11affiliationtext: INL – International Iberian Nanotechnology Laboratory,
Braga, Portugal22affiliationtext: Department of Mathematics, Bilkent
University, Ankara, Turkey

# A bundle perspective on contextuality
Empirical models and simplicial distributions on bundle scenarios

Rui Soares<EMAIL_ADDRESS>Aziz
<EMAIL_ADDRESS>Cihan<EMAIL_ADDRESS>
###### Abstract

This paper provides a bundle perspective to contextuality by introducing new
categories of contextuality scenarios based on bundles of simplicial complexes
and simplicial sets. The former approach generalizes earlier work on the
sheaf-theoretic perspective on contextuality, and the latter extends
simplicial distributions, a more recent approach to contextuality formulated
in the language of simplicial sets. After constructing our bundle categories,
we also construct functors that relate them and natural isomorphisms that
allow us to compare the notions of contextuality formulated in two languages.
We are motivated by applications to the resource theory of contextuality,
captured by the morphisms in these categories. In this paper, we develop the
main formalism and leave applications to future work.

###### Contents

  1. 1 Introduction
  2. 2 The category of scenarios
    1. 2.1 Empirical models
  3. 3 The category of bundle scenarios
    1. 3.1 Morphisms of bundle scenarios
    2. 3.2 Empirical models on bundle scenarios
      1. 3.2.1 Push-forward empirical models for bundles
  4. 4 The category of simplicial bundle scenarios
    1. 4.1 Morphisms of simplicial scenarios
    2. 4.2 Simplicial distributions on bundles
      1. 4.2.1 Push-forward simplicial distributions
    3. 4.3 The natural isomorphism
  5. 5 Contextuality for bundle scenarios
    1. 5.1 Convexity
    2. 5.2 Contextuality
  6. A Simplicial complexes
    1. A.1 Pull-backs of bundle scenarios
    2. A.2 Bundles of nerve complexes
    3. A.3 Lemmas: Category of bundle scenarios
    4. A.4 Lemmas: Embedding scenarios into bundle scenarios
    5. A.5 Lemmas: Empirical model functor on bundle scenarios
  7. B Simplicial sets
    1. B.1 The nerve space
    2. B.2 Lemmas: Category of simplicial scenarios
    3. B.3 Lemmas: Embedding bundle scenarios into simplicial scenarios
    4. B.4 Lemmas: Simplicial distributions functor
  8. C Convex categories

##  1 Introduction

This paper develops a bundle perspective on contextuality that generalizes the
sheaf-theoretic [1] and the simplicial [2] approaches. In [3, 4], the former
approach is used to construct the category $\mathbf{Scen}$ of scenarios, whose
morphisms capture a notion of simulation corresponding to the free
transformations in a resource theory of contextuality. In this paper, we
introduce two categories: (1) the category $\mathbf{bScen}$ of bundle
scenarios based on simplicial complexes, and (2) the category $\mathbf{sScen}$
of simplicial bundle scenarios based on simplicial sets. The use of bundles of
simplicial complexes to represent measurements and outcomes in contextuality
scenarios already appears in the literature; see [5, 6, 7]. However, the
notion of “bundle” is used somewhat informally and not fully formalized. We
provide an abstract definition of a bundle scenario and define morphisms
between such bundle scenarios extending the category of scenarios
$\mathbf{Scen}$. The other category $\mathbf{sScen}$ is based on simplicial
sets, combinatorial objects representing spaces more expressive than
simplicial complexes. Our motivation for such extensions is to translate the
resource-theoretic analysis developed in [3, 8, 4, 9] to bundle scenarios of
simplicial complexes and ultimately to bundle scenarios of simplicial sets.
The extension to simplicial sets carries many advantages for studying
contextuality; see [2, 10, 11, 12]. It offers promising applications in
extending cohomological frameworks for measurement-based quantum computation
developed in [13, 14].

This paper focuses on setting the formal stage, relating the three categories
of scenarios and distributions on them. Resource-theoretic implications will
be studied in upcoming work. More formally, we construct fully faithful
functors

$\mathcal{E}\colon\mathbf{Scen}\to\mathbf{bScen}\qquad\text{ and }\qquad
N\colon\mathbf{bScen}\to\mathbf{sScen}$

relating the three categories of scenarios and natural isomorphisms $\eta$ and
$\zeta$ relating the empirical models and simplicial distributions on them:

${\mathbf{Scen}}$${\mathbf{bScen}}$${\mathbf{Set}}$$\scriptstyle{\mathcal{E}}$$\scriptstyle{\mathsf{Emp}}$$\scriptstyle{\eta}$$\scriptstyle{\mathsf{bEmp}}$
${\mathbf{bScen}}$${\mathbf{sScen}}$${\mathbf{Set}}$$\scriptstyle{N}$$\scriptstyle{\mathsf{bEmp}}$$\scriptstyle{\zeta}$$\scriptstyle{\mathsf{sDist}}$
(1)

In the sheaf-theoretic framework, a scenario is described by a pair
$(\Sigma,O)$ consisting of a simplicial complex $\Sigma$ and a family
$O=\\{O_{x}\\}_{x\in\Sigma_{0}}$ of sets indexed by the vertices of the
simplicial complex. Each vertex $x$ represents a measurement with outcomes in
the set $O_{x}$, and each simplex $\sigma\in\Sigma$ represents a measurement
context, a set of measurements that can be jointly performed. It is convenient
to represent the simplicial complex $\Sigma$ as a category
$\mathbf{C}_{\Sigma}$ whose objects are the simplices of $\Sigma$ and the
morphisms are given by inclusions. Then both measurements and the
corresponding outcomes can be described as a functor

$\mathcal{E}_{O}:\mathbf{C}_{\Sigma}\to\mathbf{Set}$

that sends a measurement context $\sigma$ to the set
$\mathcal{E}_{O}(\sigma)=\prod_{x\in\sigma}O_{x}$ of all possible outcome
assignments. A morphism $(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ between
two scenarios consists of two parts: a functor
$\pi:\mathbf{C}_{\Sigma^{\prime}}\to\mathbf{C}_{\Sigma}$ induced by a
simplicial relation that relates the measurement contexts, and a natural
transformation $\alpha:\mathcal{E}_{O}\circ\pi\to\mathcal{E}_{O^{\prime}}$
that specifies a mapping of outcomes over each context. Since measurements in
a context can be simultaneously performed, outcome statistics are represented
by a probability distribution $e_{\sigma}$ on the set
$\mathcal{E}_{O}(\sigma)$ of joint outcomes. The family
$e=\\{e_{\sigma}\\}_{\sigma\in\Sigma}$ of distributions, called an empirical
model, is required to satisfy an additional condition known as the non-
signaling (or no-disturbance) condition: the measurement statistics over two
contexts should match on their intersection. Morphisms of the category of
scenarios are defined so that an empirical model $e$ on $(\Sigma,O)$ can be
pushed forward along a morphism to produce a new empirical model on
$(\Sigma^{\prime},O^{\prime})$. Operationally this coincides with the notion
of simulating the empirical model $e$ with the new one. By the push-forward
construction, empirical models on scenarios can be turned into a functor

$\mathsf{Emp}:\mathbf{Scen}\to\mathbf{Set}$

Since we can consider probabilistic mixtures of empirical models, the target
can be replaced by the category $\mathbf{Conv}_{R}$ of convex sets for our
choice of a semiring $R$, which is usually taken to be non-negative reals.

Figure 1: A morphism between two bundle scenarios on a triangle and a square.

Our first contribution in this paper is the notion of bundle scenario: A map
$f:\Gamma\to\Sigma$ of simplicial complexes is a bundle scenario if it is
surjective and satisfies the following two properties:

  * •

Locally surjectivity: The restriction of $f$ to the star of each simplex of
$\Sigma$ is surjective.

  * •

Discrete over vertices: Every two vertex of $\Gamma$ that map to the same
vertex under $f$ does not form an edge.

To a scenario $(\Sigma,O)$ one can associate a bundle scenario

$\mathcal{E}_{O}(\Sigma)\to\Sigma$

by assembling the outcomes over each simplex into a simplicial complex that
maps down to the simplicial complex of measurement contexts. Morphisms in
bundle scenarios are described using diagrams of simplicial complexes; see
Figure (1). Empirical models can be generalized to bundle scenarios and can be
assembled into a functor

$\mathsf{bEmp}:\mathbf{bScen}\to\mathbf{Set}.$

These constructions in the category of simplicial complexes can be carried
over to simplicial sets, which are suitable for generating spaces with more
elaborate identifications. A simplicial set is specified by a sequence of sets
$\\{X_{n}\\}_{n\geq 0}$ representing the simplicies together with two kinds of
maps for gluing and collapsing simplices. The notions of local surjectivity
and being discrete over vertices can be generalized to simplicial sets giving
rise to the notion of simplicial bundle scenarios: A simplicial set map
$f:E\to X$ is called a simplicial bundle scenario if it is surjective, locally
surjective and discrete over vertices. A source of examples is obtained by
applying the nerve construction, the functor $N$ in Diagram (1), to a bundle
scenario. One of the benefits of extending the category of scenarios first to
bundle scenarios and then the simplicial scenarios is that the description of
morphisms simplifies. A morphism $f\to f^{\prime}$ between two simplicial
bundle scenarios is a commutative diagram

${E}$${\pi^{*}(E)}$${E^{\prime}}$${X}$${X^{\prime}}$${X^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi}$

where the left square is a pull-back. This diagram is similar to Figure (1).
However, in the simplicial complex case, an additional complication is added
by the use of the nerve complex functor $\hat{N}$. This functor is analogous
to the nerve functor $N$, but produces a simplicial complex instead of a
simplicial set. Intuitively its role can be understood by observing that
simplicial complex maps $\Sigma^{\prime}\to\hat{N}(\Sigma)$ coincide with
simplicial relations. We introduce simplicial distributions on simplicial
bundle scenarios generalizing empirical models even further giving us a
functor

$\mathsf{sDist}:\mathbf{sScen}\to\mathbf{Set}$

The definition of a simplicial distribution is formulated in a diagrammatic
way (see Definition 4.8) rather than using the language of sheaves.

Our contributions can be summarized as follows:

  * •

We introduce bundle scenarios in Definition 3.1. For any scenario $(\Sigma,O)$
we show that $\mathcal{E}_{O}(\Sigma)\to\Sigma$ is a bundle scenario
(Proposition 3.2). Morphisms between bundle scenarios are introduced in
Definition 3.7. An important perspective here is the use of the nerve complex
functor $\hat{N}$ given in Definition 3.4 to interpret simplicial relations.

  * •

Empirical models for bundle scenarios are introduced in Definition 3.11. Non-
signaling conditions are formulated using the property of being discrete over
vertices. We construct the push-forward distribution along a morphism in
Definition 3.15. Our main result in this section, Theorem 3.18, states that
the functor $\mathcal{E}$ in Diagram (1) is fully faithful and $\eta$ is a
natural isomorphism.

  * •

Simplicial bundle scenarios are introduced in Definition 4.1. We show that
$Nf:N\Gamma\to N\Sigma$ is a simplicial bundle scenario whenever
$f:\Gamma\to\Sigma$ is a bundle scenario (Proposition 4.2). Morphisms of
simplicial bundle scenarios are easier to define than morphisms of bundle
scenarios (Definition 4.3).

  * •

Empirical models can be generalized to the simplicial set framework to give
rise to simplicial distributions (Definition 4.8). Theorem 4.18 is our main
result on simplicial bundle scenarios stating that the functor $N$ in Diagram
(1) is fully faithful and $\zeta$ is a natural isomorphism.

  * •

For the discussion of contextuality, we use the theory of convex categories
developed in [10]. The key ingredient is the observation that in Diagram (1),
the categories of scenarios can be replaced by the corresponding free convex
categories:

${D_{R}(\mathbf{Scen})}$${D_{R}(\mathbf{bScen})}$${\mathbf{Conv}_{R}}$$\scriptstyle{D_{R}(\mathcal{E})}$$\scriptstyle{\widetilde{\mathsf{Emp}}}$$\scriptstyle{\tilde{\eta}}$$\scriptstyle{\mathsf{bEmp}}$
${D_{R}(\mathbf{bScen})}$${D_{R}(\mathbf{sScen})}$${\mathbf{Conv}_{R}}$$\scriptstyle{D_{R}(N)}$$\scriptstyle{\widetilde{\mathsf{bEmp}}}$$\scriptstyle{\tilde{\zeta}}$$\scriptstyle{\widetilde{\mathsf{sDist}}}$
(2)

In Corollary 5.4 we show that $D_{R}(\mathcal{E})$ and $D_{R}(N)$ are fully
faithful. Moreover, $\tilde{\eta}$ and $\tilde{\zeta}$ are natural
isomorphisms.

  * •

We introduce contextuality for the three categories of scenarios in Definition
5.8 and Definition 5.9. Theorem 5.10 shows that these three notions coincide
when specialized to the target of the natural isomorphisms $\tilde{\eta}$ and
$\tilde{\zeta}$.

Most of the technical results are proved in the Appendix. Section A begins
with a description of the conditions of local surjectivity (Proposition A.2)
and being discrete over vertices (Proposition A.3) in terms of liftings of
diagrams. In Section A.1, we describe pull-backs of bundle scenarios and show
that they behave particularly well (Proposition A.6). In Proposition A.7, we
show that the $\hat{N}$ functor preserves bundle scenarios. Section A.3
contains technical results on the interaction of pull-backs with the $\hat{N}$
functor and Lemma A.10, which shows that the composition rule defined for the
category of bundle scenarios is associative. Section A.4 contains technical
results that are used for proving that $\mathcal{E}$ is fully faithful
(Proposition 3.9). Section A.5 is concerned with constructing empirical models
on $\hat{N}$ of a bundle scenario and how to push forward empirical models
along morphisms. In Section B, we begin by discussing the basic properties of
the nerve functor $N$. In Section B.2, we prove that $N$ preserves local
surjectivity (Lemma B.5) and being discrete over vertices (Lemma B.6). Section
B.3 and Section B.4 contain technical results for proving that $N$ is a fully
faithful functor (Proposition 4.7) and $\zeta$ a natural isomorphism
(Proposition 4.17). In Section C we recall some facts from [10] concerning
convex categories. A basic construction is the free convex category given in
Definition C.3. An important result, which is also of independent interest, is
Proposition C.6 stating $\mathbf{Conv}_{R}$ is an $R$-convex category. Diagram
(2) is obtained by a general observation (Proposition C.7) on liftings of
natural transformations to convex categories.

##### Acknowledgments.

RSB and CO are supported by the Digital Horizon Europe project FoQaCiA, GA no.
101070558, AK and CO by the US Air Force Office of Scientific Research under
award number FA9550-21-1-0002, and RSB by FCT – Fundação para a Ciência e a
Tecnologia through CEECINST/00062/2018.

##  2 The category of scenarios

In this section we introduce the category $\mathbf{Scen}$ of scenarios and the
functor $\mathsf{Emp}:\mathbf{Scen}\to\mathbf{Set}$ of empirical models
following the presentation in [4].

A scenario consists of a pair $(\Sigma,O)$, where

  * •

$\Sigma$ is a simplicial complex whose vertex set $\Sigma_{0}$ represents
measurements and its simplices represent contexts,

  * •

$O$ is a family $\\{O_{x}\\}_{x\in\Sigma_{0}}$ of nonempty sets representing
the outcomes of each measurement.

We can think of $\Sigma$ as a category $\mathbf{C}_{\Sigma}$ whose objects are
the simplices $\sigma\in\Sigma$ and morphisms are inclusions
$\sigma^{\prime}\hookrightarrow\sigma$. On this category we can define the
event presheaf, that is, the contravariant functor

$\mathcal{E}_{O}:\mathbf{C}_{\Sigma}\to\mathbf{Set}$

defined by $\mathcal{E}_{O}(\sigma)=\prod_{x\in\sigma}O_{x}$. For
$s\in\mathcal{E}_{O}(\sigma)$ we write $s|_{\sigma^{\prime}}$ for the tuple in
$\mathcal{E}_{O}(\sigma^{\prime})$ obtained by restricting to the indices from
the subset. Sometimes for we will write $\mathcal{E}_{(\Sigma,O)}$ for this
functor to indicate the underlying simplicial complex.

To define morphisms between scenarios, we need the notion of simplicial
relation. For a relation $\pi\subset\Sigma_{0}^{\prime}\times\Sigma_{0}$ we
will write

$\pi(x^{\prime})=\\{x\in\Sigma_{0}:\,(x^{\prime},x)\in\pi\\}\;\;\text{ and
}\;\;\pi(\sigma^{\prime})=\cup_{x^{\prime}\in\sigma^{\prime}}\pi(x^{\prime}).$

A simplicial relation $\pi:\Sigma^{\prime}\to\Sigma$ is a relation
$\pi\subset\Sigma^{\prime}_{0}\times\Sigma_{0}$ such that
$\pi(\sigma^{\prime})\in\Sigma$ for all $\sigma^{\prime}\in\Sigma^{\prime}$. A
simplicial relation induces a functor
$\pi:\mathbf{C}_{\Sigma^{\prime}}\to\mathbf{C}_{\Sigma}$.

######  Definition 2.1.

The category $\mathbf{Scen}$ of scenarios consists of objects given by
scenarios $(\Sigma,O)$ and a morphism
$(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ between two scenarios is given by
a pair $(\pi,\alpha)$ where

  * •

$\pi:\Sigma^{\prime}\to\Sigma$ is a simplicial relation.

  * •

$\alpha:\mathcal{E}_{O}\circ\pi\to\mathcal{E}_{O^{\prime}}$ is a natural
transformation of functors.

The composition of the morphisms
$(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ and
$(\pi^{\prime},\alpha^{\prime}):(\Sigma^{\prime},O^{\prime})\to(\Sigma^{\prime\prime},O^{\prime\prime})$
is given by

$(\pi^{\prime},\alpha^{\prime})\circ(\pi,\alpha)=(\pi\circ\pi^{\prime},\alpha^{\prime}\circ(\alpha\ast\text{id}_{\pi^{\prime}})).$
(3)

where

  * •

$\pi\circ\pi^{\prime}:\Sigma^{\prime\prime}\to\Sigma$ denotes the simplicial
relation obtained by the composition of the corresponding functors, and

  * •

$\alpha\ast\text{id}_{\pi^{\prime}}:\mathcal{E}_{O}\circ(\pi\circ\pi^{\prime})\to{\mathcal{E}_{O^{\prime}}\circ\pi^{\prime}}$
denotes the horizontal composition of natural transformations [15, Section
1.7].

(a)

(b)

Figure 2:

######  Example 2.2.

Let $\mathbb{Z}_{2}=\\{0,1\\}$ denote the two-outcome set. We will consider a
morphism $(\pi,\alpha):(\triangle,O)\to(\square,O)$ where

  * •

$\triangle$ is the simplicial complex with vertex set
$\\{x_{0},x_{1},x_{2}\\}$ (Figure 2(a)) and maximal simplices

$\\{x_{0},x_{1}\\},\;\;\\{x_{1},x_{2}\\},\;\;\\{x_{2},x_{0}\\};$

  * •

$\square$ is the simplicial complex with vertex set
$\\{x_{0},x_{1},x_{2},x_{3}\\}$ (Figure 2(b)) and maximal simplices

$\\{x_{0},x_{1}\\},\;\;\\{x_{3},x_{0}\\},\;\;\\{x_{1},x_{2}\\},\;\;\\{x_{2},x_{3}\\}.$
(4)

The outcome sets $O_{x}$ are given by $\mathbb{Z}_{2}$ for the vertices of
$\triangle$ and $\square$. The simpicial relation $\pi$ is a simplicial
complex map defined by

$\pi(x_{i})=\left\\{\begin{array}[]{ll}x_{i}&i=0,1\\\
x_{2}&i=2,3,\end{array}\right.$

and $\alpha_{x}:\mathcal{E}_{O}(\pi(x))\to\mathcal{E}_{O}(x)$ is the identity
map on $\mathbb{Z}_{2}$.

###  2.1 Empirical models

Throughout the paper $R$ will denote a zero-sum-free commutative semiring. The
zero-sum-free condition means that $a+b=0$ implies $a=b=0$ for $a,b\in R$. Let
$D_{R}:\mathbf{Set}\to\mathbf{Set}$ denote the distribution monad [16]. Under
this functor a set $U$ is sent to the set $D_{R}(U)$ of functions $p:U\to R$
of finite support such that $\sum_{u\in U}p(u)=1$. For $f:U\to V$ the function
$D_{R}(f):D_{R}(U)\to D_{R}(V)$ is defined by

$D_{R}(f)(p)(v)=\sum_{u\in f^{-1}(v)}p(u).$

An empirical model $e$ on the scenario $(\Sigma,O)$ is a family
$\\{e_{\sigma}\in D_{R}\circ\mathcal{E}_{O}(\sigma)\\}_{\sigma\in\Sigma}$ of
distributions that satisfies the compatibility condition: for each inclusion
$i:\sigma^{\prime}\hookrightarrow\sigma$ of simplices we have

$e_{\sigma^{\prime}}=e_{\sigma}|_{\sigma^{\prime}}$

where $e_{\sigma}|_{\sigma^{\prime}}$ stands for
$(D_{R}\circ\mathcal{E}_{O})(i)(e_{\sigma})$. We denote the set of empirical
models on $(\Sigma,O)$ by $\mathsf{Emp}(\Sigma,O)$.

######  Definition 2.3.

Let $e$ be an empirical model on the scenario $(\Sigma,O)$. Given a morphism
$(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ the push-forward
empirical model $(\pi,\alpha)_{*}e$ on $(\Sigma^{\prime},O^{\prime})$ is
defined by

$((\pi,\alpha)_{*}e)_{\sigma^{\prime}}=D_{R}(\alpha_{\sigma^{\prime}})(e_{\pi(\sigma^{\prime})})$

where
$\alpha_{\sigma^{\prime}}:\mathcal{E}_{O}(\pi(\sigma^{\prime}))\to\mathcal{E}_{O^{\prime}}(\sigma^{\prime})$
is the component of the natural transformation $\alpha$ at the simplex
$\sigma^{\prime}{\in\Sigma^{\prime}}$.

The push-forward construction gives a function
$(\pi,\alpha)_{*}:\mathsf{Emp}(\Sigma,O)\to\mathsf{Emp}(\Sigma^{\prime},O^{\prime})$.

######  Proposition 2.4 ([4]).

The assignments $(\Sigma,O)\mapsto\mathsf{Emp}(\Sigma,O)$ and
$(\pi,\alpha)\mapsto(\pi,\alpha)_{*}$ define a functor
$\mathsf{Emp}:\mathbf{Scen}\to\mathbf{Set}$.

######  Example 2.5.

Let us consider the morphism $(\pi,\alpha):(\triangle,O)\to(\square,O)$ in
Example 2.2. Given an empirical model on $(\triangle,O)$, the push-forward
empirical model $(\pi,\alpha)_{*}e$ is given by
$((\pi,\alpha)_{*}e)_{\sigma^{\prime}}=e_{\pi(\sigma^{\prime})}$ when
$\sigma^{\prime}$ is one of the first three simplices in Equation (4), and for
the last one it is given by

$((\pi,\alpha)_{*}e)_{\\{x_{2},x_{3}\\}}(a,b)=\left\\{\begin{array}[]{ll}e_{\\{x_{2}\\}}(a)&a=b\\\
0&\text{otherwise.}\end{array}\right.$

##  3 The category of bundle scenarios

In this section we introduce bundle scenarios of simplicial complexes and
morphisms between them. The resulting category $\mathbf{bScen}$ is the
category of bundle scenarios. We extend empirical models to bundle scenarios
and assemble them into a functor $\mathsf{Emp}:\mathbf{bScen}\to\mathbf{Set}$.
Our main result is Theorem 3.18 proving that in the left-hand diagram in (1)
the functor $\mathcal{E}$ is fully faithful and $\eta$ is a natural
isomorphism. Most of the technical results concerning simplicial complexes are
presented in Section A.

In the context of simplicial complexes locality is captured by the star
construction: The star of a simplex $\sigma\in\Sigma$ is defined by

$\mathsf{St}(\sigma)=\\{\sigma^{\prime}\in\Sigma:\,\sigma\subset\sigma^{\prime}\\}.$

######  Definition 3.1.

A map of $f:\Gamma\to\Sigma$ simplicial complexes is called

  * •

surjective if it is surjective as a function between the set of simplices,

  * •

locally surjective if
$f|_{\gamma}:\mathsf{St}(\gamma)\to\mathsf{St}(f(\gamma))$ is surjective for
all $\gamma\in\Gamma$,

  * •

discrete over vertices if $\\{x,y\\}\notin\Gamma$ for every distinct
$x,y\in\Gamma_{0}$ with $f(x)=f(y)$.

A bundle scenario is a surjective, locally surjective map $f:\Gamma\to\Sigma$
of simplicial complexes that is discrete over vertices.

The two properties, local surjectivity and being discrete over vertices, can
be characterized using lifting conditions (Propositions A.2 and A.3).
Therefore bundle scenarios can also be described in terms of lifting
conditions (Corollary A.4). This point of view will be fruitful when extending
these notions to simplicial sets in Section 4.

Our canonical example of a bundle scenario comes from a scenario $(\Sigma,O)$.
Associated to this scenario we can construct a bundle scenario

$f_{(\Sigma,O)}:\mathcal{E}_{O}\Sigma\to\Sigma$

where $\mathcal{E}_{O}\Sigma$ is the simplicial complex with vertex set
$(\mathcal{E}_{O}\Sigma)_{0}=\sqcup_{x\in\Sigma_{0}}O_{x}$ and the set of
simplices

$\mathcal{E}_{O}\Sigma=\\{(\sigma,s):\,\sigma\in\Sigma,\;s\in\prod_{x\in\sigma}O_{x}\\}.$

Each simplex $(\sigma,s)$ is regarded as the subset
$\sqcup_{x\in\sigma}s(x)\subset(\mathcal{E}_{O}\Sigma)_{0}$. The simplicial
complex map $f_{(\Sigma,O)}$ is given by projection onto the first coordinate.

######  Proposition 3.2.

The simplicial complex map $f_{(\Sigma,O)}:\mathcal{E}_{O}\Sigma\to\Sigma$ is
a bundle scenario.

###### Proof.

The outcome sets $O_{x}$ being nonempty implies that $f=f_{(\Sigma,O)}$ is
surjective. For a simplex $(\sigma,s)$ consider the map between the stars

$f_{(\sigma,s)}:\mathsf{St}(\sigma,s)\to\mathsf{St}(\sigma).$

Given an element of $\mathsf{St}(\sigma)$, that is a simplex $\sigma^{\prime}$
containing $\sigma$, consider $s^{\prime}\in\prod_{x\in\sigma^{\prime}}O_{x}$
such that $s^{\prime}|_{\sigma}=s$. Then
$(\sigma^{\prime},s^{\prime})\in\mathsf{St}(\sigma,s)$ and it maps to
$\sigma^{\prime}$ under $f_{(\sigma,s)}$. This implies that $f$ is locally
surjective. Finally, $f$ is discrete over vertices since for $x\in\Sigma_{0}$
and distinct $o,o^{\prime}\in O_{x}$ we have
$\\{(x,o),(x,o^{\prime})\\}\notin\mathcal{E}_{O}\Sigma$. ∎

There is a category-theoretic description of $\mathcal{E}_{O}\Sigma$ that uses
the notion of the category of elements of a functor.

######  Definition 3.3.

Given a contravariant functor $F:\mathbf{C}\to\mathbf{Set}$ the category of
elements, denoted by $\mathbf{El}(F)$, has objects given by pairs $(c,x)$
where $c$ is an object of $\mathbf{C}$ and $x\in F(c)$. A morphism
$(c^{\prime},x^{\prime})\to(c,x)$ is given by a morphism $f:c^{\prime}\to c$
of $\mathbf{C}$ such that $F(f)(x)=x^{\prime}$.

When this construction is applied to the functor
$\mathcal{E}_{O}:\mathbf{C}_{\Sigma}\to\mathbf{Set}$ associated to a scenario
$(\Sigma,O)$ the category $\mathbf{El}(\mathcal{E}_{O})$ of elements coincides
with $\mathbf{C}_{\mathcal{E}_{O}\Sigma}$.

###  3.1 Morphisms of bundle scenarios

To introduce morphisms between bundle scenarios we need an alternative point
of view on the notion of simplicial relation introduced in Section 2. The key
construction is the nerve complex functor

$\hat{N}:s\mathbf{Comp}\to s\mathbf{Comp}$

defined on the category of simplicial complexes:

######  Definition 3.4.

Given a simplicial complex $\Sigma$ the nerve complex $\hat{N}\Sigma$ is the
simplicial complex defined as follows:

  * •

The vertex set is given by $(\hat{N}\Sigma)_{0}=\Sigma$.

  * •

The set of simplices is
$\hat{N}\Sigma=\\{\\{\sigma_{1},\sigma_{2},\cdots,\sigma_{k}\\}:\,\cup_{i=1}^{k}\sigma_{i}\in\Sigma\\}$.

A simplicial complex map $f:\Gamma\to\Sigma$ induces a simplicial complex map
$\hat{N}f:\hat{N}\Gamma\to\hat{N}\Sigma$ between the nerve complexes defined
on vertices by $(\hat{N}f)(\gamma)=f(\gamma)$.

The functor $\hat{N}$ is in fact a monad, i.e., comes with two natural
transformations:

  * •

$\delta:\text{id}\to\hat{N}$ whose component
$\delta_{\Sigma}:\Sigma\to\hat{N}\Sigma$ is defined by
$\delta_{\Sigma}(x)=\\{x\\}$, and

  * •

$\mu:\hat{N}\circ\hat{N}\to\hat{N}$ whose component
$\mu_{\Sigma}:\hat{N}^{2}\Sigma\to\hat{N}\Sigma$ is defined by
$\mu_{\Sigma}(\\{\sigma_{1},\dots,\sigma_{n}\\})=\cup_{i=1}^{n}\sigma_{i}$.

Associated to a monad one can consider the Kleisli category [17]. For the
nerve complex functor the Kleisli category $\mathbf{sComp}_{\hat{N}}$ consists
of simplicial complexes as objects and its morphisms are given by

$\mathbf{sComp}_{\hat{N}}(\Sigma^{\prime},\Sigma)=\mathbf{sComp}(\Sigma^{\prime},\hat{N}\Sigma).$

######  Proposition 3.5.

The set of simplicial relations $\pi:\Sigma^{\prime}\to\Sigma$ is in bijection
with the set of morphisms $\Sigma^{\prime}\to\Sigma$ in the Kleisli category
$\mathbf{sComp}_{\hat{N}}$.

###### Proof.

A simplicial complex map $\Sigma^{\prime}\to\hat{N}\Sigma$ is given by a
function $\pi:\Sigma^{\prime}_{0}\to\Sigma$ such that
$\pi(\sigma^{\prime})=\cup_{x^{\prime}\in\sigma^{\prime}}\pi(x^{\prime})$ is a
simplex of $\Sigma$ for all $\sigma^{\prime}\in\Sigma^{\prime}$. This is
precisely a simplicial relation. ∎

We will regard simplicial relations as morphisms in the Kleisli category.
Given simplicial complex maps $\pi:\Sigma^{\prime}\to\hat{N}\Sigma$ and
$\pi^{\prime}:\Sigma^{\prime\prime}\to\hat{N}\Sigma^{\prime}$ we write
$\pi\diamond\pi^{\prime}:\Sigma^{\prime\prime}\to\hat{N}\Sigma$ for the
Kleisli composition, i.e., the composition in the category
$\mathbf{sComp}_{\hat{N}}$.

The final ingredient for the definition of morphisms between bundle scenarios
is pull-backs of bundle scenarios. In the category of simplicial complexes the
pull-back of a bundle scenario has a particularly nice description
(Proposition A.5).

The morphisms of the category $\mathbf{bScen}$ are constructed from two types
of morphisms:

  * •

A type I morphism $(\pi,\text{id}):f\to f^{\pi}$ is given by a pull-back
square

${\hat{N}\Gamma}$${\Gamma^{\prime}}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{f^{\pi}}$$\scriptstyle{l}$$\scriptstyle{\pi}$
(5)

To indicate that this is a pull-back we will write
$\Gamma^{\prime}=\pi^{*}(\hat{N}\Gamma)$, and sometimes we will use the
notation $l_{f}$ or even $l_{f,\pi}$ instead of $l$.

  * •

A type II morphism $\alpha:f\to f^{\prime}$ is given by a commutative diagram

${\Gamma}$${\Gamma^{\prime}}$${\Sigma}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$
(6)

The pull-back of a bundle scenario is also a bundle scenario (Proposition
A.6). The simplicial complex map $\hat{N}f$ in Diagram (5) is a bundle
scenario when $f$ is a bundle scenario (Proposition A.7). Therefore $f^{\pi}$
is also a bundle scenario.

The composition of type II morphisms is clear, whereas for type I morphisms we
have the following definition.

######  Definition 3.6.

The composition of the following type I morphisms

${\hat{N}\Gamma}$${\Gamma^{\prime}}$${\hat{N}\Gamma^{\prime}}$${\Gamma^{\prime\prime}}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$${\hat{N}\Sigma^{\prime}}$${\Sigma^{\prime\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{l}$$\scriptstyle{\hat{N}f^{\prime}}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{l^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$

is defined to be the composition of the following squares

${\hat{N}\Gamma}$${\hat{N}^{2}\Gamma}$${\hat{N}\Gamma^{\prime}}$${\Gamma^{\prime\prime}}$${\hat{N}\Sigma}$${\hat{N}^{2}\Sigma}$${\hat{N}\Sigma^{\prime}}$${\Sigma^{\prime\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{\hat{N}^{2}f}$$\scriptstyle{\mu_{\Gamma}}$$\scriptstyle{\hat{N}l}$$\scriptstyle{\hat{N}f^{\prime}}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{l^{\prime}}$$\scriptstyle{\mu_{\Sigma}}$$\scriptstyle{\hat{N}\pi}$$\scriptstyle{\pi^{\prime}}$
(7)

We will denote this composition by $(\pi\diamond\pi^{\prime},\text{id})$.

The composition of the squares in Diagram (7) is a valid type I morphism since
the composition of the left and the middle square is a pull-back diagram
(Lemma A.8). Composing two pull-back squares gives a pull-back square hence
the composition of the resulting square with the third square is also a pull-
back square.

######  Definition 3.7.

The category $\mathbf{bScen}$ of bundle scenarios consists of objects given by
bundle scenarios $f:\Gamma\to\Sigma$. A morphism $f\to f^{\prime}$ is given by
a pair $(\pi,\alpha)$ of simplicial complex maps making the diagram commute

${\hat{N}\Gamma}$${\pi^{*}(\hat{N}\Gamma)}$${\Gamma^{\prime}}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$${\Sigma^{\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{f^{\pi}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi}$
(8)

where $\pi^{*}(\hat{N}\Gamma)$ is the pull-back along $\pi$. The composition
of $(\pi_{1},\alpha_{1}):f\to f^{\prime}$ and
$(\pi_{2},\alpha_{2}):f^{\prime}\to f^{\prime\prime}$ is defined as follows:

$(\pi_{2},\alpha_{2})\circ(\pi_{1},\alpha_{1})=(\pi_{1}\diamond\pi_{2},\alpha_{2}\circ\pi_{2}^{*}(\hat{N}\alpha_{1}))$
(9)

where the map

$\pi_{2}^{*}(\hat{N}\alpha_{1}):\pi_{2}^{*}(\hat{N}\pi_{1}^{*}(\hat{N}\Gamma))\to\pi_{2}^{*}(\hat{N}\Gamma^{\prime})$

is the pull-back of $\hat{N}\alpha_{1}$ in the diagram

${\hat{N}\pi_{1}^{*}(\hat{N}\Gamma)}$${\hat{N}\Gamma^{\prime}}$${\hat{N}\Sigma^{\prime}}$$\scriptstyle{\hat{N}\alpha_{1}}$$\scriptstyle{\hat{N}f^{\pi_{1}}}$$\scriptstyle{\hat{N}f^{\prime}}$

along the simplicial complex map
$\pi_{2}:\Sigma^{\prime\prime}\to\hat{N}\Sigma^{\prime}$.

The identity morphism of the object $f:\Gamma\to\Sigma$ is given by

${\hat{N}\Gamma}$${\Gamma}$${\Gamma}$${\hat{N}\Sigma}$${\Sigma}$${\Sigma}$$\scriptstyle{\hat{N}f}$$\scriptstyle{\delta_{\Gamma}}$$\scriptstyle{f}$$\scriptstyle{\text{id}_{\Gamma}}$$\scriptstyle{f}$$\scriptstyle{\delta_{\Sigma}}$

This is a valid morphism since the left square is a pull-back diagram (Lemma
A.9). Definition 3.7 gives a well-defined category since the composition rule
is associative (Lemma A.10).

######  Example 3.8.

The scenarios $(\triangle,O)$ and $(\square,O)$ in Example 2.2 can be realized
as bundle scenarios $f_{\triangle}:\mathcal{E}_{O}\triangle\to\triangle$ and
$f_{\square}:\mathcal{E}_{O}\square\to\square$. Recall that the simplicial
relation $\pi:\square\to\triangle$ is in fact a simplicial complex map. In
effect the simplicial relation in Diagram (8) can be replaced by this
simplicial complex map when computing the pull-back. The morphism in this
example gives a morphism $(\pi,\alpha):f_{\triangle}\to f_{\square}$ of bundle
scenarios

${\mathcal{E}_{O}\triangle}$${\pi^{*}\mathcal{E}_{O}\triangle}$${\mathcal{E}_{O}\square}$${\triangle}$${\square}$$\scriptstyle{\alpha}$$\scriptstyle{\pi}$

This is depicted in Figure 1.

A simplicial complex map $\pi:\Sigma^{\prime}\to\hat{N}\Sigma$ induces a
functor

$\overline{\pi}:\mathbf{C}_{\Sigma^{\prime}}\to\mathbf{C}_{\Sigma}$

defined by
$\overline{\pi}(\sigma^{\prime})=\cup_{\sigma\in\pi(\sigma^{\prime})}\sigma$.
Then the Kleisli composition can be seen as composition of functors

$\overline{\pi\diamond\pi^{\prime}}=\overline{\pi}\circ\overline{\pi^{\prime}}.$
(10)

Given a scenario $(\Sigma,O)$ the category of elements construction
(Definition 3.3) for the functor $\mathcal{E}_{O}\circ\overline{\pi}$
specifies a simplicial complex:

$\mathbf{C}_{(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}}=\mathbf{El}(\mathcal{E}_{O}\circ\overline{\pi}).$

Explicitly, the simplicial complex $(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}$
has

  * •

vertices $(x^{\prime},s)$, where $x^{\prime}\in\Sigma^{\prime}_{0}$ and
$s\in\mathcal{E}_{O}(\pi(x^{\prime}))$, and

  * •

simplices $(\sigma^{\prime},s)$, where $\sigma^{\prime}\in\Sigma^{\prime}$ and
$s\in\mathcal{E}_{O}(\overline{\pi}(\sigma^{\prime}))$.

Note that $\pi(x^{\prime})=\overline{\pi}(\\{x^{\prime}\\})$. Projection onto
the first coordinate gives a simplicial complex map

$f_{\mathcal{E}_{O}\circ\pi}:(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}\to\Sigma^{\prime}$

Observe that for $t\in\mathcal{E}_{O}(\pi(x^{\prime}))$, there exists a
morphism $(\\{x^{\prime}\\},t)\to(\sigma^{\prime},s)$ in
$\mathbf{C}_{(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}}$ if and only if
$x^{\prime}\in\sigma^{\prime}$ and $s|_{\pi(x^{\prime})}=t$. Therefore we can
identify $(\sigma^{\prime},s)$ with the set
$\\{(x^{\prime},s|_{\pi(x^{\prime})}):\,x^{\prime}\in\sigma^{\prime}\\}$.

Given a morphism $(\pi,\alpha):(\Sigma^{\prime},O^{\prime})\to(\Sigma,O)$ of
$\mathbf{Scen}$ we define a morphism of $\mathbf{bScen}$:

${\hat{N}(\mathcal{E}_{O}\Sigma)}$${(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}}$${\mathcal{E}_{O^{\prime}}\Sigma^{\prime}}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$${\Sigma^{\prime}}$$\scriptstyle{\hat{N}f_{(\Sigma,O)}}$$\scriptstyle{f_{\mathcal{E}_{O}\circ\pi}}$$\scriptstyle{l}$$\scriptstyle{r(\alpha)}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi}$
(11)

where

  * •

$l(x^{\prime},s)=(\pi(x^{\prime}),s)$, and

  * •

$r(\alpha)(\sigma^{\prime},s)=(\sigma^{\prime},\alpha_{\sigma^{\prime}}(s))$.

The left square is a pull-back diagram (Lemma A.11) hence it is a valid type I
morphism and together with Propositions 3.2, A.6, and A.7 this implies that
$f_{\mathcal{E}_{O}\circ\pi}$ is a bundle scenario. Observe also that
naturality of $\alpha$ implies that the map $r(\alpha)$ is a morphism of
$\mathbf{sComp}$.

######  Proposition 3.9.

The assignments $(\Sigma,O)\mapsto f_{(\Sigma,O)}$ and
$(\pi,\alpha)\mapsto(\pi,r(\alpha))$, where $(\pi,r(\alpha))$ is given as in
Diagram (11), specify a fully faithful functor
$\mathcal{E}:\mathbf{Scen}\to\mathbf{bScen}.$

###### Proof.

The simplicial complex map $f_{(\Sigma,O)}$ is a bundle scenario by
Proposition 3.2. By definition it is clear that
$\mathcal{E}(\text{id}_{(\Sigma,O)})=\text{id}_{\mathcal{E}(\Sigma,O)}$. Given
the morphisms $(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ and
$(\pi^{\prime},\alpha^{\prime}):(\Sigma^{\prime},O^{\prime})\to(\Sigma^{\prime\prime},O^{\prime\prime})$
of $\mathbf{Scen}$ we have

$\displaystyle\mathcal{E}((\pi^{\prime},\alpha^{\prime})\circ(\pi,\alpha))$
$\displaystyle=\mathcal{E}(\pi\diamond\pi^{\prime},\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}}))$
$\displaystyle=(\pi\diamond\pi^{\prime},r(\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}})))$
$\displaystyle=(\pi\diamond\pi^{\prime},r(\alpha^{\prime})\circ(\pi^{\prime})^{\ast}(\hat{N}r(\alpha)))$
$\displaystyle=(\pi^{\prime},r(\alpha^{\prime}))\circ(\pi,r(\alpha))$
$\displaystyle=\mathcal{E}(\pi^{\prime},\alpha^{\prime})\circ\mathcal{E}(\pi,\alpha).$

where in the third line we used Lemma A.12. This proves that $\mathcal{E}$ is
a functor. Finally, Lemma A.13 and A.14 implies that this functor is fully
faithful. ∎

###  3.2 Empirical models on bundle scenarios

A simplicial complex map that is discrete over vertices satisfies a very
useful property which we will refer to through out the paper:

######  Remark 3.10.

Let $f:\Gamma\to\Sigma$ be a surjective simplicial complex map that is
discrete over vertices. For simplices $\sigma^{\prime}\subset\sigma\in\Sigma$
and $\gamma\in\Gamma$ with $f(\gamma)=\sigma$, there exists a unique simplex
$\gamma^{\prime}\in\Gamma$ such that $f(\gamma^{\prime})=\sigma^{\prime}$
since $f$ is discrete over vertices.

We will write

$r_{\sigma,\sigma^{\prime}}:f^{-1}(\sigma)\to f^{-1}(\sigma^{\prime})$

for the map sending $\gamma$ to the corresponding unique simplex
$\gamma^{\prime}$ specified by Remark 3.10. For a distribution $p\in
D_{R}(f^{-1}(\sigma))$ we define the restriction
$p|_{\sigma^{\prime}}=D_{R}(r_{\sigma,\sigma^{\prime}})(p)$. With this
definition we have

$p|_{\sigma^{\prime}}(\gamma^{\prime})=\sum_{\gamma^{\prime}\subset\gamma\in
f^{-1}(\sigma)}p(\gamma).$ (12)

######  Definition 3.11.

An empirical model on a bundle scenario $f:\Gamma\to\Sigma$ is a family
$p=(p_{\sigma})_{\sigma\in\Sigma}$ of distributions, where $p_{\sigma}\in
D_{R}(f^{-1}(\sigma))$, such that
$p_{\sigma}|_{\sigma^{\prime}}=p_{\sigma^{\prime}}$ for every
$\sigma^{\prime}\subset\sigma$ in $\Sigma$. We will denote the set of
empirical models on $f$ by $\mathsf{bEmp}(f)$.

####  3.2.1 Push-forward empirical models for bundles

######  Definition 3.12.

Let $f:\Gamma\to\Sigma$ be a bundle scenario and $p\in\mathsf{bEmp}(f)$ be an
empirical model. We define an empirical model $\hat{N}p$ on the bundle
scenario $\hat{N}f:\hat{N}\Gamma\to\hat{N}\Sigma$ by the formula

$(\hat{N}p)_{\\{\sigma_{1},\cdots,\sigma_{n}\\}}(\\{\gamma_{1},\cdots,\gamma_{n}\\})=p_{\cup_{i}\sigma_{i}}(\cup_{i}\gamma_{i})$
(13)

where $\\{\sigma_{1},\cdots,\sigma_{n}\\}\in\hat{N}\Sigma$, and
$\\{\gamma_{1},\cdots,\gamma_{n}\\}\in(\hat{N}f)^{-1}(\\{\sigma_{1},\cdots,\sigma_{n}\\})$.

Equation (13) gives a well-defined empirical model (Lemma A.15).

######  Definition 3.13.

For $p\in\mathsf{bEmp}(f)$ we define the push-forward of $p$ along a type I
morphism

${\hat{N}\Gamma}$${\pi^{*}(\hat{N}\Gamma)}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{f^{\pi}}$$\scriptstyle{l}$$\scriptstyle{\pi}$

by the formula

$(\pi_{\ast}p)_{\sigma^{\prime}}(\gamma^{\prime})=(\hat{N}p)_{\pi(\sigma^{\prime})}(l(\gamma^{\prime}))$

where $\sigma^{\prime}\in\Sigma^{\prime}$ and
$\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})$. When we regard simplicial
relations as functors between the corresponding categories, we can write
$(\pi_{\ast}p)_{\sigma^{\prime}}(\gamma^{\prime})=p_{\overline{\pi}(\sigma^{\prime})}(\overline{l}(\gamma^{\prime}))$.

The push-forward $\pi_{*}p$ is an empirical model on the bundle scenario
$f^{\pi}$ as we verify in Lemma A.16.

######  Definition 3.14.

For $p\in\mathsf{bEmp}(f)$ and $\sigma\in\Sigma$ we define the push-forward of
$p$ along a type II morphism

${\Gamma}$${\Gamma^{\prime}}$${\Sigma}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\scriptstyle{g}$
(14)

by the formula

$(\alpha_{\ast}p)_{\sigma}=D_{R}(\alpha|_{f^{-1}(\sigma)})(p_{\sigma}).$

In Lemma A.17 we verify that $\alpha_{*}p$ is an empirical model on the bundle
scenario $g$. Now we are ready to define the push-forward along an arbitrary
morphism.

######  Definition 3.15.

For $p\in\mathsf{bEmp}(f)$ we define the push-forward of $p$ along a morphism
$(\pi,\alpha):f\to f^{\prime}$ to be
$(\pi,\alpha)_{\ast}p=\alpha_{\ast}(\pi_{\ast}p)$.

Next, we show that with this definition assigning the set of empirical models
to a bundle scenario gives a functor.

######  Proposition 3.16.

The assignments $f\mapsto\mathsf{bEmp}(f)$ and
$(\pi,\alpha)\mapsto(\pi,\alpha)_{*}$ define a functor
$\mathsf{bEmp}:\mathbf{bScen}\to\mathbf{Set}$.

###### Proof.

Given $f:\Gamma\to\Sigma$ and $p\in\mathsf{bEmp}(f)$, it is clear that
$(\delta_{\Sigma},\text{id}_{\Gamma})_{\ast}p=p$. Using Lemma A.16, A.17, and
A.18 for the morphisms $(\pi,\alpha):f\to f^{\prime}$ and
$(\pi^{\prime},\alpha^{\prime}):f^{\prime}\to f^{\prime\prime}$ of
$\mathbf{bScen}$ we obtain

$\displaystyle(\pi^{\prime},\alpha^{\prime})_{\ast}\left((\pi,\alpha)_{\ast}p\right)$
$\displaystyle=\alpha^{\prime}_{\ast}\left(\pi^{\prime}_{\ast}(\alpha_{\ast}(\pi_{\ast}p))\right)$
$\displaystyle=\alpha^{\prime}_{\ast}(\left((\pi^{\prime})^{\ast}(\hat{N}\alpha)\right)_{\ast}\left(\pi^{\prime}_{\ast}(\pi_{\ast}p)\right))$
$\displaystyle=(\alpha^{\prime}_{\ast}\circ(\pi^{\prime})^{\ast}(\hat{N}\alpha))_{\ast}\left((\pi\diamond\pi^{\prime})_{\ast}p\right)$
$\displaystyle=(\pi\diamond\pi^{\prime},\alpha^{\prime}_{\ast}\circ(\pi^{\prime})^{\ast}(\hat{N}\alpha))_{\ast}p$
$\displaystyle=((\pi^{\prime},\alpha^{\prime})\circ(\pi,\alpha))_{\ast}p.$

∎

######  Proposition 3.17.

For an object $(\Sigma,O)$ of $\mathbf{Scen}$ and an empirical model
$e\in\mathsf{Emp}(\Sigma,O)$, defining

$\eta_{(\Sigma,O)}(e)_{\sigma}(\sigma,s)=e_{\sigma}(s),$

where $\sigma\in\Sigma$ and $s\in\mathcal{E}_{O}(\sigma)$, gives a natural
isomorphism $\eta:\mathsf{Emp}\to\mathsf{bEmp}\circ\mathcal{E}$.

###### Proof.

Observe that
$(f_{(\Sigma,O)})^{-1}(\sigma)=\\{\sigma\\}\times\mathcal{E}_{O}(\sigma)$.
That is, one can consider $e$ as a family $\\{e_{\sigma}\in
D_{R}(f_{(\Sigma,O)}^{-1}(\sigma))\\}_{\sigma\in\Sigma}$ of distributions, and
the compatibility in both cases is the same. Now, we prove the naturality.
Given $(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ a morphism of
$\mathbf{Scen}$, we need to prove that the following diagram commutes

${\mathsf{Emp}(\Sigma,O)}$${\mathsf{Emp}(\Sigma^{\prime},O^{\prime})}$${\mathsf{bEmp}(f_{(\Sigma,O)})}$${\mathsf{bEmp}(f_{(\Sigma^{\prime},O^{\prime})})}$$\scriptstyle{(\pi{,}\alpha)_{\ast}}$$\scriptstyle{\eta_{(\Sigma,O)}}$$\scriptstyle{\eta_{(\Sigma^{\prime},O^{\prime})}}$$\scriptstyle{\mathcal{E}(\pi{,}\alpha)_{\ast}}$
(15)

Given $e\in\mathsf{Emp}(\Sigma,O)$ and
$(\sigma^{\prime},s^{\prime})\in\mathcal{E}_{O^{\prime}}\Sigma^{\prime}$, we
have

$\displaystyle\left(\mathcal{E}(\pi,\alpha)_{\ast}(\eta_{(\Sigma,O)}(e))\right)_{\sigma^{\prime}}(\sigma^{\prime},s^{\prime})$
$\displaystyle=\left((\pi,r(\alpha))_{\ast}(\eta_{(\Sigma,O)}(e))\right)_{\sigma^{\prime}}(\sigma^{\prime},s^{\prime})$
$\displaystyle=r(\alpha)_{\ast}(\pi_{\ast}(\eta_{(\Sigma,O)}(e)))_{\sigma^{\prime}}(\sigma^{\prime},s^{\prime})$
$\displaystyle=D_{R}(r(\alpha)|_{(f_{\mathcal{E}_{O}\circ\pi})^{-1}(\sigma^{\prime})})(\pi_{\ast}(\eta_{(\Sigma,O)}(e))_{\sigma^{\prime}})(\sigma^{\prime},s^{\prime})$
$\displaystyle=\sum_{r(\alpha)(\sigma^{\prime},s)=(\sigma^{\prime},s^{\prime})}\left(\pi_{\ast}(\eta_{(\Sigma,O)}(e))\right)_{\sigma^{\prime}}(\sigma^{\prime},s)$
$\displaystyle=\sum_{\alpha_{\sigma^{\prime}}(s)=s^{\prime}}(\hat{N}\eta_{(\Sigma,O)}(e))_{\pi(\sigma^{\prime})}(\pi(\sigma^{\prime}),s)$
$\displaystyle=\sum_{\alpha_{\sigma^{\prime}}(s)=s^{\prime}}(\eta_{(\Sigma,O)}(e))_{\overline{\pi}(\sigma^{\prime})}(\overline{\pi}(\sigma^{\prime}),s)$
$\displaystyle=\sum_{\alpha_{\sigma^{\prime}}(s)=s^{\prime}}e_{\overline{\pi}(\sigma^{\prime})}(s).$

On the other hand,

$\displaystyle\left(\eta_{(\Sigma^{\prime},O^{\prime})}((\pi,\alpha)_{\ast}e)\right)_{\sigma^{\prime}}(\sigma^{\prime},s^{\prime})$
$\displaystyle=((\pi,\alpha)_{\ast}e)_{\sigma^{\prime}}(s^{\prime})$
$\displaystyle=D_{R}(\alpha_{\sigma^{\prime}})(e_{\overline{\pi}(\sigma^{\prime})})(s^{\prime})$
$\displaystyle=\sum_{\alpha_{\sigma^{\prime}}(s)=s^{\prime}}e_{\overline{\pi}(\sigma^{\prime})}(s).$

∎

Combining Proposition 3.9 and 3.17 we obtain our main result in this section.

######  Theorem 3.18.

Sending a scenario $(\Sigma,O)$ to the bundle scenario
$f_{(\Sigma,O)}:\mathcal{E}_{O}\Sigma\to\Sigma$ specifies a fully faithful
functor $\mathcal{E}:\mathbf{Scen}\to\mathbf{bScen}$. Moreover, there is a
natural isomorphism $\eta:\mathsf{Emp}\to\mathsf{bEmp}\circ\mathcal{E}$.

##  4 The category of simplicial bundle scenarios

The theory of simplicial distributions is introduced in [2] as a
generalization of empirical models to scenarios consisting of a space of
measurements and outcomes. In this section we extend simplicial distributions
to simplicial bundle scenarios. First we introduce simplicial bundle scenarios
and morphisms between them to obtain the category $\mathbf{sScen}$ of
simplicial bundle scenarios. Simplicial distributions are defined in this
generality as a functor $\mathsf{sDist}:\mathbf{sScen}\to\mathbf{Set}$.
Therefore in addition to extending the notion to bundles we also describe how
to push-forward a simplicial distribution along a morphism. Theorem 4.18 is
our main result that prove the nerve functor $N$ in Diagram (1) is fully
faithful and $\zeta$ is a natural isomorphism. In Section B we present the
technical results concerning simplicial sets.

######  Definition 4.1.

A map $f:E\to X$ of simplicial sets is called

  * •

surjective if it is surjective in each degree $n\geq 0$,

  * •

locally surjective if it has the right-lifting property with respect to
$d^{i}:\Delta[n-1]\to\Delta[n]$ for $n\geq 1$, and $0\leq i\leq n$, i.e., the
diagonal map exists in the following commutative diagram

${\Delta[n-1]}$${E}$${\Delta[n]}$${X}$$\scriptstyle{d^{i}}$$\scriptstyle{f}$

  * •

discrete over vertices if it has the right-lifting property with respect to
$s^{i}:\Delta[n]\to\Delta[n-1]$ for $n\geq 1$, and $0\leq i\leq n-1$, i.e.,
the diagonal map exists

${\Delta[n]}$${E}$${\Delta[n-1]}$${X}$$\scriptstyle{s^{j}}$$\scriptstyle{f}$

A simplicial (bundle) scenario is a map $f:E\to X$ of simplicial sets that is
surjective, locally surjective and discrete over vertices.

Let $N:s\mathbf{Comp}\to\mathbf{sSet}$ denote the nerve functor that sends a
simplicial complex $\Sigma$ to the simplicial set $N\Sigma$ whose
$n$-simplices are given by

$(N\Sigma)_{n}=\\{(\sigma_{1},\cdots,\sigma_{n})\in\Sigma^{n}:\,\cup_{i=1}^{n}\sigma_{i}\in\Sigma)\\}.$

For the simplicial structure maps see Definition B.3. The nerve construction
can be used to obtain a simplicial scenario from a bundle scenario of
simplicial complexes.

######  Proposition 4.2.

If $f$ is a bundle scenario then $Nf$ is a simplicial scenario.

###### Proof.

This follows from Lemma B.5 and B.6. ∎

###  4.1 Morphisms of simplicial scenarios

Defining morphisms between simplicial scenarios is easier than defining
morphisms between bundle scenarios. In Section 3.1, we introduced morphisms
between bundle scenarios in two steps by defining type I and II morphisms. For
simplicial scenarios, we can define morphisms in one step. Therefore this
approach, while generalizing bundles of simplicial complexes, has a cleaner
description of morphisms.

######  Definition 4.3.

The category $\mathbf{sScen}$ of simplicial (bundle) scenarios consists of
objects given by simplicial (bundle) scenarios and morphisms $f\to f^{\prime}$
given by pairs $(\pi,\alpha)$ of simplicial set maps making the following
diagram commute

${E}$${\pi^{*}(E)}$${E^{\prime}}$${X}$${X^{\prime}}$${X^{\prime}}$$\scriptstyle{f}$$\scriptstyle{f^{\pi}}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi}$
(16)

where $\pi^{*}(E)$ is the pull-back along $\pi$.

######  Remark 4.4.

The map $f^{\pi}$ in Diagram (16) is also a simplicial scenario since pulling
back a map preserves surjectivity, local surjectivity, and being discrete over
vertices. The latter two properties are described in terms of the liftings of
certain diagrams, and liftings still exist under pull-backs.

The composition of $(\pi_{1},\alpha_{1}):f\to f^{\prime}$ and
$(\pi_{2},\alpha_{2}):f^{\prime}\to f^{\prime\prime}$ is defined by

$(\pi_{2},\alpha_{2})\circ(\pi_{1},\alpha_{1})=(\pi_{1}\circ\pi_{2},\alpha_{2}\circ\pi_{2}^{*}(\alpha_{1}))$
(17)

where

$\pi_{2}^{*}(\alpha_{1}):\pi_{2}^{*}(\pi_{1}^{*}(E))\to\pi_{2}^{*}(E^{\prime})$

is the pull-back of $\alpha_{1}$ along $\pi_{2}$. This gives a well-defined
category: The identity morphism of an object $f:E\to X$ is given by
$(\text{id}_{X},\text{id}_{E})$ and the composition rule is associative as we
prove next.

######  Lemma 4.5.

The composition given by Equation (17) is associative.

###### Proof.

For $(\pi_{1},\alpha_{1}):f\to f^{\prime}$,
$(\pi_{2},\alpha_{2}):f^{\prime}\to f^{\prime\prime}$, and
$(\pi_{3},\alpha_{3}):f^{\prime\prime}\to f^{\prime\prime\prime}$, we have

$\displaystyle(\pi_{3},\alpha_{3})\circ\left((\pi_{2},\alpha_{2})\circ(\pi_{1},\alpha_{1})\right)$
$\displaystyle=(\pi_{3},\alpha_{3})\circ(\pi_{1}\circ\pi_{2},\alpha_{2}\circ\pi_{2}^{\ast}(\alpha_{1}))$
$\displaystyle=\left((\pi_{1}\circ\pi_{2})\circ\pi_{3},\alpha_{3}\circ{\pi_{3}}^{\ast}\left(\alpha_{2}\circ\pi_{2}^{\ast}(\alpha_{1})\right)\right)$
$\displaystyle=\left(\pi_{1}\circ(\pi_{2}\circ\pi_{3}),\alpha_{3}\circ{\pi_{3}}^{\ast}(\alpha_{2})\circ{\pi_{3}}^{\ast}(\pi_{2}^{\ast}(\alpha_{1}))\right)$
$\displaystyle=\left(\pi_{1}\circ(\pi_{2}\circ\pi_{3}),\alpha_{3}\circ{\pi_{3}}^{\ast}(\alpha_{2})\circ(\pi_{2}\circ\pi_{3})^{\ast}(\alpha_{1})\right)$
$\displaystyle=(\pi_{2}\circ\pi_{3},\alpha_{3}\circ\pi_{3}^{\ast}(\alpha_{2}))\circ(\pi_{1},\alpha_{1}))$
$\displaystyle=\left((\pi_{3},\alpha_{3})\circ(\pi_{2},\alpha_{2})\right)\circ(\pi_{1},\alpha_{1}).$

∎

Next we turn to the nerve functor $N:\mathbf{sComp}\to\mathbf{sSet}$.

######  Definition 4.6.

Let $\pi:\Sigma^{\prime}\to\hat{N}\Sigma$ be a simplicial complex map and
$\overline{\pi}$ denote the functor
$\mathbf{C}_{\Sigma^{\prime}}\to\mathbf{C}_{\Sigma}$ associated to $\pi$. We
define

$T(\pi):N\Sigma^{\prime}\to N\Sigma$

by

$T(\pi)_{n}(\sigma_{1},\cdots,\sigma_{n})=(\overline{\pi}(\sigma_{1}),\cdots,\overline{\pi}(\sigma_{n}))$

where $(\sigma_{1},\cdots,\sigma_{n})\in(N\Sigma^{\prime})_{n}$.

Given a morphism $(\pi,\alpha):f\to f^{\prime}$ between two bundle scenarios
$f:\Gamma\to\Sigma$ and $f^{\prime}:\Gamma^{\prime}\to\Sigma^{\prime}$ we
define a morphism of $\mathbf{sScen}$ by the diagram:

${N\Gamma}$${N(\pi^{*}(\hat{N}\Gamma))}$${N\Gamma^{\prime}}$${N\Sigma}$${N\Sigma^{\prime}}$${N\Sigma^{\prime}}$$\scriptstyle{Nf}$$\scriptstyle{Nf^{\pi}}$$\scriptstyle{N\alpha}$$\scriptstyle{Nf^{\prime}}$$\scriptstyle{T(\pi)}$
(18)

This is a valid morphism of $\mathbf{sScen}$ since the left square is a pull-
back diagram (Lemma B.8), i.e., $N(\pi^{*}(\hat{N}\Gamma))\cong
T(\pi)^{*}(N{\Gamma})$.

######  Proposition 4.7.

The assignments $f\mapsto Nf$ and $(\pi,\alpha)\mapsto(T(\pi),N\alpha)$
specify a fully faithful functor $N:\mathbf{bScen}\to\mathbf{sScen}.$

###### Proof.

For a bundle scenario $f:\Gamma\to\Sigma$ the simplicial map $Nf$ is a
simplicial scenario by Proposition 4.2. In addition, by Lemma B.9 we have that
$N(\delta_{\Sigma},\text{id}_{\Gamma})=(T(\delta_{\Sigma}),N(\text{id}_{\Gamma}))=(\text{id}_{N\Sigma},\text{id}_{N\Gamma})$.
Given two morphisms $(\pi,\alpha):f\to f^{\prime}$ and
$(\pi^{\prime},\alpha^{\prime}):f^{\prime}\to f^{\prime\prime}$ of
$\mathbf{sScen}$, by Lemmas B.9 and B.10, we have

$\displaystyle N((\pi^{\prime},\alpha^{\prime})\circ(\pi,\alpha))$
$\displaystyle=N(\pi\diamond\pi^{\prime},\alpha^{\prime}\circ(\pi^{\prime})^{\ast}(\hat{N}\alpha))$
$\displaystyle=(T(\pi\diamond\pi^{\prime}),N\left(\alpha^{\prime}\circ(\pi^{\prime})^{\ast}(\hat{N}\alpha)\right))$
$\displaystyle=(T(\pi)\circ T(\pi^{\prime}),N(\alpha^{\prime})\circ
N\left((\pi^{\prime})^{\ast}(\hat{N}\alpha)\right))$
$\displaystyle=\left(T(\pi)\circ T(\pi^{\prime}),N(\alpha^{\prime})\circ
T(\pi^{\prime})^{\ast}(N\alpha)\right)$
$\displaystyle=(T(\pi^{\prime}),N\alpha^{\prime})\circ(T(\pi),N\alpha)$
$\displaystyle=N(\pi^{\prime},\alpha^{\prime})\circ N(\pi,\alpha).$

This proves that $N$ is a functor. By Lemma B.11 and B.12 this functor is
fully faithful. ∎

###  4.2 Simplicial distributions on bundles

######  Definition 4.8.

Let $f:E\to X$ be a simplicial set map. A simplicial distribution on $f$ is a
simplicial set map $p:X\to D_{R}(E)$ that makes the following diagram commute:

${D_{R}(E)}$${X}$${D_{R}(X)}$$\scriptstyle{D_{R}(f)}$$\scriptstyle{p}$$\scriptstyle{\delta}$
(19)

We write $\mathsf{sDist}(f)$ for the set of simplicial distributions on $f$.

######  Proposition 4.9.

A map $p:X\to D_{R}(E)$ belongs to $\mathsf{sDist}(f)$ if and only if the
support of $p_{n}(x)$ is contained in $f_{n}^{-1}(x)$ for all $x\in X_{n}$.

###### Proof.

The map $p$ belongs to $\mathsf{sDist}(f)$ if and only if for every $x\in
X_{n}$ we have

$\sum_{e\in E_{n}}p_{n}(x)(e)\delta^{f_{n}(e)}=\delta^{x}.$

Since $R$ is a zero-sum-free semiring, this condition holds if and only if
$p_{n}(x)(e)=0$ for every $x\in X_{n}$ and $e\in E_{n}$ satisfying
$f_{n}(e)\neq x$. ∎

Given simplicial sets $X$ and $Y$ we define

$m:D_{R}(X)\times D_{R}(Y)\to D_{R}(X\times Y)$

by sending $(p,q)\in D_{R}(X_{n})\times D_{R}(Y_{n})$ to the distribution
$p\cdot q$ defined by

$p\cdot q(\sigma,\theta)=p(\sigma)q(\theta)$ (20)

for $\sigma\in X_{n}$ and $\theta\in Y_{n}$. It is straightforward to verify
that Equation (20) respects the face and the degeneracy maps. For instance,
for the face maps we have

$\displaystyle D_{R}(d_{i}^{X}\times
d_{i}^{Y})(m_{{n+1}}(p,q))(\sigma,\theta)$
$\displaystyle=\sum_{d^{X}_{i}(\sigma^{\prime})=\sigma,\,d^{Y}_{i}(\theta^{\prime})=\theta}m_{{n+1}}(p,q)(\sigma^{\prime},\theta^{\prime})$
$\displaystyle=\sum_{d^{X}_{i}(\sigma^{\prime})=\sigma,\,d^{Y}_{i}(\theta^{\prime})=\theta}p(\sigma^{\prime})q(\theta^{\prime})$
$\displaystyle=\sum_{d^{X}_{i}(\sigma^{\prime})=\sigma}p(\sigma^{\prime})\sum_{d^{Y}_{i}(\theta^{\prime})=\theta}q(\theta^{\prime})$
$\displaystyle=D_{R}(d_{i}^{X})(p)(\sigma)\cdot D_{R}(d_{i}^{Y})(q)(\theta)$
$\displaystyle=m_{{n}}(D_{R}(d_{i}^{X})(p),D_{R}(d_{i}^{Y})(q))(\sigma,\theta).$

The canonical map that goes in the other direction

$D_{R}(\text{pr}_{1})\times D_{R}(\text{pr}_{2}):D_{R}(X\times Y)\to
D_{R}(X)\times D_{R}(Y)$

splits $m$, i.e., $(D_{R}(\text{pr}_{1})\times D_{R}(\text{pr}_{2}))\circ
m=\text{id}$.

######  Remark 4.10.

Given a pair $(X,Y)$ of simplicial sets consider the projection map

$f_{(X,Y)}:X\times Y\to X$

Simplicial distributions on $f_{(X,Y)}$ coincide with simplicial distributions
on $(X,Y)$ in the sense of [2], that is, with simplicial set maps $X\to
D_{R}(Y)$. To see this correspondence, let $p$ be a simplicial distribution in
the sense of Definition 4.8. For every $\sigma\in X_{n}$ we have that

${\sum_{\theta\in Y_{n}}p_{n}(\sigma)(-,\theta)=\delta^{\sigma}.}$ (21)

This implies that $\text{supp}({p_{n}(\sigma)})$ consists of pairs
$(\sigma,\theta)$ where $\theta\in\text{supp}({p_{n}(\sigma)(\sigma,-)})$.
Therefore we can identify ${p_{n}(\sigma)}$ with the distribution
${p_{n}(\sigma)}(\sigma,-)\in D_{R}(Y_{n})$. Given $p$, the desired simplicial
set map is the composite

$X\xrightarrow{p}D_{R}(X\times Y)\xrightarrow{D_{R}(\text{pr}_{2})}D_{R}(Y)$

Conversely, a simplicial set map $q:X\to D_{R}(Y)$ can be lifted to
$D_{R}(X\times Y)$ by the following composite:

$X\xrightarrow{\text{id}\times q}X\times
D_{R}(Y)\xrightarrow{\delta\times\text{id}}D_{R}(X)\times
D_{R}(Y)\xrightarrow{m}D_{R}(X\times Y),$

which belongs to $\mathsf{sDist}(f_{(X,Y)})$ by Proposition 4.9. Note that
$f_{(X,Y)}$ fails to be a simplicial scenario (Definition 4.1): In general, it
is surjective and locally surjective, but not discrete over vertices.

####  4.2.1 Push-forward simplicial distributions

In this section we define the push-forward of a simplicial distribution along
a morphism of $\mathbf{sScen}$. As a first step we define push-forward along
$\pi:X^{\prime}\to X$ by introducing a distribution on the middle scenario
$f^{\pi}$ in Diagram (16).

######  Definition 4.11.

Given a simplicial distribution $p$ on $f:E\to X$ and a simplicial set map
$\pi:X^{\prime}\to X$, we define a simplicial distribution on $f^{\pi}$ by the
composition

$\pi_{*}(p):X^{\prime}\xrightarrow{p\circ\pi\times\delta_{X^{\prime}}}D_{R}(E)\times
D_{R}(X^{\prime})\xrightarrow{m}D_{R}(E\times X^{\prime})$

whose image lands in the simplicial subset $D_{R}(\pi^{*}E)$.

Let us unravel this construction. Consider a simplicial distribution $p:X\to
D_{R}(E)$ on $f$. The first observation which makes this construction work is
that

$D_{R}(f)\circ
p\circ\pi=\delta_{X}\circ\pi=D_{R}(\pi)\circ\delta_{X^{\prime}}.$

Therefore $p\circ\pi\times\delta_{X^{\prime}}$ lands inside the pull-back
$(D_{R}\pi)^{*}(D_{R}E)$. The second observation is that for $x^{\prime}\in
X^{\prime}_{n}$ and $(e,x)\in E_{n}\times X_{n}$ such that
$f_{n}(e)\neq\pi_{n}(x)$, Proposition 4.9 implies that
$p_{n}(\pi_{n}(x^{\prime}))(e)\cdot\delta^{x^{\prime}}(x)=0\cdot\delta^{x^{\prime}}(x)=0$.
Therefore $m\circ(p\circ\pi\times\delta_{X^{\prime}})$ lands inside the space
of distributions $D_{R}(\pi^{*}E)$ on the pull-back. In general, for
$x^{\prime}\in X^{\prime}_{n}$, we have the following formula

$(\pi_{\ast}p)_{n}(x^{\prime})=p_{n}(\pi_{n}(x^{\prime}))\cdot\delta^{x^{\prime}},$
(22)

which shows that $\pi_{\ast}p(x^{\prime})\in(f^{\pi})^{-1}(x^{\prime})$.
Therefore $\pi_{\ast}(p)\in\mathsf{sDist}(f^{\pi})$ by Proposition 4.9.

######  Definition 4.12.

Given a simplicial distribution $p$ on $f:E\to X$ and a morphism
$(\pi,\alpha):f\to f^{\prime}$ of simplicial scenarios we define the push-
forward distribution, a simplicial distribution on $f^{\prime}$, by the
following composite:

${(\pi,\alpha)_{*}p:X^{\prime}\xrightarrow{\pi_{\ast}(p)}D_{R}(\pi^{*}E)\xrightarrow{D_{R}(\alpha)}D_{R}(E^{\prime})}$
(23)

To prove that the push-forward construction is functorial we need two
preliminary results.

######  Lemma 4.13.

Let $p$ be a simplicial distribution on $f:E\to X$. For simplicial set maps
$\pi:X^{\prime}\to X$ and $\pi^{\prime}:X^{\prime\prime}\to X^{\prime}$, we
have $(\pi\circ\pi^{\prime})_{\ast}p={\pi^{\prime}}_{\ast}(\pi_{\ast}p)$.

###### Proof.

Given $x\in X_{n}^{\prime\prime}$ Equation (22) implies that

$(\pi^{\prime}_{\ast}(\pi_{\ast}p))_{n}(x)=(\pi_{\ast}p)_{n}(\pi^{\prime}_{n}(x))\cdot\delta^{x}=p_{n}\left(\pi_{n}(\pi^{\prime}_{n}(x))\right)\cdot\delta^{\pi^{\prime}_{n}(x)}\cdot\delta^{x}.$

Therefore if we apply this map to
$(e,\pi^{\prime}_{n}(x),x)\in((\pi^{\prime})^{\ast}\pi^{\ast}(E))_{n}$, we
obtain $p_{n}\left(\pi_{n}(\pi^{\prime}_{n}(x))\right)(e)$. On the other hand,

$(\pi\circ\pi^{\prime})_{\ast}p(x)=p_{n}(\pi_{n}\circ\pi^{\prime}_{n}(x))\cdot\delta^{x}.$

If we apply this map to the corresponding element
$(e,x)\in(\pi\circ\pi^{\prime})^{\ast}(E)_{n}$, we obtain the same result. ∎

######  Lemma 4.14.

Given a simplicial set map $\pi:X^{\prime}\to X$, a commutative diagram of
simplicial set maps

${E}$${E^{\prime}}$${X}$$\scriptstyle{f}$$\scriptstyle{\alpha}$$\scriptstyle{g}$

and a distribution $p\in\mathsf{sDist}(f)$, we have

$\pi_{\ast}(D_{R}(\alpha)\circ p)=D_{R}(\pi^{\ast}(\alpha))\circ\pi_{\ast}p.$

###### Proof.

Given $x^{\prime}\in X^{\prime}_{n}$ and
$(e^{\prime},x^{\prime})\in\pi^{\ast}(E^{\prime})_{n}$, using Equation (22) we
have

$\displaystyle\pi_{\ast}(D_{R}(\alpha)\circ
p)_{n}(x^{\prime})(e^{\prime},x^{\prime})$ $\displaystyle=(D_{R}(\alpha)\circ
p)_{n}\left(\pi_{n}(x^{\prime})\right)(e^{\prime})$
$\displaystyle=D_{R}(\alpha_{n})\left(p_{n}(\pi_{n}(x^{\prime}))\right)(e^{\prime})$
$\displaystyle=\sum_{e\,:\,\,\alpha_{n}(e)=e^{\prime}}p_{n}\left(\pi_{n}(x^{\prime})\right)(e).$

On the other hand,

$\displaystyle
D_{R}(\pi_{\ast}(\alpha)_{n})\left((\pi_{\ast}p)_{n}(x^{\prime})\right)(e^{\prime},x^{\prime})$
$\displaystyle=\sum_{\pi^{\ast}(\alpha)_{n}(e,x^{\prime})=(e^{\prime},x^{\prime})}(\pi_{\ast}p)_{n}(x^{\prime})(e,x^{\prime})$
$\displaystyle=\sum_{e\,:\,\,\alpha_{n}(e)=e^{\prime}}p_{n}\left(\pi_{n}(x^{\prime})\right)(e).$

∎

######  Proposition 4.15.

The push-forward construction in Definition 4.12 gives a functor

$\mathsf{sDist}:\mathbf{sScen}\to\mathbf{Set}.$

###### Proof.

For a simplicial scenario $f:E\to X$ and $p\in\mathsf{sDist}(f)$, it is clear
that $(\text{id}_{X},\text{id}_{E})_{\ast}p=p$. Applying Lemma 4.13 and 4.14
to the morphisms $(\pi,\alpha):f\to f^{\prime}$ and
$(\pi^{\prime},\alpha^{\prime}):f^{\prime}\to f^{\prime\prime}$ of
$\mathbf{sScen}$ we obtain

$\displaystyle(\pi^{\prime},\alpha^{\prime})_{\ast}\left((\pi,\alpha)_{\ast}p\right)$
$\displaystyle=D_{R}(\alpha^{\prime})\circ\pi^{\prime}_{\ast}((\pi,\alpha)_{\ast}p)$
$\displaystyle=D_{R}(\alpha^{\prime})\circ\pi^{\prime}_{\ast}(D_{R}(\alpha)\circ\pi_{\ast}p)$
$\displaystyle=D_{R}(\alpha^{\prime})\circ
D_{R}((\pi^{\prime})^{\ast}(\alpha))\circ\pi^{\prime}(\pi_{\ast}p)$
$\displaystyle=D_{R}\left(\alpha^{\prime}\circ(\pi^{\prime})^{\ast}(\alpha)\right)\circ\left((\pi\circ\pi^{\prime})_{\ast}p\right)$
$\displaystyle=\left(\pi\circ\pi^{\prime},\alpha^{\prime}\circ(\pi^{\prime})^{*}(\alpha)\right)_{\ast}p$
$\displaystyle=((\pi^{\prime},\alpha^{\prime})\circ(\pi,\alpha))_{\ast}p.$

∎

###  4.3 The natural isomorphism

In this section we construct a natural isomorphism
$\zeta:\mathsf{bEmp}\to\mathsf{sDist}\circ N$. The first step is to construct
a simplicial distribution on the nerve of a bundle scenario from a given
empirical model on the bundle.

######  Definition 4.16.

Let $f:\Gamma\to\Sigma$ be a bundle scenario and $p\in\mathsf{bEmp}(f)$ be an
empirical model. We define a simplicial distribution $Np$ on the simplicial
scenario $Nf:N\Gamma\to N\Sigma$ as follows:

$(Np)_{n}(\sigma_{1},\cdots,\sigma_{n})(\gamma_{1},\cdots,\gamma_{n})=\begin{cases}p_{\cup_{i=1}^{n}\sigma_{i}}(\cup_{i=1}^{n}\gamma_{i})&(\gamma_{1},\cdots,\gamma_{n})\in(Nf)_{n}^{-1}(\sigma_{1},\cdots,\sigma_{n})\\\
0&\text{otherwise,}\end{cases}$ (24)

where $(\sigma_{1},\cdots,\sigma_{n})\in(N\Sigma)_{n}$.

The fact that Equation (24) gives a well-defined simplicial distribution is
proved in Lemma B.13.

######  Proposition 4.17.

For a bundle scenario $f:\Gamma\to\Sigma$ and $p\in\mathsf{bEmp}(f)$, defining
$\zeta_{f}(p)=Np$ gives a natural isomorphism
$\zeta:\mathsf{bEmp}\to\mathsf{sDist}\circ N$.

###### Proof.

Given $p,q\in\mathsf{bEmp}(f)$ such that $\zeta_{f}(p)=\zeta_{f}(q)$, we have

$p_{\sigma}(\gamma)=(Np)_{1}(\sigma)(\gamma)=(Nq)_{1}(\sigma)(\gamma)=q_{\sigma}(\gamma).$

Therefore $\zeta_{f}$ is injective. To prove surjectivity, let
$\tilde{p}\in\mathsf{sDist}(Nf)$. For every $\sigma\in\Sigma$ we define
$p_{\sigma}=\tilde{p}_{1}(\sigma)$. To see that $p\in\mathsf{bEmp}(f)$, let
$\sigma^{\prime}\subset\sigma$ and $\gamma^{\prime}\in
f^{-1}(\sigma^{\prime})$. Then by Equation (12) and Lemma B.14 we have

$\displaystyle p_{\sigma}|_{\sigma^{\prime}}(\gamma^{\prime})$
$\displaystyle=\sum_{\gamma^{\prime}\subset\gamma\in
f^{-1}(\sigma)}p_{\sigma}(\gamma)$
$\displaystyle=\sum_{\gamma^{\prime}\subset\gamma\in
f^{-1}(\sigma)}\tilde{p}_{1}({\sigma})(\gamma)$ $\displaystyle=\sum_{\tau\in
f^{-1}(\sigma-\sigma^{\prime}):\,\gamma^{\prime}\cup\tau\in\Gamma}\tilde{p}_{1}({\sigma^{\prime}\cup(\sigma-\sigma^{\prime})})(\gamma^{\prime}\cup\tau)$
$\displaystyle=\sum_{\tau\,:\,\,(\gamma^{\prime},\tau)\in(Nf)^{-1}(\sigma^{\prime},\sigma-\sigma^{\prime})}\tilde{p}_{2}(\sigma^{\prime},\sigma-\sigma^{\prime})(\gamma^{\prime},\tau)$
$\displaystyle=D_{R}(d_{2})(\tilde{p}_{2}\left(\sigma^{\prime},\sigma-\sigma^{\prime})\right)(\gamma^{\prime})$
$\displaystyle=\tilde{p_{1}}\left(d_{2}(\sigma^{\prime},\sigma-\sigma^{\prime})\right)(\gamma^{\prime})$
$\displaystyle=\tilde{p_{1}}(\sigma^{\prime})(\gamma^{\prime})=p_{\sigma^{\prime}}(\gamma^{\prime}).$

Again, using Lemma B.14 we obtain that $\zeta_{f}(p)=\tilde{p}$.

Now, we prove the naturality of $\zeta$. Given a morphism $(\pi,\alpha):f\to
f^{\prime}$ of $\mathbf{bScen}$, we prove that the following diagram commutes

${\mathsf{bEmp}(f)}$${\mathsf{bEmp}(f^{\prime})}$${\mathsf{sDist}(Nf)}$${\mathsf{sDist}(Nf^{\prime})}$$\scriptstyle{(\pi{,}\alpha)_{\ast}}$$\scriptstyle{\zeta_{f}}$$\scriptstyle{\zeta_{f^{\prime}}}$$\scriptstyle{(T(\pi){,}N\alpha)_{\ast}}$
(25)

For this let $p\in\mathsf{bEmp}(f)$. By Lemma B.16 and B.15 we have

$\displaystyle\zeta_{f^{\prime}}((\pi,\alpha)_{\ast}p)$
$\displaystyle=N(\alpha_{\ast}(\pi_{\ast}p))=D_{R}(N\alpha)\circ
N(\pi_{\ast}p)$ $\displaystyle=D_{R}(N\alpha)\circ T(\pi)_{\ast}(Np)$
$\displaystyle=(T(\pi),N\alpha)_{\ast}(Np)$
$\displaystyle=(T(\pi),N\alpha)_{\ast}(\zeta_{f}(p)).$

∎

Combining Proposition 4.7 and 4.17 we obtain our main result of Section 4.

######  Theorem 4.18.

Sending a bundle scenario $f:\Gamma\to\Sigma$ to the simplicial scenario
$Nf:N\Gamma\to N\Sigma$ specifies a fully faithful functor
$N:\mathbf{bScen}\to\mathbf{sScen}$. Moreover, there is a natural isomorphism
$\zeta:\mathsf{bEmp}\to\mathsf{sDist}\circ N$.

##  5 Contextuality for bundle scenarios

In Sections 3 and 4 we have introduced distributions on scenarios of bundle
scenarios based on simplicial complexes and simplicial sets, respectively. In
this section we use the theory of convex categories [10] to introduce
contextuality for bundle scenarios in two versions. Again using this theory we
obtain the upgraded Diagram (2) together with the natural isomorphisms
$\tilde{\eta}$ and $\tilde{\zeta}$ relating empirical models and simplicial
distributions. Our main result is Theorem 5.10 relating different versions of
contextuality via these natural isomorphisms.

###  5.1 Convexity

Convexity can be studied in an abstract way for arbitrary semirings using the
notion of an algebra over a monad. The relevant monad in this case is the
distribution functor $D_{R}:\mathbf{Set}\to\mathbf{Set}$. In Section C we
recall basic properties of convexity in this abstract language. We begin by
observing that the empirical model and simplicial distribution functors
defined from the three kinds of categories of scenarios into the category of
sets actually land in the category $\mathbf{Conv}_{R}$ of $R$-convex sets.

The set of simplicial distributions on $f_{(X,Y)}:X\times Y\to X$ can be
identified with the set of simplicial set maps from $X\to D_{R}(Y)$ as
discussed in Remark 4.10. In [10] it is shown that $U=\mathbf{sSet}(X,D_{R}Y)$
is an $R$-convex set with the structure map $\nu=\nu^{U}:D_{R}(U)\to U$
defined as follows:

$\nu(Q)_{n}(x)=\sum_{p\in U}Q(p)\,p_{n}(x).$ (26)

where $Q\in D_{R}(U)$ and $x\in X_{n}$. Next, we generalize this observation.

######  Proposition 5.1.

Given a simplicial set map $f:E\to X$, the set $\mathsf{sDist}(f)$ of
simplicial distributions on $f$ is an $R$-convex set.

###### Proof.

The set $\mathsf{sDist}(f)$ is contained in $U=\mathbf{sSet}(X,D_{R}(E))$. It
suffices to show that for every $Q\in D_{R}(\mathsf{sDist}(f))$, the
distribution $\nu(Q)$ lies in $\mathsf{sDist}(f)$. Given $x\in X_{n}$ and
$e\in E_{n}$ such that $f_{n}(e)\neq x$, using Equation (26) and Proposition
4.9 we have

$\nu(Q)_{n}(x)(e)=\sum_{p\in
U}Q(p)\,p_{n}(x)(e)=\sum_{p\in\mathsf{sDist}(f)}Q(p)\,p_{n}(x)(e)=0.$

Again, by Proposition 4.9 this implies $\nu(Q)\in\mathsf{sDist}(f)$. ∎

Using this result we can also equip the set of empirical models on bundle
scenarios with an $R$-convex set structure using the natural isomorphism
$\zeta$. Furthermore, the set of empirical models on scenarios also inherit an
$R$-convex set structure via the other natural isomorphism $\eta$. We
summarize this observation:

######  Corollary 5.2.

Let $(\Sigma,O)$ be a scenario and $f:\Gamma\to\Sigma$ be a bundle scenario.

  1. 1.

The set $\mathsf{Emp}(\Sigma,O)$ of empirical models on $(\Sigma,O)$ is an
$R$-convex set and the isomorphism

$\eta_{(\Sigma,O)}:\mathsf{Emp}(\Sigma,O)\to\mathsf{bEmp}({f_{(\Sigma,O)}})$

of Proposition 3.17 is an $R$-convex map.

  2. 2.

The set $\mathsf{bEmp}(f)$ of empirical models on $f$ is an $R$-convex set and
the isomorphism

$\zeta_{f}:\mathsf{bEmp}(f)\to\mathsf{sDist}(Nf)$

of Proposition 4.17 is an $R$-convex map.

######  Proposition 5.3.

Given a morphism $(\pi,\alpha):f\to f^{\prime}$ of $\mathbf{sScen}$, the
induced map
$(\pi,\alpha)_{\ast}:\mathsf{sDist}(f)\to\mathsf{sDist}(f^{\prime})$ between
the sets of simplicial distributions is a morphism of $\mathbf{Conv}_{R}$.

###### Proof.

Since $(\pi,\alpha)$ factors as $(\text{id},\alpha)\circ(\pi,\text{id})$ as a
consequence of Equation (17), it suffices to show that
$(\text{id},\alpha)_{\ast}$ and $(\pi,\text{id})_{\ast}$ are morphisms of
$\mathbf{Conv}_{R}$. The map $D_{R}(\alpha):D_{R}(\pi^{\ast}(E))\to
D_{R}(E^{\prime})$ is a morphism of $\mathbf{sConv}_{R}$, thus by Proposition
C.1 the map
$D_{R}(\alpha)_{\ast}:\mathbf{sSet}(X,D_{R}\left(\pi^{\ast}(E)\right))\to\mathbf{sSet}(X,D_{R}(E^{\prime}))$
is a morphism of $\mathbf{Conv}_{R}$. The map $(\text{id},\alpha)_{\ast}$ is
obtained by the restriction of $D_{R}(\alpha)_{\ast}$ to
$\mathsf{sDist}(f^{\pi})$, hence it is a morphism of $\mathbf{Conv}_{R}$. In
order to prove that $(\pi,\text{id})_{\ast}=\pi_{\ast}$ belongs to
$\mathbf{Conv}_{R}$ we show that the following diagram commutes:

${D_{R}(\mathsf{sDist}(f))}$${D_{R}(\mathsf{sDist}(f^{\pi}))}$${\mathsf{sDist}(f)}$${\mathsf{sDist}(f^{\pi})}$$\scriptstyle{D_{R}(\pi_{\ast})}$$\scriptstyle{\nu^{\mathsf{sDist}(f)}}$$\scriptstyle{\nu^{\mathsf{sDist}(f^{\pi})}}$$\scriptstyle{\pi_{\ast}}$

Given $Q\in D_{R}(\mathsf{sDist}(f))$, $x^{\prime}\in X^{\prime}_{n}$ and
$(e,x^{\prime})\in\pi^{\ast}(E)_{n}$, we have

$\displaystyle\nu(D_{R}(\pi_{\ast})(Q))_{n}(x^{\prime})(e,x^{\prime})$
$\displaystyle=\sum_{q\in\mathsf{sDist}(f^{\pi})}D_{R}(\pi_{\ast})(Q)(q)q_{n}(x^{\prime})(e,x^{\prime})$
$\displaystyle=\sum_{q\in\mathsf{sDist}(f^{\pi})}\;\sum_{p:\,\pi_{\ast}p=q}Q(p)(\pi_{\ast}p)_{n}(x^{\prime})(e,x^{\prime})$
$\displaystyle=\sum_{p\in\mathsf{sDist}(f)}Q(p)p_{n}(\pi_{n}(x^{\prime}))(e)$
$\displaystyle=\nu(Q)_{n}(\pi_{n}(x^{\prime}))(e)$
$\displaystyle=\pi_{\ast}(\nu(Q))_{n}(x^{\prime})(e,x^{\prime}).$

∎

This result implies that the functor $\mathsf{sDist}$ lands in the category of
convex sets:

$\mathsf{sDist}:\mathbf{sScen}\to\mathbf{Conv}_{R}.$

Similarly, the functors $\mathsf{Emp}$ and $\mathsf{bEmp}$ land in
$\mathbf{Conv}_{R}$. In more details, we have the following:

  1. 1.

Given a morphism $(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ of
$\mathbf{Scen}$, the induced map
$(\pi,\alpha)_{\ast}:\mathsf{Emp}(\Sigma,O)\to\mathsf{Emp}(\Sigma^{\prime},O^{\prime})$
is a morphism of $\mathbf{Conv}_{R}$.

  2. 2.

Given a morphism $(\pi,\alpha):f\to f^{\prime}$ of $\mathbf{bScen}$, the
induced map $(\pi,\alpha)_{\ast}:\mathsf{bEmp}(f)\to\mathsf{bEmp}(f^{\prime})$
is a morphism of $\mathbf{Conv}_{R}$.

The first statement can be proved as follows: By Diagram (15) we have
$(\pi,\alpha)_{\ast}=\eta_{(\Sigma^{\prime},O^{\prime})}^{-1}\circ\mathcal{E}(\pi,\alpha)_{\ast}\circ\eta_{(\Sigma,O)}$.
Therefore by Corollary 5.2 and Proposition 5.3 we obtain that
$(\pi,\alpha)_{\ast}$ is an $R$-convex map. The second part is similar.

In Proposition C.6 we show that $\mathbf{Conv}_{R}$ is an $R$-convex category.
This fact allows us to lift the empirical model and simplicial distribution
functors to the free $R$-convex categories (Definition C.3) on the associated
category of scenarios using the transposes of the functors $\mathsf{Emp}$,
$\mathsf{bEmp}$, and $\mathsf{sDist}$ with respect to the adjunction
$\mathbf{Cat}\dashv\mathbf{ConvCat}_{R}$ to obtain the following functors:

$\displaystyle\widetilde{\mathsf{Emp}}:D_{R}(\mathbf{Scen})$
$\displaystyle\to\mathbf{Conv}_{R}$
$\displaystyle\widetilde{\mathsf{bEmp}}:D_{R}(\mathbf{bScen})$
$\displaystyle\to\mathbf{Conv}_{R}$
$\displaystyle\widetilde{\mathsf{sDist}}:D_{R}(\mathbf{sScen})$
$\displaystyle\to\mathbf{Conv}_{R}.$

######  Corollary 5.4.

The free convex categories of scenarios are related by the fully faithful
functors

$\displaystyle D_{R}(\mathcal{E}):D_{R}(\mathbf{Scen})$ $\displaystyle\to
D_{R}(\mathbf{bScen})$ $\displaystyle D_{R}(N):D_{R}(\mathbf{bScen})$
$\displaystyle\to D_{R}(\mathbf{sScen}).$

Moreover, the natural isomorphisms $\eta$ and $\zeta$ lift to the natural
isomorphisms

$\displaystyle\tilde{\eta}:\widetilde{\mathsf{Emp}}$
$\displaystyle\to\widetilde{\mathsf{bEmp}}\circ D_{R}(\mathcal{E})$
$\displaystyle\tilde{\zeta}:\widetilde{\mathsf{bEmp}}$
$\displaystyle\to\widetilde{\mathsf{sDist}}\circ D_{R}(N).$

###### Proof.

The first one follows from the fact that $\mathcal{E}$ and $N$ are fully
faithful (Theorem 3.18 and Theorem 4.18), and $D_{R}$ sends an isomorphism of
sets to an isomorphism. Note that transposes for the functors
$\mathsf{bEmp}\circ\mathcal{E}$ and $\mathsf{sDist}\circ N$ with respect to
the adjunction $\mathbf{Cat}\dashv\mathbf{ConvCat}_{R}$ are
$\widetilde{\mathsf{bEmp}}\circ D_{R}(\mathcal{E})$ and
$\widetilde{\mathsf{sDist}}\circ D_{R}(N)$; respectively. Therefore, we obtain
the second part directly from Theorem 3.18, Theorem 4.18, and Proposition C.7.
∎

###  5.2 Contextuality

In [2] the notion of contextuality is defined for simplicial distributions on
pairs $(X,Y)$ of simplicial sets. In this section we extend this definition to
simplicial distributions on simplicial bundle scenarios. First we introduce
deterministic distributions in the bundle picture and then define a comparison
map that sends a probabilistic mixture of deterministic distributions to a
simplicial distribution. Analogously we define contextuality for empirical
models on scenarios $(\Sigma,O)$ and bundles scenarios $f:\Gamma\to\Sigma$ of
simplicial complexes. The natural isomorphisms $\eta$ and $\zeta$ relating
empirical models and simplicial distributions on there categories of scenarios
can be used to compare these different notions of contextuality.

######  Definition 5.5.

Let $f:E\to X$ be a simplicial set map. The set of sections of $f$, i.e.,
simplicial set maps $s:X\to E$ such that $f\circ s=\text{id}_{X}$, will be
denoted by $\mathsf{sSect}(f)$. A _deterministic distribution_ is a simplicial
distribution of the form

$\delta^{s}:X\xrightarrow{s}E\xrightarrow{\delta_{E}}D_{R}(E)$

where $s$ is a section of $f$.

Sending a section $s$ to the associated deterministic distribution
$\delta^{s}$ specifies an injective map

$\delta_{f}:\mathsf{sSect}(f)\to\mathsf{sDist}(f)$ (27)

######  Lemma 5.6.

Given a morphism $(\pi,\alpha):f\to f^{\prime}$ of $\mathbf{sScen}$, the
induced map $(\pi,\alpha)_{\ast}$ sends a deterministic distribution in
$\mathsf{sDist}(f)$ to a deterministic distribution in
$\mathsf{sDist}(f^{\prime})$.

###### Proof.

Given $s\in\mathsf{sSect}(f)$, the following diagram commutes:

${E}$${X}$${X^{\prime}}$${X}$${X^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\text{id}_{X}}$$\scriptstyle{s}$$\scriptstyle{\pi}$$\scriptstyle{\text{id}_{X^{\prime}}}$$\scriptstyle{\pi}$

and induces a map
$(s\circ\pi)\times\text{id}_{X^{\prime}}:X^{\prime}\to\pi^{\ast}(E)$, which
gives a section of $f^{\pi}$. In addition, for $x^{\prime}\in X^{\prime}_{n}$
and $(e,x^{\prime})\in\pi^{\ast}(E)_{n}$, we have

$\displaystyle(\delta^{(s\circ\pi)\times\text{id}_{X^{\prime}}})_{n}(x^{\prime})(e,x^{\prime})$
$\displaystyle=\delta^{(s_{n}\circ\pi_{n}(x^{\prime}),x^{\prime})}(e,x^{\prime})$
$\displaystyle=\delta^{(s_{n}(\pi_{n}(x^{\prime}))}(e)$
$\displaystyle=(\delta^{s})_{n}(\pi_{n}(x^{\prime}))(e)$
$\displaystyle=\pi_{\ast}(\delta^{s})_{n}(x^{\prime})(e,x^{\prime}).$

Therefore
$\pi_{\ast}(\delta^{s})=\delta^{(s\circ\pi)\times\text{id}_{X^{\prime}}}$.
Now, for a section $\tilde{s}$ of $f^{\pi}$, we have

$D_{R}(\alpha)\circ\delta^{\tilde{s}}=D_{R}(\alpha)\circ\delta_{\pi^{\ast}(E)}\circ\tilde{s}=\delta_{E^{\prime}}\circ\alpha\circ\tilde{s}=\delta^{\alpha\circ\tilde{s}}.$

This implies that

$(\pi,\alpha)_{\ast}\delta^{s}=\delta^{\alpha\circ\left((s\circ\pi)\times\text{id}_{X^{\prime}}\right)}.$
(28)

∎

######  Proposition 5.7.

Sending a scenario to the set of sections of that scenario gives a functor

$\mathsf{sSect}:\mathbf{sScen}\to\mathbf{Set}$

and $\delta:\mathsf{sSect}\to\mathsf{sDist}$ defined by $\delta_{f}$ in (27)
at each simplicial scenario $f$, is a natural transformation.

###### Proof.

Given a morphism $(\pi,\alpha):f\to f^{\prime}$ and a section $s$ of $f$ the
induced section

${(\pi,\alpha)_{*}(s):X^{\prime}\xrightarrow{(s\circ\pi)\times{\text{id}_{X^{\prime}}}}\pi^{*}(E)\xrightarrow{\alpha}E^{\prime}}$

By Equation (28) and since $\mathsf{sDist}$ is a functor this assignment gives
a functor. Moreover, again the same equation implies that
$\delta:\mathsf{sSect}\to\mathsf{sDist}$ is a natural transformation. ∎

Let $f:E\to X$ be a simplicial set map. We define

$\Theta_{f}:D_{R}(\mathsf{sSect}(f))\to\mathsf{sDist}(f)$ (29)

to be the transpose of the morphism
$\delta_{f}:\mathsf{sSect}(f)\to\mathsf{sDist}(f)$ of $\mathbf{Conv}_{R}$ with
respect to the adjunction $\mathbf{Set}\dashv\mathbf{Conv}_{R}$. More
explicitly, for $d=\sum_{s}{d(s)}s\in D_{R}(\mathsf{sSect}(f))$, we have

$\Theta_{f}(d)_{n}(x)(e)=\sum_{s}{d(s)}\delta^{s_{n}(x)}(e),$

where $x\in X_{n}$ and $e\in E_{n}$. The naturality of $\delta$ implies that

$\Theta:D_{R}\circ\mathsf{sSect}\to\mathsf{sDist}$ (30)

defined by $\Theta_{f}$ in (29) at each object of $\mathbf{sScen}$ is a
natural transformation.

######  Definition 5.8.

A simplicial distribution $p\in\mathsf{sDist}(f)$ is called non-contextual if
it lies in the image of $\Theta_{f}$. Otherwise, it is called contextual.

Note that by Remark 4.10 this definition specializes to Definition 3.10 in [2]
when the scenario is of the form $f_{(X,Y)}:X\times Y\to X$.

As in the simplicial case, we define the functors $\mathsf{Sect}$ and
$\mathsf{bSect}$ on $\mathbf{Scen}$ and $\mathbf{bScen}$, respectively:

  1. 1.

We define

$\mathsf{Sect}:\mathbf{Scen}\to\mathbf{Set}$

by sending a scenario $(\Sigma,O)$ to the set $\mathcal{E}_{O}(\Sigma_{0})$.
There is a natural transformation

$\delta:\mathsf{Sect}\to\mathsf{Emp}$

whose component at $(\Sigma,O)$ is given by the map
$\delta_{(\Sigma,O)}:\mathsf{Sect}(\Sigma,O)\to\mathsf{Emp}(\Sigma,O)$ which
sends a section $s\in\mathcal{E}_{O}(\Sigma_{0})$ to the empirical model
$\\{\delta^{s|_{\sigma}}\\}_{\sigma\in\Sigma}$.

  2. 2.

We define

$\mathsf{bSect}:\mathbf{Scen}\to\mathbf{Set}$

by sending a bundle scenario $f:\Gamma\to\Sigma$ to the set of sections of
$f$, i.e., simplicial complex maps $s:\Sigma\to\Gamma$ such that $f\circ
s=\text{id}_{\Sigma}$. There is a natural transformation

$\delta:\mathsf{bSect}\to\mathsf{bEmp}$

whose component at $f$ is given by the map
$\delta_{f}:\mathsf{bSect}(f)\to\mathsf{bEmp}(f)$ which sends a section
$s\in\mathsf{bSect}(f)$ to the empirical model
$\\{\delta^{s(\sigma)}\\}_{\sigma\in\Sigma}$.

Moreover, we can define natural transformations

$\displaystyle\Theta:D_{R}\circ\mathsf{Sect}$ $\displaystyle\to\mathsf{Emp}$
$\displaystyle\Theta:D_{R}\circ\mathsf{bSect}$ $\displaystyle\to\mathsf{bEmp}$

whose components $\Theta_{(\Sigma,O)}$ and $\Theta_{f}$ are given by the
transpose of $\delta_{(\Sigma,O)}$ and $\delta_{f}$ with respect to the
adjunction $\mathbf{Set}\dashv\mathbf{Conv}_{R}$, respectively.

######  Definition 5.9.

Let $(\Sigma,O)$ be a scenario and $f:\Gamma\to\Sigma$ be a bundle scenario.

  1. 1.

An empirical model $e\in\mathsf{Emp}(\Sigma,O)$ on the scenario $(\Sigma,O)$
is called non-contextual if it lies in the image of $\Theta_{(\Sigma,O)}$.

  2. 2.

An empirical model $p\in\mathsf{bEmp}(f)$ on the bundle scenario $f$ is called
non-contextual if it lies in the image of $\Theta_{f}$.

Otherwise, the empirical model is called contextual in both cases.

Note that part (1) of Definition 5.9 for contextuality agrees with the
original definition given in [1]. Our main result of Section 5 is the
compatibility of the notion of contextuality for empirical models and
distributions on the three kinds of scenarios.

######  Theorem 5.10.

Let $(\Sigma,O)$ be a scenario and $f:\Gamma\to\Sigma$ be a bundle scenario.

  1. 1.

An empirical model $e\in\mathsf{Emp}(\Sigma,O)$ is contextual if and only if
$\eta_{(\Sigma,O)}(e)\in\mathsf{bEmp}(f_{(\Sigma,O)})$ is contextual.

  2. 2.

An empirical model $p\in\mathsf{bEmp}(f)$ is contextual if and only if
$\zeta_{f}(p)\in\mathsf{sDist}(Nf)$ is contextual.

###### Proof.

We will prove part (2). The proof of part (1) is similar. Sending
$s\in\mathsf{bSect}(f)$ to $Ns\in\mathsf{sSect}(Nf)$ gives a function
$\mathsf{bSect}(f)\to\mathsf{sSect}(Nf)$ that makes the following diagram
commute:

${\mathsf{bSect}(f)}$${\mathsf{bEmp}(f)}$${\mathsf{sSect}(Nf)}$${\mathsf{sDist}(Nf)}$$\scriptstyle{\delta_{f}}$$\scriptstyle{\cong}$$\scriptstyle{\zeta_{f}}$$\scriptstyle{\delta_{Nf}}$
(31)

We know that $\delta_{Ng}$ is injective for any morphism $g$ of
$\mathbf{bScen}$. Using this fact, the naturality of $\zeta$ and both of
horizontal $\delta$ maps, we see that the left vertical map is natural. In
addition, it’s clear that this map is injective. For a section
$\tilde{s}\in\mathsf{sSect}(Nf)$ and $\sigma\in\Sigma$ we have
$f(\tilde{s}_{1}(\sigma))=(Nf)_{1}(\tilde{s}_{1}(\sigma))=\sigma$. Since $f$
is a discrete over vertices, we obtain that $\tilde{s}_{1}(\sigma)$ is a
vertex in $\Gamma$. Thus $\tilde{s}_{1}$ is a simplicial complex map. More
precisely, it belongs to $\mathsf{bSect}(f)$, and by part (1) of Proposition
B.4 we obtain that $N\tilde{s}_{1}=\tilde{s}$. Therefore Diagram (31) induces
the following commutative diagram:

${D_{R}(\mathsf{bSect}(f))}$${\mathsf{bEmp}(f)}$${D_{R}(\mathsf{sSect}(Nf))}$${\mathsf{sDist}(Nf)}$$\scriptstyle{\Theta_{f}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\zeta_{f}}$$\scriptstyle{\Theta_{Nf}}$
(32)

∎

## References

  * [1] S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New Journal of Physics, vol. 13, no. 11, p. 113036, 2011.
  * [2] C. Okay, A. Kharoof, and S. Ipek, “Simplicial quantum contextuality,” arXiv preprint arXiv:2204.06648, 2022.
  * [3] M. Karvonen, “Categories of empirical models,” in 15th International Conference on Quantum Physics and Logic (QPL 2018) (P. Selinger and G. Chiribella, eds.), vol. 287 of Electronic Proceedings in Theoretical Computer Science, pp. 239–252, 2019.
  * [4] R. S. Barbosa, M. Karvonen, and S. Mansfield, “Closing Bell: Boxing black box simulations in the resource theory of contextuality,” in Samson Abramsky on Logic and Structure in Computer Science and Beyond (A. Palmigiano and M. Sadrzadeh, eds.), vol. 25 of Outstanding Contributions to Logic, Springer, 2023.
  * [5] S. Abramsky, R. S. Barbosa, K. Kishida, R. Lal, and S. Mansfield, “Contextuality, cohomology and paradox,” in 24th EACSL Annual Conference on Computer Science Logic (CSL 2015) (S. Kreutzer, ed.), vol. 41 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 211–228, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2015.
  * [6] K. Beer and T. J. Osborne, “Contextuality and bundle diagrams,” Physical Review A, vol. 98, no. 5, p. 052124, 2018.
  * [7] M. Terra Cunha, “On measures and measurements: a fibre bundle approach to contextuality,” Philosophical Transactions of the Royal Society A, vol. 377, no. 2157, p. 20190146, 2019.
  * [8] S. Abramsky, R. S. Barbosa, M. Karvonen, and S. Mansfield, “A comonadic view of simulation and quantum resources,” in 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LiCS 2019), pp. 1–12, IEEE, 2019.
  * [9] M. Karvonen, “Neither contextuality nor nonlocality admits catalysts,” Physical Review Letters, vol. 127, p. 160402, 2021.
  * [10] A. Kharoof and C. Okay, “Simplicial distributions, convex categories and contextuality,” arXiv preprint arXiv:2211.00571, 2022.
  * [11] C. Okay, H. Y. Chung, and S. Ipek, “Mermin polytopes in quantum computation and foundations,” arXiv preprint arXiv:2210.10186, 2022.
  * [12] A. Kharoof, S. Ipek, and C. Okay, “Topological methods for studying contextuality: $N$-cycle scenarios and beyond,” arXiv preprint arXiv:2306.01459, 2023.
  * [13] R. Raussendorf, “Cohomological framework for contextual quantum computations,” Quantum Information and Computation, vol. 19, no. 13&14, pp. 1141–1170, 2019.
  * [14] R. S. Barbosa, M. Karvonen, and S. Mansfield, “Putting paradoxes to work: contextuality in measurement-based quantum computation,” in Samson Abramsky on Logic and Structure in Computer Science and Beyond (A. Palmigiano and M. Sadrzadeh, eds.), vol. 25 of Outstanding Contributions to Logic, Springer, 2023.
  * [15] E. Riehl, Category theory in context.  Courier Dover Publications, 2017.
  * [16] B. Jacobs, “Convexity, duality and effects,” in IFIP International Conference on Theoretical Computer Science, pp. 1–19, Springer, 2010.
  * [17] S. Mac Lane, Categories for the working mathematician, vol. 5.  Springer Science & Business Media, 2013.
  * [18] P. G. Goerss and J. F. Jardine, Simplicial homotopy theory.  Springer Science & Business Media, 2009.

##  Appendix A Simplicial complexes

Simplicial complexes are one type of combinatorial model we use to represent
spaces in this paper. This section introduces the simplicial complex that
represents the standard topological $n$-simplex. These simplices can be
related by maps of simplicial complexes that come from a certain category,
called the simplex category. Using these morphisms, we provide alternative
characterizations of local surjectivity and being discrete over vertices.

Let $\Delta:\mathbf{Set}\to\mathbf{sComp}$ denote the functor that sends a set
$U$ to the simplicial complex $\Delta^{U}$ given by the power set of $U$,
i.e., $(\Delta^{U})_{0}=U$ and simplicial are all nonempty subsets of $U$. A
set map $g:U^{\prime}\to U$ induces a map of simplicial complexes, which will
be denoted by $g:\Delta^{U^{\prime}}\to\Delta^{U}$, defined by
$g(\sigma^{\prime})=\\{g(x^{\prime}):\,x^{\prime}\in\sigma^{\prime}\\}$ for
$\sigma^{\prime}\subset U^{\prime}$. Next we introduce an important
subcategory known as the simplex category and consider the restriction of the
functor on this category. The simplex category $\mathbf{\Delta}$ is defined as
follows [18]:

  * •

Objects are given by $[n]=\\{0,1,\cdots,n\\}$ where $n\geq 0$.

  * •

A morphism $\theta:[m]\to[n]$ is an order preserving function, that is,

$\theta(i)\leq\theta(j),\;\;\forall i\leq j\in[m].$

There are two kinds of distinguished maps:

$d^{i}:[n-1]\to[n]\;\;\;\text{ and }\;\;\;s^{j}:[n+1]\to[n],$

where the face maps $d^{i}$ skips $i$ in the image, and the degeneracy maps
$s^{j}$ has a double preimage at $j$. An arbitrary morphism $\theta$ can be
written as a composite of face and degeneracy maps.

We can extend this category to the augmented simplex category
$\mathbf{\Delta}_{+}$ by adding the initial object $[-1]=\emptyset$. We will
write $\Delta^{n}$ for the simplicial complex $\Delta^{[n]}$. The assignment
$[n]\mapsto\Delta^{n}$ gives a functor

$\Delta:\mathbf{\Delta}_{+}\to s\mathbf{Comp}.$

######  Definition A.1.

Let $\mathbf{C}$ be a category. We say that $f$ has the right lifting property
with respect to $g$ if for every commutative diagram in $\mathbf{C}$
consisting of the solid arrows

${A}$${X}$${B}$${Y}$$\scriptstyle{g}$$\scriptstyle{f}$

the dashed arrow exists making both triangles commute.

######  Proposition A.2.

For a map $f:\Gamma\to\Sigma$ in $s\mathbf{Comp}$ of simplicial complexes the
following conditions are equivalent:

  1. 1.

$f$ is locally surjective (Definition 3.1).

  2. 2.

$f$ has the right lifting property with respect to every injective map
$\Delta^{m}\hookrightarrow\Delta^{n}$ where $m\geq 0$.

  3. 3.

$f$ has the right lifting property with respect to every
$d^{n}:\Delta^{n-1}\hookrightarrow\Delta^{n}$ where $n\geq 1$.

###### Proof.

It is obvious that (2) implies (3). To prove that (3) implies $(1)$, for
$\gamma=\\{v_{0},v_{1},\cdots,v_{m}\\}\in\Gamma$ and
$\tau=\\{f(v_{0}),f(v_{1}),\cdots,f(v_{m}),u_{m+1},\cdots,u_{n}\\}\in\mathsf{St}(f(\gamma))$,
we define $g:\Delta^{m}\to\Gamma$ by $g(i)=v_{i}$ for every $0\leq i\leq m$,
and $k:\Delta^{n}\to\Sigma$ by

$k(i)=\begin{cases}f(v_{i})&\text{if}~{}0\leq i\leq m\\\
u_{i}&\text{if}~{}i>m.\end{cases}$

Let $d:\Delta^{m}\to\Delta^{n}$ be the composition $d^{n}\circ\cdots\circ
d^{m+2}\circ d^{m+1}$ (which sends $i\in{(\Delta^{m})_{0}}$ to
$i\in{(\Delta^{n})_{0}}$). Then the following diagram commutes:

${\Delta^{m}}$${\Gamma}$${\Delta^{n}}$${\Sigma}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{f}$$\scriptstyle{k}$
(33)

By applying part (3) $n-m$ times we obtain a map $h:\Delta^{n}\to\Gamma$ that
satisfies $h\circ d=g$ and $f\circ h=k$. Thus
$\gamma=\text{Im}(g)\subset\text{Im}(h)$, which means that
$\text{Im}(h)\in\mathsf{St}(\gamma)$. Also we obtain that
$f(\text{Im}(h))=\text{Im}(k)=\tau$.

Finally, we prove that (1) implies (2). Consider the following diagram

${\Delta^{m}}$${\Gamma}$${\Delta^{n}}$${\Sigma}$$\scriptstyle{g}$$\scriptstyle{l}$$\scriptstyle{f}$$\scriptstyle{k}$
(34)

Let $\gamma=\text{Im}(g)$ and $\tau=\text{Im}(k)$. By the commutativity of
Diagram (34) we have that $f(\gamma)\subseteq\tau$, hence
$\tau\in\mathsf{St}(f(\gamma))$. By condition (1) there exists
$\sigma\in\mathsf{St}(\gamma)$ such that $f(\sigma)=\tau$. We will denote by
$i_{m+1},\cdots,i_{n}$ the vertices in $\Delta^{n}$ that do not belong to
$\text{Im}(l)$. Since $f(\sigma)=\tau=\text{Im}(k)$, there exists
$z_{m+1},\cdots,z_{n}\in\sigma$ such that $f(z_{j})=k(i_{j})$ for every
$m+1\leq j\leq n$. We define $h:\Delta^{n}\to\Gamma$ by

$h(r)=\begin{cases}g(i)&r=l(i)\\\ z_{j}&r=i_{j}.\end{cases}$

Then $h$ is well-defined since
$\operatorname{Im}(h)=\\{g(0),\cdots,g(m),z_{m+1},\cdots,z_{n}\\}=\gamma\cup\\{z_{m+1},\cdots,z_{n}\\}\subseteq\sigma$,
thus $\operatorname{Im}(h)\in\Gamma$. In addition, by the definition of $h$,
we have $h(l(i))=g(i)$ for $0\leq i\leq m$, and

$f(h(r))=\begin{cases}f(g(i))=k(l(i))=k(r)&r=l(i)\\\
f(z_{j})=k(i_{j})=k(r)&r=i_{j}.\end{cases}$

Therefore $h$ is a lifting for Diagram (34). ∎

######  Proposition A.3.

For a map $f:\Gamma\to\Sigma$ in $s\mathbf{Comp}$ of simplicial complexes the
following conditions are equivalent:

  1. 1.

$f$ is discrete over vertices (Definition 3.1).

  2. 2.

$f$ has the right lifting property with respect to every surjective map
$\Delta^{n}\twoheadrightarrow\Delta^{m}$ where $m\geq 0$.

  3. 3.

$f$ has the right lifting property with respect to
$s^{0}:\Delta^{1}\twoheadrightarrow\Delta^{0}$.

###### Proof.

It is obvious that (2) implies (3). To prove that (3) implies (1), for
$x,y\in\Gamma_{0}$ such that $f(x)=f(y)$ suppose that $\\{x,y\\}\in\Gamma$. We
define $g:\Delta^{1}\to\Gamma$ by $g(0)=x$, $g(1)=y$, and
$k:\Delta^{0}\to\Sigma$ by $k(0)=f(x)$. We obtain the following commutative
diagram

${\Delta^{1}}$${\Gamma}$${\Delta^{0}}$${\Sigma}$$\scriptstyle{g}$$\scriptstyle{s^{0}}$$\scriptstyle{f}$$\scriptstyle{k}$
(35)

By condition (3) there exists $h:\Delta^{0}\to\Gamma$ that gives a lifting for
Diagram (35). In particular,

$x=g(0)=h(s^{0}(0))=h(s^{0}(1))=g(1)=y.$

Finally, we prove that $(1)$ implies $(2)$. We have the following commutative
diagram

${\Delta^{n}}$${\Gamma}$${\Delta^{m}}$${\Sigma}$$\scriptstyle{g}$$\scriptstyle{s}$$\scriptstyle{f}$$\scriptstyle{k}$
(36)

If $s(i_{1})=s(i_{2})$ then by Diagram (36) we have

$f(g(i_{1}))=k(s(i_{1}))=k(s(i_{2}))=f(g(i_{2})).$

On the other hand, $\\{g(i_{1}),g(i_{2})\\}\subset\text{Im}(g)$ and thus
$\\{g(i_{1}),g(i_{2})\\}\in\Gamma$. By condition (1), we conclude that
$g(i_{1})=g(i_{2})$. We define $h:\Delta^{m}\to\Gamma$ by $h(i)=g(s^{-1}(i))$.
As a map between the vertices, $h$ is well-defined since $s$ is surjective and
we proved that the preimage of $i$ under $s$ maps to a single point under $g$.
In addition, the map $h$ is a morphism of $\mathbf{sComp}$ since
$\text{Im}(h)=\text{Im}(g)\in\Gamma$. It remains to prove that $h$ is a
lifting for the Diagram (36). Given $i\in(\Delta^{n})_{0}$, by the definition
of $h$, we have $h(s(i))=g(i)$. Now given $i\in(\Delta^{m})_{0}$, we have

$f(h(i))=f(g(s^{-1}(i)))=k(s(s^{-1}(i)))=k(i).$

∎

######  Corollary A.4.

A simplicial complex map $f:\Gamma\to\Sigma$ is a bundle scenario (Definition
3.1) if and only if $f$ has the right lifting property with respect to
$\theta:\Delta^{m}\to\Delta^{n}$ for all morphisms $\theta$ of the augmented
simplex category $\mathbf{\Delta}_{+}$.

###  A.1 Pull-backs of bundle scenarios

The key observation in this section is that pull-backs along bundle scenarios
behave particularly nice and they can be described easily. In addition, we
show that pull-back of a bundle scenario is also a bundle scenario.

Let $f:\Gamma\to\Sigma$ be a bundle scenario, and let
$\pi:\Sigma\to\Sigma^{\prime}$ be a simplicial complex map. The pull-back
diagram associated to $f$ and $\pi$ consists of a commutative diagram of
simplicial complex maps

${\Gamma}$${\pi^{\ast}(\Gamma)}$${\Sigma}$${\Sigma^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$
(37)

where $\pi^{*}(\Gamma)$ is the pull-back: A simplicial complex with vertex set
consisting of pairs $(x,y)\in\Gamma_{0}\times\Sigma^{\prime}_{0}$ such that
$f(x)=\pi(y)$ and simplex set

$\pi^{\ast}\Gamma=\\{\\{(x_{1},y_{1}),\cdots,(x_{k},y_{k})\\}\subset(\pi^{\ast}\Gamma)_{0}:\,\\{x_{1},\cdots,x_{k}\\}\in\Gamma,\;\\{y_{1},\cdots,y_{k}\\}\in\Sigma^{\prime}\\}.$

The maps $\tilde{f}$ and $\tilde{\pi}$ are given by projection.

######  Proposition A.5.

Diagram (37) is a pull-back in the category of simplicial complexes.

###### Proof.

Let $\gamma\in\Gamma$ and $\sigma^{\prime}\in\Sigma^{\prime}$ be such that
$f(\gamma)=\pi(\sigma^{\prime})=\\{z_{1},\cdots,z_{n}\\}$ where $z_{i}\neq
z_{j}$ for $i\neq j$. Since $f$ is discrete over vertices there exists
distinct vertices $x_{1},\cdots,x_{n}$ in $\Gamma$ such that
$\gamma=\\{x_{1},\cdots,x_{n}\\}$ and $f(x_{i})=z_{i}$. On the other hand, for
$1\leq i\leq n$, we write $\\{y_{i1},\cdots,y_{im_{i}}\\}$ for the simplex
$\pi^{-1}(z_{i})\cap\sigma^{\prime}$. Then we have
$\sigma^{\prime}=\cup_{i=1}^{n}\\{y_{i1},\cdots,y_{im_{i}}\\}$. We conclude
that $\tau=\cup_{i=1}^{n}\\{(x_{i},y_{i1}),\cdots,(x_{i},y_{im_{i}})\\}$ is
the unique simplex in $\pi^{\ast}(\Gamma)$ such that
$\tilde{f}(\tau)=\sigma^{\prime}$ and $\tilde{\pi}(\tau)=\gamma$. ∎

######  Proposition A.6.

The map $\tilde{f}$ in Diagram (37) is a bundle scenario.

###### Proof.

Let $\sigma^{\prime}=\\{y_{1},\cdots,y_{n}\\}\in\Sigma^{\prime}$. Since $f$ is
surjective, there exists $\gamma\in\Gamma$ such that
$f(\gamma)=\pi(\sigma^{\prime})$. For $1\leq i\leq n$, we choose
$x_{i}\in\gamma$ such that $f(x_{i})=\pi(y_{i})$. Then
$\\{(x_{1},y_{1}),\cdots,(x_{n},y_{n})\\}\in\pi^{\ast}(\Gamma)$ and
$\tilde{f}(\\{(x_{1},y_{1}),\cdots,(x_{n},y_{n})\\})=\sigma^{\prime}$. This
shows that $\tilde{f}$ is surjective. To prove that $\tilde{f}$ is locally
surjective, let
$\\{(x_{1},y_{1}),\cdots,(x_{k},y_{k})\\}\in\pi^{\ast}(\Gamma)$ and
$\sigma^{\prime}=\\{y_{1},\cdots,y_{k},y_{k+1},\cdots,y_{n}\\}\in\mathsf{St}(\\{y_{1},\cdots,y_{k}\\})$.
Then $\pi(\sigma^{\prime})\in\mathsf{St}(\pi(\\{y_{1},\cdots,y_{k}\\}))$. Note
that $f(\\{x_{1},\dots,x_{k}\\})=\\{y_{1},\dots,y_{k}\\}$. Since $f$ is
locally surjective, there exists
$\gamma\in\mathsf{St}(\\{x_{1},\cdots,x_{k}\\})$ such that
$f(\gamma)=\pi(\sigma^{\prime})$. Thus for every $k+1\leq i\leq n$ there
exists $x_{i}\in\gamma$ such that $f(x_{i})=\pi(y_{i})$. We conclude that
$\\{(x_{1},y_{1}),\cdots,(x_{n},y_{n})\\}\in\pi^{\ast}(\Gamma)$ and
$\tilde{f}(\\{(x_{1},y_{1}),\cdots,(x_{n},y_{n})\\})=\sigma^{\prime}$. To
prove that $\tilde{f}$ is discrete over vertices, consider
$(x_{1},y_{1}),(x_{2},y_{2})\in\pi^{\ast}(\Gamma)_{0}$ such that
$\tilde{f}(x_{1},y_{1})=\tilde{f}(x_{2},y_{2})$, that is $y_{1}=y_{2}$. If
$\\{(x_{1},y_{1}),(x_{2},y_{2})\\}\in\pi^{\ast}(\Gamma)$ then
$\\{x_{1},x_{2}\\}\in\Gamma$. But $f(x_{1})=\pi(y_{1})=\pi(y_{2})=f(x_{2})$.
Thus we conclude that $x_{1}=x_{2}$ since $f$ is discrete over vertices.
Therefore $(x_{1},y_{1})=(x_{2},y_{2})$. ∎

###  A.2 Bundles of nerve complexes

The nerve complex functor $\hat{N}:\mathbf{sComp}\to\mathbf{sComp}$ given in
Definition 3.4 is used in the description of morphisms of $\mathbf{bScen}$. We
show that this functor preserves bundle scenarios.

######  Proposition A.7.

If $f:\Gamma\to\Sigma$ is a bundle scenario then
$\hat{N}f:\hat{N}\Gamma\to\hat{N}\Sigma$ is also a bundle scenario.

###### Proof.

Consider $\\{\sigma_{1},\cdots,\sigma_{n}\\}\in\hat{N}\Sigma$. We have
$\cup_{i=1}^{n}\sigma_{i}\in\Sigma$. Surjectivity of $f$ implies that there
exists $\gamma\in\Gamma$ such that $f(\gamma)=\cup_{i=1}^{n}\sigma_{i}$. Thus
we have $\gamma_{1},\cdots,\gamma_{n}\subseteq\gamma$ such that
$f(\gamma_{i})=\sigma_{i}$. This shows that
$\\{\gamma_{1},\cdots,\gamma_{n}\\}\in\hat{N}\Gamma$ and
$\hat{N}f(\\{\gamma_{1},\cdots,\gamma_{n}\\})=\\{\sigma_{1},\cdots,\sigma_{n}\\}$.
Now, we prove that $\hat{N}f$ is discrete over vertices. Consider distinct
vertices $\gamma_{1},\gamma_{2}$ in $\hat{N}\Gamma$ such that
$\hat{N}f(\gamma_{1})=\hat{N}f(\gamma_{2})$. This means that
$f(\gamma_{1})=f(\gamma_{2})$. Since $f$ is discrete over vertices we get that
$|\gamma_{1}|=|\gamma_{2}|$. On the other hand,
$f(\gamma_{1}\cup\gamma_{2})=f(\gamma_{1})\cup f(\gamma_{2})=f(\gamma_{1})$
and $|\gamma_{1}\cup\gamma_{2}|>|\gamma_{1}|$. We conclude that
$\gamma_{1}\cup\gamma_{2}\notin\Gamma$, hence
$\\{\gamma_{1},\gamma_{2}\\}\notin\hat{N}\Gamma$. Finally, we prove that
$\hat{N}f$ is locally surjective. Given
$\\{\gamma_{1},\cdots,\gamma_{n}\\}\in\hat{N}\Gamma$, we want to prove the
surjectivity of the restriction of $f$ on
$\mathsf{St}(\\{\gamma_{1},\cdots,\gamma_{n}\\})$:

$f|_{\\{\gamma_{1},\cdots,\gamma_{n}\\}}:\mathsf{St}(\\{\gamma_{1},\cdots,\gamma_{n}\\})\to\mathsf{St}(\\{f(\gamma_{1}),\cdots,f(\gamma_{n})\\}).$

For
$\\{f(\gamma_{1}),\cdots,f(\gamma_{n}),\sigma_{1},\cdots,\sigma_{m}\\}\in\mathsf{St}(\\{f(\gamma_{1}),\cdots,f(\gamma_{n})\\})$,
the union

$\cup_{i=1}^{n}f(\gamma_{i})\;\bigcup\;\cup_{i=1}^{m}\sigma_{i}$

belongs to $\Sigma$, which implies that the union

$f(\cup_{i=1}^{n}\gamma_{i})\bigcup\cup_{i=1}^{m}\sigma_{i}$

belongs to $\mathsf{St}(f(\cup_{i=1}^{n}\gamma_{i}))$. Since $f$ is locally
surjective, there exists $\cup_{i=1}^{n}\gamma_{i}\subset\gamma\in\Gamma$ such
that
$f(\gamma)=f(\cup_{i=1}^{n}\gamma_{i})\;\bigcup\;\cup_{i=1}^{m}\sigma_{i}$. In
particular, there exists $\tau_{1},\cdots,\tau_{m}\subset\gamma$ such that
$f(\tau_{i})=\sigma_{i}$. Thus
$\\{\gamma_{1},\cdots,\gamma_{n},\tau_{1},\cdots,\tau_{m}\\}\in\mathsf{St}(\\{\gamma_{1},\cdots,\gamma_{n}\\})$
and

$\hat{N}f(\\{\gamma_{1},\cdots,\gamma_{n},\tau_{1},\cdots,\tau_{m}\\})=\\{f(\gamma_{1}),\cdots,f(\gamma_{n}),\sigma_{1},\cdots,\sigma_{m}\\}.$

∎

###  A.3 Lemmas: Category of bundle scenarios

In $\mathbf{bScen}$ there are two types of morphisms. In this section we prove
two results about type I morphisms. First one of these is used in the
definition of composition of type I morphisms. The second result constitutes
part of the identity morphism of an object in $\mathbf{bScen}$. Finally we
show that the composition rule in this category is associative.

######  Lemma A.8.

For a type I morphism given in Diagram (5), the composition of the following
squares is a pull-back

${\hat{N}\Gamma}$${\hat{N}^{2}\Gamma}$${\hat{N}\pi^{\ast}(\hat{N}\Gamma)}$${\hat{N}\Sigma}$${\hat{N}^{2}\Sigma}$${\hat{N}\Sigma^{\prime}}$$\scriptstyle{\hat{N}f}$$\scriptstyle{\mu_{\Gamma}}$$\scriptstyle{{\hat{N}}^{2}f}$$\scriptstyle{\hat{N}f^{\pi}}$$\scriptstyle{\hat{N}l}$$\scriptstyle{\mu_{\Sigma}}$$\scriptstyle{\hat{N}\pi}$
(38)

###### Proof.

By Proposition A.5 and Proposition A.7
$(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\Gamma)$ is the pull-back of
${\mu}_{\Sigma}\circ\hat{N}\pi$ along $\hat{N}f$. Given a vertex
$(\gamma,\sigma^{\prime})$ in
$(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\Gamma)$, we have
$f(\gamma)=\overline{\pi}(\sigma^{\prime})$. Let
$\sigma^{\prime}=\\{y_{1},\cdots,y_{n}\\}$. Then
$f(\gamma)=\cup_{j=1}^{n}\pi(y_{j})$. By Remark 3.10 there is a unique simplex
$\\{\gamma_{1}\cdots,\gamma_{n}\\}\in{\hat{N}\Gamma}$ such that

$\cup_{j=1}^{n}\gamma_{j}=\gamma\,\,\text{and}\,\,f(\gamma_{j})=\pi(y_{j})\,\,,\,\forall
1\leq j\leq n.$

Let $S(\gamma,\sigma^{\prime})$ denote the vertex
$\\{(\gamma_{1},y_{1}),\cdots,(\gamma_{n},y_{n})\\}$ in
$\hat{N}(\pi^{\ast}(\hat{N}\Gamma))$. Using this we can construct a simplicial
complex map
$S:(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\Gamma)\to\hat{N}(\pi^{\ast}(\hat{N}\Gamma))$.
In addition, we have

$\displaystyle\hat{N}f^{\pi}\left(S(\gamma,\sigma^{\prime})\right)=(\hat{N}f^{\pi})(\\{(\gamma_{1},y_{1}),\cdots,(\gamma_{n},y_{n})\\})=\sigma^{\prime}$

and

$\displaystyle{\mu}_{\Gamma}$ $\displaystyle\circ(\hat{N}l)\circ
S\left(\gamma,\sigma^{\prime}\right)$
$\displaystyle={\mu}_{\Gamma}\circ(\hat{N}l)(\\{(\gamma_{1},y_{1}),\cdots,(\gamma_{n},y_{n})\\})$
$\displaystyle={\mu}_{\Gamma}(\\{\gamma_{1},\cdots,\gamma_{n}\\})$
$\displaystyle=\cup_{j=1}^{n}\gamma_{j}=\gamma.$

Finally, one can check that the induced map from
$\hat{N}(\pi^{\ast}(\hat{N}\Gamma))$ to
$(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\Gamma)$ is the inverse of $S$.
Therefore the composition of the squares in Diagram (38) is a pull-back. ∎

######  Lemma A.9.

Let $f:\Gamma\to\Sigma$ be a simplicial complex map that is discrete over
vertices. Then the following diagram

${\hat{N}\Gamma}$${\Gamma}$${\hat{N}\Sigma}$${\Sigma}$$\scriptstyle{\hat{N}f}$$\scriptstyle{f}$$\scriptstyle{\delta_{\Gamma}}$$\scriptstyle{\delta_{\Sigma}}$
(39)

is a pullback square.

###### Proof.

Given simplices $\\{\gamma_{1},\cdots,\gamma_{n}\\}\in\hat{N}\Gamma$ and
$\sigma\in\Sigma$ such that
$\\{f(\gamma_{1}),\cdots,f(\gamma_{n})\\}=\delta_{\Sigma}(\sigma)$, the images
$f(\gamma_{1}),\cdots,f(\gamma_{n})$ are the vertices of $\sigma$. The map $f$
is discrete over vertices, thus every $\gamma_{i}$ is a vertex in $\Gamma$. We
conclude that $\gamma=\cup_{i=1}^{n}\gamma_{i}$ is the unique simplex in
$\Gamma$ with $\delta_{\Sigma}(\gamma)=\\{\gamma_{1},\cdots,\gamma_{n}\\}$ and
$f(\gamma)=\sigma$. ∎

######  Lemma A.10.

The composition given by Equation (9) is associative.

###### Proof.

Given a type II morphism $\alpha:f\to f^{\prime}$ and relations
$\pi:\Sigma^{\prime}\to\hat{N}\Sigma$,
$\pi^{\prime}:\Sigma^{\prime\prime}\to\hat{N}\Sigma^{\prime}$, we have

$(\pi\diamond\pi^{\prime})^{\ast}(\hat{N}\alpha)=(\pi^{\prime})^{\ast}(\hat{N}\pi^{\ast}(\hat{N}\alpha)).$
(40)

In Lemma A.8 we proved that
$(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\Gamma)=\hat{N}\pi^{\ast}(\hat{N}\Gamma)$.
Using this we conclude that
$(\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\alpha)=\hat{N}\pi^{\ast}(\hat{N}\alpha)$.
Then we have
$(\pi\diamond\pi^{\prime})^{\ast}(\hat{N}\alpha)=(\mu_{\Sigma}\circ\hat{N}\pi\circ\pi^{\prime})^{\ast}(\hat{N}\alpha)=(\pi^{\prime})^{\ast}((\mu_{\Sigma}\circ\hat{N}\pi)^{\ast}(\hat{N}\alpha))=(\pi^{\prime})^{\ast}(\hat{N}\pi^{\ast}(\hat{N}\alpha))$.
Given $(\pi_{1},\alpha_{1}):f\to f^{\prime}$,
$(\pi_{2},\alpha_{2}):f^{\prime}\to f^{\prime\prime}$, and
$(\pi_{3},\alpha_{3}):f^{\prime\prime}\to f^{\prime\prime\prime}$, we have

$\displaystyle(\pi_{3},\alpha_{3})\circ\left((\pi_{2},\alpha_{2})\circ(\pi_{1},\alpha_{1})\right)$
$\displaystyle=(\pi_{3},\alpha_{3})\circ(\pi_{1}\diamond\pi_{2},\alpha_{2}\circ\pi_{2}^{\ast}(\hat{N}\alpha_{1}))$
$\displaystyle=\left((\pi_{1}\diamond\pi_{2})\diamond\pi_{3},\alpha_{3}\circ{\pi_{3}}^{\ast}(\hat{N}\left(\alpha_{2}\circ\pi_{2}^{\ast}(\hat{N}\alpha_{1})\right))\right)$
$\displaystyle=\left((\pi_{1}\diamond\pi_{2})\diamond\pi_{3},\alpha_{3}\circ{\pi_{3}}^{\ast}(\hat{N}\alpha_{2}\circ\hat{N}\pi_{2}^{\ast}(\hat{N}\alpha_{1}))\right)$
$\displaystyle=\left((\pi_{1}\diamond\pi_{2})\diamond\pi_{3},\alpha_{3}\circ{\pi_{3}}^{\ast}(\hat{N}\alpha_{2})\circ{\pi_{3}}^{\ast}(\hat{N}\pi_{2}^{\ast}(\hat{N}\alpha_{1}))\right).$

On the other hand,

$\displaystyle\left((\pi_{3},\alpha_{3})\circ(\pi_{2},\alpha_{2})\right)\circ(\pi_{1},\alpha_{1})$
$\displaystyle=(\pi_{2}\diamond\pi_{3},\alpha_{3}\circ\pi_{3}^{\ast}(\hat{N}\alpha_{2}))\circ(\pi_{1},\alpha_{1})$
$\displaystyle=\left(\pi_{1}\diamond(\pi_{2}\diamond\pi_{3}),\alpha_{3}\circ{\pi_{3}}^{\ast}(\hat{N}\alpha_{2})\circ(\pi_{2}\diamond\pi_{3})^{\ast}(\hat{N}\alpha_{1})\right).$

By Equation (40) and the associativity of the Kleisli composition we obtain
the equality. ∎

###  A.4 Lemmas: Embedding scenarios into bundle scenarios

In this section we prove key lemmas that are used in the construction of the
functor $\mathcal{E}:\mathbf{Scen}\to\mathbf{bScen}$ and showing that it is a
fully-faithful embedding.

######  Lemma A.11.

The following diagram is a pull-back square

${\hat{N}(\mathcal{E}_{O}\Sigma)}$${(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}}$${\hat{N}\Sigma}$${\Sigma^{\prime}}$$\scriptstyle{\hat{N}f_{(\Sigma,O)}}$$\scriptstyle{f_{{\mathcal{E}}_{O}\circ\pi}}$$\scriptstyle{l}$$\scriptstyle{\pi}$
(41)

where $l(x^{\prime},s)=(\pi(x^{\prime}),s)$.

###### Proof.

First, we prove that the map $l$ is a valid morphism in $\mathbf{sComp}$. Note
that for $s\in\mathcal{E}_{O}(\pi(x^{\prime}))$, the pair
$(\pi(x^{\prime}),s)$ is a simplex in $\mathcal{E}_{O}\Sigma$, that is, a
vertex in $\hat{N}(\mathcal{E}_{O}\Sigma)$. For a simplex
$(\sigma^{\prime},s)\in(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}$, we have
$l(\sigma^{\prime},s)=\\{l(x^{\prime},s|_{\pi(x^{\prime})}):\,x^{\prime}\in\sigma^{\prime}\\}=\\{(\pi(x^{\prime}),s|_{\pi(x^{\prime})}):\,x^{\prime}\in\sigma^{\prime}\\}$.
Then the union

$\cup_{x^{\prime}\in\sigma^{\prime}}(\pi(x^{\prime}),s|_{\pi(x^{\prime})})=(\cup_{x^{\prime}\in\sigma}\pi(x^{\prime}),s)=(\overline{\pi}(\sigma^{\prime}),s)$

is a simplex in $\mathcal{E}_{O}\Sigma$.

Next we prove that Diagram (41) is a pullback square. It is obvious that the
square commutes. Given
$\\{(\sigma_{1},s_{1}),\cdots,(\sigma_{n},s_{n})\\}\in\hat{N}(\mathcal{E}_{O}\Sigma)$
and $\sigma^{\prime}\in\Sigma^{\prime}$ such that
$\hat{N}f_{(\Sigma,O)}((\sigma_{1},s_{1}),\cdots,(\sigma_{n},s_{n}))=\pi(\sigma^{\prime})$.
Since $\cup_{i=1}^{n}(\sigma_{i},s_{i})\in\mathcal{E}_{O}\Sigma$ we have a
unique $s\in\mathcal{E}_{O}(\cup_{i=1}^{n}\sigma_{i})$ such that
$s|_{\sigma_{i}}=s_{i}$. In addition, we have
$\pi(\sigma^{\prime})=\\{\sigma_{1},\cdots,\sigma_{n}\\}$, and thus
$\overline{\pi}(\sigma^{\prime})=\cup_{i=1}^{n}\sigma_{i}$. Therefore
$(\sigma^{\prime},s)$ is the unique element in
$(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}$ with
$f_{\mathcal{E}_{O}\circ\pi}(\sigma^{\prime},s)=\sigma^{\prime}$ and
$l(\sigma^{\prime},s)=\\{(\sigma_{1},s_{1}),\cdots,(\sigma_{n},s_{n})\\}$. ∎

######  Lemma A.12.

For morphisms $(\pi,\alpha):(\Sigma,O)\to(\Sigma^{\prime},O^{\prime})$ and
$(\pi^{\prime},\alpha^{\prime}):(\Sigma^{\prime},O^{\prime})\to(\Sigma^{\prime\prime},O^{\prime\prime})$
of $\mathbf{Scen}$ we have

$r(\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}}))=r(\alpha^{\prime})\circ(\pi^{\prime})^{\ast}(\hat{N}r(\alpha))$

where $r$ is defined in Diagram (11).

###### Proof.

Note that
$\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}})$ is a
natural transformation
$\mathcal{E}_{O}\circ\overline{\pi\diamond\pi^{\prime}}\to\mathcal{E}_{O^{\prime\prime}}$.
Therefore we have a map of simplicial complexes
$r(\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}})):(\mathcal{E}_{O}\circ(\pi\diamond\pi^{\prime}))\Sigma^{\prime\prime}\to\mathcal{E}_{O^{\prime\prime}}\Sigma^{\prime\prime}$.
Given $\sigma^{\prime\prime}\in\Sigma^{\prime\prime}$ and
$s\in\mathcal{E}_{O}(\overline{\pi\diamond\pi^{\prime}}(\sigma^{\prime\prime}))$,
we have

$r(\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}}))(\sigma^{\prime\prime},s)=(\sigma^{\prime\prime},(\alpha^{\prime}\circ(\alpha\ast\text{id}_{\overline{\pi^{\prime}}}))_{\sigma^{\prime\prime}}(s))=(\sigma^{\prime\prime},\alpha^{\prime}_{\sigma^{\prime\prime}}(\alpha_{\overline{\pi^{\prime}}(\sigma^{\prime\prime})}(s))).$

Observe that by Lemma A.11

$(\pi^{\prime})^{\ast}(\hat{N}((\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}))=(\pi^{\prime})^{\ast}(\hat{N}\pi^{\ast}(\hat{N}(\mathcal{E}_{O}\Sigma))).$

Using Lemma A.8 we obtain that
$(\pi^{\prime})^{\ast}(\hat{N}\pi^{\ast}(\hat{N}(\mathcal{E}_{O}\Sigma)))=(\pi\diamond\pi^{\prime})^{\ast}(\hat{N}(\mathcal{E}_{O}\Sigma))$,
which again by Lemma A.11 is equal to
$(\mathcal{E}_{O}\circ(\pi\diamond\pi^{\prime}))\Sigma^{\prime\prime}$. Thus
we can see $(\sigma^{\prime\prime},s)$ as an element of
$(\pi^{\prime})^{\ast}(\hat{N}((\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}))$ and
by applying $(\pi^{\prime})^{\ast}(\hat{N}r(\alpha))$ to this element we
obtain $(\sigma^{\prime\prime},\alpha_{\overline{\pi^{\prime}}}(s))$ in
$(\pi^{\prime})^{\ast}(\hat{N}\mathcal{E}_{O^{\prime}}\Sigma^{\prime})$.
According to Lemma A.11 we can identify this element with
$(\mathcal{E}_{O^{\prime}}\circ\pi^{\prime})\Sigma^{\prime\prime}$. Finally,
by applying $r(\alpha^{\prime})$ on
$(\sigma^{\prime\prime},\alpha_{\overline{\pi^{\prime}}}(s))$ we obtain
$(\sigma^{\prime\prime},\alpha^{\prime}_{\sigma^{\prime\prime}}(\alpha_{\overline{\pi^{\prime}}(\sigma^{\prime\prime})}(s)))$.
∎

######  Lemma A.13.

Let $\pi:\Sigma^{\prime}\to\hat{N}\Sigma$ be a simplicial complex map. Any
family of maps $\\{\alpha_{x}:\mathcal{E}_{O}(\pi(x))\to
O^{\prime}_{x}\\}_{x\in\Sigma^{\prime}_{0}}$ can be assembled into a natural
transformation
$\alpha:\mathcal{E}_{O}\circ\overline{\pi}\to\mathcal{E}_{O^{\prime}}$.

###### Proof.

A natural transformation
$\alpha:\mathcal{E}_{O}\circ\overline{\pi}\to\mathcal{E}_{O^{\prime}}$ is a
collection of maps

$\\{\alpha_{\sigma}:\mathcal{E}_{O}(\overline{\pi}(\sigma))\to\mathcal{E}_{O^{\prime}}(\sigma)\\}_{\sigma\in\Sigma^{\prime}}$

such that for $\sigma_{1}\subset\sigma_{2}$ the following diagram commutes

${\mathcal{E}_{O}(\overline{\pi}(\sigma_{2}))}$${\mathcal{E}_{O^{\prime}}(\sigma_{2})}$${\mathcal{E}_{O}(\overline{\pi}(\sigma_{1}))}$${\mathcal{E}_{O^{\prime}}(\sigma_{1})}$$\scriptstyle{\alpha_{\sigma_{2}}}$$\scriptstyle{\alpha_{\sigma_{1}}}$
(42)

In particular, for $\tilde{x}\in\sigma$, the following diagram commutes

${\prod_{x\in\overline{\pi}(\sigma)}O_{x}}$${\prod_{x\in\sigma}O^{\prime}_{x}}$${\prod_{x\in\pi(\tilde{x})}O_{x}}$${O^{\prime}_{\tilde{x}}}$$\scriptstyle{\alpha_{\sigma}}$$\scriptstyle{\alpha_{\tilde{x}}}$
(43)

This means that for $s\in\prod_{x\in\overline{\pi}(\sigma)}O_{x}$, we have
$\alpha_{\sigma}(s)(\tilde{x})=\alpha_{\tilde{x}}(s|_{\pi(\tilde{x})})$. Thus
$\alpha$ is defined by the maps $\\{\alpha_{x}\\}_{x\in\Sigma^{\prime}_{0}}$.
On the other hand, for a collection of maps
$\\{\alpha_{x}:\prod_{y\in\pi(x)}O_{y}\to
O^{\prime}_{x}\\}_{x\in\Sigma^{\prime}_{0}}$, one can construct a natural
transformation
$\alpha:\mathcal{E}_{O}\circ\overline{\pi}\to\mathcal{E}_{O^{\prime}}$ by
defining $\alpha_{\sigma}(s)=\\{\alpha_{x}(s|_{\pi(x)})\\}_{x\in\sigma}$ for
$\sigma\in\Sigma^{\prime}$ and $s\in\prod_{x\in\overline{\pi}(\sigma)}O_{x}$.
∎

######  Lemma A.14.

Let $\pi:\Sigma^{\prime}\to\hat{N}\Sigma$ be a simplicial complex map. Any
family of maps

$\\{\gamma_{x}:\\{x\\}\times\mathcal{E}_{O}(\pi(x))\to\\{x\\}\times
O^{\prime}_{x}\\}_{x\in\Sigma^{\prime}_{0}}$

can be assembled into a type II morphism
$\gamma:(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}\to\mathcal{E}_{O^{\prime}}\Sigma^{\prime}$.

###### Proof.

A simplicial complex map
$\gamma:(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}\to\mathcal{E}_{O^{\prime}}\Sigma^{\prime}$
is determined by its restriction to the vertices of
$(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}$. Hence it is defined by the maps
$\gamma|_{\\{x\\}\times\mathcal{E}_{O}(\pi(x))}$, where
$x\in\Sigma^{\prime}_{0}$. On the other hand, if we have a collection of maps
$\\{\gamma_{x}:\\{x\\}\times\mathcal{E}_{O}(\pi(x))\to\\{x\\}\times
O^{\prime}_{x}\\}_{x\in\Sigma^{\prime}_{0}}$ then we can construct a
simplicial complex map
$\gamma:(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}\to\mathcal{E}_{O^{\prime}}\Sigma^{\prime}$
by defining $\gamma(x,s)=\gamma_{x}(x,s)$ for every vertex
$(x,s)\in(\mathcal{E}_{O}\circ\pi)\Sigma^{\prime}$. This is because, for a
simplex $(\sigma,s)$ in $\mathcal{E}\circ\pi(\Sigma^{\prime})$, where
$\sigma=\\{x_{1},\cdots,x_{n}\\}$, we have that

$\displaystyle\gamma(\sigma,s)$
$\displaystyle={\gamma}(\\{(x_{1},s|_{\pi(x_{1})}),\cdots,(x_{n},s|_{\pi(x_{n})})\\})$
$\displaystyle=\\{(\gamma_{x_{1}}(s|_{\pi(x_{1})}),\cdots,\gamma_{x_{n}}(s|_{\pi(x_{n})})\\}\in\\{\sigma\\}\times\prod_{x\in\sigma}O^{\prime}_{x}.$

Therefore $\gamma(\sigma,s)$ is a simplex in
$\mathcal{E}_{O^{\prime}}\Sigma^{\prime}$. In addition,
$f_{(\Sigma^{\prime},O^{\prime})}\circ\gamma=f_{\mathcal{E}_{O}\circ\pi}$. ∎

###  A.5 Lemmas: Empirical model functor on bundle scenarios

In this section we provide important results for our study of push-forward
empirical models. First we show that our construction $p\mapsto\hat{N}p$ given
in Definition 3.12 produces a well-defined empirical model. Then we prove
results concerning push-forwards along type I and type II morphisms.

######  Lemma A.15.

The family $\hat{N}p$ of distributions constructed in Definition 3.12 is a
well-defined empirical model on $\hat{N}f$.

###### Proof.

First, we prove that $(\hat{N}p)_{\\{\sigma_{1}\cdots\sigma_{n}\\}}$ is a
well-defined distribution on
$(\hat{N}f)^{-1}(\\{\sigma_{1}\cdots\sigma_{n}\\})$. We have

$\displaystyle\sum_{\\{\gamma_{1}\cdots\gamma_{n}\\}\in(\hat{N}f)^{-1}(\\{\sigma_{1}\cdots\sigma_{n}\\})}(\hat{N}p)_{\\{\sigma_{1}\cdots\sigma_{n}\\}}(\\{\gamma_{1},\cdots,\gamma_{n}\\})$
$\displaystyle=\sum_{\\{\gamma_{1},\cdots,\gamma_{n}\\}:\,f(\gamma_{i})=\sigma_{i}}p_{\cup_{i}\sigma_{i}}(\cup_{i}\gamma_{i})$
$\displaystyle=\sum_{\gamma:\,f(\gamma)\cup_{i}\sigma_{i}}p_{\cup_{i}\sigma_{i}}(\gamma)$
$\displaystyle=\sum_{\gamma\in
f^{-1}(\cup_{i}\sigma_{i})}p_{\cup_{i}\sigma_{i}}(\gamma)=1$

since $f$ is discrete over vertices. Now, we prove the compatibility:

$\displaystyle(\hat{N}p)_{\\{\sigma_{1},\cdots,\sigma_{k},\sigma_{k+1},\cdots,\sigma_{n}\\}|_{\\{\sigma_{1},\cdots,\sigma_{k}\\}}}(\\{\gamma_{1},\cdots,\gamma_{k}\\})$
$\displaystyle=\sum_{\\{\gamma_{k+1},\cdots,\gamma_{n}\\}:\,f(\gamma_{i})=\sigma_{i}}(\hat{N}p)_{\\{\sigma_{1},\cdots,\sigma_{n}\\}}(\\{\gamma_{1},\cdots,\gamma_{n}\\})$
$\displaystyle=\sum_{\\{\gamma_{k+1},\cdots,\gamma_{n}\\}:\,f(\gamma_{i})=\sigma_{i}}p_{\cup_{i=1}^{n}\sigma_{i}(\cup_{i=1}^{n}\gamma_{i})}$
$\displaystyle=\sum_{\cup_{i=1}^{k}\gamma_{i}\subseteq\gamma\in
f^{-1}(\cup_{i=1}^{n}\sigma_{i})}p_{\cup_{i=1}^{n}\sigma_{i}}(\gamma)$
$\displaystyle=p_{\cup_{i=1}^{n}\sigma_{i}}|_{\cup_{i=1}^{k}\sigma_{i}}(\cup_{i=1}^{k}\gamma_{i})$
$\displaystyle=p_{\cup_{i=1}^{k}\sigma_{i}}(\cup_{i=1}^{k}\gamma_{i})$
$\displaystyle=(\hat{N}p)_{\\{\sigma_{1},\cdots,\sigma_{k}\\}}(\\{\gamma_{1},\cdots,\gamma_{k}\\}).$

∎

######  Lemma A.16.

The push-forward empirical model $\pi_{\ast}(p)$ in Definition 3.13 belongs to
$\mathsf{bEmp}(f^{\pi})$. Moreover, for
$\pi^{\prime}:\Sigma^{\prime\prime}\to\hat{N}\Sigma^{\prime}$, we have
$(\pi\diamond\pi^{\prime})_{\ast}p={\pi^{\prime}}_{\ast}(\pi_{\ast}p)$.

###### Proof.

Let $\sigma^{\prime}\in\Sigma^{\prime}$ and
$\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})$. Observe that
$\pi(\sigma^{\prime})=\pi(f^{\pi}(\gamma^{\prime}))=\hat{N}f(l(\gamma^{\prime}))$.
Thus $l(\gamma^{\prime})\in(\hat{N}f)^{-1}(\pi(\sigma^{\prime}))$. In
addition,

$\sum_{\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})}(\pi_{\ast}p)_{\sigma^{\prime}}(\gamma^{\prime})=\sum_{\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})}(\hat{N}p)_{\pi(\sigma^{\prime})}(l(\gamma^{\prime})).$

By Proposition A.5 this is equal to
$\sum_{\gamma\in(\hat{N}f)^{-1}(\pi(\sigma^{\prime}))}(\hat{N}p)_{\pi(\sigma^{\prime})}(\gamma)$,
which is equal to $1$ according to Lemma A.15. Now, given
$\sigma^{\prime}_{1}\subseteq\sigma^{\prime}$ in $\Sigma^{\prime}$ and
$\gamma^{\prime}_{1}\in(f^{\pi})^{-1}(\sigma^{\prime}_{1})$, we have

$\displaystyle(\pi_{\ast}p)_{\sigma^{\prime}}|_{\sigma^{\prime}_{1}}(\gamma_{1}^{\prime})$
$\displaystyle=\sum_{\gamma_{1}^{\prime}\subseteq\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})}(\pi_{\ast}p)_{\sigma^{\prime}}(\gamma^{\prime})$
$\displaystyle=\sum_{\gamma_{1}^{\prime}\subseteq\gamma^{\prime}\in(f^{\pi})^{-1}(\sigma^{\prime})}(\hat{N}p)_{\pi(\sigma^{\prime})}(l(\gamma^{\prime}))$
$\displaystyle=\sum_{l(\gamma^{\prime}_{1})\subseteq\gamma\in(\hat{N}f)^{-1}(\pi(\sigma^{\prime}))}(\hat{N}p)_{\pi(\sigma^{\prime})}(\gamma)$
$\displaystyle=(\hat{N}p)_{\pi(\sigma^{\prime})}|_{\pi(\sigma_{1}^{\prime})}(l(\gamma_{1}^{\prime}))$
$\displaystyle=(\hat{N}p)_{\pi(\sigma_{1}^{\prime})}(l(\gamma^{\prime}_{1}))=(\pi_{\ast}p)_{\sigma^{\prime}_{1}}(\gamma^{\prime}_{1}).$

Now, given $\sigma\in\Sigma^{\prime\prime}$ and
$\gamma\in\left((f^{\pi})^{\pi^{\prime}}\right)^{-1}(\sigma)$, we have

$\displaystyle\left({\pi^{\prime}}_{\ast}(\pi_{\ast}p)\right)_{\sigma}(\gamma)$
$\displaystyle=(\pi_{\ast}p)_{\overline{\pi^{\prime}}(\sigma)}(\overline{l}_{f^{\pi}}(\gamma))$
$\displaystyle=p_{\overline{\pi}(\overline{\pi^{\prime}}(\sigma))}\left(\overline{l}_{f}(\overline{l}_{f^{\pi}}(\gamma))\right)$
$\displaystyle=p_{\overline{\pi\diamond\pi^{\prime}}(\sigma)}(\overline{l}_{f,\pi\diamond\pi^{\prime}}(\gamma))$
$\displaystyle=\left((\pi\diamond\pi^{\prime})_{\ast}p\right)_{\sigma}(\gamma).$

∎

######  Lemma A.17.

The push-forward empirical model $\alpha_{\ast}p$ in Definition 3.14 belongs
to $\mathsf{bEmp}(g)$. Moreover, given another type II morphism
$\alpha^{\prime}:g\to h$, we have
$(\alpha^{\prime}\circ\alpha)_{\ast}p={\alpha^{\prime}}_{\ast}(\alpha_{\ast}p)$.

###### Proof.

Because of the commutativity of Diagram $(\ref{AAlpDiag})$, the image of
$\alpha|_{f^{-1}(\sigma)}$ lies in $g^{-1}(\sigma)$. Thus
$(\alpha_{\ast}p)_{\sigma}\in D_{R}(g^{-1}(\sigma))$. Now, we prove the
compatibility. Given $\sigma_{1}\subseteq\sigma$ in $\Sigma$. Using the fact
that
$r_{\sigma,\sigma_{1}}\circ\alpha|_{f^{-1}(\sigma)}=\alpha|_{f^{-1}(\sigma_{1})}\circ
r_{\sigma,\sigma_{1}}$ we obtain

$\displaystyle(\alpha_{\ast}p)_{\sigma}|_{\sigma_{1}}$
$\displaystyle=D_{R}(r_{\sigma,\sigma_{1}})(D_{R}(\alpha|_{f^{-1}(\sigma)})(p_{\sigma}))$
$\displaystyle=D_{R}(r_{\sigma,\sigma_{1}}\circ\alpha|_{f^{-1}(\sigma)})(p_{\sigma})$
$\displaystyle=D_{R}(\alpha|_{f^{-1}(\sigma_{1})}\circ
r_{\sigma,\sigma_{1}})(p_{\sigma})$
$\displaystyle=D_{R}(\alpha|_{f^{-1}(\sigma_{1})})(D_{R}\left(r_{\sigma,\sigma_{1}})(p_{\sigma})\right)$
$\displaystyle=D_{R}(\alpha|_{f^{-1}(\sigma_{1})})(p_{\sigma}|_{\sigma_{1}})$
$\displaystyle=D_{R}(\alpha|_{f^{-1}(\sigma_{1})})(p_{\sigma_{1}})=(\alpha_{\ast}p)_{\sigma_{1}}.$

For the second part of the lemma, we have

$\displaystyle(\alpha^{\prime}\circ\alpha)_{\ast}(p)_{\sigma}$
$\displaystyle=D_{R}(\alpha^{\prime}\circ\alpha|_{f^{-1}(\sigma)})(p_{\sigma})$
$\displaystyle=D_{R}(\alpha^{\prime}|_{g^{-1}(\sigma)}\circ\alpha|_{f^{-1}(\sigma)})(p_{\sigma})$
$\displaystyle=D_{R}(\alpha^{\prime}|_{g^{-1}(\sigma)})\left(D_{R}(\alpha|_{f^{-1}(\sigma)})(p_{\sigma})\right)$
$\displaystyle=(\alpha^{\prime}_{\ast}(\alpha_{\ast}p))_{\sigma}.$

∎

######  Lemma A.18.

Given a type II morphism $\alpha:f\to g$, a simplicial complex map
$\pi:\Sigma^{\prime}\to\hat{N}\Sigma$, and an empirical model $p\in bEmp(f)$,
we have

$\pi_{\ast}(\alpha_{\ast}p)=\left(\pi^{\ast}(\hat{N}\alpha)\right)_{\ast}(\pi_{\ast}p).$

###### Proof.

Given $\sigma^{\prime}\in\Sigma^{\prime}$ and
$\tilde{\tau}\in(g^{\pi})^{-1}(\sigma^{\prime})$, we have

$\displaystyle\pi_{\ast}(\alpha_{\ast}p)_{\sigma^{\prime}}(\tilde{\tau})$
$\displaystyle=(\alpha_{\ast}p)_{\overline{\pi}(\sigma^{\prime})}(\overline{l}_{g}(\tilde{\tau}))$
$\displaystyle=D_{R}(\alpha|_{f^{-1}(\overline{\pi}(\sigma^{\prime}))})(p_{\overline{\pi}(\sigma^{\prime})})(\overline{l}_{g}(\tilde{\tau}))$
$\displaystyle=\sum_{\gamma:\,\alpha(\gamma)=\overline{l}_{g}(\tilde{\tau})}p_{\overline{\pi}(\sigma^{\prime})}(\gamma).$

On the other hand,

$\displaystyle(\left(\pi^{\ast}(\hat{N}\alpha)\right)_{\ast}(\pi_{\ast}p))_{\sigma^{\prime}}(\tilde{\tau})$
$\displaystyle=D_{R}(\pi^{\ast}(\hat{N}\alpha)|_{(f^{\pi})^{-1}(\sigma^{\prime})})\left((\pi_{\ast}p)_{\sigma^{\prime}}\right)(\tilde{\tau})$
$\displaystyle=\sum_{\tau:\,\pi^{\ast}(\hat{N}\alpha)(\tau)=\tilde{\tau}}(\pi_{\ast}p)_{\sigma^{\prime}}(\tau)$
$\displaystyle=\sum_{\tau:\,\pi^{\ast}(\hat{N}\alpha)(\tau)=\tilde{\tau}}p_{\overline{\pi}(\sigma^{\prime})}(\overline{l}_{f}(\tau)).$

Observe that if $\tau\in\pi^{\ast}(\hat{N}\Gamma)$ such that
$\pi^{\ast}(\hat{N}\alpha)(\tau)=\tilde{\tau}$ then
$\overline{l}_{f}(\tau)\in\Gamma$ and
$\alpha(\overline{l}_{f}(\tau))=\overline{l}_{g}(\pi^{\ast}(\hat{N}\alpha)(\tau))=\overline{l}_{g}(\tilde{\tau}))$.
In fact, using the universal property of pullbacks, one can see that the
assignment $\tau\mapsto\overline{l}_{f}(\tau)$ gives a one-to-one
corresponding between
$\\{\tau:\,\pi^{\ast}(\hat{N}\alpha)(\tau)=\tilde{\tau}\\}$ and
$\\{\gamma:\,\alpha(\gamma)=\overline{l}_{g}(\tilde{\tau})\\}$. ∎

##  Appendix B Simplicial sets

Simplicial sets are combinatorial models of spaces which have better
expressive power than simplicial complexes. In this section we give basic
definitions and introduce the simplicial set representing the standard
topological simplex.

######  Definition B.1.

A simplicial set is a functor $X:\mathbf{\Delta}^{\text{op}}\to\mathbf{Set}$.
Explicitly, a simplicial set consists of the following data:

  * •

A sequence of sets $X_{0},X_{1},\cdots,X_{n},\cdots$ for $n\geq 0$ where each
$X_{n}$ represents the set of $n$-simplices.

  * •

Face maps

$d_{i}:X_{n}\to X_{n-1}\;\;\;\text{ for }0\leq i\leq n$

representing the faces of a given simplex.

  * •

Degeneracy maps

$s_{j}:X_{n}\to X_{n+1}\;\;\;\text{ for }0\leq j\leq n$

representing the degenerate simplices.

The face and the degeneracy maps are subject to the simplicial identities
[18]. A map $f:X\to Z$ of simplicial sets consists of a sequence of functions
$f_{n}:X_{n}\to Z_{n}$, where $n\geq 0$, compatible with the face and the
degeneracy maps in the sense that

$d_{i}f_{n}(\sigma)=f_{n-1}(d_{i}\sigma)\;\;\text{ and
}\;\;s_{j}f_{n}(\sigma)=f_{n+1}(s_{j}\sigma)$

for all $0\leq i,j\leq n$ and $\sigma\in X_{n}$.

######  Example B.2.

The $d$-simplex $\Delta[d]$ is the simplicial set whose set $n$-simplices are
given by

$\\{\sigma^{a_{0}\cdots a_{n}}:\,0\leq a_{0}\leq\cdots\leq a_{d}\leq
n,\,a_{i}\in\mathbb{Z}\\}.$

The face maps deletes an index: $d_{i}(\sigma^{a_{0}\cdots
a_{n}})=\sigma^{a_{0}\cdots a_{i-1}a_{i+1}\cdots a_{n}}$, and the degeneracy
maps copies an index: $s_{j}(\sigma^{a_{0}\cdots a_{n}})=\sigma^{a_{0}\cdots
a_{j}a_{j}\cdots a_{n}}$. The simplex $\sigma^{01\cdots d}$ is the generating
simplex of the simplicial set in the sense that any other simplex can be
obtained by applying a sequence of face and degeneracy maps to this simplex.

###  B.1 The nerve space

There is a simplicial set version of the nerve complex $\hat{N}(\Sigma)$
construction which we simply denote by $N(\Sigma)$. In this section we
introduce the nerve space associated to a simplicial complex and study maps
between two such simplicial sets.

######  Definition B.3.

Let $N:s\mathbf{Comp}\to\mathbf{sSet}$ denote the functor that sends a
simplicial complex $\Sigma$ to a simplicial set $N\Sigma$, called the nerve
space of $\Sigma$, whose $n$-simplices are given by

$(N\Sigma)_{n}=\\{(\sigma_{1},\sigma_{2},\cdots,\sigma_{n})\in\Sigma^{n}:\,\cup_{i=1}^{n}\sigma_{i}\in\Sigma\\}.$

where $(N\Sigma)_{0}=\\{()\\}$. The simplicial structure maps are given by

$d_{i}(\sigma_{1},\sigma_{2},\cdots,\sigma_{n})=\left\\{\begin{array}[]{ll}(\sigma_{2},\cdots,\sigma_{n})&i=0\\\
(\sigma_{1},\cdots,\sigma_{i}\cup\sigma_{i+1},\cdots,\sigma_{n})&0<i<n\\\
(\sigma_{1},\cdots,\sigma_{n-1})&i=n\end{array}\right.$

and

$s_{j}(\sigma_{1},\sigma_{2},\cdots,\sigma_{n})=(\sigma_{1},\cdots,\sigma_{j},\emptyset,\sigma_{j+1},\cdots,\sigma_{n})\;\;\;0\leq
j\leq n.$

A simplicial complex map $f:\Gamma\to\Sigma$ induces a simplicial set map
between the nerves $Nf:N\Gamma\to N\Sigma$ defined in degree $n$ by

$(Nf)_{n}(\gamma_{1},\gamma_{2},\cdots,\gamma_{n})=(f(\gamma_{1}),f(\gamma_{2}),\cdots,f(\gamma_{n})).$

######  Proposition B.4.

Given simplicial complexes $\Gamma$ and $\Sigma$, a simplicial set map
$f:N\Gamma\to N\Sigma$ between the nerves satisfies the following properties:

  1. 1.

$f_{n}(\gamma_{1},\cdots,\gamma_{n})=(f_{1}(\gamma_{1}),\cdots,f_{1}(\gamma_{n}))$
for every $(\gamma_{1},\cdots,\gamma_{n})\in(N\Gamma)_{n}$.

  2. 2.

$f_{1}(\gamma)=\cup_{i=1}^{n}f_{1}({x_{i}})$ for every
$\gamma=\\{x_{1},\cdots,x_{n}\\}\in\Gamma$.

###### Proof.

Part 1: Given $1\leq i\leq n$, let $d^{\Gamma}:(N\Gamma)_{n}\to(N\Gamma)_{1}$
denote the composition of $i-1$ times $d_{0}$ with $(n-i)$ times $d_{n}$,
i.e., $d^{\Gamma}(\gamma_{1},\cdots,\gamma_{n})=\gamma_{i}$. Similarly, we
define $d^{\Sigma}$. Since $f$ respects the simplicial structure we have
$d^{\Sigma}(f_{n}(\gamma_{1},\cdots,\gamma_{n}))=f_{1}(d^{\Gamma}(\gamma_{1},\cdots,\gamma_{n}))=f_{1}(\gamma_{i})$.

Part 2: Given $\gamma=\\{x_{1},\cdots,x_{n}\\}\in\Gamma$, we have
$(\\{x_{1}\\},\cdots,\\{x_{n}\\})\in(N\Gamma)_{n}$. Using part 1 and the fact
that $f$ respects the simplicial structure we obtain that

$\displaystyle f_{1}(\gamma)$ $\displaystyle=f_{1}(\\{x_{1},\cdots,x_{n}\\})$
$\displaystyle=f_{1}\left(d_{1}\circ\cdots\circ
d_{1}(\\{x_{1}\\},\cdots,\\{x_{n}\\})\right)$
$\displaystyle=d_{1}\circ\cdots\circ
d_{1}\left(f_{n}(\\{x_{1}\\},\cdots,\\{x_{n}\\})\right)$
$\displaystyle=d_{1}\circ\cdots\circ d_{1}(f_{1}(x_{1}),\cdots,f_{1}(x_{n}))$
$\displaystyle=\cup_{i=1}^{n}f_{1}(x_{i}).$

∎

###  B.2 Lemmas: Category of simplicial scenarios

Similar to the $\hat{N}$ construction its simplicial set version $N$ also
preserves bundle scenarios. In this section we prove the two parts that go
into the proof, that is, this construction preserves both local surjectivity
and being discrete over vertices.

######  Lemma B.5.

If $f:\Gamma\to\Sigma$ is (locally) surjective then $Nf:N\Gamma\to N\Sigma$ is
(locally) surjective.

###### Proof.

Assume that $f$ is surjective and let
$(\sigma_{1},\cdots,\sigma_{n})\in(N\Sigma)_{n}$. There exists
$\gamma\in\Gamma$ such that $f(\gamma)=\cup_{i=1}^{n}\sigma_{i}\in\Sigma$ and
we can choose $\gamma_{1},\cdots,\gamma_{n}\subset\gamma$ such that
$f(\gamma_{i})=\sigma_{i}$. This means that
$(\gamma_{1},\cdots,\gamma_{n})\in(N\Gamma)_{n}$ and
$(Nf)_{n}(\gamma_{1},\cdots,\gamma_{n})=(\sigma_{1},\cdots,\sigma_{n})$.

Let us write $\tau=\sigma^{01\cdots(n-1)}$ and $\sigma=\sigma^{01\cdots n}$
for the generating simplices of $\Delta[n-1]$ and $\Delta[n]$, respectively;
see Example B.2. Now, assume that $f$ is locally surjective and consider the
following commutative diagram:

${\Delta[n-1]}$${N\Gamma}$${\Delta[n]}$${N\Sigma}$$\scriptstyle{\tilde{\alpha}}$$\scriptstyle{d^{i}}$$\scriptstyle{Nf}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$
(44)

where $\alpha_{\sigma}=(\sigma_{1}{,}\cdots{,}\sigma_{n})$ and
$\tilde{\alpha}_{\tau}=(\gamma_{1}{,}\cdots{,}\gamma_{n-1})$. If $0<i<n$ then

$(\sigma_{1},\cdots,\sigma_{i}\cup\sigma_{i+1},\cdots,\sigma_{n})=(f(\gamma_{1}),\cdots,f(\gamma_{n-1})).$

where $f(\gamma_{i})=\sigma_{i}\cup\sigma_{i+1}$. Let