A possible alternative model for $\infty$-groupoids

Question

I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is perhaps that of Kan complexes, but I find it very hard to work with. Therefore I've been trying to modify the definition of Kan complexes to find something easier to reason about, and recently I think I might have found something that just might work. Here's a brief summary of the definition I found:

Definition

For each $n\in\mathbb N$, let $[n]:=\{k\in\mathbb N\mid k<n\}$. Let $\mathbf{Fin}$ be the full subcategory of $\mathbf{Set}$ generated by all such $[n]$. Let $\mathbf{Fin_+}$ be the full subcategory of $\mathbf{Fin}$ generated by the non-empty sets, i.e. excluding $[0]$.

Let $\Delta^n:=\mathbf{Fin}(-,[n])$, which defines a presheaf over $\mathbf{Fin}$. Let $\Xi^n:\mathbf{Fin}^\mathrm{op}\to\mathbf{Set}$ be the presheaf that on objects, maps each $[m]$ to the set $\{h:[m]\to[n]\mid\mathrm{im}(h)\subsetneqq[n]\}$, where $\mathrm{im}(h)$ denotes the image of $h$, and on morphisms maps each $g$ to the function $h\mapsto h\circ g$. One can easily verify that this indeed defines a functor. Obviously each $\Xi^n$ is a subpresheaf of $\Delta^n$, and let this inclusion be denoted with $\iota_n:\Xi^n\hookrightarrow\Delta^n$.

We say that a presheaf $P$ over $\mathbf{Fin}$ has the filling property if for every $n\geq2$, $\mathbf{Set}^{\mathbf{Fin}^\mathrm{op}}(i_n,P):\mathbf{Set}^{\mathbf{Fin}^\mathrm{op}}(\Delta^n,P)\to\mathbf{Set}^{\mathbf{Fin}^\mathrm{op}}(\Xi^n,P)$ is surjective, i.e. for every $\phi:\Xi^n\to P$, there is a $\psi:\Delta^n\to P$ such that $\phi=\psi\circ\iota_n$. Finally, we say a presheaf $G$ over $\mathbf{Fin_+}$ is an $\infty$-groupoid if $G([1]\uplus-):\mathbf{Fin}^\mathrm{op}\to\mathbf{Set}$ has the filling property, where $\uplus$ denotes the binary coproduct in $\mathbf{Fin}$.

My Observations

So far, I have been able to prove that one can assign to every topological space $S$ an $\infty$-groupoid $G$ according to this definition such that each $G([n+1])$ is the set of $n$-simplices in $S$, and that this assignment is in fact a functor that preserves both binary products and coproducts.

Furthermore, I have found that given two presheaves $P$ and $Q$ over $\mathbf{Fin}$, it is pretty straightforward to construct a new presheaf $[P;Q]$ that corresponds to the space of natural transformations from $P$ to $Q$ (Altough whether this presheaf also has the filling property is something I'm still working on).

I've also formalized part of what I have so far here in Agda. In fact, this was my original goal: to construct models of HoTT within some version of MLTT.

My Questions

• Is there already some work similar to mine being done in homotopy theory? If so, where can I read about them?
• How does the definition I came up with relate to that of Kan complexes? Can one construct one from the other, or are they even equivalent?

Answer 1

It is known that the category of finite non-empty set is a test category, in particular there exists a model structure on the category of presheaf on $Fin_+$ whose cofibrations are the monomorphisms and which is Quillen equivalent to the model category on simplicial sets or spaces.

I haven't look at the details, but I wouldn't be surprised if the fibrant objects of that model structure would be related to the sort of condition you are talking about - but in any case these fibrants objects produces an acceptable notion of $\infty$-groupoids.

The basic theory of test category is covered in Maltsiniotis' book "La théorie de l’homotopie de Grothendieck" (Astérisque, 301) the results I'm refering too above are in Cisinski's subsequent book "Les préfaisceaux comme modèles des types d’homotopie" Asterisque 308.

This specific example of test category is discussed in section 8.3 of Cisinski' book, but I couldn't find an explicit description of the fibrant object given in the book (but it should be possible to get one if you are familiar with the general theory of Cisinski model structures).