# Abstract Orientable Incidence Structure and Algorithms for Finite Bounded
Acyclic Categories. I. Incidence Structure

Yu-Wei Huang<EMAIL_ADDRESS>
###### Abstract

A generalization of incidence relations in abstract polytope has been
explored, and parameterized surfaces are used as primers. The abstract
orientable incidence structure is defined as an algebraic model of incidence
relations, in which some algebraic properties in abtract polytope theory are
generalized. The geometric interpretation of abstract orientable incidence
structure are also discussed. The orientable incidence structure in a semi-
regular normal CW complex are briefly investigated.

##  1 Introduction

The concept of incidence relations can be traced back to the face lattice of
classical polytopes, which is a combinatorial structure among facets [3]. This
structure can be studied as an abstract polytope, which is a binary relation
describing whether a facet is contained by another facet. However, abstract
polytope is limited in its ability to describe certain types of incidence
relations. For instance, the convexity of polytopes ensures that the relation
between two facets is unique, leading to a poset structure in the abstract
polytope. The flatness of geometric objects also makes the shape uniquely
determined by the boundaries. By relaxing these constraints, more fundamental
structures of incidence relations can be revealed. Some geometric complexes
like simplicial complex and CW complex can be seen as a generalization of
polytopes [2, 1]. However, these approaches require the introduction of
auxiliary geometric objects and are not well-suited for studying incidence
structures. Although it is well known that incidence relations can be
manifested by posets, there has been little discussion of generalizing this
concept to acyclic categories in the same manner. Therefore, this article will
re-describe incidence relations between geometric objects, and show that it
has the structure of acyclic category.

##  2 Incidence Structure

###  2.1 Facet and Incidence Relation

Below we will introduce incidence structure via parameterized $n$-dimensional
surfaces, which are referred to as facets. Roughly speaking, a parameterized
$n$-dimensional surface is a surface defined by a function of $n$ parameters.
In order to rule out some strange geometries, some constraints are adopted.
The points on the surface are described by the parameters called coordinates
faithfully, so that the topological properties of this surface are fully
described by the coordinate space. The coordinate space is not limit to an
Euclidean space, so that it can describe non-trivial topology like torus. The
values of coordinates are restricted to a given range in the coordinate space,
which should be a compact and connected open subset. For example, the Figure 1
is a facet of a fragment of spherical shell described by
$\\{S(\theta,\phi)\mid 0<\theta<\pi/4,0<\phi<\pi/4\\}$, where
$S(\theta,\phi)\equiv\cos\theta\cos\phi\,\hat{x}+\sin\theta\cos\phi\,\hat{y}+\sin\phi\,\hat{z}$
is a point on the surface with coordinate $(\theta,\phi)$. The topology of
this surface can be described by the coordinate range
$(0,\pi/4)\times(0,\pi/4)\subset\mathbb{R}^{2}$. To describe a full sphere,
the coordinate space cannot be an Euclidean space because it cannot faithfully
refer to all points on the sphere. It is useful for doing actual geometric
calculations if the coordinate space has a metric, which is important when it
is not a Cartesian coordinate system. A good coordinate space should have
small distortion, such that geometric calculations can be performed
accurately. A cone can be described by $\\{Y(x,y)|0<x^{2}+y^{2}<1\\}$ with
$Y(x,y)\equiv x\,\hat{x}+y\,\hat{y}+\,\sqrt{x^{2}+y^{2}}\hat{z}$, where the
origin of the coordinate space $(0,0)$ has been removed since such point is
not differentiable. This coordinate range is valid although it is not
homeomorphic to a disk; there is no need to cut it off to make it homeomorphic
to the disk. These examples shows how the flexibility of coordinate helps us
to deal with real geometry more easily. As a special case, a point, described
by a singleton set $\\{P\\}$, has a coordinate space with zero parameter,
which is just a coordinate space that has only one valid coordinate.

If the coordinate space is orientable, the orientation of a facet can be
defined. For convenience, we assume they are always orientable. The
orientation of a facet at the point $F(u,v,\dots)$, denoted as
$\hat{d}F(u,v,\dots)$, is defined as the normalization of the wedge product of
the partial derivatives $dF(u,v,\dots)=\frac{\partial}{\partial
u}F(u,v,\dots)\wedge\frac{\partial}{\partial v}F(u,v,\dots)\wedge\dots$. In
the example of Figure 1, the orientation is
$\hat{d}S(\theta,\phi)=\cos\theta\cos\phi\,\hat{y}\wedge\hat{z}+\sin\theta\cos\phi\,\hat{z}\wedge\hat{x}+\sin\phi\,\hat{x}\wedge\hat{y}$,
which can be seen as a radial pseudovector. Each point in a $n$-dimensional
facet has an orientation as a normalized multivector of degree $n$ (more
precisely, a $n$-blade). In the same sense, the orientation of a line is just
a multivector of degree 1, which is an unit tangent vector; the orientation of
a point is just a multivector of degree 0, which only can be a sign. The
orientation of a point cannot be derived naturally, so one should attach a
sign to define its orientation, which is called a signed point.

Figure 1: A fregment of spherical shell. The arrows on the surface represent
its orientation. The arc on its bottom boundary, whose orientation is
represented as the arrow on the line, is positively-oriented with respect to
the spherical shell.

A valid facet should be possible to be extended by including the boundary of
the coordinate range, which is called closed facets. The image of the open
coordinate range is called the inner of the facet, and the image of the
boundaries of the coordinate range is called the boundaries of the facet. The
inner and the boundaries of a facet should be disjoint. Unlike the inner of
the facet, the boundaries are not faithfully described by coordinates, but are
at least locally faithfully. To describe the boundaries of a given facet, a
set of disjoint facets are introduced such that the union of them is equal to
the boundaries, and these facets are called subfacets of the given facet. The
relations between a facet and its subfacets are called incidence relations,
which are described by embed function and local connectedness. Embed function
is an mapping between coordinate spaces of facets, which shows how a facet is
embedded into another one. For example, an arc $\\{C(\theta)\mid
0<\theta<\pi/4\\}$ with
$C(\theta)\equiv\cos\theta\,\hat{x}+\sin\theta\,\hat{y}$ is covered by the
surface of the example of Figure 1, which can be described by the equation
$C(\theta)=S(\theta,0)$. This embed function is denoted as $\phi:C\to S$,
where $C$ is a subfacet of the facet $S$. Notice that there can be multiple
embed functions between two facets. For example, the circle, described by
$\\{C(\theta)|0<\theta<2\pi\\}$ with
$C(\theta)\equiv\cos\theta\,\hat{x}+\sin\theta\,\hat{y}$, covers the point
$P\equiv 1\,\hat{x}+0\,\hat{y}$ in two ways: $P=C(0)$ and $P=C(2\pi)$. This is
different from the conventional polytope, in which there is only one relation
between incident facets.

The definiton of subfacets also implies two special facets. The null face
$\varnothing$ is defined as an empty set, which has no valid coordinate. The
null face is usually considered as a $(-1)$-dimensional facet. The null face
$\varnothing$ is covered by all facets, because there is a map from an empty
set to any set. Consider multiple disjoint facets with shared subfacets, the
space containing these facets, called the universe $\mathbb{U}$, can also be
considered as a facet. For example, 3D Euclidean space is defined as
$\mathbb{U}(x,y,z)=x\,\hat{x}+y\,\hat{y}+z\,\hat{z}$. The universe
$\mathbb{U}$ covers all facets because it is the codomain of functions
defining facets. The null face $\varnothing$ and the universe $\mathbb{U}$ are
called improper facets.

Figure 2: The relative orientation of signed points, lines, and planes. Their
orientations are represented as a chain of arrows (drawn as simplexes), so
that the wedge product of these vectors forms the orientation. The shaded
regions represent positive side of facets.

Another property of an incidence relation called the local connectedness is
manifested by the relative position between two incident facets. If their
dimensions differ by only $1$, the relative position can always be described
by a sign (denoted as $|\phi|$), determining which the side of the subfacet
faces the inner of the facet. In the example of Figure 1, the spherical shell
is on the positive side of the arc according to the right-hand rule, so we
define $|\phi|=+1$ (see Figure 1). Generalizing to incidence relation between
$n$-dimensional and $(n{+}1)$-dimensional facets, we define the orientation
between them as the difference of wedge products: if the $n$-dimentional
subfacet $B$ is embedding into the $(n{+}1)$-dimentional facet $F$ by mapping
$\phi:B\to F$, define $|\phi|$ as a sign such that
$\hat{d}B\wedge\hat{n}=|\phi|\hat{d}F$, where $\hat{d}F$ and $\hat{d}B$ are
their orientations at a point in the subfacet $B$, and $\hat{n}$ is a vector
on this point pointing to the inner of the facet $F$. $|\phi|=+1$
($|\phi|=-1$) means the subfacet $B$ is positively- (negatively-) oriented
with respect to the facet $F$ (see Figure 2). Note that all points in the
subfacet $B$ should give the same value of $|\phi|$. Under this definition,
one can say: a line segment has one positive-oriented subfacet and one
negative-oriented subfacet, and both of them are positive points at the
endpoints; a circle with counter-clockwise tangent vectors is a positive-
oriented subfacet of the enclosed disk; a sphere defined like the above
example is negatively-oriented with respect to the enclosed ball. The
incidence relation between the null face and a signed point is a special case:
positive (negative) point always is positively- (negatively-) oriented with
respect to the null face. A subfacet can be covered by a facet with multiple
orientations. For example, consider a plane $S$ and a line segment $L$ which
is in the middle of the plane. Since the plane is on the left and right of
this line, there are two choices of the inward pointing vectors. Then the
incidence relations between them are described by $\phi_{+}$ and $\phi_{-}$,
which are the two identical embed functions with different orientations by
letting $|\phi_{+}|=+1$ and $|\phi_{-}|=-1$.

Figure 3: The incidence relations between a square and its vertex. The
incidence relations from vertex to edge and from edge to square can be
composed, resulting in the incidence relation from the vertex to the square.
These incidence relations form a diamond shape (right diagram), which is
filled up to indicate the equivalence of composition. The equivalence of
composition represents the completeness of the vertex, drawn as a fan shape on
the vertex. Figure 4: Multiple incidence relations on a crescent shape. There
are two diamonds on the point $P$, which represents two directions (drawn as
two red fan shapes) go from this point to the inner of the crescent shape. The
right diagram represents the embed functions between facets, while it’s
difficult to be filled up to represent their diamond properties.

It is obviously that incidence relations are composable: if a point can embed
into a line, and the line can embed into another surface, then this point can
also embed into this surface. The equivalence relations between composed embed
functions manifest the local connectedness, which is not just a sign. The
incidence relations and their composition rules constitute an incidence
structure. Figure 3 shows the incidence relations between a square and its
vertex. The vertex point embeds into a square by mapping $\phi$, which is
equal to composition $\phi_{L}^{\prime}\circ\phi_{L}$ and
$\phi_{R}^{\prime}\circ\phi_{R}$, where $\phi_{L}$ ($\phi_{R}$) is the
incidence relation from this point to the left (right) adjacent edge, and
$\phi_{L}^{\prime}$ ($\phi_{R}^{\prime}$) is the incidence relation from the
left (right) adjacent edge to the square. Such equivalence relation is just
the diamond property in abstract polytope, and this property describes the
completeness of the vertex. In abstract polytope, this diamond always exists
and is unique for any two incident facets with dimension differing by $2$. If
there is a shape that has two vertices sharing the same point, this point is
covered by this area in two directions. For example, consider the area between
two internally tangent circles called crescent shape (see Figure 4), the
tangent point is covered by outer circle in two directions, say $\phi_{L}$ and
$\phi_{R}$, and outer circle is also covered by the crescent shape, say
$\phi_{S}$. Then one can go from the tangent point to the inner of the
crescent shape in two ways, which is equal to $\phi_{S}\circ\phi_{L}$ and
$\phi_{S}\circ\phi_{R}$. These two incidence relations are not equivalent, so
they don’t form a diamond; the angle between these two edges has been
separated by the inner circle. Instead, the diamonds are formed by
$\phi_{S}\circ\phi_{L}=\phi_{S}^{\prime}\circ\phi_{L}^{\prime}$ and
$\phi_{S}\circ\phi_{R}=\phi_{S}^{\prime}\circ\phi_{R}^{\prime}$, where
$\phi_{L}^{\prime}$, $\phi_{R}^{\prime}$ and $\phi_{S}^{\prime}$ are the same
as above but relative to the inner circle. These two diamonds represent two
different directions of incidence relations between the tangent point and the
crescent shape; there are two separated ranges of angles that can go from this
point to the inner. It shows the difference between diamonds indicates the
local connectedness around the point.

Not all incidence relations can be defined continuously. The Riemann surface
$R(z)\equiv(z,f(z))$ for the function $f(z)=\sqrt{z}(1-|z|)$ with $0<|z|<1$ is
a $2$-dimensional facet in a $4$-dimensional space, which doubly cover the
unit circle $C(z)\equiv(z,0)$ with $|z|=1$. Their are two incidence relations:
any point on the circle, say $C(e^{i\theta})$, should be mapped to
$R(e^{i\theta})$ and $R(e^{i(2\pi+\theta)})$, representing two embed functions
respectively. Depending on the choice of the branch cut, the embed functions
are discontinuous at that point. It is impossible to write such embed
functions as continuous functions. Breaking of continuity may cause some
geometric calculation problems. To solve this problem, the calculation should
be limited to a small range, such that in this range the embed functions can
be written as continuous functions depending on where it is. A facet with a
non-orientable subfacet will also encounter similar problems, in which the
incidence relations cannot have a continuous sign. These kind of incidence
relations are said to be non-orientable. To simplify the discussion, we only
focus on the incidence structure with orientable incidence relations, which we
call an orientable incidence structure.

###  2.2 Vertex Figure and Local Connectedness

To explain the concept of local connectedness more clearly, we shall introduce
vertex figures, which is defined in the same manner as polytope. To make a
vertex figure on a $0$-dimensional facet (a positive point), one puts a ball
$D^{n}$ on it, which is small enough so that only the facets that cover this
point intersect this ball. Then the sliced images of facets (the intersection
of the boundary of the ball $\partial D^{n}$ and facets) are defined as the
vertex figure of the corresponding facets. For example, the vertex figure on a
vertex of a cube is a spherical triangle, which is an intersection of the cube
and a sphere centered at this vertex (see Figure 5). The orientation of the
sliced image $L$ of the facet $F$ should obey
$\hat{n}\wedge\hat{d}L=\hat{d}F$, where $\hat{n}$ is outward pointing vector
from the center of the slicing ball. The incidence relations of this vertex
figure can be trivially derived by restricting the domain and codomain of
mapping. It’s easy to prove that the sign of incidence relations are the same.
More general, the vertex figure on any facet can be defined as: pick a point
on given $m$-dimensional facet $V$, and put a ball $D^{n-m}$ on this point,
which is perpendicular to its orientation $\hat{d}V$ at this point. Then the
sliced images of facets (the intersections of the boundary of this ball
$\partial D^{n-m}$ and facets) are defined as the vertex figure of the
corresponding facets on the facet $V$. All points on the facet $V$ should give
the same vertex figure. The orientation of the sliced image $L$ of the facet
$F$ should obey $\hat{d}V\wedge\hat{n}\wedge\hat{d}L=\hat{d}F$, where
$\hat{n}$ is outward pointing vector from the center of the the slicing ball.
For example, the vertex figure on an edge (i.e., the edge figure) of a cube is
a segment of arc, which is the intersection of the cube and a circle with this
edge as its axis (see Figure 5). Its obviously the operations of taking vertex
figure are composable. For example, the edge figure of a cube is the vertex
figure of the vertex figure, as shown in Figure 5.

Figure 5: A vertex figure (top left) and edge figure (top right) of a cube
(bottom). Where $U$ and $F$ denote two faces, $E$ denotes an edge, and $V$
denote a vertex. The vertex figure on the vertex $V$ is formed by sliced
images of those facets. Further taking vertex figure on the vertex $E$ is the
same as taking vertex figure on the edge $E$ in the original cube.

Some sliced images on a subfacet have multiple connected components, each of
them is defined as an individual facet in this vertex figure. This kind of
facets are said to be locally separated by this subfacet. For example, the
vertex figure of a plane on a point lying in the middle of the plane is a
circle, which is connected, so we say the plane is not locally separated by
this point. But the vertex figure of a plane on a line lying in the middle of
the plane becomes two points, which have two connected components, so we say
the plane is locally separated by this line. The crescent shape (Figure 4) is
a non-trivial example, whose vertex figures on the tangent point are two line
segments, so we say the crescent shape is locally separated by the tangent
point. It shows the local connectedness property described by incidence
relation becomes connectedness property of the vertex figure, and such
correspondence is naturally required if one considers the incidence structure
of the vertex figure. Also, the local connectedness property can be propagated
via connectivity, so incidence relations can be tested in every points in the
connected subfacet. These descriptions manifest the local and global
properties of incidence relations.

The orientable incidence structure can be drawn as directed acyclic
multigraphs, where nodes are facets, and edges are nondecomposable embed
functions. In this diagram, called the Hasse diagram, all incident facets can
be connected by directed paths, manifesting the poset structure of incidence
relations. The Hasse diagram is not enough to describe an orientable incidence
structure, since the composition rules for incidence relations are not shown.
Because sliced images of connected facets may become disconnected, the
incidence structure of a vertex figure is not just a subgraph of the Hasse
diagram. For example, the vertex figure on the intersection point of two
circles on a torus is much more complicated to itself (see Figure 6), but the
composition rules of incidence relations are always the same.

Figure 6: The vertex figure of a torus. The top is the torus (left) and its
Hasse diagram (right); the bottom is the vertex figure on point $P$ (left) and
the corresponding Hasse diagram (right), which is much more complicated.

In our theory, the incidence relations are not unique to two incident facets.
The difference of two incidence relations represents the local connectedness
of the facet, which is the main difference to the conventional incidence
structure. The existence of multiple incidence relations forces us to consider
the composition rules between them. Unlike abstract polytope, where the
structure can be described by a poset, the orientable incidence structure act
like a category. We will extract out such structure in the next section.

##  3 Abstract Orientable Incidence Structure

###  3.1 Bounded Acyclic Category and Bounded Poset

The abstract orientable incidence structure is an algebraic structure which
represents the orientable incidence relations between facets. Unlike previous
section, it only captures its combinatorial nature without specifying
geometric properties. An abstract orientable incidence structure, denoted as
$\mathcal{F}$, is a graded bounded acyclic category satisfying semi-diamond
property. An acyclic category is a category without non-trivial cycles. It is
impossible to have two non-identity morphisms in opposite directions at the
same time, so morphisms can only go in one direction. For a finite acyclic
category, there is a rank function that maps an object to an integer, so that
all morphisms increase the rank of object except identity. One can define the
rank of an object as the dimension of corresponding facet. The acyclic
category with a rank function is called graded acyclic category. Bounded means
it has exactly one initial object and terminal object. The initial object,
which has the lowest rank, represents the null face, denoted as $\varnothing$;
the terminal object, which has the highest rank, represents the universe,
denoted as $\mathbb{U}$. Morphisms represent incidence relations between the
facets. The morphism $\phi:F\rightarrow S$ means that facet $F$ is covered by
facet $S$. There may have multiple morphisms between two objects, and this
property represents the local connectedness of the incidence relation, as we
discussed above. Below we mainly focus on the property of bounded acyclic
category, and compare it with the bounded poset.

Recall the definition of a bounded poset (a poset which includes its lower and
upper bounds), which is the base of abstract polytope:

  1. 1.

_irreflexivity_ : $a\nless a$.

  2. 2.

_asymmetry_ : if $a<b$ then $b\nless a$.

  3. 3.

_transitivity_ : if $a<b$ and $b<c$ then $a<c$.

  4. 4.

_least_ : there exists exactly one $\varnothing$ such that forall
$a\neq\varnothing$ there is $\varnothing<a$.

  5. 5.

_greatest_ : there exists exactly one $\mathbb{U}$ such that forall
$a\neq\mathbb{U}$ there is $a<\mathbb{U}$.

Now compare to the definition of bounded acyclic category:

  1. 1.

_irreflexivity_ : there is no $\phi:F\rightarrow F$ for any
$\phi\neq\operatorname{id}$.

  2. 2.

_asymmetry_ : if there is $\phi:F\rightarrow S$ for some
$\phi\neq\operatorname{id}$, then there is no $\phi^{\prime}:S\rightarrow F$
for any $\phi^{\prime}\neq\operatorname{id}$.

  3. 3.

_transitivity_ : if there are $\phi:P\rightarrow F$ and
$\phi^{\prime}:F\rightarrow S$ for some $\phi\neq\operatorname{id}$ and
$\phi^{\prime}\neq\operatorname{id}$, then
$\phi^{\prime}\circ\phi:P\rightarrow S$.

  4. 4.

_initial_ : there exists exactly one $\varnothing$, such that forall
$F\neq\varnothing$ there is $\phi:\varnothing\rightarrow F$ for _exactly one_
$\phi\neq\operatorname{id}$.

  5. 5.

_terminal_ : there exists exactly one $\mathbb{U}$, such that forall
$F\neq\mathbb{U}$ there is $\phi:F\rightarrow\mathbb{U}$ for _exactly one_
$\phi\neq\operatorname{id}$.

The key different is that, there can be multiple relations between two objects
in the bounded acyclic category, and the transitivity of order relations
becomes the composition of morphisms. A bounded poset is just a thin bounded
acyclic category. It is natural to define an induced bounded poset for a
bounded acyclic category by forgetting differences between multiple relations.
In this sense, one says acyclic category is a generalization of poset.
Abstract orientable incidence structure also loosens three constraints of
abstract polytope, which will be discussed in the end of this section. Notice
that the rules _least_ and _greatest_ in a bounded poset becomes _initial_ and
_terminal_ , which looks stricter in the bounded acyclic category. The initial
object and the terminal object are called improper objects.

###  3.2 Upper Category and Downward Functor

The upper closure of an element $x$ in a bounded poset $\mathcal{P}$ is a
subset whose element is greater than or equal to $x$. Such subset is also a
bounded poset, where the order relation is inherited from the host poset
$\mathcal{P}$. To emphasize this, it is denoted as
$\mathcal{P}\operatorname{\uparrow}x$. Similarly, one can construct a full
subcategory $\mathcal{F}^{\prime}$ by only including the upper closure of an
object $F_{m}$. There is no non-trivial morphism that point to the object
$F_{m}$, but it is not an initial object since the morphism $\phi:F_{m}\to S$
may not be unique for the object $S$, so that it is not a bounded acyclic
category. To fix it, one should split objects such that there is only one
initial morphism for each object. Construct upper category of object $F_{m}$
in a bounded acyclic category $\mathcal{F}$, denoted as
$\mathcal{F}\operatorname{\uparrow}F_{m}$:
$\operatorname{Obj}(\mathcal{F}\operatorname{\uparrow}F_{m})$ is a set of
$F_{s}|\phi_{sm}\rangle$ for all objects $F_{s}$ and morphisms
$\phi_{sm}:F_{m}\rightarrow F_{s}$.
$\operatorname{Hom}(F_{s}|\phi_{sm}\rangle,F_{t}|\phi_{tm}\rangle)$ is a set
of $\phi_{ts}|\phi_{sm}\rangle:F_{s}|\phi_{sm}\rangle\rightarrow
F_{t}|\phi_{tm}\rangle$ for all morphisms $\phi_{ts}:F_{s}\rightarrow F_{t}$
and $\phi_{sm}:F_{m}\to F_{s}$ such that $\phi_{ts}\circ\phi_{sm}=\phi_{tm}$.
The composition rule is
$\phi_{pt}|\phi_{tm}\rangle\circ\phi_{ts}|\phi_{sm}\rangle=\phi_{ps}|\phi_{sm}\rangle$
with $\phi_{ps}=\phi_{pt}\circ\phi_{ts}$. The object $F_{t}$ have been marked
with morphism $\phi_{tm}$, so that it is splitted into multiple objects
corresponding to each initial morphism $\phi_{tm}$. Although the object
splits, the composition rule is inherited from the host category
$\mathcal{F}$. In terms of incidence structure, the upper category
$\mathcal{F}\operatorname{\uparrow}F_{m}$ just corresponds to the vertex
figure on an object $F_{m}$: in upper category, an object
$F_{n}|\phi_{nm}\rangle$ represents a connected component of sliced images,
and a morphism $\phi_{ts}|\phi_{sm}\rangle$ represents an incidence relation
between facets $F_{s}|\phi_{sm}\rangle$ and $F_{t}|\phi_{tm}\rangle$.

There is a functor
$F_{m}^{\downarrow}:\mathcal{F}\operatorname{\uparrow}F_{m}\rightarrow\mathcal{F}$
that maps object $F_{s}|\phi_{sm}\rangle$ to $F_{s}$, and maps morphism
$\phi_{ts}|\phi_{sm}\rangle:F_{s}|\phi_{sm}\rangle\rightarrow
F_{t}|\phi_{tm}\rangle$ to $\phi_{ts}:F_{s}\rightarrow F_{t}$. This functor is
a reversed operation of taking upper category, so it is named downward
functor. Drawn as Hasse diagrams, the downward functor becomes graph
homomorphism between two graphs. Unlike the abstract polytope, this is not
injective on nodes and edges, which manifests the local connectedness of
incidence relations.

Since an upper category itself is also a bounded acyclic category, one can
further construct an upper category of an object in the upper category. The
category
$\mathcal{F}\operatorname{\uparrow}F_{m}\operatorname{\uparrow}F_{n}|\phi_{nm}\rangle$
is an upper category of object $F_{n}|\phi_{nm}\rangle$, where
$\operatorname{Obj}(\mathcal{F}\operatorname{\uparrow}F_{m}\operatorname{\uparrow}F_{n}|\phi_{nm}\rangle)$
is a set of object $F_{s}|\phi_{sm}\rangle|\phi_{sn}|\phi_{nm}\rangle\rangle$,
or abbreviated as $F_{s}|\phi_{sn},\phi_{nm}\rangle$, for all objects $F_{s}$
and morphisms $\phi_{sn}:F_{n}\rightarrow F_{s}$;
$\operatorname{Hom}(F_{t}|\phi_{tn},\phi_{nm}\rangle,F_{s}|\phi_{sn},\phi_{nm}\rangle)$
is a set of
$\phi_{ts}|\phi_{sm}\rangle|\phi_{sn}|\phi_{nm}\rangle\rangle:F_{s}|\phi_{sm}\rangle|\phi_{sn}|\phi_{nm}\rangle\rangle\rightarrow
F_{t}|\phi_{tm}\rangle|\phi_{tn}|\phi_{nm}\rangle\rangle$, or abbreviated as
$\phi_{ts}|\phi_{sn},\phi_{nm}\rangle:F_{s}|\phi_{sn},\phi_{nm}\rangle\rightarrow
F_{t}|\phi_{tn},\phi_{nm}\rangle$ for all morphisms
$\phi_{ts}:F_{s}\rightarrow F_{t}$ and $\phi_{sn}:F_{n}\rightarrow F_{s}$ such
that $\phi_{ts}\circ\phi_{sn}=\phi_{tn}$. Noting that we assume
$\phi_{sm}=\phi_{sn}\circ\phi_{nm}$ and $\phi_{tm}=\phi_{tn}\circ\phi_{nm}$,
and the abbreviation works because $\phi_{sm}$ can be determined uniquely. It
is isomorphic to the upper category $\mathcal{F}\operatorname{\uparrow}F_{n}$,
which shows that taking upper category is composable. This rule is consistent
with above discussion: taking vertex figure is composable. Similarly, downward
functors are also composable. Downward functor can be reduced down to a
functor
$\phi_{mn}^{\downarrow}:\mathcal{F}\operatorname{\uparrow}F_{m}\rightarrow\mathcal{F}\operatorname{\uparrow}F_{n}$
for morphism $\phi_{mn}:F_{n}\rightarrow F_{m}$, in which object
$F_{s}|\phi_{sm}\rangle$ is mapped to $F_{s}|\phi_{sn}\rangle$, and morphism
$\phi_{ts}|\phi_{sm}\rangle:F_{s}|\phi_{sm}\rangle\rightarrow
F_{t}|\phi_{tm}\rangle$ is mapped to
$\phi_{ts}|\phi_{sn}\rangle:F_{s}|\phi_{sn}\rangle\rightarrow
F_{t}|\phi_{tn}\rangle$, where $\phi_{sn}=\phi_{sm}\circ\phi_{mn}$. Upper
closures of a poset form a poset by inclusion relation, its opposite category
is isomorphic to the host poset. Similarly, upper categories
$\\{\mathcal{F}\operatorname{\uparrow}F_{m}|F_{m}\in\operatorname{Obj}\\}$ and
downward functors
$\\{\phi_{mn}^{\downarrow}|\phi_{mn}\in\operatorname{Hom}\\}$ also form a
bounded acyclic category, its opposite category is isomorphic to the host
category $\mathcal{F}$ (see Figure 7).

Figure 7: The category of upper categories (right) of a cone (left). It is
drawn as a set of Hasse diagrams with graph homomorphisms between them (purple
arrows). This representation is sufficiently to describe a bounded acyclic
category.

The Hasse diagram of a bounded acyclic category can easily see which part is
splittable. Two proper objects are said to be linked if they are connected in
the Hasse diagram excluding improper objects. If not all proper objects are
linked, this category is said to be splittable. It is the generalization of
connectedness condition of abstract polytope. The linked objects may form
multiple clusters, and they can be splitted into separated bounded acyclic
categories. The linked clusters can be determined by linkages between
minimal/maximal objects, which are defined as minimal/maximal elements of the
induced poset of all proper objects. A bounded acyclic category is said to be
strongly unsplittable iff all sections are unsplittable, which is an analogue
of strongly connectedness of posets. The definition of sections in an acyclic
category will be introduced latter. The words “linked” and “splittable” are
used because “disjoint” and “disconnected” are reserved for actual geometric
properties.

###  3.3 Section Category and Morphism Chain

There is a dual concept of upper category called lower category, which is a
generalization of lower closure of a poset. Upper category is constructed by
treating one object as the initial object, and split objects such that
outgoing morphisms are initial morphisms. In the opposite, lower category,
denoted as $\mathcal{F}\operatorname{\downarrow}F_{m}$, is constructed by
treating one object as the terminal object, and split objects such that
incoming morphisms are terminal morphisms:
$\operatorname{Obj}(\mathcal{F}\operatorname{\downarrow}F_{m})$ is a set of
$\langle\phi_{mt}|F_{t}$ for all objects $F_{t}$ and morphisms
$\phi_{mt}:F_{t}\rightarrow F_{m}$;
$\operatorname{Hom}(\langle\phi_{ms}|F_{s},\langle\phi_{mt}|F_{t})$ is a set
of
$\langle\phi_{mt}|\phi_{ts}:\langle\phi_{ms}|F_{s}\rightarrow\langle\phi_{mt}|F_{t}$
for all morphisms $\phi_{ts}:F_{s}\rightarrow F_{t}$ and
$\phi_{mt}:F_{t}\rightarrow F_{m}$ such that
$\phi_{mt}\circ\phi_{ts}=\phi_{ms}$. The composition rule is
$\langle\phi_{mp}|\phi_{pt}\circ\langle\phi_{mt}|\phi_{ts}=\langle\phi_{mp}|\phi_{ps}$
with $\phi_{ps}=\phi_{pt}\circ\phi_{ts}$. In geometry, lower category
corresponds to face figure, which is like imagining an ant living in a 2D
world. The face figure of a sphere with one meridian is equivalent to a 2D
space bounded by two lines (see Figure 8). The line is duplicated because it
is covered by the sphere in two ways. As a creature living in 3D space, we
know it is one line, but for the 2D creatures living on the sphere, they only
see a space rift, and they cannot confirm if there is a wider space in this
rift.

Figure 8: A 2D ant lives on a sphere with a meridian (left), which is the same
as living on its face figure from the ant’s point of view (right).

There is also a categorical analogue of the interval of a poset:
$[a,b]\equiv\\{c\in\mathcal{P}|a\leq c\leq b\\}$, which can be seen as a lower
closure of an upper closure. Given a poset, by taking non-trivial intervals
between all minimal and maximal elements, the given poset can be covered by
multiple bounded posets. Similarly, an acyclic category can be turned into
direct sum of bounded acyclic categories by splitting objects, and those
bounded acyclic categories are called sections, whose name comes from abstract
polytope. Consider an acyclic category, say $\mathcal{F}$, define a section
category of morphism $\phi_{0}$ on $\mathcal{F}$, denoted as
$\langle\mathcal{F}\rangle_{\phi_{0}}$:
$\operatorname{Obj}(\langle\mathcal{F}\rangle_{\phi_{0}})$ is a set of
$\langle\phi_{n}^{\ast}|\phi_{n}\rangle$ for all morphisms $\phi_{n}^{\ast}$
and $\phi_{n}$ such that $\phi_{n}^{\ast}\circ\phi_{n}=\phi_{0}$.
$\operatorname{Hom}(\langle\phi_{n}^{\ast}|\phi_{n}\rangle,\langle\phi_{m}^{\ast}|\phi_{m}\rangle)$
is a set of $\langle\phi_{m}^{\ast}|\phi_{mn}|\phi_{n}\rangle$ for all
morphisms $\phi_{m}^{\ast}$, $\phi_{mn}$, $\phi_{n}$ such that
$\phi_{m}^{\ast}\circ\phi_{mn}\circ\phi_{n}=\phi_{0}$.

taking section category doesn’t break the algebra of morphisms, it just
renames some objects so that it is bounded. “The algebra of morphisms” is the
generalization of “the transitivity of order relation”. Define local-embedding
function for a poset: an order-preserving function $f$ is local-embedding iff
it is an order isomorphism between $[x,y]$ and $[f(x),f(y)]$ for any pair
$x\leq y$. That is, the element $f(x)\leq z^{\prime}\leq f(y)$ implies there
exists exactly one $x\leq z\leq y$ such that $f(z)=z^{\prime}$. Note that it
is not an one-to-one function. The generalization to an acyclic category
becomes a local-embedding functor: for any morphism $\phi$, the pair of
morphisms $\psi^{\prime}$ and $\psi^{\prime\prime}$ with constraint
$\psi^{\prime\prime}\circ\psi^{\prime}=\mu(\phi)$ implies there exists exactly
one pair of morphisms $\phi^{\prime}$ and $\phi^{\prime\prime}$ with
constraint $\phi^{\prime\prime}\circ\phi^{\prime}=\phi$ such that
$\mu(\phi^{\prime})=\psi^{\prime}$ and
$\mu(\phi^{\prime\prime})=\psi^{\prime\prime}$. The inverse of taking section
categories is the local-embedding functor. Local-embedding functor describes
the correspondence of algebra of morphisms without mention any object. It
shows that the identity of object isn’t important in a bounded acyclic
category; initial morphisms and terminal morphisms are enough to indicate
objects in a bounded acyclic category.

The chain representation provides a clear view of this. Recall the definition
of the chain of a poset, which is defined as a total ordering subset. A subset
of a chain is also total ordering, called a subchain. In an acyclic category,
the morphism between two objects is not unique, so the generalization becomes:
the morphism chain of a morphism $\phi_{n0}:F_{0}\rightarrow F_{n}$ is a non-
empty sequence of morphisms $\langle\phi_{n},\dots,\phi_{2},\phi_{1}\rangle$
that composed to this morphism. Its subchain can be constructed by composing
adjacent morphisms into one. In other words, a subchain just skips some
intermediate objects except the start and end, which is slightly different
from the conventional definition of the morphism chain. A morphism chain
containing identity morphisms is said to be degenerated. A morphism chain with
$n$ intermediate object is called $n$-chain. $0$-chain is just the host
morphism itself $\langle\phi_{n0}\rangle$, so it is said to be a trivial
chain. $1$-chain represents an object of the section of the morphism
$\phi_{n0}$, and $2$-chain represents a morphism, whose source object and
target object are just two non-trivial subchains.

Abstract orientable incidence structure can be described by only their
subchain relations, called a nerve, which is also slightly different from the
conventional definition. The nerve of an acyclic category is composed by a
collection of $n$-chains
$N_{n}=\\{\langle\phi_{n},\dots,\phi_{1},\phi_{0}\rangle\\}$ for
$n=0,1,2,\dots$ and the face maps $d_{i}:N_{n}\to N_{n-1}$ for $i=1\sim n$,
which skip the $i$-th object in the $n$-chain, and the degeneracy maps
$s_{i}:N_{n}\to N_{n+1}$ for $i=1\sim n$, which insert an identity morphism at
the $i$-th object. The laws for face maps and degeneracy maps are the same as
the conventional one, which should not be repeated here. Note that in our
definition, a $1$-chain represents an object, not a morphism. A bounded
acyclic category has nerve with only one $0$-chain, which will back to the
conventional definition by replacing $n$ with $n-1$. The upper category of a
$1$-chain $F$ becomes easy to define in this formulation, which is just
letting $N_{n}^{\prime}=d_{1}^{-n}(F)$ and $d_{i}^{\prime}=d_{i}$,
$s_{i}^{\prime}=s_{i}$, where $d_{1}^{-n}$ indicates taking preimage of
$d_{1}$ $n$ times. Lower category and section category can be defined in a
similar approach. Two acyclic categories will have the same nerve if they are
surjectively section-embedded by the same direct sum of bounded acyclic
categories. Because no relabeling is required, nerves are more natural for
describing orientable incidence structures.

A section category not only can be defined on an acyclic category, but also on
the non-recursive category, which is a category without non-trivial non-
degenerated recursive morphism chain, that is,
$\psi\circ\phi\circ\psi^{\prime}=\phi$ implies $\psi$ and $\psi^{\prime}$
being identity morphisms. This property only rules out the infinity of
morphisms, and non-trivial cycles of morphisms are still possible, while
ensuring that their section categories are bounded acyclic categories. Non-
recursiveness and finiteness naturally induce acyclicity.

###  3.4 Semi-Regular Normal CW Complex

The diamond property of an abstract orientable incidence structure is stated
as: a 2-rank morphism should be divided into _exactly_ two maximum chains.
2-rank morphism means the rank of source object and target object differ by 2.
In other words, a $1$-dimensional facet should have exactly 2 $0$-dimensional
subfacets, that means only line segments are valid $1$-dimensional facets.
Moreover, the multiplication of the sign of chains should be different, that
is, a valid diamond should obey $\phi_{+}\circ\psi_{+}=\phi_{-}\circ\psi_{-}$
and $|\phi_{+}|\times|\psi_{+}|=-|\phi_{-}|\times|\psi_{-}|$. This property
captures the topological nature of the Euclidean space. To include infinite
lines and rays as valid $1$-dimensional facets, this property should be
loosen. The semi-diamond property can be stated as: a 2-rank morphism can be
divided into _at most_ two maximum chains. In other words, a $1$-dimensional
facet should have at most 2 $0$-dimensional subfacets.

Except diamond property, the conventional abstract polytope have two more
constraints: strongly connected and uniform maximal chains. Strongly connected
means all facets in the polytope are firmly glued together. With this
property, it is not valid to glue two cubes together with one edge. Also, a
disk with a hole is also invalid, since its boundaries are not connected.
Uniform maximal chains property states that all maximal chains contain the
same number of facets. With this property, the ranks of facets of a maximal
chain are differed by $1$ adjacently, so that a rank function can be derived
naturally. That means all facets only have direct subfacets with dimensions
differing by $1$. They can be generalized as two additional properties for an
abstract orientable incidence structure. That is, strongly decomposable: all
morphisms (except terminal morphisms) can be decomposed down to 1-rank
morphisms; strongly unsplittable: all section categories of morphisms with
rank greater than 2 are unsplittable. A weaker version, called strongly
initial unsplittable, can be defined as: a bounded acyclic category is said to
be initial unsplittable iff all section categories of initial morphisms with
rank greater than 2 are unsplittable. Strongly initial unsplittable further
states that section categories of all non-initial morphisms can always be
splitted into strongly initial unsplittable categories. Strongly initial
unsplittable relaxes the constraint of shared vertices, but still limit to
connected boundaries.

If only facets homeomorphic to $n$-disks are considered, it becomes a semi-
regular and normal CW complex. A regular CW complex is a CW complex whose
gluing maps are homeomorphisms onto images. A CW complex is said to be normal
if each closed cell is a subcomplex. The incidence structure of a regular and
normal CW complex is given by inclusion relations between closed cells, which
forms a CW poset [1]. Define semi-regular CW complex as a CW complex whose
gluing maps are local homeomorphisms onto images. A semi-regular and normal CW
complex possesses an incidence structure manifested by the fibers of gluing
maps, which forms a bounded acyclic category. It obeys diamond property and is
strongly decomposable and strongly initial unsplittable: it is initial
unsplittable due to connectivity of cells, and because of local embedding
gluing maps, the vertex figure of a cell is always a disjoint union of simply-
connected components, which leads to strongly initial unsplittable property.
It seems there are more constraints on the incidence structure for a semi-
regular and normal CW complex, such that the geometric realization of an
abstract orientable incidence structure satisfying such constraints is also a
semi-regular and normal CW complex.

Under this constraint, the barycentric subdivision can be defined without
geometric property. Recall that the barycentric subdivision of a conventional
polytope is just the geometric realization (more precisely, the order complex)
of its face lattice (see Figure 9). The barycentric subdivision of a semi-
regular normal CW complex can be defined similarly, and it is just the
geometric realization of the nerve of its incidence structure. In our
definition of morphism chains, it should be defined as: the geometric
realization of a nerve is constructed by making $(n{-}1)$-simplex for each
non-degenerated $n$-chain $\langle\phi_{n},\dots,\phi_{2},\phi_{1}\rangle$
which is bounded by the corresponding simplexes of its direct subchains, where
the orientation of this simplex should be defined such that the first boundary
$\langle\phi_{n}\circ\phi_{n-1},\dots\rangle$ is positively-oriented with
respect to it, and the second boundary
$\langle\phi_{n},\phi_{n-1}\circ\phi_{n-2},\dots\rangle$ is negatively-
oriented with respect to it, and so on. The $(-1)$-simplex is the null face.
The geometric realization of the nerve of a semi-regular normal CW complex is
just its barycentric subdivision (see Figure 10). More interestingly, the
incidence structure of a vertex figure is just making a vertex figure of the
geometric realization of upper closure of its incidence structure. There is a
similar relation to the face figure. It shows “the incidence structure of a
geometric object is related to the geometry of an incidence structure”. This
series of articles will not discuss such correspondence further, since it only
appears in a restricted incidence structure. In the next article, we will
focus on finite bounded acyclic categories, and develop algorithms for them.

Figure 9: Relations between incidence structure and chains. The incidence
structure of a polytope is its face lattice (top left to top right), which is
drawn as a Hasse diagram. The barycentric subdivision of a polytope is built
by dividing facets according to the barycenters (top left to bottom left),
which results in a simplicial complex. Taking geometric realization of the
face lattice (the order complex of a poset) also results in the same complex
(top right to bottom left). The incidence structure of this complex (bottom
left to bottom right) is just the poset of chains of the face lattice (top
right to bottom right).

Figure 10: Relation between incidence structure and nerve. The incidence
structure of a semi-regular normal CW complex is the bounded acyclic category
formed by fibers of gluing maps (top left to top right). The barycentric
subdivision is defined in a similar way (top left to bottom left), which
results in a geometrical simplicial set. Taking geometric realization of the
incidence structure (the geometric realization of its nerve) also results in
the same complex (top right to bottom left). The incidence structure of this
complex (bottom left to bottom right) is just the nerve of the bounded acyclic
category (top right to bottom right).

## References

  * [1] Anders Björner  “Posets, regular CW complexes and Bruhat order”  In _European Journal of Combinatorics_ 5.1 Academic Press, 1984, pp. 7–16
  * [2] Dimitry Kozlov  “Combinatorial algebraic topology”  Springer Science & Business Media, 2008
  * [3] Peter McMullen and Egon Schulte  “Abstract regular polytopes”  Cambridge University Press, 2002


$\vartheta:=(\vartheta(\ast),\vartheta^{o},\vartheta^{u}):\mathbf{Coalg}_{\operatorname{\mathsf{id}}}^{\mathbb{T}}(p)$
is an open dynamical system over $p$ and
$\pi_{\vartheta\theta}:\vartheta(\ast)\to\theta(\ast)$ is a bundle in
$\mathcal{E}$, such that, for all time $t:{\mathbb{T}}$ and sections $\sigma$
of $p$ and $\varsigma$ of $b$, the following diagrams commute, thereby
inducing a bundle of closed dynamical systems
$\pi^{\sigma\varsigma}_{\vartheta\theta}:\vartheta^{\sigma}\to\theta^{\varsigma}$
in $\mathbf{Cat}(\mathbf{B}{\mathbb{T}},\mathcal{E})$:

$c$$\pi$$X$$Y$$c^{\dagger}_{\pi}$$\pi$$c$$X$$Y$$\rho^{u}$$\gamma^{\leftarrow}$$\mathcal{P}X$$\mathcal{P}Y$$Y$$\operatorname{{\mathcal{P}}}X$$X$$Y$${\vartheta(\ast)}$${{\sum\limits_{w:\vartheta(\ast)}p[\vartheta^{o}(t,w)]}}$${{\vartheta(\ast)}}$${\theta(\ast)}$${{\sum\limits_{x:\theta(\ast)}b[\theta^{o}(t,x)]}}$${\theta(\ast)}$$\scriptstyle{\pi_{\vartheta\theta}}$$\scriptstyle{\pi_{\vartheta\theta}}$$\scriptstyle{\vartheta^{o}(t)^{\ast}\sigma}$$\scriptstyle{\vartheta^{u}(t)}$$\scriptstyle{\theta^{o}(t)^{\ast}\varsigma}$$\scriptstyle{\theta^{u}(t)}$

######  Proposition A.6 (Category of open bundle dynamical systems over
$(p,b)$).

Let $p,b:\mathbf{Poly}_{\mathcal{E}}$ be polynomials in $\mathcal{E}$, and let
$\theta:=(\theta(\ast),\theta^{o},\theta^{u}):\mathbf{Coalg}_{\operatorname{\mathsf{id}}}^{\mathbb{T}}(b)$
be an open dynamical system over $b$. Open bundle dynamical systems over
$(p,b,\theta)$ form the objects of a category
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$. Morphisms
$f:(\pi_{\vartheta\theta},\vartheta)\to(\pi_{\varrho\theta},\varrho)$ are maps
$f:\vartheta(\ast)\to\varrho(\ast)$ in $\mathcal{E}$ making the following
diagram commute for all times $t:{\mathbb{T}}$ and sections $\sigma$ of $p$
and $\varsigma$ of $b$:

$c$$\pi$$X$$Y$$c^{\dagger}_{\pi}$$\pi$$c$$X$$Y$$\rho^{u}$$\gamma^{\leftarrow}$$\mathcal{P}X$$\mathcal{P}Y$$Y$$\operatorname{{\mathcal{P}}}X$$X$$Y$${{\vartheta(\ast)}}$${{\sum\limits_{w:\vartheta(\ast)}p[\vartheta^{o}(t,w)]}}$${{\vartheta(\ast)}}$${{\theta(\ast)}}$${{\sum\limits_{x:\theta(\ast)}b[\theta^{o}(t,x)]}}$${{\theta(\ast)}}$${{\varrho(\ast)}}$${{\sum\limits_{y:\varrho(\ast)}p[\varrho^{o}(t,y)]}}$${{\varrho(\ast)}}$$\scriptstyle{\vartheta^{o}(t)^{\ast}\sigma}$$\scriptstyle{\vartheta^{u}(t)}$$\scriptstyle{\theta^{o}(t)^{\ast}\varsigma}$$\scriptstyle{\theta^{u}(t)}$$\scriptstyle{\pi_{\vartheta\theta}}$$\scriptstyle{\pi_{\varrho\theta}}$$\scriptstyle{\pi_{\vartheta\theta}}$$\scriptstyle{\pi_{\varrho\theta}}$$\scriptstyle{\varrho^{o}(t)^{\ast}\sigma}$$\scriptstyle{\varrho^{u}(t)}$$\scriptstyle{f}$$\scriptstyle{f}$

That is, $f$ is a map on the state spaces that induces a morphism
$(\pi_{\vartheta\theta},\vartheta^{\sigma})\to(\pi_{\varrho\theta},\varrho^{\sigma})$
in $\mathbf{Cat}(\mathbf{B}{\mathbb{T}},\mathcal{E})/\theta^{\varsigma}$ of
bundles of the closures. Identity morphisms are the corresponding identity
maps, and composition is by pasting.

######  Proposition A.7 (Opindexed category of open bundle dynamical systems).

Varying the polynomials $p$ in $\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$
induces an opindexed category
$\mathbf{BunDyn}^{\mathbb{T}}({-},b,\theta):\mathbf{Poly}_{\mathcal{E}}\to\mathbf{Cat}$.
On polynomials $p$, it returns the categories
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$ of Proposition A.6. On morphisms
$\varphi:p\to q$ of polynomials, define the functors
$\mathbf{BunDyn}^{\mathbb{T}}(\varphi,b,\theta):\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)\to\mathbf{BunDyn}^{\mathbb{T}}(q,b,\theta)$
as in Proposition 2.27. That is, suppose
$(\pi_{\vartheta\theta},\vartheta):\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$
is object (open bundle dynamical system) in
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$, where
$\vartheta:=(\vartheta(\ast),\vartheta^{o},\vartheta^{u})$. Then its image
$\mathbf{BunDyn}^{\mathbb{T}}(\varphi,b,\theta)(\pi_{\vartheta\theta},\vartheta)$
is defined as the pair $(\pi_{\vartheta\theta},\varphi\vartheta)$, where
$\varphi\vartheta:=(\vartheta(\ast),\phi_{1}\circ\vartheta^{o},\vartheta^{u}\circ{\vartheta^{o}}^{\ast}\varphi^{\\#})$.
On morphisms
$f:(\pi_{\vartheta\theta},\vartheta)\to(\pi_{\varrho\theta},\varrho)$,
$\mathbf{BunDyn}^{\mathbb{T}}(\varphi,b,\theta)(f)$ is again given by the same
underlying map $f:\vartheta(\ast)\to\varrho(\ast)$ of state spaces.

###### Proof.

The proof amounts to the proof for Proposition 2.27 that
$\mathbf{RDyn}^{\mathbb{T}}(\varphi,\theta)$ constitutes an indexed category,
except that the closed base dynamical system $\theta$ of that Proposition is
here replaced, for any section $\varsigma$ of $b$, by the closure
$\theta^{\varsigma}$ by $\varsigma$ of the open dynamical system
$\theta:\mathbf{Coalg}_{\operatorname{\mathsf{id}}}^{\mathbb{T}}(b)$ of the
present Proposition. The proof goes through accordingly, since the relevant
diagrams are guaranteed to commute for any such $\varsigma$ by the conditions
in Definition A.5 and Proposition A.6. ∎

######  Proposition A.8 (Doubly-opindexed category of open bundle dynamical
systems).

Letting the base system $\theta$ also vary induces a doubly-opindexed category
$\mathbf{BunDyn}^{\mathbb{T}}({-},b,{=}):\mathbf{Poly}_{\mathcal{E}}\times\mathbf{Coalg}_{\operatorname{\mathsf{id}}}^{\mathbb{T}}(b)\to\mathbf{Cat}$.
Given a polynomial $p:\mathbf{Poly}_{\mathcal{E}}$ and morphism
$\phi:\theta\to\rho$ in
$\mathbf{Coalg}_{\operatorname{\mathsf{id}}}^{\mathbb{T}}(b)$, the functor
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\phi):\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)\to\mathbf{BunDyn}^{\mathbb{T}}(p,b,\rho)$
is defined by post-composition, as in Proposition 2.27 for the action of
$\mathbf{RDyn}^{\mathbb{T}}$ on morphisms of the base systems there. More
explicitly, such a morphism $\phi$ corresponds to a map
$\phi:\theta(\ast)\to\rho(\ast)$ of state spaces in $\mathcal{E}$. Given an
object $(\pi_{\vartheta\theta},\vartheta)$ of
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$, we define
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\phi)(\pi_{\vartheta\theta},\vartheta):=(\phi\circ\pi_{\vartheta\theta},\vartheta)$.
Given a morphism
$f:(\pi_{\vartheta\theta},\vartheta)\to(\pi_{\varrho\theta},\varrho)$ in
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\theta)$, its image
$\mathbf{BunDyn}^{\mathbb{T}}(p,b,\phi)(f):(\phi\circ\pi_{\vartheta\theta},\vartheta)\to(\phi\circ\pi_{\varrho\theta},\varrho)$
is given by the same underlying map $f:\vartheta(\ast)\to\varrho(\ast)$ of
state spaces.

###### Proof.

As for Proposition A.7, the proof here amounts to the proof for Proposition
2.27 that $\mathbf{RDyn}^{\mathbb{T}}(p,\phi)$ constitutes an indexed
category, except again the closed systems are replaced by (the appropriate
closures of) open ones, and the measure-preserving structure is forgotten. ∎

######  Proof A.9 (Proof of the Laplace approximation).

First note that the KL divergence is bounded from above by the free energy
since $\log p_{\gamma\bullet\pi}(y)$ is always negative.

Next, we can write the density functions as:

$\displaystyle\log
p_{\gamma}(y|x)=\frac{1}{2}\left\langle\epsilon_{\gamma}{,}\>{\Sigma_{\gamma}}^{-1}{\epsilon_{\gamma}}\right\rangle-\log\sqrt{(2\pi)^{|Y|}\det\Sigma_{\gamma}}$
$\displaystyle\log
p_{\rho_{\pi}}(x|y)=\frac{1}{2}\left\langle\epsilon_{\rho_{\pi}}{,}\>{\Sigma_{\rho_{\pi}}}^{-1}{\epsilon_{\rho_{\pi}}}\right\rangle-\log\sqrt{(2\pi)^{|X|}\det\Sigma_{\rho_{\pi}}}$
$\displaystyle\log
p_{\pi}(x)=\frac{1}{2}\left\langle\epsilon_{\pi}{,}\>{\Sigma_{\pi}}^{-1}{\epsilon_{\pi}}\right\rangle-\log\sqrt{(2\pi)^{|X|}\det\Sigma_{\pi}}$

where for clarity we have omitted the dependence of $\Sigma_{\gamma}$ on $x$
and $\Sigma_{\rho_{\pi}}$ on $y$, and where

$\displaystyle\epsilon_{\gamma}:Y\times X\to Y$ $\displaystyle:(y,x)\mapsto
y-\mu_{\gamma}(x)\,,$ $\displaystyle\epsilon_{\rho_{\pi}}:X\times Y\to X$
$\displaystyle:(x,y)\mapsto x-\mu_{\rho_{\pi}}(y)\,,$
$\displaystyle\epsilon_{\pi}:X\times 1\to X$ $\displaystyle:(x,\ast)\mapsto
x-\mu_{\pi}\,.$

Then, note that we can write the free energy $\operatorname{{\mathcal{F}}}(y)$
as the difference between expected energy and entropy:

$\displaystyle\operatorname{{\mathcal{F}}}(y)$
$\displaystyle=\operatorname*{\mathbb{E}}_{x\sim\rho_{\pi}(y)}\left[\log\frac{p_{\rho_{\pi}}(x|y)}{p_{\gamma}(y|x)\cdot
p_{\pi}(x)}\right]$
$\displaystyle=\operatorname*{\mathbb{E}}_{x\sim\rho_{\pi}(y)}\left[-\log
p_{\gamma}(y|x)-\log p_{\pi}(x)\right]-S_{X}\left[\rho_{\pi}(y)\right]$
$\displaystyle=\operatorname*{\mathbb{E}}_{x\sim\rho_{\pi}(y)}\left[E_{(\pi,\gamma)}(x,y)\right]-S_{X}\left[\rho_{\pi}(y)\right]$

Next, since the eigenvalues of $\Sigma_{\rho_{\pi}}(y)$ are small for all
$y:Y$, we can approximate the expected energy by its second-order Taylor
expansion around the mean $\mu_{\rho_{\pi}}(y)$:

$\displaystyle\operatorname{{\mathcal{F}}}(y)\approx$
$\displaystyle\;E_{(\pi,\gamma)}(\mu_{\rho_{\pi}}(y),y)+\frac{1}{2}\left\langle\epsilon_{\rho_{\pi}}\left(\mu_{\rho_{\pi}}(y),y\right){,}\>\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)\cdot\epsilon_{\rho_{\pi}}\left(\mu_{\rho_{\pi}}(y),y\right)\right\rangle$
$\displaystyle\;\,-S_{X}\big{[}\rho_{\pi}(y)\big{]}\,.$

where
$\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)$
is the Hessian of $E_{(\pi,\gamma)}$ with respect to $x$ evaluated at
$(\mu_{\rho_{\pi}}(y),y)$.

Note that

$\left\langle\epsilon_{\rho_{\pi}}\left(\mu_{\rho_{\pi}}(y),y\right){,}\>\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)\cdot\epsilon_{\rho_{\pi}}\left(\mu_{\rho_{\pi}}(y),y\right)\right\rangle=\tr\left[\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)\,\Sigma_{\rho_{\pi}}(y)\right]\,,$
(4)

that the entropy of a Gaussian measure depends only on its covariance,

$S_{X}\big{[}\rho_{\pi}(y)\big{]}=\frac{1}{2}\log\det\left(2\pi\,e\,\Sigma_{\rho_{\pi}}(y)\right)\,,$

and that the energy $E_{(\pi,\gamma)}(\mu_{\rho_{\pi}}(y),y)$ does not depend
on $\Sigma_{\rho_{\pi}}(y)$. We can therefore write down directly the
covariance $\Sigma_{\rho_{\pi}}^{\ast}(y)$ minimizing
$\operatorname{{\mathcal{F}}}(y)$ as a function of $y$. We have

$\partial_{\Sigma_{\rho_{\pi}}}\operatorname{{\mathcal{F}}}(y)\approx\frac{1}{2}\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)+\frac{1}{2}{\Sigma_{\rho_{\pi}}}^{-1}\,.$

Setting $\partial_{\Sigma_{\rho_{\pi}}}\operatorname{{\mathcal{F}}}(y)=0$, we
find the optimum

$\Sigma_{\rho_{\pi}}^{\ast}(y)=\left(\partial_{x}^{2}E_{(\pi,\gamma)}\right)\left(\mu_{\rho_{\pi}}(y),y\right)^{-1}\,.$

Finally, on substituting $\Sigma_{\rho_{\pi}}^{\ast}(y)$ in equation (4), we
obtain the desired expression

$\operatorname{{\mathcal{F}}}(y)\approx
E_{(\pi,\gamma)}\left(\mu_{\rho_{\pi}}(y),y\right)-S_{X}\left[\rho_{\pi}(y)\right]=:\operatorname{{\mathcal{F}}}^{L}(y)\,.$

∎