An abstracted matrix approach to category theory?

An abstracted matrix approach to category theory?

[Andrius Kulikauskas]

I'm studying the basics of Category Theory.  I appreciate help with my
questions!

My main interest is to "know everything and apply that knowledge
usefully".  I have a set of notes that reflect more than 25 years of
work on this:
http://www.worknets.org/wiki.cgi?LivingByTruth/Summary

Category theory may be relevant in describing an "algebra of views" and
the composition of views.  Consider a lost child who goes to where she
thinks her mother will look for her.  Such a child is taking up her view
of her mother's view of her view of her mother's view of her view.  In
this way, it is possible for two people to have a "good understanding"
even if they presently have no channel of communication.
http://www.worknets.org/wiki.cgi?LivingByTruth/LostChild

Has anybody ever studied an algebra of views or perspectives?

I'm studying from "Basic Category Theory for Computer Scientists" by
Benjamin C. Pierce.

I'm interested to apply the mathematics from my Ph.D. thesis "Symmetric
Functions of the Eigenvalues of a Matrix" (UCSD, 1993).
http://www.worknets.org/upload/AndriusKulikauskas/AndriusKulikauskasThesis.pdf
I am wondering if it is possible to consider a category as an
"abstracted matrix" and take its trace or determinant, etc.

Symmetric functions are those functions such as X1 + X2 + X3 or X1*X2*X3
which stay the same even if you permute the variables.  They are
ubiquitous in algebraic combinatorics because objects are generally
built up with labels (which may at some point taken off) where the order
of the labels themselves generally shouldn't matter.  I studied
algebraic combinatorics as the "basement" of mathematics where objects
are constructed, and thus in some sense, foundational.  I was also
interested that the vector space of symmetric functions had several
"natural" bases, namely the power, elementary, homogeneous, monomial,
forgotten, Schur which I thought might reflect how humans look at things.
http://mathworld.wolfram.com/SymmetricFunction.html

For my thesis, I noted that the trace and determinant of a matrix are
symmetric functions of its eigenvalues, namely the sum E1 + E2 + E3
+...  and the product E1*E2*E3...   We can calculate these and all
symmetric functions of eigenvalues straightforwardly in terms of the
edges of a matrix, which is interesting, because in general, we can't
calculate the eigenvalues themselves.  I gave combinatorial
interpretations of these functions in terms of walks, cycles, Lyndon
words, rimhook tableux, etc on a generic matrix A.  What's very
interesting is that if you set that generic matrix A to be a diagonal
matrix X (whose eigenvalues are necessarily the diagonal elements), then
the results collapse to yield the usual symmetric functions.  The matrix
A can be specialized in other ways as well.  Also, the symmetric
functions of the eigenvalues capture the combinatorics of all of the
theorems that I could find about generic matrices.

I am drawn to think of a category as a "generic matrix" from the
combinatorial point of view, if possible.  In such a view:
* the objects would be the dimensions of a square matrix A (and I allow
that the objects are unordered and perhaps uncountable, so that I speak
of an "abstracted matrix").
* each arrow F:X->Y would be a term in an (unordered, commutative) sum
that corresponds to Axy
* composition and associativity imply that each term in such a sum can
be written (perhaps in more than one way) as a path (composed of arrows)
starting at X and passing through perhaps several other nodes and
arriving at Y.
* an identity 1:X->X can be thought of as a term 1 in the series for the
diagonal cell Axx  (I suppose that the identity need not be unique? or
must it?)
* if there are no arrows from X to Y, then we write a 0 in the cell Axy

What I would then want to say is that the series Axy consists of terms
such that:
* some are distinct atoms F:X->Y
* the rest are generated by composition   F:X-> ... ... ... ->Y
* there is an equivalence relationship by which certain compositions are
equal, perhaps imagined to collapse

I suppose that this is alway possible in that, given X, Y, we can simply
take as our "atoms" the entire collection of arrows from X to Y.  Then
all compositions must collapse back into our atoms.  Indeed, there is a
partial order (by inclusion) of the sets of arrows that may be
understood as "the atoms".  In this partial order we may ask if there is
any set of arrows that is "natural", for example, depending on the
equivalence relationship, there may or not be a smallest set of atoms.
Perhaps this search for atoms might be analogous to what "representation
theory" does for groups.

This is a very "concrete" approach and when I wrote to Joseph Goguen
(who sadly passed away) about it, he didn't think that was a good way to
think about categories.  But I have in my thesis a powerful perspective
for dealing with a generic matrix.  And I think that an "abstracted
matrix" is metaphysically, conceptually about the most basic object that
math offers.

I suppose that my question is, Assuming a labeling for objects, What can
we say about the systems for labeling arrows (the ones that drop
parentheses, thanks to associativity)?

Once we have a system for labeling the arrows, then my thesis work
observes that various constructions accord with symmetric functions of
"eigenvalues" of the related matrix.  For example, the trace generates
the closed walks (from any X back to itself) that you can take within
the category.

The determinant of a category is the product of signed cycles.  The sign
is straightforward if there is an implicit order in the objects, and if
one does not exist for some of the objects, then we can simply leave
that unresolved.  Where the sign exists, then there can be some
collapsing.  Are there theorems in category theory where products of
signed cycles are generated?

Note that we may have two categories with the same objects (or we may
always extend the sets of objects and embed the categories).  Then
multiplication of matrices gives a composition of the categories.  We
can multiply (and compose) a matrix with itself.  (Indeed, a category
may be thought of as the matrix generated from its atoms Q by series
expansion matrix multiplication 1/(1-Q) ).  And the matrix/category A
may be thought as the paths accepted by an automata and Q as the basic
rules.

Has this "abstracted matrix" approach been taken?  I appreciate
suggestions on what to read and what to explore.

Thank you,

Andrius Kulikauskas
Minciu Sodas
http://www.ms.lt
ms@ms.lt
+370 699 30003
Dukiskiu km
Butrimoniu sen
Alytaus raj
Lithuania


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