# Graded homotopy structures

As I mentioned in the 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.

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

### Preliminaries

- Graphs $G$ consist of
*vertices*$V(G)$ and*edges*$E(G)$, where we write the edge between vertices $s$ and $t$ as $[s,t]$. 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*$x\in V(G)$. We write $(G,x)$ to mean the graph along with its distinguished vertex. - A
*(weak) graph homomorphism*$\varphi\colon (G,x)\to(H,y)$ is a map of sets $V(G)\to V(H)$ such that, for all $[s,t]\in E(G)$, either $\varphi(s)=\varphi(t)$ or $[\varphi(s),\varphi(t)]\in E(H)$. It is said to be*based*if $\varphi(x)=y$. - The
*cartesian product*$G\mathbin{\square} H$ of the graphs $(G,x)$ and $(H,y)$ is the graph with vertex set $V(G)\times V(H)$, with distinguished vertex $(x,y)$, and with an edge between $(s,u)$ and $(t,v)$ whenever- $s=t$ and $[u,v]\in E(H)$; or
- $u=v$ and $[s,t]\in E(G)$.

- The
*path of length $n$*, denoted by $I_n$, is the graph with vertices labelled from $0$ to $n\in\mathbb{N}$, and edges $[i,i+1]$ for $i=0,\ldots,n-1$. The*path of infinite length*, denoted by $I_\infty$, has vertices labelled by $\mathbb{Z}$. - We say that two graphs homomorphisms $\varphi,\psi\colon(G,x)\to(H,y)$ are
*A-homotopic*, written $\varphi\simeq_A\psi$, if there exists some $n\in\mathbb{N}$ and a graph homomorphism $h\colon G\mathbin{\square} I_n\to H$ such that- $h(s,0) = \varphi(s)$ for all $s\in V(G)$;
- $h(s,n) = \psi(s)$ for all $s\in V(G)$; and
- $h(x,i)=y$ for all $0\leqslant i\leqslant n$.

We say that two graphs $G$ and $H$ are

*A-homotopic*if there exist graph homomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to G$ such that $\psi\circ\varphi\simeq_A\operatorname{id}_G$ and $\varphi\circ\psi\simeq_A\operatorname{id}_H$. **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 $C_n$ and $C_m$ (for $m,n\geqslant 3$) are A-homotopic if and only if $m=n$, and $C_n$ is contractible (i.e. homotopic to the graph with a single point and no edges) for $n=3,4$, but not for any $n\geqslant 5$.- 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 $\varphi\simeq_A\psi$ we can ask for the *minimal* such $n\in\mathbb{N}$ in the definition of the A-homotopy. We then say that the A-homotopy is an *$n$-homotopy*, and we extend this definition slightly to allow for the fact that we can trivially consider an $n$-homotopy as an $(n+1)$-homotopy (in $n+1$ various ways, corresponding to the classical idea of simplicial (co)face/(co)degeneracy (depending on your choice of nomenclature) maps). That is, we say $n$-homotopic to mean “$m$-homotopic with $m\leqslant n$”.

We say that two graphs $G,H$ are *$n$-homotopic* if there exist graph homomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to G$ such that $\psi\circ\varphi$ is $m_1$-homotopic to $\operatorname{id}_G$ and $\varphi\circ\psi$ is $m_2$-homotopic to $\operatorname{id}_H$, with $\operatorname{max}{m_1,m_2}=n$.

Then we can consider the category $\mathsf{Grph}_n$, which has objects being equivalence classes of $n$-homotopic graphs, and morphisms being equivalence classes of $n$-homotopic graph homomorphisms. This gives us the following structure:

- $\mathsf{Grph}_0 = \mathsf{Grph}$;
- $\mathsf{Grph}_\infty = \mathrm{Ho}(\mathsf{Grph})$;
- functors $\mathsf{Grph}_n \to \mathsf{Grph}_{n+1}$ that are surjective on objects, where functoriality relies on the fact that $n$-homotopies can be considered as $(n+1)$-homotopies.

We can think of the number $n$ as some sort of “complexity” of the homotopy: small $n$ 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 $\mathbb{R}^{\geqslant0}$ instead of $\mathbb{N}$. 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).