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\begin_layout Title
Topological Spaces and Sheaf Axioms
\end_layout
\begin_layout Author
Linas Vepstas
\end_layout
\begin_layout Date
29 August 2020
\end_layout
\begin_layout Abstract
This chapter presents the conventional (pre-)sheaf axioms and then demonstrates
how these describe the graphical/tensor structures that are the primary
topic of this series of chapters.
\end_layout
\begin_layout Abstract
This is a part of a sequence of articles exploring inter-related ideas;
it is meant to provide details for a broader context.
The current working title for the broader text is
\begin_inset Quotes eld
\end_inset
Connectors and Variables
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Section*
Pre-sheaves
\end_layout
\begin_layout Standard
This text reviews the concept of a pre-sheaves and sheaves, starting from
the conventional definition, and then demonstrating how this applies to
the system of knowledge representation being developed in these texts.
It is primarily an exercise in examining the axioms using different kinds
of notation, as notational differences appear to present one of the primary
stumbling blocks to the development of AGI.
\end_layout
\begin_layout Standard
The conventional definition of a (pre-)sheaf begins as follows: Let
\begin_inset Formula $X$
\end_inset
be a topological space and let
\begin_inset Formula $C$
\end_inset
be a category.
A presheaf
\begin_inset Formula $F$
\end_inset
on
\begin_inset Formula $X$
\end_inset
is a functor with values in
\begin_inset Formula $C$
\end_inset
having a certain set of properties ...
(to be reviewed below).
The conception of
\begin_inset Quotes eld
\end_inset
seeds
\begin_inset Quotes erd
\end_inset
being advanced here play the role of a topological space; this is the primary
driver of much of the mathematical machinery.
Thus, the re review here starts with topological spaces.
\end_layout
\begin_layout Subsection*
Topological space
\end_layout
\begin_layout Standard
Let's review the conventional definition of a topological space.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
This entire section painfully belabors something that is relatively obvious
to those who know algebraic topology: it is describing something very much
like a
\begin_inset Quotes eld
\end_inset
simplicial complex
\begin_inset Quotes erd
\end_inset
, and points out that it is a topological space.
Perhaps we could go farther: describe the analog of
\begin_inset Quotes eld
\end_inset
CW-complexes
\begin_inset Quotes erd
\end_inset
, by describing gluing and a weak topology.
Yet there are differences: the connectors on seeds are typed, whereas the
vertexes of simplicial complexes are simply-typed.
The seeds also resemble dual-graphs, in that the connectors connect up
neighbors, and so denote where two things come together at a boundary.
So we cannot just make this jump, as simplicial complexes are almost but
not quite the correct concept.
It would also put too great a burden on the reader to tell them that they
must first study algebraic topology.
The goals here are entirely different: the goals are to gain insight into
knowledge representation and AGI.
\end_layout
\end_inset
During the review, keep in mind the conception of
\begin_inset Quotes eld
\end_inset
seeds
\begin_inset Quotes erd
\end_inset
, as presented earlier.
As a reminder, here are diagrams of a single seed, a pair of connected
seeds, and a hedge-hog: four interconnected seeds.
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename graphics/seed-unlabelled.eps
lyxscale 60
width 15col%
\end_inset
\begin_inset space \hspace{}
\length 15col%
\end_inset
\begin_inset Graphics
filename graphics/seeds-two.eps
lyxscale 60
width 15col%
\end_inset
\begin_inset space \hspace{}
\length 15col%
\end_inset
\begin_inset Graphics
filename graphics/hedgehog.eps
lyxscale 60
width 15col%
\end_inset
\end_layout
\begin_layout Standard
These begin to look like the open sets of a topology, if one considers the
open circles to be the
\begin_inset Quotes eld
\end_inset
open
\begin_inset Quotes erd
\end_inset
boundary of the set, and the black circles to be the interior.
The primary conceptual issue is that topologies are conventionally framed
in terms of
\begin_inset Quotes eld
\end_inset
sets
\begin_inset Quotes erd
\end_inset
, which then presumes
\begin_inset Quotes eld
\end_inset
set theory
\begin_inset Quotes erd
\end_inset
underneath it all.
And yet nothing about a single seed looks particularly set-like.
Yes, the middle diagram above is obviously a
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
of two seeds, and the diagram on the right is the
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
of four seeds.
But they are not sets.
\end_layout
\begin_layout Standard
No matter.
A topology is defined in terms of a collection of axioms, and one should
argue that anything obeying those axioms is a topology, irregardless of
whether the underlying objects are actually sets, or not.
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
In a certain sense, this is preciely what the Yoneda lemma states.
But that sense is
\begin_inset Quotes eld
\end_inset
sideways
\begin_inset Quotes erd
\end_inset
from the direction we wish to travel in.
The Yoneda lemma is also sufficiently abstract that it cannot be made a
pre-requisite to understanding this text.
It does, however, provide the sign-post to a
\begin_inset Quotes eld
\end_inset
classical
\begin_inset Quotes erd
\end_inset
formalization of the discussion here.
\end_layout
\end_inset
What are the axioms of a topology? Let's quote directly from Wikipedia,
again:
\begin_inset Quotes eld
\end_inset
\emph on
A topological space is an ordered pair
\begin_inset Formula $\left(X,\tau\right)$
\end_inset
, where
\begin_inset Formula $X$
\end_inset
is a set and
\begin_inset Formula $\tau$
\end_inset
is a collection of subsets of
\begin_inset Formula $X$
\end_inset
, satisfying the following axioms:...
\emph default
\begin_inset Quotes erd
\end_inset
.
In the current context, the goal is to cross out the word
\begin_inset Quotes eld
\end_inset
set
\begin_inset Quotes erd
\end_inset
and replace it by
\begin_inset Quotes eld
\end_inset
hedgehog
\begin_inset Quotes erd
\end_inset
or
\begin_inset Quotes eld
\end_inset
mesh
\begin_inset Quotes erd
\end_inset
(the diagram above shows four interconnected seeds; in general it could
be a mesh of trillions, or even an infinite mesh.
An infinite mesh might be countable, if it was constructed recursively,
or even uncountable, if one is willing to entertain a mesh that interconnects
elements in a Cantor set.) One point to be careful about, in this analogy:
here,
\begin_inset Formula $X$
\end_inset
is meant to be the fully-assembled mesh; it is NOT a collection of seeds!
\begin_inset Foot
status collapsed
\begin_layout Plain Layout
Of course,
\begin_inset Formula $X$
\end_inset
could be just a collection of seeds, if that is the topological space that
one wished to work with; but that would be a very special case, and not
the general case.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Let's continue:
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
\emph on
The empty set and
\begin_inset Formula $X$
\end_inset
itself belong to
\begin_inset Formula $\tau$
\end_inset
.
\emph default
\begin_inset Quotes erd
\end_inset
Of course: the empty seed is just nothing at all: the seed with no connectors,
and no center.
And sure, let us assign the the hedgehog to
\begin_inset Formula $\tau.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
\emph on
Any arbitrary (finite or infinite) union of members of
\begin_inset Formula $\tau$
\end_inset
still belongs to
\begin_inset Formula $\tau$
\end_inset
.
\emph default
\begin_inset Quotes erd
\end_inset
Of course, although one must be careful in reading the word
\emph on
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
\emph default
, here.
There are two warring senses.
In one sense, the word
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
can be taken to mean
\begin_inset Quotes eld
\end_inset
the connection or connecting-together of two or more seeds, if they are
connectable
\begin_inset Quotes erd
\end_inset
.
This is meant to be the re-gluing together of pieces of a mesh that were
cut out: if one takes some scissors, and cuts out pieces from the mesh,
and then joins them back together again, it is still a sub-portion of the
mesh.
In the other sense, the union is a disjoint union: supposed the two scissored
pieces of the mesh were far apart from each other.
One is then left with a
\emph on
set
\emph default
of two pieces, that cannot be joined together.
This is still a union.
If one carefully examines the concept of a
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
in conventional set theory, it
\emph on
always
\emph default
means
\begin_inset Quotes eld
\end_inset
glue them together if you can
\begin_inset Quotes erd
\end_inset
.
Take, for example, a Venn diagram shown to elementary-school students:
the union is always the gluing-together, the overlapping bits being the
joint.
The overlapping bits are mashed together: this is the
\begin_inset Quotes eld
\end_inset
union
\begin_inset Quotes erd
\end_inset
of set theory: it is a gluing-together of the common, shared parts.
\end_layout
\begin_layout Itemize
\begin_inset Quotes erd
\end_inset
\emph on
The intersection of any finite number of members of
\begin_inset Formula $\tau$
\end_inset
still belongs to
\begin_inset Formula $\tau$
\end_inset
.
\emph default
\begin_inset Quotes erd
\end_inset
The reader was already invited to think of taking scissors to the mesh
to disconnect it, and also of gluing together pieces.
How, exactly, can the word
\begin_inset Quotes eld
\end_inset
intersection
\begin_inset Quotes erd
\end_inset
be interpreted? The intent is that it be read in the intuitively-obvious
sense.
If we have two fragments of a mesh, and they have points and links in common,
then the intersection are those points and links that are in common between
those two fragments.
Cut edges then become open connectors: as always, a connector is a
\begin_inset Quotes eld
\end_inset
half-edge
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
\emph on
The elements of
\begin_inset Formula $\tau$
\end_inset
are called
\series bold
\shape italic
\emph default
open sets
\series default
\shape default
\emph on
and the collection
\begin_inset Formula $\tau$
\end_inset
is called a
\series bold
\shape slanted
topology
\series default
\shape default
on
\begin_inset Formula $X$
\end_inset
.
\emph default
\begin_inset Quotes eld
\end_inset
That is the conclusion, with emphasis in the original.
\end_layout
\begin_layout Standard
It would appear that a mesh: that is, a graph whose edges can be scissored
into half-edges aka connectors forms a topology, in that it obeys the axioms
of a topology (ignoring the fact that hedge-hogs are not actually sets).
This is meant to be self-evident, so that it becomes straight-forward to
develop notational devices.
For example, given a hedgehog
\begin_inset Formula $U,$
\end_inset
the boundary
\begin_inset Formula $\partial U$
\end_inset
of
\begin_inset Formula $U$
\end_inset
is the collection of connectors on
\begin_inset Formula $U$
\end_inset
that have not been connected up, yet.
The interior then are the vertexes that are fully connected (and so also,
the edges connecting them.)
\end_layout
\begin_layout Subsubsection*
Non-topological concerns
\end_layout
\begin_layout Standard
Of course, the mesh is a fair bit richer than the set: the edges are meant
to be typed, whereas in set theory, all sets have the same type: set theory
is
\begin_inset Quotes eld
\end_inset
simply typed
\begin_inset Quotes erd
\end_inset
, there is only one type, and that is the type of the set.
\end_layout
\begin_layout Standard
Sets, as commonly presented in textbooks on general topology, are all about
uncountable collections of
\begin_inset Quotes eld
\end_inset
points
\begin_inset Quotes erd
\end_inset
.
The imposition of uncountablility leads to a wide variety of careful distinctio
ns, e.g.
between Hausdorf spaces, which behave in a generally intuitive fashion,
and other spaces, which often do not.
In the current texts, the number of connectors is meant to be finite, or
at least always countable.
The size of the graph is meant to be finite, or at most recursively infinite.
Expanding beyond this would, of course, lead to a wild bestiary of strange
things.
\end_layout
\begin_layout Standard
Another, more arcance distinction is that between Boolean algebras and Heyting
algebras, which is foundational for the category of Frames and Locales,
which in turn form the basic examples for Topoi.
\end_layout
\begin_layout Standard
If one cuts up a graph into disjoint, unconnected parts, by cutting a collection
of edges, then by naive reasoning, the cut edges belong to neither the
one nor the other subgraph.
Thus, applying naive notions about cutting result in an algebra of graphs
that is not a Boolean algebra: the cut edges are neither in the set, nor
in its complement.
The sophisticated way of tackling this is with the concept of a
\begin_inset Quotes eld
\end_inset
Subobject Classifier
\begin_inset Quotes erd
\end_inset
, which, for Boolean algebras, just boils down to the ordinary membership
predicates: functions that take a value in
\begin_inset Formula $2=\left\{ 0,1\right\} =\left\{ F,T\right\} $
\end_inset
.
Either something belongs or doesn't.
It is, or it ain't.
The subobject classifier is conceptually heavyweight, though.
The whole point of having half-edges is that it is conceptually simple.
A jigsaw-puzzle piece with unconnected tabs is a
\begin_inset Quotes eld
\end_inset
thing
\begin_inset Quotes erd
\end_inset
, and there is no particular problem of asking whether the unconnected tabs
belong to it, or to it's set-complelement.
\end_layout
\begin_layout Standard
TODO: Refining the discussion of Frames and Locales, as it applies to the
jigsaw-puzzle metaphor, is arguably an important pre-requiste for the discussio
n of pre-sheaves and sheaves.
However, this is held off until later, as the level of abstraction is more
demanding.
\end_layout
\begin_layout Subsubsection*
Topological conclusions
\end_layout
\begin_layout Standard
At any rate, the goal here would seem to have been met: meshes can be made
to look like a topology.
This means that many of the mechanisms and concepts intended for topologies
can be carried forward, transported and applied to meshes.
\end_layout
\begin_layout Subsection*
Presheaf axioms
\end_layout
\begin_layout Standard
The conventional definition begins as follows.
Let
\begin_inset Formula $X$
\end_inset
be a topological space and let
\begin_inset Formula $C$
\end_inset
be a category.
A presheaf
\begin_inset Formula $F$
\end_inset
on
\begin_inset Formula $X$
\end_inset
is a functor with values in
\begin_inset Formula $C$
\end_inset
having the following properties (quoted from Wikipedia):
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
For each open set
\begin_inset Formula $U$
\end_inset
of
\begin_inset Formula $X$
\end_inset
, there corresponds an object
\begin_inset Formula $F\left(U\right)$
\end_inset
in
\begin_inset Formula $C$
\end_inset
.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
For each inclusion of open sets
\begin_inset Formula $V\subseteq U$
\end_inset
, there is a corresponding morphism
\begin_inset Formula $\mbox{res}_{V,U}:F\left(U\right)\to F\left(V\right)$
\end_inset
in the category
\begin_inset Formula $C$
\end_inset
.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
For every open set
\begin_inset Formula $U$
\end_inset
of
\begin_inset Formula $X$
\end_inset
, the restriction morphism
\begin_inset Formula $\mbox{res}_{U,U}:F\left(U\right)\to F\left(U\right)$
\end_inset
is the identity morphism on
\begin_inset Formula $F\left(U\right)$
\end_inset
.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
If we have three open sets
\begin_inset Formula $W\subseteq V\subseteq U$
\end_inset
, then the composite
\begin_inset Formula $\mbox{res}_{W,V}\circ\mbox{res}_{V,U}$
\end_inset
equals
\begin_inset Formula $\mbox{res}_{W,U}$
\end_inset
.
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Standard
These are rather abstract statements; before they can be given a concrete
meaning, an exploration of what the category
\begin_inset Formula $C$
\end_inset
might be is required.
Since, again, the task at hand is that of providing meaning, the category
\begin_inset Formula $C$
\end_inset
can be taken to be
\begin_inset Quotes eld
\end_inset
the category of meanings
\begin_inset Quotes erd
\end_inset
.
But what is a meaning? To prevent the discussion from devolving into abstract
nonsense, it is best to make
\begin_inset Formula $C$
\end_inset
concrete.
\end_layout
\begin_layout Standard
To that end, a suitable example might be the vectorial/tensorial meanings
provided by word-sense disambiguation (
\emph on
e.g.
\emph default
the
\begin_inset Quotes eld
\end_inset
Mihalcea algorithm
\begin_inset Quotes erd
\end_inset
), or the compositional concept of meaning that is commonly invoked in artificia
l neural nets (
\emph on
e.g.
\emph default
Skipgram models of language.)
\end_layout
\begin_layout Standard
To recap these two different models of meaning ...
(is this wise here or does this belong in a different chapter?)
\end_layout
\begin_layout Standard
TODO this section is obviously, wildly incomplete.
This needs to be started with a realistic non-abstract example.
\end_layout
\begin_layout Standard
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btprint "btPrintCited"
bibfiles "lang"
options "plain"
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