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Category theory See also: QuiverMatrix, AlgebraOfViews

AndriusKulikauskas July 9, 2009 9:02 CET It is possible to define a Category, in the sense of CategoryTheory, as an AbstractedMatrix. I should then be able to apply the results of my thesis work on the SymmetricFunctions of the Eigenvalues of a Matrix. In particular, I can define the Trace and Determinant of the Category, and the Power, Elementary, Homogeneous, Monomial, Forgotten and Schur functions of the eigenvalues of the Category, or at least, their combinatorial equivalents.

===July 9, 2009===

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
• 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

===Notes===

The identity terms for each object are necessarily unique (because if I:A->A and J:A->A are both identities, then I*J = J because I is an identity, and I*J = I because J is an identity, so I = J.)

In doing matrix multiplication, because of terms collapsing, it may be that in the result the same term appears more than once. Given categories A and B, it's not necessarily the case that A*B is a category.