The n-Category Café

I’ll try to say a few words about these machines before touching upon your question.

First, there are general ‘machines’ for writing down the polyhedral shapes of higher categorical data, which were directly inspired by Stasheff’s classic work on the combinatorics of $A_\infty$-spaces and $A_\infty$-maps. A difference is that units (and how weak maps are supposed to interact with units) don’t appear in Stasheff’s work, but they are not too hard to work in.

Before I describe one such machine, let me mention Stefan Forcey as someone who has also thought a lot about the nature of these shapes and their geometric descriptions as subsets of Euclidean space defined by specified linear inequalities.

If you allow me to talk not about the shapes directly but about their barycentric subdivisions, which are simplicial sets, then I can say very quickly what those are in operadic language. Let $O$ be the monad for the monadic underlying functor

$$U: Operad \to Set^{\mathbb{N}}$$

from nonpermutative (aka ‘Stasheff’) operads to graded sets. Let $t$ be the terminal operad. Then the bar resolution of $t$, which in the language of two-sided bar constructions is denoted

$$B(O, O, t),$$

is a simplicial operad, a universal cofibrant replacement of $t$, and the elements of this operad $M$, which I call the ‘monoidahedral operad’ (after the associahedral operad $K$ of Stasheff), are exactly the simplices which appear in the subdivisions of all those classical shapes like Stasheff polytopes. In other words, if you carefully work through the combinatorics, then out pops the ten triangles in the barycentric subdivision of the pentagon; they reside in the 2-dimensional part of the simplicial set which consists of operations of arity 4, just as one would expect from Stasheff’s work.

The subdivisions of the polyhedral shapes which show up for $A_\infty$ maps can also be described by a bar resolution or universal cofibrant replacement, this time starting with the terminal graded simplicial set $t$ as bimodule over this monoidahedral operad $M$.

Since these techniques are simplicial, I’ll bet that it wouldn’t be that hard to make explicit contact with the algebraic Kan complexes being explored by Thomas. The guiding idea is to consider universal cofibrant replacement of the terminal operad, which is contractible. The guiding philosophy I had all along (since 1995) is that a reasonable (algebraic) theory of weak $n$-categories should be governed by a universal contractible operad in some sense. This seems to be close to Batanin’s philosophy, who worked it all out nicely in terms of globular operads.

I was about to write some paragraphs on why my own progress in implementing this simple philosophy was blocked, and I can still do that if desired, but you asked about the connection with so-called ‘Trimble $n$-categories’, and to what extent Trimble 4-categories correspond to tetracategories. I still feel odd using my name here; I originally called them “flabby $n$-categories” and I’ll do so again for this comment.

As I think you know (Urs), flabby $n$-categories were introduced to give a rigorous algebraic meaning to the term ‘fundamental $n$-groupoid’. The original definition was based on topology, and took advantage of a specific curious feature of $Top$ itself and the interval object therein: that there’s a contractible operad $E$ whose $n$-ary operations are maps

$$I \to I^{\vee n} = I \vee \ldots \vee I$$

(where $\vee$ is a join on bipointed spaces, basically tensor product of cospans). How is this specific to $Top$? Because in a lot of model categories, there might not be any such maps (as maps between bipointed objects)! For example, there are no such maps for the interval object in simplicial sets.

The relevance of this particular operad $E$ is that we have, for any space $X$, a natural action

$$E(n) \times P X(x_0, x_1) \times \ldots \times P X(x_{n-1}, x_n) \to P X(x_0, x_n)$$

where $P X(x, y)$ is the space of paths $\alpha: I \to X$ such that $\alpha(0) = x$ and $\alpha(1) = y$. (Full details have been written up by Tom Leinster and by Eugenia Cheng & Aaron Lauda; see for instance here.) Given that, we define

$$\Pi_n(X) := \Pi_{n-1}(P X)$$

and note that by applying the product-preserving functor $\Pi_{n-1}$ to this natural action, $\Pi_n(X)$ carries an action by $\Pi_{n-1}(E)$, which is a pithy way of inductively defining what a flabby $n$-category is.

The ‘machine’ for defining flabby $n$-categories can be generalized away from $Top$, under certain conditions. We could work with other model categories $V$ (for example), provided that a few basic ingredients are in place:

• There’s a product-preserving functor $\Pi_0: V \to Set$, to get the induction started;
• There’s a contractible operad $M$ valued in $V$;
• $M$ acts naturally on “enough” path spaces $P X$ to push the induction through.

I’m not sure what I mean by this last; I need to think about it more. But if some such set of conditions applied to the monoidahedral operad $M$ in simplicial sets, then the connection between, on the one hand, my hoped-for approach to things like tetracategories, and on the other hand, flabby $n$-categories, could be made a great deal tighter I think. (There are however strict interchange laws for flabby $n$-categories, so this is not a “fully weak” notion of $n$-category, which is what I was originally hoping to describe in my various approaches to $n$-categories.)

If this comment is hard to follow, then I’m happy to try to make it clearer.

Let me slightly modify that question: does anyone know an interesting and easily describable model category where this assumption is not satisfied? I feel like there must be trivial examples as well, but I’m also guessing that it will fail in some Bousfield localizations. Note in particular that in any such example, it must be the case that not all objects are fibrant, since any acyclic cofibration with fibrant domain is necessarily split monic.

Here’s a pretty trivial example. There’s a model structure on the category of groupoids which Bousfield localizes the usual model structure and which is Quillen equivalent to $Set$ (with its trivial model structure where the only weak equivalences are isomorphisms). The cofibrations are the functors which are injective on objects, the weak equivalences are the functors inducing bijections on $\pi_0$, and the fibrations are the isofibrations which additionally induce isomorphisms on $\pi_1$ with all basepoints. The fibrant objects are the disjoint unions of cliques, and there can be lots of acyclic cofibrations (with non-fibrant domain) that are not injective on arrows.

In a simplicial T-complex, there is a requirement that each horn has a unique ‘thin’ filler. The properties of thin elements are described (3 axioms), thus there is a choice of thin filler specified by the choice of the thin elements, thus a certain ‘algebraic’ quality about the notion.

In a group T-complex, there is even an algorithm to specify the fillers. The strictness of the model is obtained by the T-complex condition which is simply that in each dimension the subgroup generated by the degenerate elements intersects the Moore complex trivially - that simple. Again this is algebraic.

In a complicial set (not the weak version) there is a very similar condition on the thin elements and again this amounts to a choice of composite via a choice of filler. Dominic’s work predates the re-emergence of quasi-categories, as it feeds off Ross Streets earlier work and John Robert’s notion of a model for non-Abelian cohomology.

In Jack Duskin’s hypergroupoid again the condition is algebraic, given by an isomorphism from a space of horns. (The situation is slightly different as in the usual form the models are often truncated.)

Gary Nan Tie’s work explored the links between Duskin’s models and Nick Ashley’s simplicial T-complexes. I should also mention Pilar Carassco’s hyper-crossed complex.

I do not say that what Thomas has done is not different, no, merely that I am very surprised not to see some mention of this extensive earlier work.

(I can provide references for the above articles if people would like, but in fact most are referenced in versions of the Menagerie, and can be sought there.)

In a simplicial T-complex, there is a requirement that each horn has a unique ‘thin’ filler.

Oh, I see.

So after we have chosen all fillers in a Kan complex as Thomas considers, if we declare all chosen fillers as thin, do we get a simplicial T-complex?

(Have to run now, will look into this in more detail later.)

That is why I replied! There are nice results that I go through in the Menagerie. (I think that the present version in the Lab has that stuff in but if not I can send you the on going one (which is a lot longer!))
I will put some more references on the Lab. Both simplicial T and complicial are poorly represented. The Menagerie treatment is meant as an introduction, so any comments as to where more detail or examples or …. would be welcome.

I have yet to read your paper so don’t thank me for that yet! One obvious question is whether the algorithmic algebraic fillers given for a simplicial group in May’s book for instance, give it the structure of an algebraic Kan complex, and if they do, does thaat structure give an induced one on the classifying space. (This may be in your paper in which case I should have read it before asking the questions!)

One thought that you probably have some ideas about (and I ask this of all readers of the café postings) if one has any version of T-complex, or algebraic Kan complex, then it is natural to ask for the T-fibrations (and there may be an algebraic Kan fibration in your paper). Has anyone a candidate for such a thing? I have asked Ronnie Brown and he does not remember seeing such an idea. The point is that the classifying space of a crossed complex classifies fibre bundles with some thinness conditions in the fibres …, but what form do they most naturally take?