Exponentiable Virtual Double Categories and Representability of Exponentials

Ea E Thompson; Kevin D. Carlson

arxiv: 2605.20586 · v1 · pith:TPNQSK5Inew · submitted 2026-05-20 · 🧮 math.CT

Exponentiable Virtual Double Categories and Representability of Exponentials

Kevin D. Carlson , Ea E Thompson This is my paper

Pith reviewed 2026-05-21 02:36 UTC · model grok-4.3

classification 🧮 math.CT

keywords virtual double categoriesexponentiabilitycell decompositionslimit sketchesmulticategoriessemicategoriescospansfunctors

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The pith

A virtual double category is exponentiable precisely when its cells admit specific decompositions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes several equivalent conditions under which a virtual double category is exponentiable, so that morphisms out of it themselves assemble into a virtual double category. These conditions are expressed in terms of the existence of decompositions that break cells into compatible sequences respecting their domains and codomains. The characterizations rest on a reduction to the general theory of exponentiability for categories of models of limit sketches, which is then applied uniformly to virtual double categories and to the related cases of semicategories, categories, and multicategories. The work also identifies broad classes of examples that satisfy the conditions automatically.

Core claim

Exponentiable virtual double categories are exactly those in which every cell admits a decomposition into a composite of cells whose source and target data match in a prescribed way, with several such decomposition conditions shown to be equivalent. The same general criteria imply that the virtual double category of cospans is always exponentiable, that virtual double categories arising from pseudo double categories or from exponentiable multicategories are exponentiable, and that the virtual double category of virtual double functors admits composites under stated conditions on cells. The same method yields fresh proofs of exponentiability for semicategories, ordinary categories, multicatg,

What carries the argument

Decompositions of cells that respect domain-codomain data, serving as the equivalent conditions that determine exponentiability.

If this is right

Virtual double categories of cospans are always exponentiable.
Virtual double categories coming from pseudo double categories are exponentiable.
Virtual double categories coming from exponentiable multicategories are exponentiable.
The virtual double category of virtual double functors admits composites once the cell-decomposition conditions hold.
The same criteria give uniform proofs of exponentiability for semicategories, categories, and multicategories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The cell-decomposition criteria may make it routine to check exponentiability for new families of virtual double categories built from familiar data.
Similar decomposition conditions could be sought for other notions of higher morphism between virtual double categories.
Concrete small examples, such as the virtual double category whose arrows are relations, can be checked directly to test whether the listed conditions hold.

Load-bearing premise

The general theory of exponentiability for categories of models of limit sketches applies directly to virtual double categories without any extra conditions.

What would settle it

Exhibit one virtual double category whose cells fail every listed decomposition condition yet whose morphisms still form a virtual double category, or conversely one whose cells decompose as required but whose morphisms do not.

read the original abstract

Virtual double categories provide an effective framework for formal category theory. Recent work has investigated the question of higher morphisms between virtual double categories, following on from work on higher morphisms between double categories, and building up to Arkor's recent conjecture on exponentiable virtual double categories--those virtual double categories, morphisms out of which can themselves be enriched to a whole virtual double category. In this paper we resolve Arkor's conjecture by providing a number of equivalent characterizations of the exponentiable virtual double categories in terms of existence of decompositions of cells. We also show that virtual double categories of cospans are always exponentiable, as are the virtual double categories arising from pseudo double categories or from exponentiable multicategories, as studied by Pisani. We give conditions under which the virtual double category of virtual double functors admits composites, following work of Par\'e for the non-virtual case. We base our work on a general approach to exponentiability for categories of models of limit sketches, which we apply to give new treatments of exponentiability for semicategories, categories, multicategories, and their functors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper resolves Arkor's conjecture by giving cell-decomposition characterizations of exponentiable virtual double categories and applies a limit-sketch method to get uniform results across related structures.

read the letter

The main point is that they settle Arkor's conjecture with several equivalent conditions for when a virtual double category is exponentiable, all phrased around existence of cell decompositions. They also prove that cospan virtual double categories are always exponentiable, as are those arising from pseudo double categories or from exponentiable multicategories, and they give conditions for composites in the functor virtual double category.

Referee Report

1 major / 2 minor

Summary. The paper resolves Arkor's conjecture by giving several equivalent characterizations of exponentiable virtual double categories in terms of the existence of cell decompositions. It develops a general theory of exponentiability for categories of models of limit sketches and applies this framework to virtual double categories, semicategories, categories, and multicategories. The authors further show that virtual double categories of cospans are always exponentiable, that those arising from pseudo double categories or exponentiable multicategories are exponentiable, and they give conditions under which the virtual double category of virtual double functors admits composites.

Significance. If the characterizations hold, the work supplies concrete, checkable criteria for exponentiability in a setting central to formal category theory and higher morphisms between double categories. The general limit-sketch approach is a clear strength, yielding new treatments of exponentiability for semicategories, categories, and multicategories alongside the main result on virtual double categories. The explicit examples (cospans, pseudo double categories) and the conditions for functor composites add practical value.

major comments (1)

[§3 and §4] §3 (general theory) and §4 (application to virtual double categories): the reduction of exponentiability for virtual double categories to the general limit-sketch theorem assumes without further verification that the virtual (non-strict) composition and cell structure satisfy all hypotheses of the general result on local presentability and exactness needed for representability of exponentials. If this step requires an implicit completion or equivalence that does not hold verbatim, the claimed cell-decomposition characterizations would not follow automatically.

minor comments (2)

[§4.2] Notation for cells and decompositions in §4.2 is introduced without a summary table; a small table listing the equivalent conditions would improve readability.
[§5] The citation to Paré's work on the non-virtual case is used in §5 but the precise statement being generalized is not restated; a one-sentence recap would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the application of our general theorem. We provide a point-by-point response below.

read point-by-point responses

Referee: [§3 and §4] §3 (general theory) and §4 (application to virtual double categories): the reduction of exponentiability for virtual double categories to the general limit-sketch theorem assumes without further verification that the virtual (non-strict) composition and cell structure satisfy all hypotheses of the general result on local presentability and exactness needed for representability of exponentials. If this step requires an implicit completion or equivalence that does not hold verbatim, the claimed cell-decomposition characterizations would not follow automatically.

Authors: We are grateful for this comment, which helps us strengthen the presentation. The general result in §3 applies to any category of models of a limit sketch that is locally presentable and exact in the sense required for the exponential to be representable. For virtual double categories, the structure is encoded by a limit sketch whose models are precisely the virtual double categories with their non-strict composition and cells. In §4, we describe this sketch explicitly and note that the category of such models is locally presentable by virtue of being a category of models of a limit sketch (which are always locally presentable when the sketch is small). The exactness conditions are satisfied because the composition operations are defined via limits that are preserved under the relevant functors, as the virtual nature allows for the necessary decompositions without strictness. Thus, the reduction holds verbatim, and the cell-decomposition characterizations are a direct consequence. To make this fully transparent, we will insert a short subsection or a series of remarks in §4 verifying each hypothesis of the general theorem for this specific case. revision: partial

Circularity Check

0 steps flagged

Minor citations to prior sketch and double category results without load-bearing reduction of main claims

full rationale

The paper resolves Arkor's conjecture via equivalent characterizations of exponentiable virtual double categories in terms of cell decompositions, based on applying a general theory of exponentiability for limit-sketch model categories. It also treats related cases such as semicategories, multicategories, and cospan virtual double categories. Although the work references earlier results on double categories, sketches, and related structures, these function as background tools rather than forcing the central characterizations by construction. The cell-decomposition analysis supplies independent content, and the derivation remains self-contained against the general framework without reducing predictions or uniqueness claims to fitted inputs or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely inside established category theory and the theory of limit sketches; no new free parameters, ad-hoc axioms, or postulated entities are introduced.

axioms (1)

standard math Standard axioms of category theory and the theory of limit sketches
The reduction to models of limit sketches presupposes the usual definitions and properties of sketches and their models.

pith-pipeline@v0.9.0 · 5721 in / 1126 out tokens · 29519 ms · 2026-05-21T02:36:31.509014+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

48550 / arXiv

doi: 10 . 48550 / arXiv . 2311 . 12746. arXiv: 2311 . 12746 [math]. (Visited on 05/22/2025) (cit. on p. 2). [ABK24] N. Arkor, J. Bourke, and J. Ko. Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

work page 2025
[2]

Compositional Modeling with Stock and Flow Dia- grams

arXiv: 2412.07475 [math.CT] . url: https://arxiv.org/ abs/2412.07475 (cit. on p. 65). [Adu+23] R. Aduddell et al. A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks. 2023. arXiv: 2301. 01445 [q-bio.MN] (cit. on p. 2). [AR94] J. Ad´ amek and J. Rosick´ y. Locally Presentable and Accessible Cat- egories. London Ma...

work page doi:10.4204/eptcs 2023
[3]

Homotopy-coherent algebra via Segal conditions

doi: 10 . 48550 / arXiv . 2310 . 02852. arXiv: 2310 . 02852 [math]. (Visited on 04/13/2025) (cit. on p. 2). [CH21] H. Chu and R. Haugseng. “Homotopy-coherent algebra via Segal conditions”. In: Advances in Mathematics 385.107733 (2021). issn: 0001-8708. doi: https://doi.org/10.1016/j.aim.2021.107733 . url: https://www.sciencedirect.com/science/article/pii/...

work page doi:10.1016/j.aim.2021.107733 2025
[4]

For x ∈ Ob(E), objects F ∈ Ob(ΠP (C × D))/x ∼= Vdc(Ob ×E D, C) over x correspond to functors F : P −1 0 (x) → C0

work page
[5]

For a : x → x′ in E, tight arrows f ∈ ArrT(ΠP (C × D))/a ∼= Vdc(Tight ×E D, C) over a correspond to functors α : P −1 0 (x) +P −1 0 (a) P −1 0 (x′) → C0 extending those on objects

work page
[6]

For φ : x 7 →x′ in E, Loose arrowsΦ ∈ Loose(ΠP (C×D))/φ ∼= Vdc(Loose×E D, C) over φ correspond to maps of spans of categories (P −1 0 (x) ← P −1 1 (φ) → P −1 0 (x′)) → (C0 ← C1 → C0)

work page
[7]

To begin characterizing exponentiable VDC morphisms, we begin by rela- tivizing Definition 4.9

For a multicell α in E, multicells Γ ∈ Sqn(ΠP (C × D))/α ∼= Vdc(Sqn ×E D, C), over α are given by a map of profunctors MCelln(D)/α → ∂∗MCelln(C) over C2 0, where ∂ : fcn(D1)op /s(α) ×(D1)/t(α) → fcn(C1)op ×C1 is the functor which together with the natural transformations putting MCelln(D)/α over C2 0 determines the boundary data of Γ. To begin characteriz...

work page
[8]

When [ n] = [0], fc0(D1) = Ob(D) consists of the objects of D

An object in D is a pair of an object [ n] ∈ ∆ together with a sequence of loose arrows ( φ : x0 7 →xn) ∈ fcn(D1). When [ n] = [0], fc0(D1) = Ob(D) consists of the objects of D

work page
[9]

A morphism ([ n], φ : x0 7 →xn) → ([m], ψ : y0 7 →ym) in D consists of a pair of a morphism σ : [m] → [n] in ∆, together with a chain of compatible multicells χ = (χ1 · · · χm) in D where χi has loose target ψi : yi−1 7 →yi and loose source ( φσ(i−1)+1 · · · φσ(i)) : xσ(i−1) 7 →xσ(i): xσ(i−1) xσ(i−1)+1 · · · xσ(i) χi yi−1 yi φσ(i−1)+1 p φσ(i−1)+2 p φσ(i) ...

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[10]

q admits a functorial choice of cocartesian lifts of inert morphisms in ∆op

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[11]

, δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1| {z } n which is induced by the cocartesian lifts of the inert morphisms δ[i,i+1] :

For each integer n ≥ 0, the functor (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1| {z } n which is induced by the cocartesian lifts of the inert morphisms δ[i,i+1] :

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[12]

∼= [i, i + 1] ⊆ [n] and δ{i} : [0] ∼= {i} ⊆ [n], is an isomorphism

work page
[13]

Before proving the proposition let’s explicate the three conditions appearing in the statement

For each σ : [m] → [n] in ∆, and each x ∈ An, y ∈ Am, the function Aσ(x, y) Aσδ [0,1](x, δ[0,1] ∗ y) × Aσδ{1}(x,δ{1} ∗ y) · · · × Aσδ{m−1}(x,δ{m−1} ∗ y) Aσδ [m−1,m](x, δ[m−1,m] ∗ y) induced by post-composing with the cocartesian lifts is a bijection, where the subscript on A indicates what morphism in ∆op the morphisms in A are lying over. Before proving ...

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[14]

multicells ( α1 · · · αn)) to the subsequence (φl+1 · · · φl+k) (resp

If i : [ k] ∼= [l, l + k] ⊆ [n] is an inert morphism, then we have a natural functor i∗ : Dn → Dk given by projecting from a sequence of loose arrows (φ1 · · · φn) (resp. multicells ( α1 · · · αn)) to the subsequence (φl+1 · · · φl+k) (resp. ( αl+1 · · · αl+k)). For a sequence of loose arrows φ = (φ1 · · · φn) in D, the chosen cocartesian lift is then the...

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[15]

Since the functors i∗ are defined to be projections, (2) follows

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[16]

Next, suppose q : A → ∆op is a functor satisfying properties (1)-(3), and write (−)∗ for the functorial choice of cocartesian lifts for inert morphisms

Similarly, the functors i∗ being the natural projections also implies (3). Next, suppose q : A → ∆op is a functor satisfying properties (1)-(3), and write (−)∗ for the functorial choice of cocartesian lifts for inert morphisms. In particular, this choice fixes an isomorphism Φ n := (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1 for each integ...

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[17]

The underlying tight category of A is the fiber category A0

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[18]

The loose arrows of A are the objects of the fiber category A1

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[19]

For n ≥ 0, the set of n-ary multicells with loose source φ : x0 7 →xn and loose target ψ : y0 7 →y1 is given by Aan(Φ−1 n (φ1 · · · φn), ψ), while post composing with δ{0} ∗ , δ{1} ∗ : A1 → A0 serve as the tight source and target maps for the n-ary multicells

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[20]

The unitality and associativity axioms for composition come from the associated axioms for A

For a sequence of compatible multicells ( α1 · · · αn), where αi is ki-ary, and an n-ary multicell β whose loose source is given by the loose targets of α, their composite is defined as (α1 · · · αn) β := β ◦ (Ψτ(ki ))−1(α1 · · · αn) where τ(ki) : [n] → [k1 + · · · + kn] is given by j 7→Pj i=1 kj. The unitality and associativity axioms for composition com...

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[21]

For each integer n ≥ 0, the object [ n] is mapped to the fiber P(D)n := Dn

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[22]

For each morphism σ : [n] → [m] in the simplex category, the profunctor P(D)(σ) : Dn × Dop m → Set is given by Dσ(−, −)

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[23]

For each composable pair of morphisms [ n] f − →[m] g − →[k] in the simplex category, the image of the cocartesian cell for g ◦ f is ℓf,g : P(D)(f) ⊙ P(D)(g) ⇒ P(D)(g ◦ f) given by composing sequences of multicells

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[24]

In particular, the description of the operator categories for VDCs translates to those normal VDFs Q : L(∆) → Prof satisfying the following:

The rest of the data of the VDF follows, since every cell in L(∆) arises from a sequence of composable morphisms. In particular, the description of the operator categories for VDCs translates to those normal VDFs Q : L(∆) → Prof satisfying the following:

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[25]

for inert morphisms [ k] i − →[n] j − →[m], i∗j∗ = ( j ◦ i)∗, and (id[k])∗ = idQ[k])

For i : [ k] ,→ [n] inert, the profunctor Q(i) : Q[k] × Qop [n] → Set is rep- resented by a functor i∗ : Q[n] → Q[k] so that Q(i) ∼= Q[k](i∗(−), −), and these representations can be chosen to be functorial in the inert mor- phisms (i.e. for inert morphisms [ k] i − →[n] j − →[m], i∗j∗ = ( j ◦ i)∗, and (id[k])∗ = idQ[k]). 102

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[26]

For i : [k] ,→ [n] inert and g : [l] → [k] arbitrary, the laxator cell Q[l] Q[k] Q[n] ℓg,f Q[l] Q[n] Q(g) p Q(i) p Q(i◦g) p is opcartesian

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[27]

, δ[n−1,n] ∗ ) : Q[n] → Q[1] ×Q[0] · · · ×Q[0] Q[1] in Cat = Prof0 induced by the functors from (1), for the inert morphisms δ[i,i+1] : [1] → [n], is an isomorphism

For each [ n] ∈ ∆, the natural map (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : Q[n] → Q[1] ×Q[0] · · · ×Q[0] Q[1] in Cat = Prof0 induced by the functors from (1), for the inert morphisms δ[i,i+1] : [1] → [n], is an isomorphism

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[28]

One of the benefits of this profunctor description of VDCs is the ability to translate it into the theory of pullback sketches in VDCs

For each morphism ϕ : [n] → [m], the natural map of profunctors Q(ϕ) Q(ϕ[0,1])(−, δ[0,1] ∗ ) × Q(ϕ{1})(−,δ1∗) · · · × Q(ϕn−1)(−,δ{n−1} ∗ ) Q(ϕ[n−1,n])(−, δ[n−1,n] ∗ ) in Prof1 induced by the projections in (1) is an isomorphism. One of the benefits of this profunctor description of VDCs is the ability to translate it into the theory of pullback sketches i...

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[29]

An object φ in D (i.e. a sequence of loose arrows in D) is sent to the fiber category F −1(φ) ⊆ E|φ| whose objects are length |φ|-sequences of loose arrows in E lying over φ, and whose morphisms are sequences of unary multicells between such loose arrows lying over id φ

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[30]

A map β : ( H1 · · · Hm) → (J1 · · · Jn) lying over f : [ n] → [m] in D is sent to the profunctor F −1(β, f) : F −1(J) 7 →F −1(H) (i.e. F −1(J) × F −1(H)op → Set) that maps an object ( I, K) to the set of maps α : (K1, ..., Km) → (I1, ..., In) lying over β, and maps sequences of unary multicells to the set function given by pasting the unary multicells ab...

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[31]

A graph of categories s, t : D1 ⇒ D0, from which we can define Dn = D1 ×D0 · · · ×D0 D1

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[32]

Two graphs of profunctors D1 D1 D0 D0 D0 D2 D0 D1 D1 D0 D0 D0 s P(D)(a0) p t s P(D)(a2) p t D0(−,−) p D0(−,−) p π1 π2 D0(−,−)p s t s t s t For f : [n] → [m] a degeneracy or inner face map, we define P(D)(f) := P(D)(ak1) ×D0(−,−) · · · ×D0(−,−) P(D)(akn) where k = 1 ± δi, so that all but one of the profunctors is D1(−, −), and the remaining profunctor is e...

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[33]

For any two binary-nullary trees T and T ′ with the same number of leaves, a natural isomorphism αT,T ′ : P(D)(T ) → P(D)(T ′), satisfying the cocycle 107 identities αT,T = id, α T ′,T ′′ ◦ αT,T ′ = αT,T ′′ This presentation exhibits exponentiable VDCs as a kind of unbiased pro- double categories. Namely, we need only specify the data for nullary, unary, ...

work page

[1] [1]

48550 / arXiv

doi: 10 . 48550 / arXiv . 2311 . 12746. arXiv: 2311 . 12746 [math]. (Visited on 05/22/2025) (cit. on p. 2). [ABK24] N. Arkor, J. Bourke, and J. Ko. Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

work page 2025

[2] [2]

Compositional Modeling with Stock and Flow Dia- grams

arXiv: 2412.07475 [math.CT] . url: https://arxiv.org/ abs/2412.07475 (cit. on p. 65). [Adu+23] R. Aduddell et al. A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks. 2023. arXiv: 2301. 01445 [q-bio.MN] (cit. on p. 2). [AR94] J. Ad´ amek and J. Rosick´ y. Locally Presentable and Accessible Cat- egories. London Ma...

work page doi:10.4204/eptcs 2023

[3] [3]

Homotopy-coherent algebra via Segal conditions

doi: 10 . 48550 / arXiv . 2310 . 02852. arXiv: 2310 . 02852 [math]. (Visited on 04/13/2025) (cit. on p. 2). [CH21] H. Chu and R. Haugseng. “Homotopy-coherent algebra via Segal conditions”. In: Advances in Mathematics 385.107733 (2021). issn: 0001-8708. doi: https://doi.org/10.1016/j.aim.2021.107733 . url: https://www.sciencedirect.com/science/article/pii/...

work page doi:10.1016/j.aim.2021.107733 2025

[4] [4]

For x ∈ Ob(E), objects F ∈ Ob(ΠP (C × D))/x ∼= Vdc(Ob ×E D, C) over x correspond to functors F : P −1 0 (x) → C0

work page

[5] [5]

For a : x → x′ in E, tight arrows f ∈ ArrT(ΠP (C × D))/a ∼= Vdc(Tight ×E D, C) over a correspond to functors α : P −1 0 (x) +P −1 0 (a) P −1 0 (x′) → C0 extending those on objects

work page

[6] [6]

For φ : x 7 →x′ in E, Loose arrowsΦ ∈ Loose(ΠP (C×D))/φ ∼= Vdc(Loose×E D, C) over φ correspond to maps of spans of categories (P −1 0 (x) ← P −1 1 (φ) → P −1 0 (x′)) → (C0 ← C1 → C0)

work page

[7] [7]

To begin characterizing exponentiable VDC morphisms, we begin by rela- tivizing Definition 4.9

For a multicell α in E, multicells Γ ∈ Sqn(ΠP (C × D))/α ∼= Vdc(Sqn ×E D, C), over α are given by a map of profunctors MCelln(D)/α → ∂∗MCelln(C) over C2 0, where ∂ : fcn(D1)op /s(α) ×(D1)/t(α) → fcn(C1)op ×C1 is the functor which together with the natural transformations putting MCelln(D)/α over C2 0 determines the boundary data of Γ. To begin characteriz...

work page

[8] [8]

When [ n] = [0], fc0(D1) = Ob(D) consists of the objects of D

An object in D is a pair of an object [ n] ∈ ∆ together with a sequence of loose arrows ( φ : x0 7 →xn) ∈ fcn(D1). When [ n] = [0], fc0(D1) = Ob(D) consists of the objects of D

work page

[9] [9]

A morphism ([ n], φ : x0 7 →xn) → ([m], ψ : y0 7 →ym) in D consists of a pair of a morphism σ : [m] → [n] in ∆, together with a chain of compatible multicells χ = (χ1 · · · χm) in D where χi has loose target ψi : yi−1 7 →yi and loose source ( φσ(i−1)+1 · · · φσ(i)) : xσ(i−1) 7 →xσ(i): xσ(i−1) xσ(i−1)+1 · · · xσ(i) χi yi−1 yi φσ(i−1)+1 p φσ(i−1)+2 p φσ(i) ...

work page

[10] [10]

q admits a functorial choice of cocartesian lifts of inert morphisms in ∆op

work page

[11] [11]

, δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1| {z } n which is induced by the cocartesian lifts of the inert morphisms δ[i,i+1] :

For each integer n ≥ 0, the functor (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1| {z } n which is induced by the cocartesian lifts of the inert morphisms δ[i,i+1] :

work page

[12] [12]

∼= [i, i + 1] ⊆ [n] and δ{i} : [0] ∼= {i} ⊆ [n], is an isomorphism

work page

[13] [13]

Before proving the proposition let’s explicate the three conditions appearing in the statement

For each σ : [m] → [n] in ∆, and each x ∈ An, y ∈ Am, the function Aσ(x, y) Aσδ [0,1](x, δ[0,1] ∗ y) × Aσδ{1}(x,δ{1} ∗ y) · · · × Aσδ{m−1}(x,δ{m−1} ∗ y) Aσδ [m−1,m](x, δ[m−1,m] ∗ y) induced by post-composing with the cocartesian lifts is a bijection, where the subscript on A indicates what morphism in ∆op the morphisms in A are lying over. Before proving ...

work page

[14] [14]

multicells ( α1 · · · αn)) to the subsequence (φl+1 · · · φl+k) (resp

If i : [ k] ∼= [l, l + k] ⊆ [n] is an inert morphism, then we have a natural functor i∗ : Dn → Dk given by projecting from a sequence of loose arrows (φ1 · · · φn) (resp. multicells ( α1 · · · αn)) to the subsequence (φl+1 · · · φl+k) (resp. ( αl+1 · · · αl+k)). For a sequence of loose arrows φ = (φ1 · · · φn) in D, the chosen cocartesian lift is then the...

work page

[15] [15]

Since the functors i∗ are defined to be projections, (2) follows

work page

[16] [16]

Next, suppose q : A → ∆op is a functor satisfying properties (1)-(3), and write (−)∗ for the functorial choice of cocartesian lifts for inert morphisms

Similarly, the functors i∗ being the natural projections also implies (3). Next, suppose q : A → ∆op is a functor satisfying properties (1)-(3), and write (−)∗ for the functorial choice of cocartesian lifts for inert morphisms. In particular, this choice fixes an isomorphism Φ n := (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : An → A1 ×A0 · · · ×A0 A1 for each integ...

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[17] [17]

The underlying tight category of A is the fiber category A0

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[18] [18]

The loose arrows of A are the objects of the fiber category A1

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[19] [19]

For n ≥ 0, the set of n-ary multicells with loose source φ : x0 7 →xn and loose target ψ : y0 7 →y1 is given by Aan(Φ−1 n (φ1 · · · φn), ψ), while post composing with δ{0} ∗ , δ{1} ∗ : A1 → A0 serve as the tight source and target maps for the n-ary multicells

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[20] [20]

The unitality and associativity axioms for composition come from the associated axioms for A

For a sequence of compatible multicells ( α1 · · · αn), where αi is ki-ary, and an n-ary multicell β whose loose source is given by the loose targets of α, their composite is defined as (α1 · · · αn) β := β ◦ (Ψτ(ki ))−1(α1 · · · αn) where τ(ki) : [n] → [k1 + · · · + kn] is given by j 7→Pj i=1 kj. The unitality and associativity axioms for composition com...

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[21] [21]

For each integer n ≥ 0, the object [ n] is mapped to the fiber P(D)n := Dn

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[22] [22]

For each morphism σ : [n] → [m] in the simplex category, the profunctor P(D)(σ) : Dn × Dop m → Set is given by Dσ(−, −)

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[23] [23]

For each composable pair of morphisms [ n] f − →[m] g − →[k] in the simplex category, the image of the cocartesian cell for g ◦ f is ℓf,g : P(D)(f) ⊙ P(D)(g) ⇒ P(D)(g ◦ f) given by composing sequences of multicells

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[24] [24]

In particular, the description of the operator categories for VDCs translates to those normal VDFs Q : L(∆) → Prof satisfying the following:

The rest of the data of the VDF follows, since every cell in L(∆) arises from a sequence of composable morphisms. In particular, the description of the operator categories for VDCs translates to those normal VDFs Q : L(∆) → Prof satisfying the following:

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[25] [25]

for inert morphisms [ k] i − →[n] j − →[m], i∗j∗ = ( j ◦ i)∗, and (id[k])∗ = idQ[k])

For i : [ k] ,→ [n] inert, the profunctor Q(i) : Q[k] × Qop [n] → Set is rep- resented by a functor i∗ : Q[n] → Q[k] so that Q(i) ∼= Q[k](i∗(−), −), and these representations can be chosen to be functorial in the inert mor- phisms (i.e. for inert morphisms [ k] i − →[n] j − →[m], i∗j∗ = ( j ◦ i)∗, and (id[k])∗ = idQ[k]). 102

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[26] [26]

For i : [k] ,→ [n] inert and g : [l] → [k] arbitrary, the laxator cell Q[l] Q[k] Q[n] ℓg,f Q[l] Q[n] Q(g) p Q(i) p Q(i◦g) p is opcartesian

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[27] [27]

, δ[n−1,n] ∗ ) : Q[n] → Q[1] ×Q[0] · · · ×Q[0] Q[1] in Cat = Prof0 induced by the functors from (1), for the inert morphisms δ[i,i+1] : [1] → [n], is an isomorphism

For each [ n] ∈ ∆, the natural map (δ[0,1] ∗ , . . . , δ[n−1,n] ∗ ) : Q[n] → Q[1] ×Q[0] · · · ×Q[0] Q[1] in Cat = Prof0 induced by the functors from (1), for the inert morphisms δ[i,i+1] : [1] → [n], is an isomorphism

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[28] [28]

One of the benefits of this profunctor description of VDCs is the ability to translate it into the theory of pullback sketches in VDCs

For each morphism ϕ : [n] → [m], the natural map of profunctors Q(ϕ) Q(ϕ[0,1])(−, δ[0,1] ∗ ) × Q(ϕ{1})(−,δ1∗) · · · × Q(ϕn−1)(−,δ{n−1} ∗ ) Q(ϕ[n−1,n])(−, δ[n−1,n] ∗ ) in Prof1 induced by the projections in (1) is an isomorphism. One of the benefits of this profunctor description of VDCs is the ability to translate it into the theory of pullback sketches i...

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[29] [29]

An object φ in D (i.e. a sequence of loose arrows in D) is sent to the fiber category F −1(φ) ⊆ E|φ| whose objects are length |φ|-sequences of loose arrows in E lying over φ, and whose morphisms are sequences of unary multicells between such loose arrows lying over id φ

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[30] [30]

A map β : ( H1 · · · Hm) → (J1 · · · Jn) lying over f : [ n] → [m] in D is sent to the profunctor F −1(β, f) : F −1(J) 7 →F −1(H) (i.e. F −1(J) × F −1(H)op → Set) that maps an object ( I, K) to the set of maps α : (K1, ..., Km) → (I1, ..., In) lying over β, and maps sequences of unary multicells to the set function given by pasting the unary multicells ab...

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[31] [31]

A graph of categories s, t : D1 ⇒ D0, from which we can define Dn = D1 ×D0 · · · ×D0 D1

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[32] [32]

Two graphs of profunctors D1 D1 D0 D0 D0 D2 D0 D1 D1 D0 D0 D0 s P(D)(a0) p t s P(D)(a2) p t D0(−,−) p D0(−,−) p π1 π2 D0(−,−)p s t s t s t For f : [n] → [m] a degeneracy or inner face map, we define P(D)(f) := P(D)(ak1) ×D0(−,−) · · · ×D0(−,−) P(D)(akn) where k = 1 ± δi, so that all but one of the profunctors is D1(−, −), and the remaining profunctor is e...

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[33] [33]

For any two binary-nullary trees T and T ′ with the same number of leaves, a natural isomorphism αT,T ′ : P(D)(T ) → P(D)(T ′), satisfying the cocycle 107 identities αT,T = id, α T ′,T ′′ ◦ αT,T ′ = αT,T ′′ This presentation exhibits exponentiable VDCs as a kind of unbiased pro- double categories. Namely, we need only specify the data for nullary, unary, ...

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