aa r X i v : . [ m a t h . C T ] J a n Discrete Double Fibrations
Michael LambertJanuary 19, 2021
Abstract
Presheaves on a small category are well-known to correspond via a category of elementsconstruction to ordinary discrete fibrations over that same small category. Work of R. Par´eproposes that presheaves on a small double category are certain lax functors valued in thedouble category of sets with spans. This paper isolates the discrete fibration concept cor-responding to this presheaf notion and shows that the category of elements constructionintroduced by Par´e leads to an equivalence of virtual double categories.
Contents Introduction
A discrete fibration is functor F : F → C such that for each arrow f : C → F Y in C , there is aunique arrow f ∗ Y → Y in F whose image under F is f . Such functors correspond to ordinarypresheaves DFib ( C ) ≃ [ C op , Set ]via a well-known “category of elements” construction [Mac98]. This is a “representation theorem”in the sense that discrete fibrations over C correspond to set-valued representations C .Investigation of higher-dimensional structures leads to the question of analogues of such well-established developments in lower-dimensional settings. In the case of fibrations, for example,ordinary fibrations have their analogues in “2-fibrations” over a fixed base 2-category [Buc14].Discrete fibrations over a fixed base 2-category have their 2-dimensional version in “discrete 2-fibrations” [Lam20]. The justification that the notion of (discrete) fibration is the correct one ineach case consists in the existence of a representation theorem taking the form of an equivalencewith certain representations of the base structure via a category of elements construction.In the double-categorical world, R. Par´e proposes [Par´e11] that certain span-valued, lax func-tors on a double category B are presheaves on B . This paper aims to isolate the notion of “discretedouble fibration” corresponding to this notion of presheaf. As justification of the proposed defini-tion, Par´e’s double category of elements will be used to exhibit a representation theorem on thepattern of those reviewed above. The technically interesting part of the results is that ultimatelyan equivalence of virtual double categories is achieved. For this the language of monoids andmodules in virtual double categories as in [Lei04] will be used. To give away the answer completely, a discrete double fibration should be a category object inthe category of discrete fibrations. This follows the approach to double category theorizing that a“double ” is a category object in the category of ’s. The point of the paper is thus to show thatthis expected definition comports with Par´e’s notion of presheaf by exhibiting the representationtheorem as discussed above. This will be proved directly for all the 1-categorical structure involved.However, lax functors and their transformations are fragments of a higher double-dimensionalstructure. That is, modules and their multimodulations make lax functors into a virtual doublecategory. The question, then, arises as to the corresponding virtual double category structureon the discrete fibration side of the desired equivalence. It turns out that there is already well-established language for these structures from [Lei04].For a category with finite limits C , the category of monoids in the bicategory of spans in C isequivalent to the category of category objects in C . In fact the correspondence is much strongerthan this, since – under the assumption of a fixed universe of concrete sets – the data on eitherside of the equivalence is the same modulo rearrangement of tuples. This general result leads tothe special case for C = DFib , yielding the equivalence
Mon ( Span ( DFib )) ≃ Cat ( DFib ) . This is the main clue leading to a natural candidate for virtual double category structures ondiscrete double fibrations. For monoids in the bicategory part of any double category D form the1-categorical part of the virtual double category of monoids and modules in D . That is, there is avirtual double category M od ( D ) of modules whose underlying 1-category is precisely monoids in2he bicategory part of D . This leads to the natural candidate for virtual double category structureon discrete double fibrations as M od ( S pan ( DFib )). The ultimate objective of the paper is thusto show that an appropriate slice of such modules is equivalent to the virtual double category oflax presheaves via the double category of elements construction.
Section 2 reviews lax functors, their transformations and the elements construction from Par´e’s[Par´e11]. The main definition of “discrete double fibration” is given as a double functor P : E → B for which both P and P are discrete fibrations. This is equivalently a category object in thecategory of discrete fibrations. The section culminates in the first main contribution of the paper,proved as Theorem 2.4.3 below. This result says that for any strict double category B , there is anequivalence of 1-categories DFib ( B ) ≃ Lax ( B op , S pan )between the category of discrete double fibrations over B and span-valued lax functors on B ,induced by the double category of elements construction. This is achieved by constructing apseudo-inverse for the double category of elements.Section 3 introduces the virtual double category structure on lax span-valued functors as in[Par´e11] and [Par´e13] with the goal of extending the theorem above to an equivalence of virtualdouble categories. It turns out that a convenient language for setting up the virtual double categorystructure on discrete double fibrations is that of monoids and modules in virtual double categoriesas in [Lei04] and later [CS10].Section 4 extends the elements construction and its pseudo-inverse to functors of virtual doublecategories and culminates in the second result, proved as Theorem 4.3.5. That is, for any strictdouble category B , there is an equivalence of virtual double categories D Fib ( B ) ≃ L ax ( B op , S pan )induced by the elements functor. Double categories go back to Ehresmann [Ehr63]. Other references include [GP99], [GP04], and[Shul08]. By the phrase “double category” we will always mean a strict double category, which isa category object in categories. Adaptations for the pseudo-case in some cases are easy to makebut tedious; in other cases these are more subtle and require separate treatment.One quirk of our presentation is that, owing to different conventions concerning which arrows are“vertical” and which are “horizontal,” we prefer to use the more descriptive language of “arrows,”“proarrows,” and “cells.” Arrows are of course the ordinary arrows between objects (the horizontalarrows in [GP99] and [GP04] vs. the vertical ones in [Shul08] and [CS10]); whereas the proarrowsare the objects of the bicategory part (that is, the vertical ones of [GP99] and [GP04] vs. thehorizontal ones of [Shul08] and [CS10]). In this language, the arrows of the canonical example ofcategories and profunctors would be the ordinary functors while the proarrows are the profunctors.This choice is inspired by the language of a “proarrow equipment” in [Wood82] and [Wood85],which sought to axiomatize this very situation as a 2-category “equipped with proarrows.”Throughout use blackboard letters A , B , C , D for double categories. The 0-part of such D isthe category of objects and arrows, denoted by ‘ D ’. The 1-part is the category of proarrows and3ells, denoted by ‘ D ’. External composition is written with a tensor ‘ ⊗ ’. The external unit isdenoted with u : D → D . External source and target functors are src , tgt : D ⇒ D . Internalstructure is denoted by juxtaposition in the usual order. The terms “domain” and “codomain”refer to internal structure, whereas “source” and “target” refer to external structure. One resultrequires the notion of the “transpose” of a double category D , denoted here by D † . When D isstrict, D † is a strict double category and transposing extends to a functor ( − ) † : Dbl → Dbl onthe 1-category of strict double categories and double functors.Script letters C , D , X , denote ordinary categories with objects sets C and arrow sets C .Well-known 1-, 2-, and bi-categories referred to throughout are given in boldface such as Set , Cat , Span , and
Rel . We always refer to the highest level of structure commonly associated with theobjects of the category. Thus,
Cat is the 2-category of categories, unless specified otherwise. Themain exception is that
DFib is the ordinary category of discrete fibrations. Throughout we takefor granted the notions of monoid and category internal to a given category with finite limits.Some 2-categorical concepts are referenced but not in a crucial way. Common double-categoriesare presented in mixed typeface and named by their proarrows. In particular P rof is the doublecategory of profunctors; R el is the double category of sets and relations; S pan (rather than S et )is the double category of sets and spans. The main objects of study are lax functors between double categories. The definitions are recalledin this section. The double category of elements is recalled, and it is seen how its constructionleads naturally to the definition of a discrete double fibration.
As groundwork, recall here the definitions of lax functor and their natural transformations.
Definition 2.1.1 (See e.g. § § . A lax functor between (pseudo-)double categories F : A → B consists of assignments A F A f F f m F m α F α on objects, arrows, proarrows and cells, respecting domains, codomains, sources and targets; andlaxity cells
F A φ A u F A ✤ / / F A F A φ n,m F m ✤ / / F B
F n ✤ / / F CF A
F u A ✤ / / F A F A F ( n ⊗ m ) ✤ / / F C for each object A and composable proarrows m : A −7−→ B and n : B −7−→ C , all subject to the followingaxioms.1. [Internal Functoriality] The object and arrows assignments A F A , f F f above definean ordinary functor F : A → B ; and the proarrow and cell assignments define an ordinaryfunctor F : A → B with respect to internal identities and composition of cells.4. [Naturality] For any f : A → B , there is an equality · u F f
F f (cid:15) (cid:15) u F A ✤ / / · F f (cid:15) (cid:15) · φ A u F A ✤ / / ·· φ B ✤ / / · = · F u f F f (cid:15) (cid:15) ✤ / / · F f (cid:15) (cid:15) · F u B ✤ / / · · F u B ✤ / / · and for any externally composable cells α : m ⇒ p and β : n ⇒ q , there is an equality · F αF f (cid:15) (cid:15)
F m ✤ / / · F β (cid:15) (cid:15)
F n ✤ / / · F h (cid:15) (cid:15) · φ n,m F m ✤ / / · F n ✤ / / ·· φ q,p F p ✤ / / · F q ✤ / / · = · F ( β ⊗ α ) F f (cid:15) (cid:15) F ( n ⊗ m ) / / · F f (cid:15) (cid:15) · F ( q ⊗ p ) ✤ / / · · F ( q ⊗ p ) ✤ / / · of composite cells.3. [Unit and Associativity] Given a proarrow m : A −7−→ B , the unit laxity cells satisfy · φ A u F A ✤ / / · F u m (cid:15) (cid:15) F m ✤ / / · · ∼ = u F A ✤ / / · F m ✤ / / ·· φ m,uA F u A ✤ / / · F m ✤ / / · = · ∼ = F m / / ·· F ( m ⊗ u A ) ✤ / / · · F ( m ⊗ u A ) ✤ / / · and · F u m F m ✤ / / · φ B (cid:15) (cid:15) u F B ✤ / / · · ∼ = F m ✤ / / · u F B ✤ / / ·· φ uB,m F m ✤ / / · F u B ✤ / / · = · ∼ = F m / / ·· F ( u B ⊗ m ) ✤ / / · · F ( u B ⊗ m ) ✤ / / · and for any three composable proarrows m : A −7−→ B , n : B −7−→ C , and p : C −7−→ D , the laxitycells are associative in the sense that · F u m F m ✤ / / · φ p,n F n ✤ / / · F p ✤ / / · · φ n,m F m ✤ / / · F n ✤ / / · F u p F p ✤ / / ·· φ p ⊗ n,m ✤ / / · ✤ / / · = · φ p,n ⊗ m ✤ / / · ✤ / / ·· F (( p ⊗ n ) ⊗ m ) ✤ / / · · F ( p ⊗ ( n ⊗ m )) ✤ / / · F of the associativity iso cell( p ⊗ n ) ⊗ m ∼ = p ⊗ ( n ⊗ m )given with the structure of A .Notice that the structural isomorphism reduce to strict identities when A and B are strict dou-ble categories. A pseudo double functor is a lax double functor where the laxity cells areisomorphisms; a strict double functor is one whose laxity cells are identities. Remark . Generally speaking, use lower-case Greek letters for the laxity cells correspondingto the Latin capital letter for the functor. Thus, F is a lax functor with laxity cells denoted by ‘ φ ’with subscripts; ‘ γ ’ is used for a lax functor G . Example 2.1.3 (Cf. § . The usual object functor ob( − ) : Cat → Set extends to alax double functor Ob ( − ) : P rof → S pan in the following way. On a category C take the objectset C to be the image as usual; likewise take the object part F : C → D to be the image of agiven functor F : C → D . On a profunctor P : C −7−→ D – that is, a functor P : C op × D → Set –take the image to be the disjoint union Ob ( P ) := a C,D P ( C, D ) . The assignment on a given transformation of profunctors is induced by the universal property ofthe coproduct. This is a bona fide lax functor
Definition 2.1.4 (Cf. § . Let
F, G : A ⇒ B denote lax functors with laxity cells φ and γ . A natural transformation τ : F → G assigns to each object A an arrow τ A : F A → GA and to each proarrow m : A −7−→ B a cell F A τ m F m ✤ / / τ A (cid:15) (cid:15) F B τ B (cid:15) (cid:15) GA Gm ✤ / / GB in such a way that the following axioms are satisfied.1. [Naturality] Given any arrow f : A → B , the usual naturality square F A
F f / / τ A (cid:15) (cid:15) F B τ B (cid:15) (cid:15) GA Gf / / GB commutes; and given any cell α : m ⇒ n , the corresponding naturality square commutes inthe sense that the compositions · F α (cid:15) (cid:15)
F m ✤ / / · (cid:15) (cid:15) · τ m F m ✤ / / ·· τ n ✤ / / · = · Gα (cid:15) (cid:15) ✤ / / · (cid:15) (cid:15) · Gn ✤ / / · · Gn ✤ / / · are equal. 6. [Functoriality] For any object A , the compositions · u τA (cid:15) (cid:15) u F A ✤ / / · (cid:15) (cid:15) · φ A u F A ✤ / / ·· γ A ✤ / / · = · τ uA (cid:15) (cid:15) ✤ / / · (cid:15) (cid:15) · Gu A ✤ / / · · Gu A ✤ / / · are equal; and for any composable proarrows m : A −7−→ B and n : B −7−→ C , the composites · τ m τ A (cid:15) (cid:15) F m ✤ / / · τ n (cid:15) (cid:15) F n ✤ / / · τ C (cid:15) (cid:15) · φ n,m F m ✤ / / · F n ✤ / / ·· γ n,m Gm ✤ / / · Gn ✤ / / · = · τ n ⊗ m τ A (cid:15) (cid:15) ✤ / / · τ C (cid:15) (cid:15) · G ( n ⊗ m ) ✤ / / · · G ( n ⊗ m ) ✤ / / · are equal.Let Lax ( A , B ) denote the category of lax functors A → B and their transformations. The com-position is “component-wise” and identity transformations are those with identity morphisms ineach of their components.In a few simple cases, this category can be described in terms of well-known structures. Theseequivalences are given by the “elements” construction at the beginning of the next subsection. Example 2.1.5.
A lax functor → S pan is essentially a small category. In fact, taking elementsinduces an equivalence Lax ( , S pan ) ≃ Cat . In terms of monoids and their morphisms (i.e. a 1-category of monoids and their morphisms in abicategory), this means that
Lax ( , S pan ) ≃ Mon ( Span ( Set )) . as 1-categories. This is a primordial example or special case of our main results, Theorem 2.4.3and Theorem 4.3.5. Example 2.1.6 (See § . If C is an ordinary category, let V C denote the “verticaldouble category” formed from the objects of C and using the morphisms of C as the proarrowswith only identity arrows and cells. There is then an equivalence Lax ( V C , S pan ) ≃ Cat / C of ordinary categories given by taking elements. 7 .2 Discrete Double Fibrations The fibration properties of the double category of elements construction lead to the axiomatizationof our notion of “discrete double fibration.” Let us recall the details of this double category andits associated projection functor.
Construction 2.2.1 ( § . Let F : B op → S pan denote a lax functor with laxity cells φ . The double category of elements E l ( F ) has1. as objects those pairs ( B, x ) with x ∈ F B ;2. as morphisms f : ( B, x ) → ( C, y ) those arrows f : B → C of B such that f ∗ y = x holds underthe action of the transition function f ∗ : F C → F B ;3. as proarrows (
B, x ) −7−→ ( C, y ) those pairs ( m, s ) consisting of a proarrow m : B −7−→ C and anelement s ∈ F m such that s = x and s = y both hold; and4. as cells ( m, s ) ⇒ ( n, t ) those cells θ of B as at left below for which the associated morphismof spans on the right A θf (cid:15) (cid:15) m ✤ / / B g (cid:15) (cid:15) F C f ∗ (cid:15) (cid:15) F n ( − ) o o θ ∗ (cid:15) (cid:15) ( − ) / / F D g ∗ (cid:15) (cid:15) C n ✤ / / D F A F m ( − ) o o ( − ) / / F B satisfies θ ∗ ( t ) = s .The 1-category structure on objects and morphisms is the same as the ordinary category of ele-ments. Internal composition of cells uses the internal composition in B and the strict equalitiesof the form θ ∗ ( t ) = s in the definition. External composition uses the given laxity morphisms.For example, given composable proarrows ( m, s ) : ( B, x ) −7−→ ( C, y ) and ( n, t ) : (
C, y ) −7−→ ( D, z ), thecomposite is defined as ( n, t ) ⊗ ( m, s ) := ( n ⊗ m, φ n,p ( s, t )) . Of course this is a well-defined proarrow. Identities and external composition of cells is similar.This makes E l ( F ) a double category. It is only as strict as B . There is a projection doublefunctor Π : E l ( F ) → B taking the indexing objects, morphisms, proarrows and cells to B . This isstrict even if B is pseudo.It is remarked in [Par´e11] that taking elements extends to a functor. Here are the details. Construction 2.2.2 (Morphism from a Transformation of Lax Functors) . Let τ : F → G denotea transformation of lax functors F, G : B op ⇒ S pan as in Definition 2.1.4. Define what will be astrict functor of double categories E l ( τ ) : E l ( F ) → E l ( G ) over B . On objects and arrows, take( D, x ) ( D, τ D x ) f f The arrow assignment is well-defined because f ∗ τ D = τ C f ∗ holds by the strict naturality conditionin the definition. This assignment is functorial by construction. Now, for proarrows, send( m, s ) ( m, τ m s ) α α similarly to the object and arrow assignments. Well-definition again follows from naturality. Func-toriality is by construction. The rest of the important content is in the following result.8 emma 2.2.3. Given a transformation τ : F → G of span-valued, lax functors, the assignmentsabove yield a morphism from Π : E l ( F ) → B to Π : E l ( G ) → B over B . This association isfunctorial, meaning that the elements functor E l ( − ) : Lax ( B op , S pan ) −→ Dbl / B valued in the slice of double categories over B is well-defined.Proof. It is left to see that E l ( τ ) : E l ( F ) → E l ( G ) preserves external composites and units. Thedefinition of external composition in the elements construction incorporates the laxity morphismscoming with F and G . So, the preservation of external composition reduces to the statementthat these laxity morphisms interact in the proper way with the components of τ used in thedefinition. But this is precisely what the “Functoriality” condition of Definition 2.1.4 axiomatizes.Unit preservation follows similarly.As above, the elements functor is valued in the slice of the category of double categories over B . However, those double functors strictly in its image possess certain fibration properties leadingto the definition of a “discrete double fibration.” First the properties: Lemma 2.2.4.
Let F : B op → S pan denote a lax double functor. The projection functors Π and Π underlying the canonical projection Π : E lt ( F ) → B are discrete fibrations.Proof. Here are the required lifts. Given (
D, x ) and f : C → D , the lift is f : ( C, f ∗ x ) → ( D, x )where f ∗ : F D → F C is the transition function. Similarly, given ( n, s ) for a proarrow n and s ∈ F n and a cell α : m ⇒ n , the cartesian cell above α with codomain ( n, s ) is( A, f ∗ x ) αf (cid:15) (cid:15) ( m,α ∗ s ) ✤ / / ( B, g ∗ y ) g (cid:15) (cid:15) ( C, x ) ( n,s ) ✤ / / ( D, y )where α ∗ : F n → F m is the transition function between the vertices of the spans. Notice that bythe fact that α ∗ induces a morphism of spans, the source and target of ( m, α ∗ s ) are well-defined.Such functors are equivalently characterized with a distinctly double categorical flavor. Proposition 2.2.5.
A double functor P : E → B between strict double categories for which P and P are discrete fibrations is equivalently a category object in DFib , and thus equivalently a monoidin the bicategory of spans in
DFib .Proof.
In the first place
DFib has strict pullbacks, so the statement itself makes sense. On theone hand, a category object in any subcategory of an arrow category closed under finite limits isan internal functor. Thus, a category in
DFib is a double functor. Its object and arrow parts mustbe objects of
DFib , that is, discrete fibrations. On the other hand, a double functor is a categoryobject in the arrow category of
Cat . But if P and P are discrete fibrations, then such P lives in Cat ( DFib ). A monoid in spans in categories is equivalently a double category; similarly a monoidin spans in the arrow category of
Cat is equivalently a double functor. Thus, the components ofsuch a double functor are discrete fibrations if and only if the double functor is in fact a monoidin spans in
DFib . 9 efinition 2.2.6. A discrete double fibration is a category object in DFib . A morphism ofdiscrete double fibrations is a pair of double functors (
H, K ) making a commutative square. Take
Cat ( DFib ) to be the category of discrete double fibrations. Let
DFib ( B ) denote the subcategoryof discrete double fibrations with fixed target B and morphisms with K = 1 B . This is equivalentlythe fiber of the codomain projection cod : Cat ( DFib ) → Dbl over B . Remark . Although the official definition has a succinct and philosophically appropriate phras-ing, as it is convenient, any of the equivalent characterizations of discrete double fibrations inProposition 2.2.5 may be used throughout.
Remark . The name used is “ discrete double fibration” because this is a discretization ofa more general concept of “double fibration.” This will be a double functor P : E → B whoseunderlying functors P and P are fibrations but that satisfy some further compatibility conditions. Remark . The results so far means that by fiat the elements functor is valued in discretedouble fibrations E l ( − ) : Lax ( B op , S pan ) −→ DFib ( B )over B . The purpose of this section is now to exhibit the pseudo-inverse yielding an equivalenceof categories. First end this subsection with some general results. Example 2.2.10 (Domain Projection) . For any double category, the domain projection functordom : B /X −→ B from the double slice B /X is a discrete double fibration. This is the image under the elementsfunctor E l ( − ) : Lax ( B op , S pan ) → DFib ( B ) of the canonical representable functor on X . Proposition 2.2.11.
A double functor P : E → B is a discrete double fibration over a strict doublecategory B if, and only if, the transpose square E † P † (cid:15) (cid:15) cod / / E † P † (cid:15) (cid:15) B † / / B † is a pullback in Cat .Proof.
Straightforward verification.
Remark . Lemma 2.2.11 is the analogue of the characterization of ordinary discrete fibrations,saying that a functor F : F → C is a discrete fibration if, and only if, the square F F (cid:15) (cid:15) d / / F F (cid:15) (cid:15) C d / / C is a pullback in Set . This characterization will be important in the monadicity developments inforthcoming work. 10 .3 Pseudo-Inverse on Objects and Horizontal Morphisms
The present goal is to construct a pseudo-inverse for the elements functor. This will be a functorfrom discrete double fibrations back to lax span-valued double functors. For a discrete doublefibration P : E → B , begin correspondences leading to a lax double functor F P : B op → S pan inthe following way. Construction 2.3.1 (Transition Morphisms) . On objects, take B E B , the set of objects of E over B ∈ B via P . Call this the “fiber” over B . Since P is a discrete fibration, for everyhorizontal morphism f : B → C , there is a corresponding transition function f ∗ : E C → E B givenon x ∈ E C by taking the domain f ∗ x of the unique horizontal morphism over f with codomain x .Given a proarrow m : A −7−→ B , required is a span from E A to E B . For this, let E m denote the fiberof P over m , and send m to the span E A src ←− E m tgt −→ E B . Finally, associated to a cell α of B of the form A αf (cid:15) (cid:15) m ✤ / / B g (cid:15) (cid:15) C n ✤ / / D will be a cell of S pan , that is, a morphism of the spans associated to m and n . In light of thedefinitions so far, required is a function E n → E m making the diagram E C (cid:15) (cid:15) E n o o α ∗ (cid:15) (cid:15) ✤✤✤ / / E D (cid:15) (cid:15) E A E m o o / / E B commute. But such a function E n → E m is given by the fact that P is a discrete fibration. Thatis, given u ∈ E n , there is a unique cell of E with codomain u over α , which, by the fact that P isa discrete fibration, must be of the form f ∗ x ⇓ (cid:15) (cid:15) ✤ / / g ∗ y (cid:15) (cid:15) x ✤ u / / y with source and target the unique lifts over f and g respectively. Thus, send u to the verticalsource of this lifted cell, which, by construction, is over m , hence well-defined. The cell diagramin S pan displayed above then commutes by construction of E n → E m . Construction 2.3.2 (Laxity Cells I) . Let D denote an object of B . The laxity cell for thecorresponding unit proarrow is of the form E D E D o o (cid:15) (cid:15) ✤✤✤ / / E D E D E u D src o o tgt / / E D The functor between vertices λ D : E D → E u D is given by X u X . Notice that the unit conditionof Definition 2.1.1 is satisfied because the external composition for E is strict.11 onstruction 2.3.3 (Laxity Cells II) . Let m : A −7−→ B and n : B −7−→ C denote proarrows of B .Required for F P are laxity cells for such m and n of the form E A φ n,m E m src o o tgt / / E B E n src o o tgt / / E C E A E n ⊗ m src o o tgt / / E C between spans. The composed span in the domain of the cell has vertex given by the pullback.The cell amounts to a functor between vertices respecting the source and target functors. This isgiven by external composition of proarrows and cells over m and n respectively: E m × E B E nφ n,m (cid:15) (cid:15) u : x −7−→ y, v : y −7−→ z ❴ (cid:15) (cid:15) E n ⊗ m v ⊗ u This is strictly functorial and a well-defined span morphism by the assumptions on P : E → B .The laxity coherence law follows from the fact that external composition for E is associative. Lemma 2.3.4.
Let P : E → B denote a double fibration. The assignments D E D f f ∗ m E m α α ∗ with unit and laxity cells as above define a lax functor F P : B op → S pan .Proof. What remains to check is the naturality conditions in Definition 2.1.1. Take cells α : m ⇒ p and β : n ⇒ q of B . The first condition amounts to the commutativity of the square made by thefunctors between vertices: E p × E N E q φ / / α ∗ × β ∗ (cid:15) (cid:15) E q ⊗ p ( β ⊗ α ) ∗ (cid:15) (cid:15) E m × E B E n φ / / E n ⊗ m But this follows by uniqueness assumptions. Naturality for the unit cells follows similarly.
Construction 2.3.5 (Pseudo-Inverse on Morphisms of Discrete Double Fibrations) . Start with amorphism H : P → B of discrete double fibrations P : E → B and Q : G → B . Define correspon-dences for what will be a transformation F H : F P → F Q between lax double functors F P and F Q arising as in Lemma 2.3.4. On objects B , define ( F H ) B to be the restriction of H to the fibers H : E B → G B . This is well defined since QH = P holds. Similarly, for a proarrow m : B −7−→ C ,take the component proarrow ( F H ) v in S pan to be the span E BH (cid:15) (cid:15) E m o o H (cid:15) (cid:15) / / E CH (cid:15) (cid:15) G B G m o o / / G C Again this is well-defined since QH = P holds. 12 roposition 2.3.6. The assignments in Construction 2.3.5 make a transformation F H : F P → F Q of lax functors as in Definition 2.1.4. These assignments result in a functor F ( − ) : Lax ( B op , S pan ) −→ DFib ( B ) . Proof.
The first naturality condition holds because H commutes with P and Q and because Q is a discrete fibration; the second naturality condition holds again because QH = P is true andbecause Q is also a discrete fibration. Proarrow functoriality follows from the fact that H is astrict double functor. The pseudo-inverse construction of the last subsection in fact induces an equivalence of categories,leading to the first representation theorem.
Construction 2.4.1.
To define a natural isomorphism η : 1 ∼ = F E l ( − ) between functors, neededare component transformations of lax functors indexed by lax H : B op → S pan . For such a laxfunctor H , the required transformation of lax functors η H : H → F E l ( H ) is given by η H,A : HA −→ E l ( H ) A X ( A, x )on objects A ∈ | B | and by η H,m : Hm −→ E l ( F ) m s ( m, s )for a proarrow m : B −7−→ C , giving a morphism of spans HB η (cid:15) (cid:15) Hm o o η (cid:15) (cid:15) / / HC η (cid:15) (cid:15) E l ( H ) B E l ( H ) m o o / / E l ( H ) C . Since the maps η H,A and η H,v just add in indices, they are both bijections of sets. Moreover,these components result in a transformation of lax functors as in Definition 2.1.4 by construction.Thus, such η H is the vertical component of a supposed transformation of functors of virtual doublecategories. Naturality in H is proved in the theorem below. Construction 2.4.2.
On the other hand, a natural isomorphism ǫ : Elt ( F ( − ) ) ∼ = 1 is given bycomponents ǫ P : E l ( F P ) −→ E where P : E → B is a discrete double fibration. On objects andarrows take ( D, x ) x ( C, x ) f −→ ( D, y ) f ∗ x ! −→ y where f ∗ x → y is the unique arrow above f with codomain y . It is well-defined because f ∗ y = x holds. These are bijections by uniqueness of these lifts; they are functorial and respect the fiberingover B . On proarrows and cells, take( m, s ) s α α ∗ s ⇒ s where α ∗ s ⇒ s is the unique lift in E of α with codomain s . Again these are bijections, functorialand fiber-respecting. Naturality in P is proved in the following result.13 heorem 2.4.3. For any strict double category B , there is an equivalence of categories DFib ( B ) ≃ Lax ( B op , S pan ) induced by the double category of elements construction.Proof. It remains to check the naturality of the isomorphisms in Constructions 2.4.1 and 2.4.2. If H and G are lax functors with a transformation τ : H → G , the diagram H τ (cid:15) (cid:15) η / / F E l ( H ) (cid:15) (cid:15) G η / / F E l ( G ) commutes because indexing commutes with applying the components of τ . Naturality in P alsofollows. Let H : P → Q be a morphism of double fibrations P : E → B and Q : G → B . Fornaturality, it is required that the square E l ( F P ) E l ( F H ) (cid:15) (cid:15) ǫ / / E H (cid:15) (cid:15) E l ( F Q ) ǫ / / G commutes. Chasing an object ( D, X ) or a proarrow ( m, s ) around each side of the square, theresult either way is HX in the former case and is Hs in the latter case. Thus, to check are thearrows and cells. Given an arrow f : ( C, X ) → ( D, Y ), the counter-clockwise direction gives theleft arrow below whereas the clockwise direction gives the one on the right:! : f ∗ Hy → y H (!) : Hf ∗ y → Hy These are strictly equal, however, by uniqueness of lifts. A formally similar argument works toshow that the square commutes also at the level of cells.
The main representation theorem of the paper asserts an equivalence of certain virtual doublecategories. To define these first recall the definition, presented here in the form of [CS10]. Virtualdouble categories have been known under the name “ f . c . -multicategories,” that is, as an exampleof a “generalized multicategory,” in this case relative to the free-category monad in [Lei04]. Definition 3.1.1 (See § . A virtual double category D consists of an underlyingcategory D , giving the objects and morphisms of D , together with, for any two objects C and D ,a class of proarrows v : C −7−→ D and for any “arity” k multicells of the form A µf (cid:15) (cid:15) m ✤ / / A m ✤ / / · · · m k ✤ / / A kg (cid:15) (cid:15) B n ✤ / / B m , m , . . . , m k ) is a “ k -ary multisource.” The definition allows nullary multisourceswith k = 0. A functor of virtual double categories F : C → D sends objects to objects, arrows toarrows, proarrows to proarrows and multicells to multicells in such a way as to preserve domains,codomains, multisources (so arities in particular), targets, identities and compositions. Example 3.1.2.
Every double category is a virtual double category by forgetting external com-position. In particular, for C with pullbacks Span ( C ) is a virtual double category. Definition 3.1.3 (Cf. § . A unit proarrow for an object D ∈ D in a virtual doublecategory is a proarrow u D : D −7−→ D and a nullary opcartesian cell D ⇓ DD u D ✤ / / D. A virtual double category has units if it is equipped with a choice of unit for each object.
Remark . The unit proarrow of any honest double category is a unit in this sense. Existence ofunits is part of the requirements for a virtual equipment as described in [CS10]. The full structurewill not be needed here. All the interesting examples possess all units, so without further ado allvirtual double categories will be assumed to have units. Functors of virtual double categories willbe assumed to be normal in the sense that they preserve nullary opcartesian cells.
Remark . The universal property of any unit u D on a given object D implies that D possessesgeneric multicells · ⇓ u D ✤ / / · u D ✤ / / · · · u D ✤ / / ·· u D ✤ / / · of any arity k . These are given from the unique factorization. Moreover, by uniqueness of thesefactorizations, a correct multicomposite of any such cells gives the generic multicell of the properarity determined in this way. Example 3.1.6 (Terminal Object) . The terminal object in vDbl is peculiar and is needed later.It has a single object • , an identity arrow 1 • and an “identity proarrow” u • . Whereas one mightexpect there to be only a single multicell u • ⇒ u • , in fact required are generic multicells withmultisources of all arities k • µ k • (cid:15) (cid:15) u • ✤ / / • u • ✤ / / · · · u • ✤ / / • • (cid:15) (cid:15) • u • ✤ / / • as otherwise the natural definitions on objects, arrows and proarrows will not extend to a uniquefunctor of virtual double categories D → . Multi-composition is defined to give the genericmulticell with the appropriate arity. Notice, then, that in particular has units by fiat. Example 3.1.7. A point of a virtual double category is a (normalized) functor D : → D . Apoint thus consists of an object D , its identity arrow 1 D , its unit proarrow u D and the correspondinggeneric multicells.A notion of transformation gives the 2-categorical structure on virtual double categories.15 efinition 3.1.8. Let
F, G : C ⇒ D denote functors of virtual double categories. A transfor-mation τ : F → G assigns to each object C of C an arrow τ C : F C → GC and to each proarrow m : C −7−→ D a cell F C τ m τ C (cid:15) (cid:15) F m ✤ / / F D θ D (cid:15) (cid:15) GC Gm ✤ / / GD in such a way that1. [Arrow Naturality] for each arrow f : C → D , the square F C τ C (cid:15) (cid:15) F f / / F D θ D (cid:15) (cid:15) GC Gf / / GD commutes; and2. [Cell Naturality] for each multicell · µf (cid:15) (cid:15) m ✤ / / · m ✤ / / · · · m k ✤ / / · g (cid:15) (cid:15) · n ✤ / / · the composed multicells on either side of · τ m (cid:15) (cid:15) F m ✤ / / · τ m (cid:15) (cid:15) F m ✤ / / · · · τ mk F m k ✤ / / · (cid:15) (cid:15) · F µ (cid:15) (cid:15)
F m ✤ / / · F m ✤ / / · · · F m k ✤ / / · (cid:15) (cid:15) · Gµ (cid:15) (cid:15) Gm ✤ / / · Gm ✤ / / · · · Gm k ✤ / / · (cid:15) (cid:15) = · τ n (cid:15) (cid:15) F n ✤ / / · (cid:15) (cid:15) · Gn ✤ / / · · Gn ✤ / / · are equal.Denote the 2-category of virtual double categories, their functors and transformations by vDbl . Proposition 3.1.9. vDbl has (strict 2-)pullbacks.Proof.
Given two functors F : A → C and G : B → C , the pullback has as its underlying category( A × C B ) = A × B C . Proarrows are pairs ( m, n ) for a proarrow m of A and one n of B satisfying F m = Gn . Similarly,multicells are pairs ( µ, ν ) with µ in A and ν in B such that F µ = Gν . Composition uses compositionin A and B . This is a virtual double category fitting into a commutative square A × C B d (cid:15) (cid:15) d / / B G (cid:15) (cid:15) A F / / C with the expected 2-categorical universal property [Str76]..16urther limits of a 2-categorical variety abound in vDbl . The comma category is of interest.These are well-known in Cat (e.g. § I.6 [Mac98]). The formal abstraction and and elementaryphrasing of its universal property in an arbitrary 2-category appear in § Proposition 3.1.10.
The 2-category vDbl has comma objects.Proof.
Given functors F : A → C and G : B → C , the (purported) comma F/G has its underlying1-category as (
F/G ) = F /G . Proarrows are triples ( m, α, n ) with m and n proarrows of A and B , respectively, and α a cell α : F m ⇒ Gn of C . A multicell is thus a pair of multicells · µ (cid:15) (cid:15) m ✤ / / · m ✤ / / · · · m k ✤ / / · (cid:15) (cid:15) · ν (cid:15) (cid:15) n ✤ / / · n ✤ / / · · · n k ✤ / / · (cid:15) (cid:15) · p ✤ / / · · q ✤ / / · from A and B , respectively, of the same arity and satisfying the equation · α (cid:15) (cid:15) F m ✤ / / · α (cid:15) (cid:15) F m ✤ / / · · · α k F m k ✤ / / · (cid:15) (cid:15) · F µ (cid:15) (cid:15)
F m ✤ / / · F m ✤ / / · · · F m k ✤ / / · (cid:15) (cid:15) · Gν (cid:15) (cid:15) Gn ✤ / / · Gn ✤ / / · · · Gn k ✤ / / · (cid:15) (cid:15) = · β (cid:15) (cid:15) F p ✤ / / · (cid:15) (cid:15) · Gq ✤ / / · · Gq ✤ / / · Composition is given by that in A and B . So defined, F/G comes with evident projection functorsto A and B . The expected transformation F/G ⇒ d (cid:15) (cid:15) d / / B G (cid:15) (cid:15) A F / / C has components f for objects ( A, f, B ) and α for proarrows ( m, α, n ). It satisfies the required2-categorical universal property in the reference by construction. Remark . This makes vDbl into a “representable 2-category” [Gray74] and [Str74].
Example 3.1.12 (Slice Virtual Double Category) . Let D denote an object in a virtual doublecategory D . The slice virtual double category over D is defined as the comma 1 /D /D ⇒ (cid:15) (cid:15) d / / D (cid:15) (cid:15) D / / D Denote this as usual by D /D . The coslice is defined analogously.17 .2 Virtual Double Category Structure on Double Presheaves The virtual double category structure on
Lax ( B op , S pan ) will be given by taking so-called “mod-ules” as proarrows and “multimodulations” as the multicells. Here we revisit the the defini-tions for lax functors between arbitrary double categories. A path of proarrows is a sequence m = ( m , . . . , m k ) such that, reading left to right, the target of one proarrow is the source of thenext. The external composite is denoted by [ m ]. Definition 3.2.1 (Cf. § . A module between lax functors M : F −7−→ G : A ⇒ B ofdouble categories assigns1. to each proarrow m : A −7−→ B of A , a proarrow M m : F A −7−→ GB of B ;2. to each cell of A as at left, one of B as at right A θm ✤ / / f (cid:15) (cid:15) B g (cid:15) (cid:15) F A
MθMm ✤ / / F f (cid:15) (cid:15) GB Gg (cid:15) (cid:15) C n ✤ / / D F C Mn ✤ / / GD
3. for each pair of proarrows m : A −7−→ B and n : B −7−→ C , action multicells of B · λ m,n F m ✤ / / · Mn ✤ / / · · ρ m,n Mm ✤ / / · Gn ✤ / / ·· M ( n ⊗ m ) ✤ / / · · M ( n ⊗ m ) ✤ / / · in such a way that the following axioms hold.1. [Functoriality] For any (internal) composite of monocells βα and any proarrow m , the equa-tions M βM α = M ( βα ) and M m = 1 Mm hold;2. [Naturality] For any external composite β ⊗ α , the equalities · λF m ✤ / / · Mn ✤ / / · · F αF m ✤ / / (cid:15) (cid:15) · MβMn ✤ / / (cid:15) (cid:15) · (cid:15) (cid:15) · M ( β ⊗ α ) ✤ / / (cid:15) (cid:15) · (cid:15) (cid:15) = · λ ✤ / / · ✤ / / ·· M ( q ⊗ p ) ✤ / / · · M ( q ⊗ p ) ✤ / / · and · ρMm ✤ / / · Gn ✤ / / · · MαMm ✤ / / (cid:15) (cid:15) · GβGn ✤ / / (cid:15) (cid:15) · (cid:15) (cid:15) · M ( β ⊗ α ) ✤ / / (cid:15) (cid:15) · (cid:15) (cid:15) = · ρ / / · / / ·· M ( q ⊗ p ) ✤ / / · · M ( q ⊗ p ) ✤ / / · both hold. 18. [Associativity] For any composable sequence of proarrows ( m, n, p ) of A , the equalities (mod-ulo suppressed associativity isomorphisms) · F m ✤ / / · λF n ✤ / / · Mw ✤ / / · · F γF m ✤ / / · F n ✤ / / · Mw ✤ / / ·· λ ✤ / / · ✤ / / · = · λ ✤ / / · ✤ / / ·· M ( p ⊗ ( n ⊗ m )) ✤ / / · · M (( p ⊗ n ) ⊗ m ) ✤ / / · and · ρMm ✤ / / · Gn ✤ / / · Gp ✤ / / · · Mm ✤ / / · GγGn ✤ / / · Gp ✤ / / ·· ρ ✤ / / · ✤ / / · = · ρ ✤ / / · ✤ / / ·· M ( p ⊗ ( n ⊗ m )) ✤ / / · · M (( p ⊗ n ) ⊗ m ) ✤ / / · are valid.4. [Compatibility] The composites · λF m ✤ / / · Mn ✤ / / · Gp ✤ / / · · F m ✤ / / · ρMn ✤ / / · Gp ✤ / / ·· ρ ✤ / / · ✤ / / · = · λ ✤ / / · ✤ / / ·· M ( p ⊗ ( n ⊗ m )) ✤ / / · · M (( p ⊗ n ) ⊗ m ) ✤ / / · are equal.5. [Unit] For any proarrow m : A −7−→ B of A , the composites · γu F A ✤ / / · Mm ✤ / / · · Mm ✤ / / (cid:15) (cid:15) · γu GB ✤ / / (cid:15) (cid:15) · (cid:15) (cid:15) · λF u A ✤ / / (cid:15) (cid:15) · ✤ / / · (cid:15) (cid:15) and · ρ ✤ / / · Gu B ✤ / / ·· M ( m ⊗ u A ) ✤ / / · · M ( u B ⊗ m ) ✤ / / · are equal to the respective canonical composition multicells for ( m F A , M m ) and (
M u, m GA ).Multicells are given by the notion of a “multimodulation,” recalled next. A path of cells θ : ( θ , . . . , θ k ) a sequence of cells θ : m ⇒ n , . . . , θ k : m k ⇒ n k has the target of a given cellequal to the source of the next. Externally composable sequences of modules M , . . . , M k betweenlax functors will be thought of as proarrows F i −7−→ F i +1 from a lax functor F i to one F i +1 , notatedwith superscripts so as not to confuse these with the components of the functors.19 efinition 3.2.2 (See § § . A multimodulation · µτ (cid:15) (cid:15) M ✤ / / · M ✤ / / · · · M k ✤ / / · σ (cid:15) (cid:15) · N ✤ / / · from modules M i to N with source τ and target σ assigns to each path m = ( m , . . . , m k ) ofproarrows of A , a multicell · µ m τ (cid:15) (cid:15) M m ✤ / / · M m ✤ / / · · · M k m k ✤ / / · σ (cid:15) (cid:15) · N [ m ] ✤ / / · in such a way that the following axioms are satisfied.1. [Naturality] for any path of cells θ : m ⇒ n , . . . , θ k : m k ⇒ n k , the two composites · (cid:15) (cid:15) M θ M m ✤ / / · (cid:15) (cid:15) M θ M m ✤ / / · · · M k θ k M k m k / / · (cid:15) (cid:15) · µ m (cid:15) (cid:15) M m ✤ / / · M m ✤ / / · · · M k m k / / · (cid:15) (cid:15) · (cid:15) (cid:15) µ n M n ✤ / / · M n / / · · · M k n k / / · (cid:15) (cid:15) = · (cid:15) (cid:15) N [ θ ] N [ m ] ✤ / / · (cid:15) (cid:15) · N [ n ] / / · · N [ n ] ✤ / / · are equal.2. The following equivariance axioms are satisfied:(a) [Left and Right Equivariance] For any paths of proarrows y m = ( m , . . . , m k , y ) and m x = ( x, m , . . . , m k ), the composites on either side of · τ x F x ✤ / / (cid:15) (cid:15) · µ [ m ] M m ✤ / / (cid:15) (cid:15) · M m ✤ / / · · · M k m k ✤ / / · (cid:15) (cid:15) · λF x ✤ / / · M m ✤ / / · M m ✤ / / · · · M k m k ✤ / / ·· λG x ✤ / / · N [ m ] ✤ / / · = · µ [ m x ] M ( m ⊗ x ) ✤ / / (cid:15) (cid:15) · M m ✤ / / · · · M k m k ✤ / / · (cid:15) (cid:15) · N [ m x ] ✤ / / · · N [ m x ] ✤ / / · and · µ [ m ] (cid:15) (cid:15) M m ✤ / / · · · ✤ / / · M k m k ✤ / / · τ y (cid:15) (cid:15) F n y ✤ / / · (cid:15) (cid:15) · M m ✤ / / · · · ✤ / / · ρM k m k ✤ / / · F n y ✤ / / ·· ρN [ m ] ✤ / / · G n y ✤ / / · = · µ [ y m ] (cid:15) (cid:15) M m ✤ / / · · · ✤ / / · M k ( y ⊗ m k ) ✤ / / · (cid:15) (cid:15) · N [ y m ] ✤ / / · N [ y m ] ✤ / / · are equal. 20b) [Inner Equivariance] For any path of proarrows ( m , . . . , m i , x, m i +1 , . . . , m k ), the com-posite · M m ✤ / / · · · ✤ / / · ρM i m i ✤ / / · F i x ✤ / / · ✤ / / · · · M k m k ✤ / / ·· µ [ m ] (cid:15) (cid:15) ✤ / / · · · ✤ / / · ✤ / / · ✤ / / · · · ✤ / / · (cid:15) (cid:15) · N [ m ] ✤ / / · is equal to · M m ✤ / / · · · ✤ / / · λF i x ✤ / / · M i +1 m i +1 ✤ / / · ✤ / / · · · M k m k ✤ / / ·· µ [ m ] (cid:15) (cid:15) ✤ / / · · · ✤ / / · ✤ / / · ✤ / / · · · ✤ / / · (cid:15) (cid:15) · N [ m ] ✤ / / · for i = 1 , . . . , k − Theorem 3.2.3 (Par´e) . Double functors F : A → B and horizontal transformations, together withmodules and their multi-modulations giving the proarrows and multicells, comprise a virtual doublecategory denoted by L ax ( A , B ) .Proof. See the lead-up to Theorem 1.2.5 of [Par´e13].
Remark . Our main interest is of course in the virtual double category L ax ( B op , S pan ). Bythe Theorem, in general, this is not a genuine double category. This is because composition ofproarrows need not exist. The paper [Par´e13] is a dedicated study of this issue. The additional structure on
DFib ( B ) making it a virtual double category goes by well-knownterminology from another context. We already know that DFib ( B ) could have been defined as Cat ( DFib ) / B . Another way of look at Cat ( DFib ) is that it is the category
Mon ( Span ( DFib ))of monoids in spans in discrete fibrations. In general
Mon ( Span ( C )) for a category C with finitelimits is the underlying category of the virtual double category P rof ( C ) = M od ( Span ( C )) ofmodules in spans in C as in [Lei04] or [CS10]. So, we take the modules and their multicells fromthis context as the virtual double category structure on DFib ( B ). Here are the definitions. Definition 3.3.1 (Cf. § § . Let D denote a virtual double category.Define its virtual double category of monoids and modules , denoted by M od ( D ), by taking1. objects: monoids , namely, triples ( r, µ, η ) consisting of a proarrow r : A −7−→ A and cells A µr ✤ / / A r ✤ / / A A η AA r ✤ / / A A r ✤ / / A satisfying the usual axioms for a monoid, namely, the multiplication law µ (1 , µ ) = µ ( µ, µ (1 , η ) = 1 and µ ( η,
1) = 1;21. arrows: monoid homomorphisms ( r, µ, η ) → ( s, ν, ǫ ), namely, those pairs ( f, φ ) consisting ofan arrow f : A → B and a cell A φf (cid:15) (cid:15) r ✤ / / A f (cid:15) (cid:15) B s ✤ / / B satisfying the unit axiom φη = ǫf and multiplication axiom ν ( φ, φ ) = φµ .3. proarrows: so-called modules ( r, µ, η ) −7−→ ( s, ν, ǫ ), namely, triples ( m, λ, ρ ) with m : A −7−→ B a proarrow and λ , ρ left and right action cells A λr ✤ / / A m ✤ / / B A ρm ✤ / / B s ✤ / / BA m ✤ / / B A m ✤ / / B satisfying the module axioms λ ( µ,
1) = λ (1 , λ ) and ρ (1 , µ ) = ρ ( ρ,
1) for the multiplicationand λ ( η,
1) = 1 and ρ (1 , η ) = 1 for the units; a sequence of modules consists of finitely manymodules ( m i , λ i , ρ i ) for which src m i +1 = tgt m i and s i +1 = r i both hold;4. multicells from a sequence of modules ( m i , λ i , ρ i ) to one ( n, λ, ρ ) consist of those multicellsin A · γf (cid:15) (cid:15) m ✤ / / · · · m p ✤ / / · g (cid:15) (cid:15) · n ✤ / / · satisfying the equivariance axioms expressed by the equalities of composite cells:(a) [Left] · φr ✤ / / f (cid:15) (cid:15) · γm ✤ / / f (cid:15) (cid:15) · m ✤ / / · · · m p ✤ / / · g (cid:15) (cid:15) · λr ✤ / / · m ✤ / / · m ✤ / / · · · m p ✤ / / ·· λs ✤ / / · n ✤ / / · = · γm ✤ / / f (cid:15) (cid:15) · m ✤ / / · · · m p ✤ / / · g (cid:15) (cid:15) · n ✤ / / · · n ✤ / / · (b) [Right] · γf (cid:15) (cid:15) m ✤ / / · · · m p − ✤ / / · m p ✤ / / · ψg (cid:15) (cid:15) r p ✤ / / · g (cid:15) (cid:15) · m ✤ / / · · · m p − ✤ / / · ρm p ✤ / / · r p ✤ / / ·· ρn ✤ / / · s q ✤ / / · = · γf (cid:15) (cid:15) m ✤ / / · · · m p − ✤ / / · m p ✤ / / · g (cid:15) (cid:15) · n ✤ / / · n ✤ / / · · m ✤ / / · · · m i − ✤ / / · ρm i ✤ / / · r i ✤ / / · m i +1 ✤ / / · · · m p ✤ / / ·· γf (cid:15) (cid:15) ✤ / / · · · ✤ / / · ✤ / / · ✤ / / · · · ✤ / / · g (cid:15) (cid:15) · n ✤ / / · is equal to · m ✤ / / · · · m i ✤ / / · λr i ✤ / / · m i +1 ✤ / / · m i +2 ✤ / / · · · m p ✤ / / ·· γf (cid:15) (cid:15) ✤ / / · · · ✤ / / · ✤ / / · ✤ / / · · · ✤ / / · g (cid:15) (cid:15) · n ✤ / / · for i = 1 , . . . , p − A . Proposition 3.3.2.
For any virtual double category D , M od ( D ) has units.Proof. Any object in M od ( D ) is a monoid. Its equipped multiplication gives the unit proarrow.For more see § Remark . The definition above omits most of the diagrams and states just the equations outof space considerations. However, upon writing down all the diagrams, one might notice a formalsimilarity between these axioms and those for modules and multimodulations in L ax ( B op , S pan ).It is the point of the next section to show that this is in fact an equivalence of virtual doublecategories. First, however, let us consider some examples. Let C denote a category with finite limits. Then Span ( C ) is a double category. As in [CS10],denote M od ( Span ( C )) by P rof ( C ). Several choices of C are of interest. Modules in P rof ( C ) arealready known by well-established terminology. Definition 3.4.1 (Cf. § . Let C denote a category with finite limits; and let C and D denote internal categories. An internal profunctor M : C −7−→ D is a module, i.e. a proarrowin P rof ( C ). A multicell of internal profunctors is thus a multicell as above. Example 3.4.2.
The virtual double category P rof ( Set ) is P rof . That is, a monoid in Span ( set )is a category. A unit proarrow for such C is thus the span C ← C → C formed from the domainand codomain maps with actions given by composition. Example 3.4.3.
Letting C = Cat as a 1-category, P rof ( Cat ) consists of usual double categoriesand double functors as the objects and arrows. Internal profunctors M : A −7−→ B between doublecategories consist of a span A ∂ ←− M ∂ −→ B and left and right action functors L : A × A M −→ M R : M × B B −→ M M , . . . , M k ) ⇒ N thus consistsof a functor m : M × A · · · × A k M k −→ N from the vertex of the composite of ( M , . . . , M k ), making a morphism of spans, and satisfyingthe various equivariance requirements as in the definition. Notice that owing to the peculiaritiesof the cell structure of as in Example 3.1.6, a point D : → P rof ( Cat ) is a double category D , with the identity double functor 1 : D → D , the unit proarrow u : D −7−→ D , namely, the spanformed by the external source and target functors with actions given by external composition, andfinally multicells of all arities given by iterated external composition. This can all be generalizedto Cat ( C ) for arbitrary C with finite limits. Example 3.4.4.
Let C = Cat , the “arrow category” of Cat and consider P rof ( Cat ). Amonoid is then a double functor, a morphism is a commutative square of double functors. Amodule consists of two modules in the former sense – one between domains of the two doublecategories and one between the codomains; the vertices of these modules are related by a functormaking a morphism of spans. Multicells have a similar “two-tiered” structure. Example 3.4.5.
Letting C = DFib , the virtual double category P rof ( DFib ) is the sub-virtualdouble category of the previous example where all the objects are not just double functors but areinstead discrete double fibrations . There is a codomain functorcod : P rof ( DFib ) −→ P rof ( Cat )taking an object P : E → B its codomain B and every proarrow M : P −7−→ Q to the module betweendouble categories giving the codomains of M . Take P rof ( DFib ) / B to be the pullback of cod in vDbl along the point B : → P rof ( Cat ). Definition 3.4.6.
The virtual double category of discrete double fibrations over a double category B is P rof ( DFib ) / B . Denote this by D Fib ( B ). Remark . A module between discrete double fibrations M : P −7−→ Q thus consists of a discretefibration M : M → B and a morphism of spans E P (cid:15) (cid:15) M ∂ o o ∂ / / M (cid:15) (cid:15) G Q (cid:15) (cid:15) B B o o tgt / / B and left and right actions functors making commutative squares E × E M L / / P × M (cid:15) (cid:15) M M (cid:15) (cid:15) M × G G R / / M × Q (cid:15) (cid:15) M M (cid:15) (cid:15) B × B B −⊗− / / B B × B B −⊗− / / B that satisfy the action requirements as in the definition. A multicell µ : ( M , . . . , M k ) ⇒ N betweensuch modules consists of a functor µ making a commutative square M × E · · · × E k M k (cid:15) (cid:15) µ / / N N (cid:15) (cid:15) B × B · · · × B B −⊗−···−⊗− / / B Remark . As in § M od ( − ) defines an endo-2-functor M od ( − ) : vDbl → vDbl . Another way to look at the codomain functor the previous example isthat it is induced from the codomain functor cod : DFib → Cat , passing first through
Span ( − )and then M od ( − ). This section extends the result of Theorem 2.4.3, culminating in a proof that elements constructionextends to an equivalence of virtual double categories D Fib ( B ) ≃ L ax ( B op , S pan )This appears below as Theorem 4.3.5. The elements functor of Lemma 2.2.3 extends to one between virtual double categories. Neededare assignments on modules and multimodulations.
Construction 4.1.1 (Elements from a Module) . Let M : F −7−→ G denote a module between laxdouble functors as in Definition 3.2.1. Construct a category E l ( M ) in the following way. Objectsare pairs ( m, s ) with m : B −7−→ C a proarrow of B and s ∈ M m . A morphism ( m, s ) → ( n, t ) is acell α with source m and target n for which the equation M α ( t ) = s holds. So defined, E l ( M ) isa category since M is strictly functorial on cells. Notice that there are thus projection functors E l ( F ) ∂ ←− E l ( M ) ∂ −→ E l ( G ) taking an object ( m, s ) to ∂ ( v, s ) = ( B, ∂ s ) and ∂ ( v, s ) = ( C, ∂ s ) and extended to morphismsas follows. Given a cell A αf (cid:15) (cid:15) m ✤ / / B g (cid:15) (cid:15) C n ✤ / / D take ∂ α to be the morphism f : ( A, ∂ s ) → ( B, ∂ t ) and analogously for ∂ α . These are well-definedby the commutativity conditions coming with the morphism of spans M α . The assignments arethen functorial by that assumed for M . Construction 4.1.2 (Actions) . Form the pullback of ∂ : E l ( M ) → E l ( F ) along the targetprojection E l ( F ) → E l ( F ) and give assignments L : E l ( F ) × E l ( F ) E l ( M ) → E l ( M )that will amount to an action. Summarize these assignments on objects and arrows at once by thepicture: ( A, x ) αf (cid:15) (cid:15) ( m,u ) ✤ / / ( B, y ) g (cid:15) (cid:15) ( p, r ) β (cid:15) (cid:15) ( p ⊗ m, λ ( u, r )) β ⊗ α (cid:15) (cid:15) ( C, z ) ( n,v ) ✤ / / ( D, w ) ( q, s ) ( q ⊗ n, λ ( v, s ))25here λ is the action cell coming with M . Of course α and β are composable by the constructionof the pullback, but it needs to be seen that the composite β ⊗ α does give a morphism of E l ( M )But this is equivalent to the validity of the equation M ( β ⊗ α )( λ ( v, s )) = λ ( u, r )But this holds by the naturality condition for λ in Definition 3.2.1, since u = F α ( v ) and M β ( s ) = r both hold by the construction of morphisms in E l ( M ). So defined, L is a functor by the strictinterchange law in B ; by the fact that M is strictly horizontally functorial; and by the normalizationhypothesis for units. A functor R for a right action of E l ( G ) on E l ( M ) is constructed analogously.It remains to see that the action axioms are satisfied and that they are suitably compatible, yieldingan internal profunctor. Proposition 4.1.3.
The assignments of Construction 4.1.2 are well-defined functors yielding aninternal profunctor between discrete double fibrations E l ( M ) : E l ( F ) −7−→ E l ( G ) .Proof. The action functors L and R are unital by the normalization assumption for vertical com-position with units in B . Required are action iso cells such as E l ( F ) × E l ( F ) E l ( F ) × E l ( F ) E l ( M ) ⊗× (cid:15) (cid:15) × L / / E l ( F ) × E l ( F ) E l ( M ) L (cid:15) (cid:15) E l ( F ) × E l ( F ) E l ( M ) L / / E l ( M )and similarly for R . But chasing an object of the domain around either of the square as aboveand comparing, commutativity is given by associativity of proarrow composition in B . Lastly, theactions L and R should be compatible in the sense that E l ( F ) × E l ( F ) E l ( M ) × E l ( G ) E l ( G ) L × (cid:15) (cid:15) × R / / E l ( F ) × E l ( F ) E l ( M ) L (cid:15) (cid:15) E l ( M ) × E l ( G ) E l ( G ) R / / E l ( M )commutes. But again chasing objects and arrows around each side of the square shows thatcommutativity follows from the compatibility assumption in Definition 3.2.1. Construction 4.1.4 (Elements from a Multimodulation) . Start with a multimodulation of con-travariant lax S pan -valued functors F µτ (cid:15) (cid:15) M ✤ / / F M ✤ / / · · · M k ✤ / / F kσ (cid:15) (cid:15) G N ✤ / / G as in Definition 3.2.2. This means that there are projection spans E l ( F i − ) ← E l ( M i ) → E l ( F i )and one for N , each with appropriate left and right actions as in Construction 4.1.1. Define whatwill be a functor E l ( µ ) : E l ( M ) × E l ( F ) · · · × E l ( F k − ) E l ( M k ) −→ E l ( N )26n the following way. On objects take(( m , s ) , . . . , ( m k , s k )) ([ m ] , µ m ( s ))where µ m is the given function coming with µ and s = ( s , . . . s k ). An arrow of the supposed sourceis a sequence of externally composable cells θ i : ( m i , s i ) → ( n i , t i ). Assign to such a sequence themorphism of N represented by their composite( θ , . . . , θ k ) θ k ⊗ θ k − ⊗ · · · ⊗ θ . This does define a morphism ([ m ] , µ m ( s )) → ([ n ] , µ n ( t )) of E l ( N ) by the strict composition for themodule N as in Definition 3.2.1. This functor has several naturality and equivariance properties,coming from the assumed properties of the original multimodulation µ . For example, notice that E l ( µ ) commutes with the projections and the 0-level of the induced double functors E l ( τ ) and E l ( σ ) by construction. Further properties are summarized in the next result. Proposition 4.1.5.
The functor E l ( µ ) of Construction 4.1.1 defines a multicell between internalprofunctors of the form E l ( F ) E l ( µ ) (cid:15) (cid:15) E l ( M ) ✤ / / E l ( F ) E l ( M ) ✤ / / · · · E l ( M k ) ✤ / / E l ( F k ) (cid:15) (cid:15) E l ( G ) E l ( N ) ✤ / / E l ( G ) This completes assignments for the elements functor E l ( − ) : L ax ( B op , S pan ) → D Fib ( B ) betweenvirtual double categories.Proof. The appropriate commutativity at the 0-level was observed above. That the action is leftequivariant is the statement that the square E l ( F ) × E l ( F ) E l ( M ) × E l ( F ) · · · × E l ( F k − ) E l ( M k ) E l ( τ ) × E l ( µ ) / / L × (cid:15) (cid:15) E l ( G ) × E l ( G ) E l ( N ) L (cid:15) (cid:15) E l ( M ) × E l ( F ) · · · × E l ( F k − ) E l ( M k ) E l ( µ ) / / E l ( N )commutes. But chasing an object of the upper left corner around both sides of the square revealsthat commutativity at the object level is precisely the left equivariance condition in Definition 3.2.2.Right and inner equivariance follow by the same type of argument. The assignments are alreadyknown to be well-defined on objects and morphisms. It follows easily that these assignments aresuitably functorial in the sense of virtual double categories. Likewise, the pseudo-inverse of Lemma 2.3.6 extends to a functor of virtual double categories.
Construction 4.2.1 (Pseudo-Inverse for Modules) . Let M : P −7−→ Q denote an internal profunctorbetween discrete double fibrations P : E → B and Q : G → B as in Remark 3.4.7. Construct whatwill be a module F M : F P −7−→ F Q between the associated lax functors F P and F Q from Lemma 2.3.427n the following way. Let M m denote the inverse image of the proarrow m : B −7−→ C of B under thediscrete fibration Π : M → B coming with M . To each such proarrow m assign the span of sets E B ∂ ←− M m ∂ −→ G C Note that this is well-defined by the first assumed commutativity condition for M . To each cell of B , assign the morphism of spans A θf (cid:15) (cid:15) m ✤ / / B g (cid:15) (cid:15) E Bf ∗ (cid:15) (cid:15) M n o o θ ∗ (cid:15) (cid:15) / / G Cg ∗ (cid:15) (cid:15) C n ✤ / / D E A M m o o / / G D with θ ∗ : M n → M m given by taking an object of M over n to the domain of the unique morphismof M above θ via Π : M → B . This is well-defined and makes a span morphism. To completethe data, start with composable vertical arrows m : A −7−→ B and n : B −7−→ B and give assignments λ and ρ by using the given actions L and R , taking λ m,n : E m × E D M n → M n ⊗ m ( ˜ m, ˜ n ) L ( ˜ m, ˜ n )and similarly for ρ . These are well-defined by the second row of commutativity conditions inDefinition 3.2.1. Additionally, these maps commute with the projections, in the sense that thediagrams E A M m × G B G n o o ρ (cid:15) (cid:15) / / G C E A E u × E B M n o o λ (cid:15) (cid:15) / / G C E A M n ⊗ m o o / / G C E A M n ⊗ m o o / / G C both commute. Proposition 4.2.2.
The assignments of Construction 4.2.1 make F M : F P −7−→ F Q a module in thesense of Definition 3.2.1.Proof. All of the requirements for F M to be a module between lax functors are met by the cor-responding properties of the original module M , together with the fact that Π : M → B is adiscrete fibration. Construction 4.2.3 (Pseudo-Inverse Assignment on Modulations) . Start with a modulation U in D Fib ( B ) as in Remark 3.4.7. Thus, in particular, we have a functor U : M × E · · · × E k − M k −→ N commuting with the projections to the end factors and commuting with the ( k − B . Required is a multi-modulation F P F U F M ✤ / / F H (cid:15) (cid:15) F P F M ✤ / / · · · F Mk ✤ / / F P k F K (cid:15) (cid:15) F Q F N ✤ / / F Q m = ( m , . . . m k ), this is just to ask for acorresponding set function M m × E A × · · · × E k − Ak − M m k −→ N [ m ] which is given by the arrow part of U , namely, U . Well-definition follows from the fact that U commutes with the projections to B and the ( k − B . Proposition 4.2.4.
The choice of ( F U ) m = U in Construction 4.2.3 results in a multimodulationof modules between lax functors as in Definition 3.2.2. This extends the functor F ( − ) to a functorof virtual double categories F ( − ) : D Fib ( B ) −→ L ax ( B op , S pan ) . Proof.
The horizontal naturality condition holds by construction of the transition functions cor-responding to cells θ and because Π : N → B is a discrete fibration. Right, left and innerequivariance then follow from the corresponding properties of the original functor U . The extended elements construction and the purported pseudo-inverse induce an equivalence ofvirtual double categories, leading to the full representation theorem, namely, Theorem 4.3.5 below.
Construction 4.3.1.
Extend the isomorphism of Construction 2.4.1 to an isomorphism of functorsof virtual double categories. The required multimodulation for the cell-components is straightfor-ward to produce. Given M : H −7−→ G and a proarrow m : B −7−→ C of B , define a function η M,m : M m −→ E l ( M ) m s ( m, s )again just adding in an index. This is a bijection fitting into a morphism of spans HB η H,B (cid:15) (cid:15)
M m o o η M,m (cid:15) (cid:15) / / GC η G,C (cid:15) (cid:15) E l ( H ) B E l ( M ) m o o / / E l ( G ) C that defines the required invertible modulation H η M η H (cid:15) (cid:15) M ✤ / / G η G (cid:15) (cid:15) E l ( F H ) E l ( M ) ✤ / / E l ( F G )as in Definition 3.2.2 by construction of the elements functor and its purported pseudo-inverse.This is easy to check from the definitions. Proposition 4.3.2.
The assignments in Construction 4.3.1 yield a natural isomorphism of func-tors of virtual double categories η : 1 ∼ = E l ( F ( − ) ) . roof. As discussed above, the components of the purported transformation are all well-defined,so it remains only to check the “cell naturality” condition of Definition 3.1.8. Start with a genericmultimodulation H µτ (cid:15) (cid:15) M ✤ / / H M ✤ / / · · · M k / / H kσ (cid:15) (cid:15) G N ✤ / / G in L ax ( B op , S pan ). By construction, the composite on right side of the condition sends an k -tuple s = ( s , . . . , s k ) to (( m , s ) , . . . , ( m k , s k )) and then to ([ m ] , µ [ m ] ( s )); where as that on theleft side sends the same element to µ [ m ] ( s ) first and then to ([ m ] , µ [ m ] ( s )). The point is that byconstruction evaluating and indexing commute. In any case, the two sides are equal, and η sodefined is a natural isomorphism as claimed. Construction 4.3.3.
Extend the natural isomorphism of Construction 2.4.2 to one of functorsof virtual double categories ǫ : E l ( F ( − ) ) ∼ = 1. Take an internal profunctor M : P −7−→ Q of discretedouble fibrations P : E → B and Q : G → B . The required span morphism E E l ( F M ) o o ǫ M (cid:15) (cid:15) ✤✤✤ / / G ( m, s ) α −→ ( n, t ) → E M o o / / G s ! −→ t is given by the fact that M : M → B is a discrete fibration. It is a functor and an isomorphismby uniqueness and equivariant by construction, making a morphism of internal profunctors. Proposition 4.3.4.
The assignments in Construction 4.3.3 are a natural isomorphism of functorsof virtual double categories.Proof.
The one condition to check is the “Cell Naturality” of Definition 3.1.8. Taking a multicellbetween internal profunctors P µH (cid:15) (cid:15) M ✤ / / P M ✤ / / · · · M k / / P kK (cid:15) (cid:15) Q N ✤ / / Q the statement of the condition reduces to checking that the equation µ ◦ ( ǫ M × · · · ǫ M k ) = ǫ N ◦ E l ( F µ )holds. But this is true by definition of E l ( F µ ) and the components of ǫ . For on the one hand, a k -tuple (( m , s ) , . . . , ( m k , s k )) is sent to s = ( s , . . . , s k ) and then to µ ( s ). On the other hand, thesame k -tuple is sent to ([ m ] , s ) by E l ( F µ ) and then to µ ( s ) by ǫ N . The same kind of check worksat the level of arrows. Thus, ǫ is a natural isomorphism. Theorem 4.3.5.
There is an equivalence of virtual double categories L ax ( B op , S pan ) ≃ D Fib ( B ) for any double category B induced by the elements functor E l ( − ) .Proof. This is proved by Propositions 4.3.1 and 4.3.3.30
Prospectus
Let us close with a preview of forthcoming work relating to the present results.
It is well-known that ordinary discrete fibrations over a fixed base are monadic over a slice of thecategory of sets. This fact is of central importance in the elementary axiomatization of resultsrelating to presheaves and sheaves in the language of an elementary topos [Dia73], [Dia75]. In thisdevelopment, “base-valued functors” (i.e. presheaves) are axiomatized as certain algebras for amonad on a slice of the ambient topos.Any parallel development in a double categorical setting of these presheaf results will require ananalogous monadicity result. Forthcoming work will establish that discrete double fibrations overa fixed base double category are monadic over a certain slice of the double category of categories.Pursing some notion of “double topos” as a forum for formal category theory, this will give a settingfor elementary axiomatization of elements of presheaf and Yoneda theory for double categories.
The main definition of the paper anticipates the natural question about whether there is a moregeneral notion of a “double fibration” of which a discrete double fibration is a special case. For recallthat each ordinary discrete fibration F : F → C between 1-categories is a (split) fibration in a moregeneral sense. Split fibrations of course have lifting properties with respect to certain compatiblychosen “cartesian arrows” and correspond via a category of elements construction to contravariantcategory-valued 2-functors on the base category. The question of the double-categorical analogueis the subject of forthcoming work with G. Cruttwell, D. Pronk and M. Szyld. The evidence ofthe correctness of the proposed definition will be a representation theorem like Theorem 2.4.3 inthe present paper, but suitably upping the dimension of the representing structure. Thanks to Dr. Dorette Pronk for supervising the author’s thesis where some of the ideas for thispaper were first conceived. Thanks also to Geoff Cruttwell, Dorette Pronk and Martin Szyld fora number of helpful conversations, comments, and suggestions on the material in this project andrelated research. Thanks in particular to Geoff Cruttwell for clarifying some questions on virtualdouble categories. Special thanks are due to Bob Par´e for his encouragement when the author wasjust getting into double-categorical Yoneda theory.
References [Buc14] M. Buckley. Fibred 2-categories and bicategories.
Journal of Pure and Applied Algebra ,218(6):1034-1074, 2014.[CS10] G.S.H. Cruttwell and M.A. Shulman. A unified framework for generalized multicategories.
Theory and Applications of Categories , 24(21):580-655, 2010.[Dia73] R. Diaconescu. Change of base for some toposes. PhD Thesis, Dalhousie University, 1973.31Dia75] R. Diaconescu. Change of base for toposes with generators.
Journal of Pure and AppliedAlgebra , 6(3):191-218.[Ehr63] C. Ehresmann. Cat´egories et structures. Dunod, Paris, 1963.[GP99] M. Grandis and R. Par´e. Limits in Double Categories.
Cahiers de Topologie et G´eom´etrieDiff´erentielle Cat´egoriques , 40(3):162-220.[GP04] M. Grandis and R. Par´e. Adjoints for Double Categories.
Cahiers de Topologie etG´eom´etrie Diff´erentielle Cat´egoriques , 45(3):193-240.[Gray74] J. Gray.
Formal Category Theory: Adjointness for 2-Categories . Volume 391 of
LectureNotes in Mathematics , Spring, Berlin, 1974.[Joh14] P.T. Johnstone.
Topos Theory . Dover, Minneapolis, 2014.[Lam20] M. Lambert. Discrete 2-fibrations. Preprint: https://arxiv.org/abs/2001.11477 ,2020.[Lei04] T. Leinster.
Higher Operads, Higher Categories . Volume 298 of
London Mathematical So-ciety Lecture Notes Series , Cambridge University Press, Cambridge, 2004.[Mac98] S. MacLane.
Category Theory for the Working Mathematician . Volume 5 of
GraduateTexts in Mathematics , Springer, Berlin, 1998.[Par´e11] R. Par´e. Yoneda theory for double categories.
Theory and Applications of Categories ,25(17):436-489, 2011.[Par´e13] R. Par´e. Composition of modules for lax functors.
Theory and Applications of Categories ,27(16):393-444, 2013.[Shul08] M. Shulman. Framed bicategories and monoidal fibrations.
Theory and Applications ofCategories , 20(18):650-738, 2008.[Str74] R. Street. Fibrations and Yoneda’s lemma in a 2-category. In G.M. Kelly ed.
CategorySeminar: Proceedings Sydney Category Theory Seminar 1972/1973 , Volume 420 of
LectureNotes in Mathematics , Spring, Berlin, pp. 104-133, 1974.[Str76] R. Street. Limits indexed by category-valued 2-functors.
Journal of Pure and Applied Al-gebra , 8(2):149-181, 1976.[Wood82] R. Wood. Abstract proarrows I.
Cahiers de Topologie et G´eom´etrie Diff´erentielleCat´egoriques , 23(3): 279-290, 1982.[Wood85] R. Wood. Proarrows II.