### Grothendieck-Maltsiniotis ∞-categories

#### Posted by Mike Shulman

Yesterday Georges Maltsiniotis posted a paper on the arXiv in which he presents a definition of $\infty$-groupoid, said to be due to Grothendieck in Pursuing Stacks, and modifies it to give a similar definition of $\infty$-categories. (These definitions have been available on his website for a while, but only in French.)

These “Grothendieck-Maltsiniotis” definitions are quite similar to that of Batanin, especially as modified by Leinster. The precise relationship between the two is studied in the thesis of D. Ara, a student of Maltsiniotis, but it’s not hard to get an intuitive idea of their similarities and differences.

Both definitions are alike in that they have, as underlying data, a globular set $X$, equipped with extra algebraic structure. Moreover, in both cases this algebraic structure consists of operations whose

**input**is a “globular pasting diagram” of cells in $X$, and whose**output**is a single cell in $X$.

Recall that a *globular pasting diagram* is “an arrangement of cells which would have a unique composite in a strict $\infty$-category.” There are lots of pictures of these in the Cheng-Lauda guidebook.

The two definitions are again similar in that they are each not really a single definition, but a family of them. In Batanin’s world you get one definition for each globular operad; in Maltsiniotis’ world you get one definition for each “globular theory.” And again in both cases, not every operad/theory gives you a good definition of “weak $\infty$-category;” you need two additional conditions:

- All composites and coherences that should exist, do exist. In particular, there is a way to compose any globular pasting diagram, and any two ways to compose a given diagram are related by specified equivalences.
- There are no unwanted
*equations*between operations. This is what gives us a definition of*weak*$\infty$-category; in both cases, there is an operad/theory for strict $\infty$-categories which satisfies the first condition, so we need to exclude it somehow.

Finally, in both cases these conditions can be formulated in terms of a weak factorization system on the category of globular operads/theories, in terms of which we are interested in *cofibrant replacements* of the operad/theory for strict $\infty$-categories. The first condition says that the map $T_{weak}\to T_{strict}$ is an “acyclic fibration,” while the second says that $T_{weak}$ is cofibrant.

So much for the similarities; how about the differences? It seems to me that there are two main differences. The first lies in the treatment of dimensions. In Batanin’s world, every operation takes a globular pasting diagram of some dimension $n$ as input and gives a cell of the same dimension $n$ as output. However, there are operations we expect to exist which do change dimension: for instance, the identity of an $n$-cell is an $(n+1)$-cell, while its source and target are $(n-1)$-cells. Coherence isomorphisms are also operations which raise dimension; for instance, the standard biased associator for 1-cell composition takes a 1-dimensional pasting diagram as input (three composable arrows) and gives a 2-cell as output (the associator relating two composites). Batanin’s definition deals with this in two ways:

- The globular-set structure of $X$, including sources and targets, is given
*a priori*before any globular operad acts on it. Every $n$-dimensional operation in a globular operad also has a source and target, which are $(n-1)$-dimensional operations, and the action of “applying an operation to a pasting diagram” respects sources and targets. - A pasting diagram comes with a
*specified*dimension, which may be greater than the highest dimension of cell occurring in it. Thus, for instance, the associator is a 2-dimensional operation, whose source is the pasting diagram $\to\to\to$*regarded as a 2-dimensional pasting diagram*which happens to contain only 1-cells.

(Of course, phrased this way it sounds very artificial, but in fact it’s all quite natural when you write it using generalized multicategories relative to the free strict $\infty$-category monad. An $n$-dimensional globular pasting diagram in a globular set $X$ is an $n$-cell in the free strict $\infty$-category on $X$, and in this strict $\infty$-category every $n$-cell comes with an identity $(n+1)$-cell, which has an identity $(n+2)$-cell, and so on; these identity pasting diagrams give a very natural way to “regard a pasting diagram as having a higher dimension.”)

In the Grothendieck-Maltsiniotis definition, on the other hand, operations are not required to preserve dimension, and in fact pasting diagrams have no specified “dimension,” although we could call their dimension equal to the highest dimension of a cell occurring in them (if we so wished). Now identities, sources and targets, and coherence equivalences are all “operations” on an equal footing, and in particular the globular set structure of $X$ is not formally separated from the action of a globular theory on it.

The second difference lies in how “contractibility” (the existence of composites, identities, and coherence) is specified. Following Leinster, let us say that a globular operad is *contractible* if for any two operations $u$ and $v$ of the same $n$-dimensional input shape $\pi$, and any $(n+1)$-dimensional input shape $\sigma$ whose source and target is $\pi$, there exists an operation $w$ of shape $\sigma$ whose source and target are $u$ and $v$. (Batanin’s notion of contractibility was more biased.) It’s easy to see how this gives us composition: given any shape $\sigma$ we want to compose, if we decide on a way to compose up its boundaries, there is some compatible way to compose it as well. We get identities from “degenerate” pasting diagrams, as remarked above. Moreover, we also get coherences from degeneracies: given any two operations $u$ and $v$ which compose the same $n$-dimensional shape, let $\sigma$ be the corresponding degenerate $(n+1)$-dimensional shape; then $w$ is a coherence cell mediating between $u$ and $v$.

On the other hand, in the Grothendieck-Maltsiniotis world, we say that a globular theory is *contractible* if given any two operations $u$ and $v$ with the same input shape $\pi$ and whose outputs are parallel $n$-cells, there exists an operation $w$ with the *same* input shape $\pi$ and whose output is an $(n+1)$-cell from $u$ to $v$. In this case, it is obvious how we get the coherences: given any two ways to compose the same shape, we get immediately a coherence cell relating them. Composition is actually also easy: if we have a pasting diagram of shape $\pi$ we want to compose, and we have ways $u$ and $v$ to compose up its source and target, then $u$ and $v$ are again operations with the same input shape and we get an operation $w$, of output one dimension higher, from one to the other. Finally, we get an identity $(n+1)$-cell for any $n$-cell by taking $u$ and $v$ both to be the identity *operation* taking an $n$-cell to itself.

Note that the fact that sources and targets are just ordinary operations in the Grothendieck-Maltsiniotis world means that *a priori*, they are completely symmetric. In particular, if we take $u$ to be the target of a single $n$-cell and $v$ its source, then we obtain an operation assigning to any $n$-cell, another $n$-cell in the opposite direction, and the coherence laws make the two coherently inverse. This is why the first version of the definition is actually a definition of $\infty$-*groupoid*. To get a definition of $\infty$-category, you have to break the symmetry between $u$ and $v$ in an appropriate way, which Maltsiniotis describes, but which I haven’t digested yet.

To conclude, here is an attempted translation of part of Theorem 6.6.8 from Ara’s thesis: *The functor $M$ induces an equivalence of categories between the category of homogeneous globular theories over $\Theta$ and the category of $\omega$-operads…. Moreover, if $C$ is a homogeneous globular theory over $\Theta$ then the categories $Mod(C)$ and $M_C$-Alg are canonically isomorphic.*

And here is part of Corollary 6.7.11: *Let $C$ be a homogeneous globular theory over $\Theta$ and $M$ the associated $\omega$-collection. The $\omega$-collection $M$ is contractible if and only if the globular theory $C$ is a pseudo-coherator.*

I haven’t yet read the definitions of “homogeneous globular theory over $\Theta$” or the functor $M$ or “pseudo-coherator,” but at least “homogeneous” sounds like a condition relating to the “treatment of dimensions” discussed above. So this seems to be saying that the Grothendieck-Maltsiniotis family of definitions essentially includes Batanin’s. I haven’t reproduced the actual definitions here, but they don’t seem particularly difficult. They do feel a little more *ad hoc* to me than (Leinster’s rephrasing of) Batanin’s definition, but possibly that’s just what I’m familiar with; I’d be interested to hear other people’s reactions.

## Re: Grothendieck-Maltsiniotis ∞-categories

Thanks, Mike. While I don’t have time for this now, here is an $n$Lab entry:

Grothendieck-Maltsiniotis $\infty$-category.

By the way, the links to the ENS-server with Maltsiniotis’ previous preprint and the thesis of Ara don’t work for me. Anyone else have this problem? Maybe it’s just my machine being stubborn.