Natural Functorial Categorical Intuition

["Ellis D. Cooper" <xtalv1@netropolis.net>]

Many thanks for responses to my initial post on the Subject.
It was partly motivated by the seeming variety of kinds of intuition
in the cited
references.

Also, I am utterly in awe of categorical intuitions codified, for example, by
adjoint pairs of functors in geometry, algebra, and logic. Until recently
I only dreamed of some day having such an intuition. I think I now have one,
and would like to know if you agree that it is specifically a
categorical intuition.
It is a separate question whether there is a rigorous explication and proof.

Define a (two-dimensional) shape to be a smooth injection of the circle
into the plane. By the Jordan Curve Theorem a shape has an exterior.
Given a point in the exterior -- call it a viewpoint -- there exists
a finite set
of intersections of the lines through the viewpoint -- call them
sightlines -- which are either tangent
to or pass through an inflection point (with respect to an orthogonal
coordinate system in which the sightline is one coordinate).

For a sufficiently remote viewpoint (maybe infinitely far away) there
exist exactly
two of these sightlines tangent to the shape between which
the angle is less than pi radians, and such that all other of these sightlines
are contained within the sweep of one of the two to the other.

I am guessing that if a cover relation (as in Hasse diagrams of
finite posets) is
defined to be an acyclic irreflexive relation, then the category of
finite posets
is a reflective subcategory of the category of cover relations, where
the adjoint
to the inclusion is given by taking the reflexive transitive closure.
Is that right?

Given a shape and a viewpoint
construct a cover relation by declaring that among the elements of the
above set of intersections, one element covers another if (1) it is
encountered earlier by a sweeping sightline as above,
and (2) there exists a segment of the shape connecting the two elements
that contains no other intersections. By the aforementioned adjunction this
construction leads to a finite poset for any given shape and remote viewpoint.

It is my intuitive guess that for a given shape there exists an
algebraic structure
comprised of the set of all finite posets corresponding to its
viewpoints, and that
this algebraic structure involves spans of poset maps among those
finite posets. I guess
moreover that there exists a functor from the isotopy category of shapes to
an appropriately defined category of these algebraic structures,
which I like to
call algebraic models of shapes.

Among the aforementioned intersections there are those which are directly
"visible" from a viewpoint in the sense that no points of the shape intervene
between such a point and the viewpoint. Call the set of directly visible points
the partial-view from the viewpoint. It is my intuitive guess that
the set of all partial-views
of a shape also comprise an algebraic structure. Call it the
partial-view model of
the shape -- clearly it forgets information contained in the algebraic model.

It seems to me that the algebraic model of a shape determines the partial-view
model. That is, if multiple intersections lie on the same sightline
in the algebraic model,
then only the one closest to the viewpoint is in its partial-view. So
there exists a functor from a category of
algebraic models to a category of partial-view models. That word
"closest" is what made my intuition click:
this functor has a left adjoint. In other words, I am guessing that
partial-views
of a shape may be "integrated" to form its algebraic model.

If these intuitions can be rigorously worked out maybe there is a
mathematical theory that is to "dents" in shapes
as algebraic topology is to holes in spaces.

Ellis D. Cooper



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