Eric Forgy Free Direct Graded Categories

Over on the n-Forum, Mike Shulman made a comment in response to some of my questions:

Yes, in the construction I described, each noninvertible $f$ goes from $a(i)$ to $b(j)$ whenever $i\le j$. Perhaps what you want is an “$N$-graded category”? In general, if $M$ is any monoid, then an $M$-graded category is a category in which every morphism is assigned an element of $M$, called its “degree,” such that identities have degree 0 and $deg(f\circ g) = deg(f) + deg(g)$. If $C$ is an $M$-graded category, then you could define a category with object set $ob(C)\times M$ and with a morphism from $(a,m)$ to $(b,n)$ being a morphism $f$ from $a$ to $b$ in $C$ such that $m+deg(f)=n$. If $M$ is the natural numbers, then this category will be direct. But in general, for an arbitrary category $C$, there needn’t be any nontrivial way to make it $N$-graded. For instance, in an $M$-graded category, any isomorphism must have a degree which is invertible in $M$, which when $M=N$ means that it must be 0; thus no groupoid admits any nontrivial $N$-grading. Likewise, any idempotent in an $N$-graded category must have degree 0, and hence so must any split monic or split epic, so you can construct lots more categories admitting no nontrivial $N$-grading.

From there, he and David helped me get to the point where I decided to try to construct a free direct graded category. These notes represent my humble attempt. As usual, comments are more than welcome.

Motivation

For various reasons, I have come to question the assumption that fundamentally spacetime is a continuum. If not a continuum, what could it be? The mathematics of differential geometry is so beautiful, it has to be somehow “right”, but just exactly how right is it?

I feel that Urs and I made good progress on developing an alternative “finitary” version of differential geometry in our paper. One major criticism of our work, which I think contributed to its lack of interest in the engineering community is the fact that it seems to be restricted to topological hypercubes modeling spacetime, i.e. it is not cut out for space alone, and it is hard to build a good model of an aircraft using cubes. It is pretty mind numbing to me how time was forced onto us from our considerations and could not be stuffed away. To do calculus, we needed time. Period.

I’ve tried, off and on when time allows, to dispel this idea via diamonation. The idea there is to “extrude” a triangulation of space to a “diamonation” of spacetime.

Of course, I think all of this will ultimately be described nicely as a simple example of space and quantity.

As an intermediate step, lately I have been thinking about extruding categories in general and groupoids in particular. I have somehow ingrained the idea in my head that groupoids relate to “space” and one way to study groupoids is to extrude them into spacetime. If we can do this, we may be able to use Urs and my stuff (or something like it) to study them.

My immediate goal here is to try to make the idea of “extruding a category” precise. For me to make anything precise would be a monumental achievement (and highly unlikely), so my hopes are not high. I am hopeful, however, that I can make enough progress, that I might raise the interest of someone who does have the ability to make the idea precise.

I think the best idea so far is the one born here, i.e. describe the category of extruded categories, which could maybe be called “direct graded categories” or N-graded categories for N the preorder of natural numbers. Then describe the forgetful functor

Maybe what I’m looking for will be the adjoint of this functor. From what little I understand about this, it means I am trying to describe a “free direct graded category”, hence the name of this page.

Anyway, the story is long and convoluted, but I think I (we? please?) can focus on the aspect of “extruding categories” here and if we make progress, then we can see if it fits into some grander scheme.

Example: Groupoid with two objects and two morphism

Consider a groupoid $G$ with two objects $a$, $b$ and two non-identity morphisms $f:a\to b$ and $f^{-1}:b\to a$.