formal results about thenature of bi-representations and bi-adjunctions. These arguments depend crucially uponthe apparatus of a 2-dimensional Yoneda lemma – see [7, § nps here. Let us first recall the Yoneda embedding2-functor. Notation . Let C be a 2-category. We denote the Yoneda embedding 2-functor by Y : C → Ps ( C op , Cat ) . This 2-functor sends an object C ∈ C to the 2-functor Y C := C ( − , C ) : C op → Cat , andacts in the obvious way on hom-categories.
We will make extensive use of the full sub-2-category on the image of the Yoneda 2-functor, but this 2-category is isomorphic to the following.
Definition 7.5.
Let C be a 2-category. We define a 2-category C Y with the same objectsas C and whose hom-categories are given by C Y ( C, C ′ ) := Ps ( C op , Cat )( Y C , Y C ′ )for all C, C ′ ∈ C . Composition operations are given by those of Ps ( C op , Cat ). Remark . The 2-category C Y is isomorphic to the full sub-2-category of Ps ( C op , Cat )on the objects of the form Y C for C ∈ C . Observe that we therefore have the followingfactorisation of 2-functors C Ps ( C op , Cat ) C Y YY where Y is the identity on objects.We have avoided defining C Y as “the full sub-2-category of bi-representables” as thisis problematic inasmuch as objects are concerned: we will need to know which object isassociated to a given bi-representable functor, but a priori any such object is only defined upto equivalence. One way to solve this is to chose, for each bi-representable, a representingobject, and the result is precisely our 2-category C Y above.The fact that the 2-functor Y is an embedding of 2-categories may be reformulated asthe fact that Y mediates an identity-on-objects pseudo-equivalence between C and C Y . Lemma 7.7 (2-dimensional Yoneda lemma) . Let C be a -category. Then the normalpseudo-functors Y : C → C Y is part of a pseudo-equivalence.Proof. Since Y is the identity on objects it is enough to show, by Remark 3.11, that itinduces equivalences between the hom-categories C ( C, C ′ ) ≃ Ps ( C op , Cat )( Y C , Y C ′ ) = C Y ( C, C ′ ) , for all objects C, C ′ ∈ C . But this is the case by [7, Lemma 8.3.12]. (cid:4) Remark . As we have shown that Y is part of a pseudo-equivalence, we have in fact con-structed a normal pseudo-functor E : C Y → C together with pseudo-natural isomorphisms η : id C ∼ = = ⇒ EY and ε : YE ∼ = = ⇒ id C Y .Should we unravel the proof, we may see that E is the identity on objects, and acts onhom-categories C Y ( C, C ′ ) by first taking the C -component of the pseudo-natural transfor-mation or modification, and then evaluating it at id C ∈ Y C ( C ) = C ( C, C ). In fact, closerinspection reveals that id C = EY . Lemma 7.9.
Let C and D be -categories. Suppose that Q : D → Ps ( C op , Cat ) is a normalpseudo-functor such that, for all objects D ∈ D , QD = Y RD for a chosen object RD ∈ C .Then there is a normal pseudo-functor R : D → C and a pseudo-natural isomorphism Y R ∼ = Q in Ps ( D , Ps ( C op , Cat )) . I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 45
Proof.
First note that the image of the normal pseudo-functor Q : D → Ps ( C op , Cat ) iscontained in the full sub-2-category C Y of Ps ( C op , Cat ). That is, we have the followingfactorisation
D Ps ( C op , Cat ) C Y QQ ,where QD = RD , for all objects D ∈ D . Using this we define the normal pseudo-functor R : D → C to be the composite D Q −→ C Y E −→ C . Observe that by Lemma 7.7 we have a pseudo-natural isomorphism Y R = YE Q ∼ = Q in Ps ( D , C Y ). By post-composing with the inclusion C Y ֒ → Ps ( C op , Cat ) this gives apseudo-natural isomorphism Y R ∼ = Q in Ps ( D , Ps ( C op , Cat )). (cid:4)
Remark . Recall from Proposition 2.8 that Ps ( D , Ps ( C op , Cat )) ∼ = Ps ( C op × D , Cat ),so that we may recast the above lemma. Let Q : C op × D → Cat be a normal pseudo-functor such that, for all objects D ∈ D , Q ( − , D ) = Y RD for a chosen object RD ∈ C .Then we may apply the above lemma to obtain that there is a normal pseudo-functor R : D → C and a pseudo-natural isomorphism C ( − , R − ) ∼ = Q in Ps ( C op × D , Cat ).Now we are in a position to give a proof of Theorem 7.3.
Proof (Theorem 7.3) . First note that Remark 7.2 directly gives that (i) implies (ii). Weshow the other implication.Suppose (ii), that is, for all objects D ∈ D , we have a bi-representation ( RD, Ψ D )of D ( L − , D ), i.e., a pseudo-natural adjoint equivalence Ψ D : C ( − , RD ) ≃ = ⇒ D ( L − , D ).We want to construct the data of a normal pseudo-functor R : D → C and a pseudo-natural adjoint equivalence Φ − , − : C ( − , R − ) ≃ −→ D ( L − , − ) in Ps ( C op × D , Cat ). Forthis, we will simultaneously construct a pseudo-functor (
Q, φ ) : C op × D → Cat such that Q ( C, D ) = C ( C, RD ) = Y RD ( C ) for all ( C, D ) ∈ C op × D along with a pseudo-naturaladjoint equivalence Γ : Q ≃ = ⇒ D ( L − , − ) in Ps ( C op × D , Cat ). Note that while our con-struction of Q below does not necessarily yield a normal pseudo-functor, we may apply anormalisation argument such as [11, Proposition 5.2] to construct a normal pseudo-functor Q n which agrees with Q on objects and a pseudo-natural isomorphism ν : Q n ∼ = = ⇒ Q . Fi-nally, by applying Remark 7.10 to Q n , we may extract a normal pseudo-functor R : D → C and a pseudo-natural isomorphism ξ : C ( − , R − ) ∼ = = ⇒ Q n in Ps ( C op × D , Cat ). Then thepseudo-natural adjoint equivalence Φ can be obtained as the following composite C ( − , R − ) ∼ = = ⇒ ξ Q n ∼ = = ⇒ ν Q ≃ = ⇒ Γ D ( L − , − ) . It remains to construct the pseudo-functor Q : C op × D → Cat and pseudo-naturaladjoint equivalence Γ : Q ≃ = ⇒ D ( L − , − ).On objects C ∈ C and D ∈ D , we define Q ( C, D ) := C ( C, RD ) where RD is thechosen representing object which exists by assumption, and we define Γ C,D as the adjoint equivalence Γ
C,D := Ψ DC : Q ( C, D ) = C ( C, RD ) ≃ −→ D ( LC, D ) . On morphisms c : C → C ′ in C and d : D → D ′ in D , we must define Q ( c, d ) and Γ c,d suchthat they fit in the following square: C ( C ′ , RD ) D ( LC ′ , D ) C ( C, RD ′ ) D ( LC, D ′ ) Γ C ′ ,D Γ C,D ′ Q ( c, d ) D ( Lc, d )Γ c,d ∼ = .To do this we, use the equivalence data (Ψ DC , (Ψ DC ) -1 , η DC , ε DC ) and set Q ( c, d ) to be thecomposite C ( C ′ , RD ) D ( LC ′ , D ) D ( LC, D ′ ) C ( C, RD ′ ) Ψ DC ′ D ( Lc, d ) (Ψ D ′ C ) -1 ,and Γ c,d to be the following pasting. C ( C ′ , RD ) D ( LC ′ , D ) D ( LC, D ′ ) D ( LC, D ′ ) C ( C, RD ′ ) Ψ DC ′ D ( Lc, d )(Ψ D ′ C ) -1 Ψ D ′ C Q ( c, d ) ( ε D ′ C ) -1 ∼ = .Next, on 2-morphisms α : c ⇒ c ′ in C and β : d ⇒ d ′ in D , we define Q ( α, β ) to be thefollowing pasting: C ( C ′ , RD ) D ( LC ′ , D ) D ( LC, D ′ ) C ( C, RD ′ ) Ψ DC ′ D ( Lc, d ) D ( Lc ′ , d ′ ) (Ψ D ′ C ) -1 D ( Lα, β ) .With this definition of Q on 2-morphisms, we can directly check that Γ is natural withrespect to this assignment. More precisely, the following pasting equality holds I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 47 C ( C ′ , RD ) D ( LC ′ , D ) C ( C, RD ′ ) D ( LC, D ′ ) Γ C ′ ,D Q ( c ′ , d ′ ) Γ C,D ′ Γ c ′ ,d ′ ∼ = D ( Lα, β ) C ( C ′ , RD ) D ( LC ′ , D ) C ( C, RD ′ ) D ( LC, D ′ ) Γ C ′ ,D Γ C,D ′ D ( c, d ) Q ( α, β ) Γ c,d ∼ = =since both sides are given by the following pasting. C ( C ′ , RD ) D ( LC ′ , D ) D ( LC, D ′ ) D ( LC, D ′ ) C ( C, RD ′ ) Ψ DC ′ D ( Lc, d ) D ( Lc ′ , d ′ )(Ψ D ′ C ) -1 Ψ D ′ C Q ( c ′ , d ′ ) ( ε D ′ C ) -1 ∼ = D ( Lα, β ) With that achieved, it remains to supply the data of the compositors and unitors of Q and verify the pseudo-naturality conditions of Γ with respect to these. We start withthe compositors. Let c : C → C ′ and c ′ : C ′ → C ′′ be composable morphisms in C and d : D → D ′ and d ′ : D ′ → D ′′ be composable morphisms in D . We define the 2-isomorphismcompositor φ ( c ′ ,d ) , ( c,d ′ ) : Q ( c, d ′ ) Q ( c ′ , d ) ∼ = = ⇒ Q ( c ′ c, d ′ d ) as the below pasting. C ( C ′′ , RD ) D ( LC ′′ , D ) D ( LC ′ , D ′ ) C ( C ′ , RD ′ ) D ( LC ′ , D ′ ) D ( LC, D ′′ ) D ( LC, D ′′ ) Ψ DC ′′ D ( Lc ′ , d ) (Ψ D ′ C ′ ) -1 Ψ D ′ C ′ D ( Lc, d ′ )(Ψ D ′′ C ) -1 D ( L ( c ′ c ) , d ′ d ) ε D ′ C ′ ∼ = δ ∗ c,c ′ ∼ = Q ( c ′ c, d ′ d ) Q ( c ′ , d ) Q ( c, d ′ ) From this definition, the definition of Q on 2-morphisms in terms of L , and the propertiesof the compositor δ of L , we may directly verify that φ is associative and is natural withrespect to 2-morphisms.We need to check that Γ is compatible with the compositors φ , namely that the followingpasting equality holds. C ( C ′′ , RD ) C ( C ′′ , RD ) D ( LC ′′ , D ) C ( C ′ , RD ′ ) D ( LC ′ , D ′ ) C ( C, RD ′′ ) D ( LC, D ′′ ) C ( C, RD ′′ ) Γ C ′′ ,D Γ C ′ ,D ′ Γ C,D ′′ Q ( c ′ , d ) Q ( c, d ′ ) Q ( c ′ c, d ′ d ) D ( Lc ′ , d ) D ( Lc, d ′ )Γ c ′ ,d ∼ = Γ c,d ′ ∼ = φ ( c ′ ,d ) , ( c,d ′ ) ∼ = C ( C ′′ , RD ) D ( LC ′′ , D ) C ( C, RD ′′ ) D ( LC, D ′′ ) Q ( c ′ c, d ′ d ) = D ( L ( c ′ c ) , d ′ d ) D ( LC ′ , D ′ ) D ( Lc ′ , d ) D ( Lc, d ′ )Γ C ′′ ,D Γ C,D ′′ Γ c ′ c,d ′ d ∼ = δ ∗ c,c ′ ∼ = By direct expansion of definitions, we see that both pastings reduce to the following pasting. C ( C ′′ , RD ) D ( LC ′′ , D ) D ( LC ′ , D ′ ) D ( LC, D ′′ ) D ( LC, D ′′ ) C ( C, RD ′′ ) Ψ DC ′′ D ( L ( c ′ c ) , d ′ d )(Ψ D ′′ C ) -1 Ψ D ′′ C Q ( c ′ c, d ′ d ) D ( Lc ′ , d ) D ( Lc, d ′ )( ε D ′′ C ) -1 ∼ = δ ∗ c,c ′ ∼ = To complete the proof it remains to deal with the unitors. Given objects C ∈ C and D ∈ D , recall that Q (id C , id D ) is given by the following composite C ( C, RD ) D ( LC, D ) C ( C, RD ) Ψ DC (Ψ DC ) -1 , I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 49 since D ( L (id C ) , id D ) = id D ( LC,D ) by normality of L . We define the 2-isomorphism unitor φ ( C,D ) as the unit C ( C, RD ) C ( C, RD ) D ( LC, D ) Ψ DC (Ψ DC ) -1 η DC ∼ = .From this definition and the triangle equalities for ( η DC , ε DC ), we may directly verify that φ is satisfies the unitality conditions and that Γ is compatible with the unitor φ . Thiscompletes the constructions of Q and Γ and proves the theorem. (cid:4) We have now successfully shown that bi-adjunctions (
L, R,
Φ) are equivalently familiesof bi-representations of D ( L − , D ), indexed by the objects D ∈ D . In the remainder of thesection our goal is to combine Theorem 6.8 and Theorem 7.3 in order to obtain the followingcharacterisation of bi-adjunctions in terms of bi-terminal objects in different pseudo-slices.Recall that (double) bi-terminal are defined as (double) bi-initial objects in the (horizontal)opposite.In the statement of the theorem below, the pseudo-slice double categories H L ↓↓ D aregiven by the following cospan in DblCat h,nps H C H L −−→ H D D ←− , and the pseudo-slice 2-categories L ↓ D and Ar ∗ L ↓ D are given by the following cospansin nps C L −→ D D ←− and Ar ∗ C Ar ∗ L −−−→ Ar ∗ D D ←− , respectively, for objects D ∈ D . Theorem 7.11.
Let C and D be -categories, and L : C → D be a normal pseudo-functor.The following statements are equivalent.(i) The normal pseudo-functor L : C → D has a right bi-adjoint R : D → C .(ii) For all objects D ∈ D , there is a chosen object RD ∈ C together with a chosenmorphism ε D : LRD → D in D such that ( RD, ε D ) is double bi-terminal in H L ↓↓ D .(iii) For all objects D ∈ D , there is a chosen object RD ∈ C together with a chosenmorphism ε D : LRD → D in D such that ( RD, ε D ) is bi-terminal in L ↓ D and ( RD, id ε D ) is bi-terminal in Ar ∗ L ↓ D .(iv) For all objects D ∈ D , there is a chosen object RD ∈ C together with a chosenmorphism ε D : LRD → D in D such that ( RD, id ε D ) is bi-terminal in Ar ∗ L ↓ D . The missing components for the proof of this theorem are canonical isomorphisms ofdouble categories el ( D ( L − , D )) ∼ = ( H L ↓↓ D ) op , as well as related canonical isomorphismsfor the 2-categories el ( D ( L − , D )) and mor ( D ( L − , D )). This is the content of the followingresults. Lemma 7.12.
Let C and D be -categories, L : C → D be a normal pseudo-functor, and D ∈ D be an object. There is a canonical isomorphism of double categories as in thefollowing commutative triangle. el ( D ( L − , D )) ( H L ↓↓ D ) op H C op ∼ = Π op Π Proof.
We describe the data of the double category el ( D ( L − , D )). Then, by a straight-forward comparison with the data in the double category H L ↓↓ D , which is the dual con-struction to the double category described in Remark 4.3 with the double functor F = H L being horizontal, we can see that the isomorphism above canonically holds.An object in el ( D ( L − , D )) is a pair ( C, f ) of objects C ∈ C and f ∈ D ( LC, D ),i.e., a morphism f : LC → D in D . A horizontal morphism ( c, ψ ) : ( C ′ , f ′ ) → ( C, f )in el ( D ( L − , D )) comprises the data of a morphism c : C → C ′ in C and an isomorphism ψ : f ∼ = −→ D ( Lc, D ) f ′ in D ( LC, D ), i.e., a 2-isomorphism in D LCLC ′ D Lc ff ′ ψ ∼ = .Note that this corresponds to a morphism ( c, ψ ) : ( C, f ) → ( C ′ , f ′ ) in H L ↓↓ D , and itis the reason why we need to take the horizontal opposite ( H L ↓↓ D ) op . A vertical mor-phism α : ( C, f ) (
C, g ) in el ( D ( L − , D )) is a morphism α : f → g in D ( LC, D ), i.e.,a 2-morphism α : f ⇒ g between morphisms f, g : LC → D in D . Finally, a square γ : ( α ′ ( c,ψ )( d,ϕ ) α ) is a 2-morphism γ : c ⇒ d in C satisfying the pasting equality in Defini-tion 6.3 (iv), which can be translated into the following pasting equality in D . LCLC ′ D LcLd Lγ ff ′ g ′ α ′ ψ ∼ = = LCLC ′ D Lc fgg ′ ϕ ∼ = α (cid:4) Corollary 7.13.
Let C and D be -categories, L : C → D be a normal pseudo-functor, and D ∈ D be an object. There are canonical isomorphisms of -categories as in the followingcommutative triangles. I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 51 el ( D ( L − , D )) ( L ↓ D ) op C op ∼ = π op π mor ( D ( L − , D )) (Ar ∗ L ↓ D ) op Ar ∗ C op ∼ = π op π Proof.
This follows directly from the definitions of el and mor , Lemma 7.12 and Proposi-tion 4.8. (cid:4) The proof of Theorem 7.11 now follows in a straightforward manner.
Proof (Theorem 7.11) . By Theorems 6.8 and 7.3 and Lemma 7.12, we see that (i) and (ii) areequivalent. The equivalences of (ii), (iii), and (iv) follow from Theorem 6.8, Lemma 7.12,and Corollary 7.13. (cid:4)
Remark . Although we have proven this result by means of formal arguments involvinga reformulation of a 2-dimensional Yoneda lemma (see Lemma 7.7), these details are nota necessary feature of the proof of this theorem. For the reader for whom such devicesare unfamiliar or otherwise constitute a significant detour, we note here that a pleasinglydirect (if somewhat lengthy) proof of this theorem is possible and follows entirely similarlines to the proof of Theorem 6.8.Much as in the case of Theorem 6.14, we may improve Theorem 7.11 by assuming that C has chosen tensors by and that L preserves them, i.e., for every object C ∈ C , thereis a chosen tensor of LC by in D and we have an isomorphism L ( C ⊗ ) ∼ = ( LC ) ⊗ in D natural with respect to the defining cones. Theorem 7.15.
Let C and D be -categories, and L : C → D be a normal pseudo-functor.Suppose that C has chosen tensors by and that L preserves them. Then the followingstatements are equivalent.(i) The normal pseudo-functor L : C → D has a chosen right bi-adjoint R : D → C .(ii) For all objects D ∈ D , there is a chosen object RD ∈ C together with a chosenmorphism ε D : LRD → D in D such that ( RD, ε D ) is bi-terminal in L ↓ D .Remark . Note that tensors are a special case of a weighted 2-colimit construction.Therefore, if a 2-category C has tensors by and L : C → D is a left bi-adjoint, it preservesin particular all tensors by . In this way, this additional hypothesis on L is entirelyanodyne in the following sense: given an L which we suspect to be a left bi-adjoint, inorder to apply the above theorem we would need to know that L preserves tensors by ,but this should be part of a “background-check” on L in the first place. Proof (Theorem 7.15) . By Theorems 7.11 and 6.14, it is enough to show that the normalpseudo-functor D ( L − , D ) : C op → Cat preserves powers by . This is indeed the case sinceit follows from the fact that L preserves tensors by that D ( L ( C ⊗ ) , D ) ∼ = D (( LC ) ⊗ , D ) ∼ = Cat ( , D ( LC, D )) . (cid:4) Weighted bi-limits.
The primary and indeed motivating application of this theoryis to the notion of 2-dimensional limits. In [3, Counter-example 2.12], we give an exampleof a 2-terminal object in the slice 2-category of cones over a 2-functor F : I → C that doesnot give a 2-limit of F . This also gives a counter-example of a bi-terminal object in thepseudo-slice 2-category of cones over F which is not a bi-limit of F , as explained in [3, § C has tensors by , then 2-limits and2-terminal objects in the slice do correspond precisely. But we deferred the correspondingresult for bi-limits to this document.With this in mind, we now apply Theorem 6.8 to the case of (weighted) bi-limits inorder to obtain a correct characterisation in terms of bi-terminal objects. We further provethe deferred results for (weighted) bi-limits involving tensors by , which are obtained asa direct application of Theorem 6.14.Let us begin by recalling the definition of a weighted bi-limit. Definition 7.17.
Let I and C be 2-categories, and let F : I → C and W : I → Cat benormal pseudo-functors. A weighted bi-limit of F by W is a pair ( X, λ ) of an object X ∈ C together with a pseudo-natural transformation λ : W ⇒ C ( X, F − ) in Ps ( I , Cat ),such that, for every object C ∈ C , pre-composition by λ induces an adjoint equivalence ofcategories λ ∗ ◦ C ( − , F ) : C ( C, X ) ≃ −→ Ps ( I , Cat )( W, C ( C, F − )) , where C ( − , F ) : C op → Ps ( I , Cat ) is the normal pseudo-functor sending an object C ∈ C to the normal pseudo-functor C ( C, F − ) : I → Cat . Remark . Note that a weighted bi-limit (
X, λ ) induces a 2-natural adjoint equivalence λ ∗ ◦ C ( − , F ) : C ( − , X ) ≃ = ⇒ Ps ( I , Cat )( W, C ( − , F )) , so that we can see that weighted bi-limits are, in particular, bi-representations of the 2-functor Ps ( I , Cat )( W, C ( − , F )) : C op → Cat . Conversely, if we have a bi-representation(
X, ρ ), with ρ : C ( − , X ) ≃ = ⇒ Ps ( I , Cat )( W, C ( − , F )) a pseudo-natural adjoint equivalencein Ps ( C op , Cat ), we may set λ := ρ X (id X ) : W ⇒ C ( X, F − ). Then by Corollary 6.10, ρ = λ ∗ ◦ C ( − , F ) is a 2-natural adjoint equivalence, that is, a weighted bi-limit of F by W .We now aim to apply Theorem 6.8 to this setting in order to obtain a characterisationof weighted bi-limits in terms of bi-initial objects in different pseudo-slices.In the statement of the theorem below, the pseudo-slice double category W ↓↓ H C ( − , F )is given by the following cospan in DblCat h,nps W −→ H Ps ( I , Cat ) H C ( − ,F ) ←−−−−−− H C op , and the pseudo-slice 2-categories W ↓ C ( − , F ) and W ↓ Ar ∗ C ( − , F ) are given by the followingcospans in nps W −→ Ps ( I , Cat ) C ( − ,F ) ←−−−−− C op and W −→ Ar ∗ Ps ( I , Cat ) Ar ∗ C ( − ,F ) ←−−−−−−− Ar ∗ C op , respectively. I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 53
Theorem 7.19.
Let I and C be -categories, and let F : I → C and W : I → Cat benormal pseudo-functors. The following statements are equivalent.(i) There is a chosen weighted bi-limit ( X, λ ) of F by W .(ii) There is a chosen object X ∈ C together with a chosen pseudo-natural transfor-mation λ : W ⇒ C ( X, F − ) in Ps ( I , Cat ) such that ( X, λ ) is double bi-initial in W ↓↓ H C ( − , F ) .(iii) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : W ⇒ C ( X, F − ) in Ps ( I , Cat ) such that ( X, λ ) is bi-initial in W ↓ C ( − , F ) and ( X, id λ ) is bi-initial in W ↓ Ar ∗ C ( − , F ) .(iv) There is a chosen object X ∈ C together with a chosen pseudo-natural trans-formation λ : W ⇒ C ( X, F − ) in Ps ( I , Cat ) such that ( X, id λ ) is bi-initial in W ↓ Ar ∗ C ( − , F ) .Remark . At a cursory reading it may surprise readers to learn that weighted bi-limits are characterised as somehow bi-initial rather than bi-terminal objects. However, such astatement belies their true nature. When we unravel definitions, we see that the doublebi-initiality in the pseudo-slice double category W ↓↓ H C ( − , F ) is expressed over H C op ,and its presence is indicative of a “mapping in” property for the limiting object in C –precisely as one might expect from bi-limits.The proof of Theorem 7.19 is deferred to the end of the section, as we need to estab-lish some technical results (Lemma 7.26 and Corollary 7.27) relating the double category el ( Ps ( I , Cat )( W, C ( − , F ))) to the pseudo-slice double category W ↓↓ H C ( − , F ), and sim-ilarly so for el ( Ps ( I , Cat )( W, C ( − , F )) and mor ( Ps ( I , Cat )( W, C ( − , F ))).Assuming Theorem 7.19, we now specialise Theorem 6.14 to the weighted bi-limit case.Here we only need to assume that the 2-category C has tensors by as these are preservedautomatically in this special case. Theorem 7.21.
Let I and C be -categories, and let F : I → C and W : I → Cat be normalpseudo-functors. Suppose that C has chosen tensors by . Then the following statementsare equivalent.(i) There is a chosen weighted bi-limit ( X, λ ) of F by W .(ii) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : W ⇒ C ( X, F − ) in Ps ( I , Cat ) such that ( X, λ ) is bi-initial in W ↓ C ( − , F ) .Proof. By Theorems 7.19 and 6.14, it is enough to show that the normal pseudo-functor Ps ( I , Cat )( W, C ( − , F )) : C op → Cat preserves powers by . Indeed we have that Ps ( I , Cat )( W, C ( C ⊗ , F − )) ∼ = Ps ( I , Cat )( W, Cat ( , C ( C, F − ))) ∼ = Cat ( , Ps ( I , Cat )( W, C ( C, F − ))as powers in Ps ( I , Cat ) are given by point-wise powers in
Cat . (cid:4) In the special case where the weight W is constant at the terminal category, i.e., W = ∆ ,the characterisation of weighted bi-limits by ∆ , called conical bi-limits , takes a morefamiliar form. In the statement of the corollary below, the pseudo-slice double category H ∆ ↓↓ F isgiven by the following cospan in DblCat h,nps H C H ∆ −−→ H Ps ( I , C ) F ←− , and the pseudo-slice 2-categories ∆ ↓ F and Ar ∗ ∆ ↓ F are given by the following cospansin nps C ∆ −→ Ps ( I , C ) F ←− and Ar ∗ C Ar ∗ ∆ −−−→ Ar ∗ Ps ( I , C ) F ←− , respectively. Corollary 7.22.
Let I and C be -categories, and F : I → C be a normal pseudo-functor.The following statements are equivalent.(i) There is a bi-limit ( X, λ ) of F .(ii) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : ∆ X ⇒ F in Ps ( I , C ) such that ( X, λ ) is double bi-terminal in H ∆ ↓↓ F .(iii) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : ∆ X ⇒ F in Ps ( I , C ) such that ( X, λ ) is bi-terminal in ∆ ↓ F and ( X, id λ ) is bi-terminal in Ar ∗ ∆ ↓ F .(iv) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : ∆ X ⇒ F in Ps ( I , C ) such that ( X, id λ ) is bi-terminal in Ar ∗ ∆ ↓ F . The proof is deferred to the end of the section, where we prove the needed technicalresults (Lemma 7.28 and Corollary 7.29) relating pseudo-slices of weighted cones for theweight W = ∆ to the usual pseudo-slices of cones. Remark . As we already mentioned, we show in [3, §
5] that the data of a bi-limit of F is not fully captured by a bi-terminal object in the usual pseudo-slice 2-category ∆ ↓ F of cones. Statement (iv) above shows that by “shifting” the pseudo-slice ∆ ↓ F to thepseudo-slice Ar ∗ ∆ ↓ F whose objects are modifications between cones, we can successfullycapture the additional data we require.In particular, by comparing Corollary 7.22 with the characterisation of bi-adjunctionsof Theorem 7.11, we can see bi-limits as a right bi-adjoint. Namely: Remark . Let I and C be 2-categories. If every normal pseudo-functor F : I → C has achosen bi-limit, then this bi-limit construction extends to a right bi-adjoint to the diagonal2-functor ∆ : C → Ps ( I , C ).Assuming Corollary 7.22 and specialising Theorem 7.21 to the case W = ∆ , we obtainthe promised results of [3, Proposition 5.5]. Corollary 7.25.
Let I and C be -categories, and F : I → C be a normal pseudo-functor.Suppose that C has chosen tensors by . Then the following statements are equivalent.(i) There is a bi-limit ( X, λ ) of F .(ii) There is a chosen object X ∈ C together with a chosen pseudo-natural transforma-tion λ : ∆ X ⇒ F in Ps ( I , C ) such that ( X, λ ) is bi-terminal in ∆ ↓ F . I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 55
Proof.
This follows directly from Corollary 7.22 and Theorem 7.21 applied to W = ∆ . (cid:4) The rest of this section will be devoted to the technical lemmas supporting the proofs ofTheorem 7.19 and Corollary 7.22 which give general characterisations of weighted bi-limitsand conical bi-limits.
Lemma 7.26.
Let I and C be -categories, and let F : I → C and W : I → Cat be normalpseudo-functors. There is a canonical isomorphism of double categories as in the followingcommutative triangle. el ( Ps ( I , Cat )( W, C ( − , F ))) W ↓↓ H C ( − , F ) H C op ∼ = ΠΠ Proof.
We describe the data of the double category el ( Ps ( I , Cat )( W, C ( − , F ))). By astraightforward comparison with the data described in Remark 4.3 of the pseudo-slicedouble category W ↓↓ H C ( − , F ), we will see that the isomorphism above canonically holds.An object in el ( Ps ( I , Cat )( W, C ( − , F ))) consists of a pair ( C, κ ) of an object C ∈ C anda pseudo-natural transformation κ : W ⇒ C ( C, F − ) in Ps ( I , Cat ). A horizontal morphism( c, Ψ) : ( C ′ , κ ′ ) → ( C, κ ) in el ( Ps ( I , Cat )( W, C ( − , F ))) consists of a morphism c : C → C ′ in C together with an invertible modification Ψ in Ps ( I , Cat ) of the form W C ( C ′ , F − ) C ( C, F − ) κ ′ κ C ( c, F − )Ψ ∼ = .A vertical morphism Θ : ( C, κ ) (
C, µ ) in el ( Ps ( I , Cat )( W, C ( − , F ))) is a modificationΘ : κ µ between pseudo-natural transformations κ, µ : W → C ( C, F − ) in Ps ( I , Cat ).Finally, a square γ : (Θ ′ ( c, Ψ)( d, Φ) Θ) is a 2-morphism γ : c ⇒ d in C satisfying the pastingequality in Definition 6.3 (iv), which can be translated into a pasting equality for themodification C ( γ, F − ) in Ps ( I , Cat ). (cid:4) Corollary 7.27.
Let I and C be -categories, and let F : I → C and W : I → Cat be normalpseudo-functors. There are canonical isomorphisms of -categories as in the followingcommutative triangles. el ( Ps ( I , Cat )( W, C ( − , F ))) W ↓ C ( − , F ) C op ∼ = π op π mor ( Ps ( I , Cat )( W, C ( − , F ))) W ↓ Ar ∗ C ( − , F )Ar ∗ C op ∼ = π op π Proof.
This follows directly from the definitions of el and mor , Lemma 7.26 and Proposi-tion 4.8. (cid:4) With Lemma 7.26 and Corollary 7.27 above established we may now give a direct proofof Theorem 7.19.
Proof (Theorem 7.19) . Recall from Remark 7.18 that a weighted bi-limit of F by W isequivalently a bi-representation of the 2-functor Ps ( I , Cat )( W, C ( − , F )). Then the resultis obtained as a direct application of Theorem 6.8 using Lemma 7.26 and Corollary 7.27. (cid:4) In the conical case we may simplify the pseudo-slices above through the below compu-tations to obtain a proof of Corollary 7.22.
Lemma 7.28.
Let I and C be -categories, and F : I → C be a normal pseudo-functor.There is a canonical isomorphism of double categories as in the following commutativetriangle. ∆ ↓↓ H C ( − , F ) ( H ∆ ↓↓ F ) op H C op ∼ =Π Π op Proof.
This follows from the fact that, given an object C ∈ C , a pseudo-natural transfor-mation κ : ∆ ⇒ C ( C, F − ) in Ps ( I , Cat ) corresponds to a pseudo-natural transformation κ : ∆ C ⇒ F in Ps ( I , C ). (cid:4) Corollary 7.29.
Let I and C be -categories, and F : I → C be a normal pseudo-functor.There are canonical isomorphisms of -categories as in the following commutative triangles. ∆ ↓ C ( − , F ) (∆ ↓ F ) op C op ∼ = π op π ∆ ↓ Ar ∗ C ( − , F ) (Ar ∗ ∆ ↓ F ) op Ar ∗ C op ∼ = π op π Proof.
This follows directly from Lemma 7.28 and Proposition 4.8. (cid:4)
Finally we obtain a straightforward proof of Corollary 7.22.
I-INITIAL OBJECTS AND BI-REPRESENTATIONS ARE NOT SO DIFFERENT 57
Proof (Corollary 7.22) . This result is obtained by applying Theorem 7.19 to the specialcase where W = ∆ and using Lemma 7.28 and Corollary 7.29. (cid:4) References [1] C. Auderset. Adjonctions et monades au niveau des 2-cat´egories.
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