# 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.

**Edit.** **As one of my supervisors recently
pointed out to me, the property of being an `n-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 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.

*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).