In this note we investigate exponentiable objects and morphisms in the 2-categories and of augmented virtual double categories and virtual double categories, respectively. In doing so we will characterize exponentiable morphisms in the 2-category of -categories for an arbitrary algebraic pattern.
2. Exponentiable VDCs
In this section we aim to extend Pisani's argument for exponentiable multicategories[1] to VDCs. First, we write for the full sub-2-category of VDCs with only unary cells. Write for the inclusion. This has a natural right adjoint given by forgetting all cells but the unary ones. Observe that for a VDC and a unary VDC , the adjunction satisfies the Frobenius law
More generally, we write for the subcategory of VDCs that only have unary and -ary cells. We have again a natural right adjoint given by forgetting all cells that aren't either unary or -ary. In particular, .
We write , , for the VDC given by a single multi-cell
so that . Further define for the VDC with no loose arrows, a single non-identity tight arrow, and no cells. Similarly, write , , for the VDC given by a single multi-cell
while is the walking object. Then VD functors are exactly the multi-cells in with -length source, while the VD functors are exactly the loosely globular multi-cells in .
Definition 1(Pro-representable VDC).
A virtual double category is said to be representable, or exponentiable, if taking its product admits a right adjoint:
Now, observe that if and are VDCs for which their exponent exists, then it must consist of the following data:
Objects are functors .
Tight arrows are natural transformations between functors
Loose arrows consist of a pair of functors together with, for each unary cell in , a unary cell
in which is functorial in the tight direction. Equivalently this is a map of (non-unital) loose bimodule . Equivalently, these are maps of spans
Multi-cells for correspond to loose arrows and , tight transformations and , and for each -multicell in an -multicell
satisfying the pasting identity:
For we a have a functor , a VD functor , tight transformations and , and for each -multicell in , a -multicell
satisfying the pasting identity:
Proposition 2(Exponentiable VDCs).
The exponentiable VDCs are those admitting unique (up to associativity) decompositions of any shape of multi-cell.
Proof. : Suppose is an exponentiable VDC. Let be as defined previously, and consider the colimit of the diagram in given below:
Then can be diagrammatically summarized by
The product consists of five copies of , denoted , and , together with (non-unital) loose hom-bimodules , , and , tight hom-bimodules and , and for any -cell in a -cell
where the subscripts are used to specify in which copy each object, tight arrow, and loose arrow lives. Similarly has nine copies of , six copies of , five copies of , along two kinds of -cells as above, one special kind of -cells, and a -cell for any -cell in .
On the other hand, the colimit of the diagram
also has nine copies of , six copies of , five copies of , and the same two kinds of -cells, and -cells, but only -cells which arise as formal composites
subject to the associativity constraint which for a pasting diagram
says that the two orders of composing, namely composing the center cells with the upper ones versus composing the center cells with the lower one, yield the same formal -cell.
As is assumed exponentiable, preserves colimits so that the canonical VD functor must be an isomorphism. Thus, any -ary cell in must decompose as above, with this decomposition being unique up to the equivalence relation generated by the associativity constraint above. More generally, for any and any with , we must have that any -ary cell in decomposes as a composite
where , and such that this decomposition is unique up to an associativity for composing cells in the center of the pasting diagram, as in the case for , just considered.
For instance, if we instead consider the pushout
so that can be summarized diagrammatically as
Then similarly, the canonical map must be an isomorphism, which stipulates that every unary cell in must decompose as
with this decomposition being unique up to the same kind of associativity condition given previously in the case of . Inductively, we get decomposition of -multicells through -multicells which are unique up to associativity.
: Conversely, suppose is a VDC such that for each with , each -multicell decomposes as a composite of -multicells with an -multicell, and the decomposition is unique up to associativity. Let be any other VDC. Define the underlying box-graph of be defined in terms of the description given previously for objects, tight arrows, loose arrows, and multicells. It remains to define a unital and associative scheme for pasting multicells. Suppose we have a pasting scheme
so that and are VD functors, where . Then we define their composite by setting its action on a -multicell
by first decomposing it through the assumed isomorphism
as
which is unique up to associativity in center pastings, and then applying the pasting scheme and composing in :
which by functoriality of the , , and is independent of the choice of decomposition. Naturality follows from functoriality of the , , and . Associativity of the composition then follows from associativity of the composition in . Further, for any loose arrow , we have the identity cell corresponding to the VD functor which sends a unary cell in to
From our definition of composition in above, and the functoriality of -multicells, it follows that these unary cells act as identities under vertical pasting.
Finally, to show that the VDC is indeed a power in , let be another VDC. Note that since tight transformations correspond to VD functors , it suffices to prove the -dimensional universal property. Then we define the map
by sending a VD functor to the VD functor given as follows:
An object is sent to the functor which sends an object to and a tight arrow to .
A tight arrow is sent to the natural transformation given at an object by , so that naturality follows from functoriality of .
A loose arrow is sent to the map of (non-unital) loose bimodules with domain and target actions and , respectively, along with carrier functor sending a loose arrow to and sending a unary cell to
An -multicell in is sent to the -multicell which acts on an -multicell in by .
The functoriality of and definition of cellular composition in ensures that this indeed produces a VD functor. Conversely, given a VD functor the above definition of implies how should be defined, with
For objects and ,
For tight arrows and ,
For loose arrows and ,
For -multicells and , .
The definition of composition in and functoriality of ensures that is a VD-functor. □
We refer to exponentiable VDCs as pro-representable. For a VDC , and for loose arrows , and in , let denote the set of cells with loose source the sequence and with loose target . Then square composition gives functions
Proof.
Let be a representable VDC. Since has all cocartesian cells, the functions
are all surjective. To show they are bijective, consider a decomposition of a multi-cell in
where . It suffices to show that the decomposition above right is equivalent to the canonical decomposition
Using the universal property of cocartesian cells we can factor the first decomposition as follows:
The uniqueness in factorizations through cocartesian cells ensures that the composite cell below the cocartesian cells on the right must be equal to , and hence the two decompositions are equivalent up to associativity. □
For instance, this says that for any category , the tight VDC is pro-representable and the unital VDC is pro-representable. For example, the walking object VDC is pro-representable, with is the chaotic VDC associated to the category (i.e. we add a unique loose arrow between all objects, and a unique cell filling all squares). This fits into a triple adjunction
In particular, we easily obtain the well-known results that limits and colimits in are given by limits and colimits in on the level of tight categories. On the other hand, because the non-unital loose arrow is not exponentiable (c.f. Exponentiating (A)VDCs > ^266992Example 11 ( is not Pro-representable)) although we have an adjunction
the functor does not admit a right adjoint, as it doesn't preserve arbitrary colimits.
Our next goal is to find some examples of pro-representable VDCs. First let's consider the following construction of new pro-representable VDCs from old ones:
Construction 4(Tight Bimodule Associated to Map of Pro-Representable VDCs).
Let be a VD functor between pro-representable VDCs. Then we can define a tight bimodule by taking its companion in the virtual equipment . Recall that this is equivalent to a VD category with a VD functor into the unital tight arrow. Then is also pro-representable as a VDC.
Proof.
DETAILS NEEDED
It suffices to show that we have essentially unique decompositions of heterocells. But heterocells are precisely multi-cells of the form
in , which can be decomposed uniquely up to associativity since is pro-representable. Since also contains a copy of , in order for to be pro-representable we also need to be pro-representable. □
If is not pro-representable, we can always restrict the tight bimodule along the inclusion of a sub-VDC which is pro-representable in order to get a pro-representable VDC . We now show that a maximal such sub-VDC always exists:
Proposition 5( is stable under filtered colimits).
is stable under filtered colimits.
Proof.
□
To give some novel examples or pro-representable VDCs, let's recall a characterization of pro-monoidal categories.
Theorem 6(Characterization of Pro-monoidal Categories).
Let be a symmetric monoidal closed category with all limits and colimits together with a normalization . A small -category underlies a pro-representable multi-category if and only if is a biclosed -category, with a choice of biclosed structure corresponding to a pro-representable multi-category with underlying category .
Proof. TBC □
Example 7(Cartesian Closed presheaf Categories).
Recall that any presheaf category has the structure of a cartesian closed category, with and , where .
Thus, any small category underlies a pro-representable multicategory. Namely, the multi-category associated to the cartesian closed structure on , , has objects those of , and has multi-morphisms defined inductively. We have a single nullary morphism for each object . Then if the -ary morphisms are defined for some , we can define the -ary morphisms as the co-end
Thus, , and so inductively
It follows that the multicategory is representable if and only if has finite coproducts.
Example 8( for a discrete category).
If is a discrete category, then is the multicategory with two objects, and , for which
If instead of being a discrete category we had be the walking parallel arrows , then
Recall that we have an embedding of the 2-category of multicategories into the 2-category of VDCs given by sending a multicategory to the VDC whose underlying tight category is the terminal category , whose loose arrows are objects in , and whose cells are multimorphisms in . The multicategories sent to pro-representable VDCs are precisely the pro-representable multicategories, so that this embedding restricts to an embedding .
Before this, observe that if is a pro-representable VDC and is any other VDC, then can be described as follows:
An object, which is a monoid, is a functor map of non-unital loose bimodules with legs , together with compatible actions on -multicells making it a VD functor
A tight arrow is a tight transformation between VD functors
A loose arrow is a VD functor
An -multicell
consists of VD functors and , tight transformations , and modules and , together with for every -cell in a cell
compatible with vertical pasting, along with an equality of the multi-cells
and
corresponding to inner equivariance, as well as equalities
and
corresponding to outer equivariance. Thus, we recover Pare's VDC of VD functors, tight transformations, modules, and multi-modulations.
2.1. Examples of Pro-representable VDCs
We will begin with an example motivated by the promonoidal category governing symmetric monoidal categories in[2]. First, we construct a promonoidal category for coloured pros. Recall that a coloured pro consists of the following data: (find reference - right now adapting hackney and robertson's definition of coloured props)
A set of colours
For every ordered list of colours , a set of operations
A specified element for each
An associative vertical composition
An associative horizontal composition
satisfying unitality and four-interchange axioms. Coloured props additionally have symmetry actions on domains and codomains which are compatible with compositions.
We can define a promonoidal category governing coloured props in a precise sense. Objects are pairs of objects in (= the simplex category with a free initial object), while the hom-set is given by
Proposition 9(Cospans give Pro-representable VDCs).
For any category , the VDC is pro-representable, and is representable if and only if admits finite pushouts.
Proof.
The second claim is well-known so let's consider the first claim. To begin consider an -cell in which equates to a commuting diagram of the form
and let be the unique maps for the middle 's. First let's show existence of decompositions. If for . If there is nothing to do. If , and , then we can factor it as
where , and for , if we insert the nullary cell
If (or similarly for ) we insert the nullary cell
on the left (resp. right) the the bottom cell becomes
So far this proves surjectivity of our laxators. To prove injectivity we need uniqueness. However, this follows from the fact that any other decomposition is equivalent up to associativity to this one by sliding the bottom cell up. For example, in the case an arbitrary decomposition below left is equivalent to the canonical decomposition through an associativity pasting in the center:
□
Proposition 10( preserves exponentiability).
If is a pro-representable VDC, then so is .
Proof.
□
Example 11( is not Pro-representable).
The non-unital walking loose arrow is not a pro-representable VDC. This can be shown by providing a colimit that doesn't preserve. Consider the diagram and resulting colimit in below:
Let denote the colimit. Then has two distinct unary non-identity cells. On the other hand, if we take the product with before taking the colimit, then the result only has a single non-identity unary cell
2.2. Exponentiating Morphisms
In this section we investigate exponentiable morphisms between VDCs. The arguments in this section are heavily inspired by the section on powerful morphisms in Street and Verity's paper on factorization and torsors[3]. In particular, we use the fact that a morphism in a finitely complete category is powerful, or exponentiable, if and only if the pullback functor admits a right adjoint. Since we will be considering the case where (and later ), it is finitely presentable so such an adjoint exists if and only if preserves all colimits by a version off the Locally Presentable Categories > ^6b717dTheorem 15 (Special Adjoint Functor Theorem).
First, suppose is an exponentiable morphism, so that we have a right adjoint . Then for a VDC , the structure of can be determined by the adjunction
First, note that if is an object in , then . If is a tight arrow in , corresponding to a map , then is a collage associated to a (non-unital) tight bimodule. If is a loose arrow in , corresponding to a map , then is the collage associated to a (non-unital) loose bimodule. Finally, for a multi-cell
in , which can be given by a map , can be thought of as a collage of the diagram
Then we can describe the structure of as follows:
An object in lying over is equivalent to a functor
A tight arrow in lying over in is equivalent to a pair of functors and along with a functor extending them
A loose arrow in lying over in is equivalent to a pair of functors and along with a map of spans
where we view .
Finally, for -multicell in lying over the multi-cell in depicted previously, if , its boundary consists of a collection of maps of spans
and
along with a functor extending and , and a functor extending and . We write
for the resulting boundary with a multi-cell filling it. Such a multi-cell assigns to each multi-cell
in lying over a multi-cell
satisfying the pasting identity
The nullary case is similar.
Given a VD functor , let be a functor over preserving cocartesian lifts of inerts obtained by un-straightening . For example, the fiber , while for , is the category whose objects are length sequences of loose arrows, , and whose morphisms are length sequences of unary cells
A morphism in , lying over a map of posets consists of a family of multi-cells:
Then is equivalent to the data of a normal VD functor , where an object is mapped to the fiber category , while a sequence of loose arrows viewed as an object of is sent to the fiber which consists of length sequences of loose arrows lying over and sequences of unary cells lying over the sequence of identity unary cells for . Finally, a map lying over in is sent to the profunctor (i.e. ) that maps an object to the set of maps lying over , and maps sequences of unary cells to the set function given by pasting.
In this language, a partition of an integer is equivalent to an order preserving map that is active (i.e. it preserves top and bottom elements). For , let denote the unique active map from to .
Theorem 12(Exponentiable VD Functors).
A VD-functor is exponentiable, or powerful, if and only if for any integer , any active map in , and any lying over and lying over in , the laxator
is an isomorphism.
Intuition: Intuitively the condition in the theorem statement says that we can uniquely lift decompositions of cells in , up to associativity of the pasting in the decomposition (with intermediate cells lying over identity cells). When is the terminal VDC this reduces to the co-end isomorphism Exponentiating (A)VDCs > ^6074f7Exponentiating (A)VDCs > (1)
Proof. : First, suppose is exponentiable, so that preserves all colimits. As in the proof of Exponentiating (A)VDCs > ^a824a1Proposition 2 (Exponentiable VDCs) let be a diagram picking out cells in the specified decomposition of the -multi-cell in . Then the colimit has domain the free -multicell with decomposition relative to the partition (since colimits in the slice are created in the underlying category), and picks out the specified decomposition of in . Since is exponentiable, the natural map
is an isomorphism. The right side contains all the -multi-cells in lying over , while the image of the map contains only those -multi-cells in that can be decomposed relative to the partition and lying over the specified decomposition of . Thus, the induced map
: Conversely, suppose the stated laxator morphisms are isomorphisms. Fix a VDC , and let the underlying box-graph of be defined in terms of the description given previously for objects, tight arrows, loose arrows, and multicells. It remains to define a unital and associative assignment for pasting multicells. Suppose we have a pasting scheme
where lies over and lies over in , so that and are VD functors, where . Write . Then we define their composite by setting its action on a -multicell
lying over by first decomposing it through the assumed isomorphism
as
which is unique up to associativity in center pastings over identity cells, and then applying the pasting scheme and composing in :
which by functoriality of the , , and is independent of the choice of decomposition. Naturality follows from functoriality of the , , and . Associativity of the composition then follows from associativity of the composition in . Further, for any loose arrow , we have the identity cell corresponding to the VD functor which sends a unary cell in lying over to
From our definition of composition in above, and the functoriality of -multicells, it follows that these unary cells act as identities under vertical pasting.
Finally, to show that the VDC is indeed a dependent product in , let be another map of VDCs. Note that since tight transformations over correspond to VD functors over , with the structure map for given by projecting onto and then using its structure map , it suffices to prove the -dimensional universal property. Then we define the map
by sending a VD functor to the VD functor over given as follows:
An object is sent to the functor which sends an object over to and a tight arrow over to .
A tight arrow is sent to the functor extending and by mapping a tight arrow over to .
A loose arrow is sent to the map of spans
which sends a loose arrow lying over to the lose arrow , and sends a unary cell lying over to .
An -multicell in is sent to the -multicell which acts on an -multicell in lying over by .
The functoriality of and the definition of cellular composition in ensures that this indeed produces a VD functor over . Conversely, given a VD functor over , the above definition of implies should be defined as follows:
For objects and ,
For tight arrows and ,
For loose arrows and ,
For -multicells and , .
The definition of composition in and functoriality of ensures that is a VD-functor. □
Explication: The exponentiable, or powerful, VD functors are precisely those for which any multicell in , below upper left, lying over a multi-cell in , below upper right, if the multi-cell in admits a decomposition, below bottom right, then the multicell in admits a decomposition lying over the decomposition of its image, below lower left:
Further, any two such decompositions are equivalent via unary cells lying over identity unary cells. That is, if we have two decompositions, as on the left and right below, we get a zig-zag of cells (of length possibly , though we just depict the length case) given by pulling out unary cells in fibers from either the top or bottom:
where for all .
3. Exponentiation for Arbitrary Algebraic Patterns
In order to extend the previous results for VDCs to AVDCs we will go one step of generality further and characterize exponentiability of arbitrary categories over algebraic patterns. Recall that an algebraic pattern is a tuple consisting of category together with an orthogonal factorization system and a full subcategory of the wide inert subcategory consisting of elementary objects (c.f. Homotopy-Coherent Algebra via Segal Conditions - Scrap NotesHomotopy-Coherent Algebra via Segal Conditions - Scrap Notes). The 2-category of -categories is the sub-2-category of the slice whose objects are the weak Segal -fibrations, whose morphisms are those that preserve cocartesian lifts of inert morphisms, and whose cells are arbitrary.
Recall that being a weak Segal -fibration means that
(i) admits cocartesian lifts of inert morphisms
(ii) The subcategory is dense (i.e. for any , the natural map
is an isomorphism.
(iii) For any in , and any lying over and respectively, the natural map
is an isomorphism.
Equivalently, under the un/straightening correspondence such a weak Segal -fibration is equivalent to a normal VD functor such that
(i) If is inert, then the profunctor is corepresented by a functor , and for any (not necessarily inert) the laxator
is an isomorphism.
(ii) By (i) we have that induces a pseudo-functor . Then we require that is the right Kan extension of
(iii) For any , we have a functor sending to the profunctor , and an inert map , viewed as a map , to the transformation which is given by following the sequence of assignments
(Show this is functorial) Then we require that the natural cone is a limit cone.
Recall that the case for VDCs corresponds to the algebraic pattern . TBC
3.1. Exponentiable AVDCs
In this section we aim to adapt the previous argument for exponentiable VDCs to AVDCs. We write , , , for the AVDC given by a single multi-cell
Further, let denote the AVDC with no loose arrows, a single non-identity tight arrow, and no cells non-identity globular cells, and similarly let denote the AVDC with no non-identity tight arrows, a single loose arrow between distinct objects, and no non-identity cells. Then AVD functors are exactly the multi-cells in with -length source and -length target, while AVD functors and are precisely tight and loose arrows in , respectively.
More generally, if is a 2-category, let denote the AVDC with tight 2-category , and no loose arrows, and let denote the AVDC with discrete tight category , one loose arrow for each morphism in , and special cells those in .
Now, suppose and are AVDCs for which the exponent exists. Then must consist of the following data:
Objects are 2-functors .
Tight arrows are 2-natural transformations between 2-functors
Loose arrows consist of a pair of 2-functors together with, for each unary cell in , a unary cell
which is functorial in the tight direction and natural with respect to globular cells. Equivalently this is a map of (non-unital) loose bimodules
Multi-cells correspond to 2-functors , , maps of (non-unital) loose bimodules if and a map of non-unital loose bimodules if , 2-natural transformations and , and for each -multicell in an -multicell
satisfying the pasting identity:
Proposition 13(Exponentiable AVDCs).
The exponentiable AVDCs are those whose cells admit unbiased decompositions that are unique up to associativity.
Proof. : Suppose is an exponentiable AVDC. Let , such that . Then let be the diagram depicted below:
where the middle vertical arrows pick out the loose target of the unique non-identity multi-cell, the top arrows pick out tight sources and targets of the multi-cells, and the bottom maps picks out the -th object in the loose source of the unique non-identity multi-cell, and if it also picks out the loose arrow with that object as source.
Let . Then can be diagrammatically summarized by
where underlines are used to indicate (possibly empty) sequences of loose arrows. Note that the product consists of copies of , denoted , and , together with copies of (non-unital) loose hom-bimodule , copies of the tight hom-bimodule , and for any -cell in , an associated -cell. CONTINUE FROM HERE