# Graded homotopy structures

###### 29th July, 2019

As I mentioned in a previous post, I recently saw a talk by Rachel Hardeman on the A-homotopy theory of graphs, and it really intrigued me. In particular, it seemed to me that there was some nice structure that could be abstractified: that of a “graded homotopy structure”, as I’ve been calling it in my head. Rather than trying to type out everything in `#math.CT:matrix.org`

, I’ve decided to post it here, in the hope that I might be able to get some answers.

**16/9/19 EDIT:** *As one of my supervisors recently pointed out to me, the property of being an `-homotopy’ is not transitive, and so this example is really a non-example. I’ll keep the post here for reference purposes, but the only useful/true bits are those quoted from [RH19].*

The main reference is [RH19] Rachel Hardeman. *Computing A-homotopy groups using coverings and lifting properties*. arXiv: 1904.12065.

# Preliminaries

- Graphs consist of
*vertices*and*edges*, where we write the edge between vertices and as . All graphs are assumed to be*simple*(no multiple edges between any two points or loops on a single point) and have a*distinguished vertex*. We write to mean the graph along with its distinguished vertex. - A
*(weak) graph homomorphism*is a map of sets such that, for all , either or . It is said to be*based*if . - The
*cartesian product*of the graphs and is the graph with vertex set , with distinguished vertex , and with an edge between and whenever- and ; or
- and .

- The
*path of length*, denoted by , is the graph with vertices labelled from to , and edges for . The*path of infinite length*, denoted by , has vertices labelled by . - We say that two graphs homomorphisms are
*A-homotopic*, written , if there exists some and a graph homomorphism such that- for all ;
- for all ; and
- for all .

We say that two graphs and are

*A-homotopic*if there exist graph homomorphisms and such that and . **N.B.**A-homotopy theory is possibly very different from what you might, at a quick first glance, expect. For example, any two cyclic graphs and (for ) are A-homotopic if and only if , and is contractible (i.e. homotopic to the graph with a single point and no edges) for , but not for any .- The
*A-homotopic fundamental group*of a graph can be defined, as well as a simplicial structure on the group of cochains, and all this sort of stuff. This (amongst other nice formalisations that we would hope for) is done in [RH19].

# “Graded homotopy” structure

Given two A-homotopic graph homomorphisms we can ask for the *minimal* such in the definition of the A-homotopy. We then say that the A-homotopy is an *-homotopy*, and we extend this definition slightly to allow for the fact that we can trivially consider an -homotopy as an -homotopy (in various ways, corresponding to the classical idea of simplicial (co)face/(co)degeneracy (depending on your choice of nomenclature) maps). That is, we say -homotopic to mean “-homotopic with ”.

We say that two graphs are *-homotopic* if there exist graph homomorphisms and such that is -homotopic to and is -homotopic to , with .

Then we can consider the category , which has objects being equivalence classes of -homotopic graphs, and morphisms being equivalence classes of -homotopic graph homomorphisms. This gives us the following structure:

- ;
- ;
- functors that are surjective on objects, where functoriality relies on the fact that -homotopies can be considered as -homotopies.

We can think of the number as some sort of “complexity” of the homotopy: small correspond to “homotopies that can be performed in a few steps” (here it is a good idea to see some of the examples in [RH19] to get an idea of how graph homotopies behave).

# Questions

If anybody has any answers to, or comments about, the following questions (or this post in general) then please don’t hesitate to get in touch!

- What is this structure? Some sort of enrichment? Does it already have a name?
- What other examples exist? For example, it would be nice to get something similar for the category of chain complexes of an abelian category, but I see no way a priori of assigning “complexity” to a homotopy for an arbitrary choice of abelian category. If things are enriched over metric spaces, however, then this is a different story…
- It seems believable that we could define something analogous with instead of . Could we do so for arbitrary (bounded-below) posets?
- Does this tie in to the idea of “approximate composition” (c.f. Walter Tholen’s talk on metagories at ACT2019).