Loose Bimodules for (Augmented) Virtual Double Categories

1. Introduction

In this note we aim to obtain notions of loose bimodules, as seen in Sophie Libkind, David Jaz Myers, and Kevin Carlson's paper Loose Biomodules and Their Construction[1], in the context of general virtual double categories.

Let denote the -category of virtual double categories, let denote the -category of augmented virtual double categories, with associated chordate -categories. We write and (resp. and ) for the -subcategories of unital and composable virtual double categories, respectively, with -unit and -composition preserving virtual double functors, respectively and transformations of virtual double functors, with tight morphisms the strictly unit preserving virtual double functors (resp. augmented virtual double categories).

In order to use the constructions of Loose Bimodules and Their Construction - Scrap Notes we will need that and admit certain -pullbacks. But this follows from Theorem 13 (Limits in Tensored Categories are Conical) and the fact that their underlying categories are finitely presentable, and hence (co)complete, while also being tensored. Indeed, for a category , if is the unital virtual double category whose underlying tight category is , with only identity pro-arrows and identity cells, then is the tensoring in .

  • TODO: Add details and do version for augmented virtual double categories

We also have comma objects.

Lemma 1 (Comma Objects in and ).

The -categories and admit all comma objects.

Proof.
Consider the cospan of virtual double categories (resp. augmented virtual double categories). I claim that the comma virtual double category in (resp. ) can be given by the (augmented) virtual double category with the following data:

  • The ordinary comma category as its category of tight morphisms
  • For any two objects , the class of cells
  • A -cell consists of a pair of -cells, as below

such that we have the pasting equality

Since and preserve identities and composites strictly, this data indeed defines a(n) (augmented) virtual double category, which comes equipped with a natural square

where the tight arrows and the cells of the transformation are given by those for the objects and loose arrows in . When all (augmented) virtual double categories involved are representable, and and are strict virtual double functors between representable double categories, with TBC
□

Recall that for virtual double categories and , a virtual profunctor consists of an ordinary profunctor (i.e. a functor ) together with the following:

  • For every string of loose arrows in , and each loose arrow in , as well as elements and , a collection of hetero-cells

which are acted on above by cells in and below by cells in , compatible with the action of tight arrows from . These are the loose morphisms in a virtual equipment where is the free category monad on , which extends to a monad on the virtual equipment . The objects are precisely (small) virtual double categories, with homomorphisms being virtual double functors, modules being virtual profunctors, and cells being generalized virtual transformations explicate this. Explicitly, given virtual double categories and with their underlying spans of graphs and , a virtual profunctor consists of a graph which is part of a span

together with maps of spans of graphs

which are compatible up to natural isomorphisms.

This recovers the notion of (virtual) tight bimodules (virtual double categories over the walking tight arrow ).

Given maps of VD categories , , and a tight bimodule , the restriction has hetero-cells

for a sequence of loose arrows in , a loose arrow in , and a hetero-cell

in . For instance, if is a VD functor, then its companion is the tight bimodule with hetero-cells given by multi-cells

in with a loose arrow in . Dually, its conjoint is the tight bimodule with hetero-cells given by multi-cells

for loose arrows in .

1.1. Recollections on Category Bimodules

Recall that category bimodules are either equivalent to profunctors , or to functors over the walking arrow , via the collage-construction. Explicitly, we get a equivalence of pseudo-double categories
where on both double categories have the same tight categories, under the correspondence in Theorem 2 (Characterization of Two-sided Discrete Fibrations), and where loose arrows on the right are functors over the walking arrow with appropriate domain and codomain.
Proof.
TBD
□

We will write when being vague about which of these two equivalent double categories we are considering. We can also represent using limit sketches. Let denote the -category of (non-weighted) sketches, sketch morphisms, and arbitrary natural transformations.

Example 2 (Simplex Limit Sketch).

We can view as a limit sketch by declaring its marked cones to be those given by all inert maps to objects and from an object . Recall that the inert maps in are those injective maps which are surjective onto a segment of . These are precisely the injective maps generated by the edge face maps . Let denote the full subcategory of spanned by and . Then the cones are precisely those functors

for each .

The models for this limit sketch in are those simplicial sets satisfying the Segal condition, which is to say the induced maps

are isomorphisms. This recovers the essential image of the nerve .

This is a special case of the construction for an algebraic pattern , which is a small category with an orthogonal factorization system and a full subcategory of elementary objects. Then has the structure of a limit sketch with marked cones
for all .

Note that if is a sketch and , then the slice category gets the structure of a sketch where a marked cone in determines a unique marked cone for each in using the natural isomorphism .

Definition 3 (Bimodule in a Category ).

A bimodule in , for a category, is a model for the limit sketch in .

Note that . This notation is used since we have an isomorphism . Recall that is the limit sketch whose objects are morphisms in (i.e. morphisms in ), and whose morphisms are commuting triangles. Note that this is precisely the opposite of the full subcategory , which under our previous isomorphism corresponds to profunctors such that is the singleton for all and . The marked cones are the composites

for each simplex map , which we can think of as a primitive category bimodule.

Note that an object of consists of the following data:

  1. For each primitive category bimodule , an object
  2. For each map of category bimodules a morphism in
  3. For each primitive category bimodule , a limit cone which takes the form

so that the natural map

is an isomorphism in . In analogy with Segal simplicial sets, we can consider the primitive category bimodules as giving and as the domain and codomain of the bimodule, as the carrier, and and as objects of left and right actions, respectively. Explicitly, using the previous isomorphism, we get action maps

Note that when the isomorphisms in Loose Bimodules for (Augmented) Virtual Double Categories > (1) realize the pair as a strict category object internal to , and similarly for realizing as a category object internal to . A morphism in is then precisely a pair of functors of -internal categories and together with a morphism which respects the bi-category actions. ADD DETAILS

In particular, when , we recover our category of category bimodules.

Theorem 4 (Slice Theorem for Sketches).

Given a limit sketch and a model , there is an isomorphism

where is given the structure of a limit sketch by marking all lifts of marked cones in along the projection .

Proof.
Recall that for a copresheaf we have a sequence of equivalences

with the middle isomorphism coming from Lemma 5 (Maps into Discrete Fibrations are Fibrations). The claim will now follow from a more general result for -sketches seen below.
□

As a special case we obtain the equivalences:
𝓎

Question

Why is it pseudo-bimodules in in the forest

2. Loose Bimodules on (Augmented) Virtual Double Categories

To begin, recall that we have strict -functors and , with the second of which being fully-faithful PROVE THIS!!. In particular, via we can consider the walking loose arrow as a unital augmented virtual double category, while then realizes it also as a unital virtual double category.

We will write for the unital VDC, while will denote the non-unital VDC with two objects and a sngle loose arrow between them. Recall we have a 2-comonad given by forgetting non-identity -ary cells, the coalgebra 2-category of which being . We will proceed using the language of AVDCs, specifying the difference between the general case and the case of diminished AVDCs to encapsulate the behaviour for ordinary VDCs.

Definition 5 (Loose Bimodule on s).

A loose bimodule is an augmented virtual double functor into the unital walking loose arrow, where is an augmented virtual double category. We write for the slice -category of loose bimodules.

Diagrammatically we can depict this situation by a pair of pullbacks

Note that a loose bimodule in this sense agrees with a loose bimodule in the sense of Loose Bimodules and Their Construction - Scrap Notes > ^4fe56e when is representable, which is to say it's in the image of . We write and for the source and target of the augmented virtual loose bimodule. Recall that if is a (strict) double category, a loose bimodule is equivalent to a bimodule between monoids in , which consists of span of categories together with functors and which satisfy the bimodule axioms.

On the other hand, when is an AVDC, a AVD-functor can be explicated as the following data:

  • An AVDC over and an AVDC over
  • For each pair of objects and , a set of loose arrows , lying over the non-identity loose arrow in
  • For each sequence of loose arrows in and in , along with tight arrows and , and loose arrows and , a collection of cells
  • For pair of cells and , along with sequences of cells and , a composite cell

If is a diminished AVDC (i.e. an ordinary VDC) then the data is identical, with the one change that and are also diminished AVDCs (i.e. VDCs).

Remark 6 (Underline Notation).

In the above work, a loose arrow with an underline, such as , represents to a sequence of loose arrows , while a cell with an underline, such as notates a sequence of - and -coary cells.

Question

Does match an existing structure?

If instead we consider AVD functors into the non-unital loose arrow, , the data is encapsulated as follows:

  • -categories and over and , respectively
  • A category together with projections and to the underlying categories of the -categories
  • If denotes the set of cells in , then we have compatible action maps and which preserve sources and targets, and are strictly compatible under vertical composition in the sense that we have the equalities

Note that if is a VDC then this forces that is a strict double category with no composable arrows, and this data is equivalent to a span of categories , with no additional data. Morphisms correspond to -functors together with a functor which is compatible with the under sources and targets, and equivariant with respect with respect to the actions on either side. Finally, -cells , given by (unital) oplax transformations (c.f. Definition 5 (Lax AVD Transformation)) correspond to lax transformations (because of the arrow convention above) together with for each object in the carrier, a cell , which form a natural family up to the action by the lax transformations, which is to say

Question

Is this a known structure?

In particular, it follows that is isomorphic to the -category whose objects are spans of categories, whose morphisms are maps of spans, and whose cells are compatible natural transformations between the morphisms in the maps of spans.

2.1. Models Approach

Recall that as in[2] a diminished AVDC (i.e. a VDC with whiskering) is equivalent to the data of a weak segal fibration for the algebraic pattern , which is to

  • (i) admits cocartesian lifts of all inert morphisms (i.e. inclusions of sub-intervals, )
  • (ii) For each the functor induced by the cocartesian arrows is an equivalence of categories
  • (iii) For every morphism in and , composition with cocartesian morphisms over the inert morphisms gives an isomorphism
    Explicate: For such a functor , the fiber is the tight category for the diminished AVDC, while the objects of are pro-arrows between objects in . By (ii), the objects of for is a sequence of pro-arrows . By (iii) a morphism lying over consists of a sequence of morphisms for each , which we picture as a cell with loose

For example, is represented

2.2. Square Bimodules and Tri-AVDC of Bimodules

Let denote the unital AVDC which has a unique non-cocartesian cell

Then we can define square bimodules in an analogous fashion to tight and loose bimodules.

Definition 7 (Square Bimodule).

Given and , a squares bimodule is an AVDC together with an AVD functor .

  • TBC

3. Stuff below needs to be changed/adjusted

The data for such a loose bimodule is equivalent to the data of a twisted virtual double functor where is the pseudo-double category of categories, functors, profunctors, and profunctor transformations, interpreted as an augmented virtual double category via .

Definition 8 (Twisted Virtual Double Functor).

For augmented virtual double categories , a twisted virtual double functor consists of the following data:

  1. An assignment on objects
  2. For each sequence of loose arrows a tight arrow which is functorial in the sense that
  3. For each tight arrow a profunctor , which is pseudo-functorial, which is to say that if is another tight arrow, then is given by composition of profunctors
  4. For each multi-cell , a transformation of profunctors
    Add coherence details

Case for VDCs (not AVDCs)
In our current situation the twisted virtual double functor is given by sending a pair of objects to the category with objects pairs of integers and sequences of tight morphisms, and with morphisms consisting of a
a poset map preserving top and bottom elements along with tight arrows and cells

Composition is given by composing poset maps and vertical composition of cells in .

Sequences of loose arrows then induce functors given by pasting on identity cells, and this assignment is functorial with respect to concatenation of loose arrows in either variable. Explicitly, an object is sent to where denotes concatenation, and morphism below left, corresponding to a sequence of cells in , is sent to the concatenation below right

For each pair of tight arrows , a profunctor given by the functor that sends pairs of objects and to the set of sequences of cells of the form

to the set of maps, analogously to those in , but with tight source and tight target . The action on morphisms is given by vertical composition of such sequences of cells.

Additionally, observe that a map of loose bimodules corresponds in this language to virtual double functors and together with a transformation of twisted virtual double functors from to given by the following data:

  1. For each pair of objects , a functor
  2. For each pair of tight arrows , a profunctor transformation

given by a natural transformation

which at is given by applying to the resulting cells in .

Definition 9 (Transformation of Twisted Virtual Double Functors).

Given twisted virtual double functors , a transformation of twisted virtual double functors consists of

  1. A tight arrow for all objects
  2. A cell, as below, for all tight arrows , satisfying cellular naturality

Finally, a virtual transformation over is equivalent to a modification between transformations of twisted virtual double functors. This consists of the following data:

  1. Virtual transformations and
  2. For each pair of objects , a profunctor transformation

given by a natural transformation

which at is given by pasting with the squares, for a loose morphism in .

  • TODO: Make this all explicit with diagram
  • Note: Technically only one loose arrow in a given sequence over the loose arrow can lie over (all other ones need to be in one of the fibers) - maybe use this in formalizing the above?

In this way we get a fully-faithful embedding

where on the right is the very large -category of locally small augmented virtual double categories.

  • TODO: Make this precise and formulate as a theorem

One of the essential benefits of this perspective is that we immediately obtain that the -categories involved are pro-arrow equipments. Indeed, has all restrictions, and these induce restrictions for

  • TODO: Generalize this and state it as a proposition

In particular, any virtual double functor induces a 2-functor(pseudo-functor?)

  • TODO: Prove this

We will write for the virtual equipment of augmented virtual double categories, tight -cells the virtual double functors between them, loose -cells the loose bimodules, and -cells of the form

3.1. -Sketches Perspective

Recall that in the sense of Leinster's Higher Operads, Higher Category Theory, the -category of pseudo-models of , which is given a -category structure by labelling its inerts as tight morphisms, is equivalent to the -category of unbiased weak double categories, -weak double functors, and tight transformations, with the tight morphisms being strict double functors. Alternatively, using Example 15 (Pseudocategory sketch) as a baseline, we could instead take as the -sketch generated by objects , for all , together with tight morphisms generated by the which satisfy the same relations as they do in , and loose morphisms generated by and under composition and the join functors , where we write for the -fold composite, along with invertible -cells

such that is a bifunctor, and subject to the equalities

and

REFERENCE BICATEGORICAL COHERENCE THEOREM to prove this is enough Then we have an equivalence of -categories
for any -sketch , and the right hand side is isomorphic to when .

  • TODO: Prove this claim

Extending slightly we obtain an equivalence of -categories

which is equivalent to when .

  • TODO: Prove this claim

3.2. Non-existence of Associative Composition and Virtual Double Category of Loose Bimodules

In this subsection we discuss the non-existence of a natural associative composition for loose bimodules, with argument following that in[3]. Let be the walking tight arrow, let be the walking tight composite, and let be the double category with two non-identity tight arrows and , and one non-identity loose arrow , with no non-identity cells. Using the barrel perspective on loose bimodules we then define bimodules , , and as in

and

Together these loose bimodules can be represented by the diagram

Now, towards a contradiction suppose we had a pseudo-associative notion of composition satisfying

  • ,

4. Pseudo-Functorial Restriction of Loose Bimodules

Firstly, we define the VDC to be the sub-VDC of whose loose arrows consist of two-sided codiscrete cofibrations:

VDC's internal to

If we can make sense of VDCs internal to another category, is a virtual double 2-category?

Question

Is a pro-representable VDC?

Question

Is there a Grothendieck type construction for VD functors into ?

4.1. Restriction Versus Collapse

In addition to restriction of loose bimodules we can also consider collapse of loose bimodules. This is most easily defined in the context of two-sided double fibrations , where we simply post-compose by the unique map 𝟙 into the terminal double category. This construction defines loose bimodule 𝟙, and in fact we get a cartesian 2-functor

In particular, we get induced 2-functors

We also have the cartesian pseudo-functor , and hence the pseudo-functors

and

We then get a natural cartesian 2-functor such that the composite lands in , the full sub--category of symmetric monoidal loose right modules. Explicitly, this sends a symmetric monoidal loose bimodule to the niche

where the left pseudo double-functor picks out the loose unit for the domain of the loose bimodule.

In the language of two-sided double fibrations these two pseudo-functors on symmetric monoidal loose bimodules correspond to

and

In particular, we see that we get a natural comparison map

  • Claim: This extends into a pseudo-natural transformation .
    Recall that given a two-sided double fibration , the associated loose bimodule has for and those loose arrows which are objects of lying over and on the left and right. Thus, is given by the following data:
  • The terminal double category lying over and the double category lying over
  • For each , and each pro-arrow in over the walking loose arrow, a pro-arrow , and for each cell in over the walking loose arrow below left, a cell in below right
  • Composition on the left is strictly unital, while composition on the right is given as in using the description of loose arrows and cells as above.
  • Vertical composition of cells over the walking loose arrow is given as below:

using the vertical composition in .

Example 10 (-categorical Example).

Let be a symmetric monoidal -category, so that we have the natural symmetric monoidal hom-two sided (discrete) fibration , which corresponds to the (discrete) bimodule , or the profunctor . The collapse corresponds to the (not necessarily discrete) two-sided fibration , or the profunctor , or the bimodule .

5. Loose Adjunctions and Loose Universal Properties

We now discuss a bimodule-oriented notion of loose adjunction.

Definition 11 (Loose Adjunction).

Given virtual double functors , a lax loose adjunction between them consists of an adjunction of loose bimodules .

5.1.1. References

  1. Libkind, S., Jaz Meyers, D., Carlson, K. (2025). Loose Bimodules and their Construction. https://forest.topos.site/ssl-0030.xml#tree-8021↩︎

  2. Gepner, D., & Haugseng, R. (2015). Enriched -categories via non-symmetric -operads. Advances in Mathematics, 279, 575–716. https://doi.org/10.1016/j.aim.2015.02.007↩︎

  3. nlab authors (2025, June). Double Profunctor. https://ncatlab.org/nlab/show/double+profunctor (Note: https://ncatlab.org/nlab/revision/double+profunctor/13)↩︎