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Stream: community: our work

Topic: Clémence Chanavat

Clémence Chanavat (May 05 2025 at 09:44):

I have a new paper on the arxiv:

• Gray products of diagrammatic $(\infty, n)$-categories

where I study the Gray product of diagrammatic sets. In a previous paper with @Amar Hadzihasanovic, we showed that diagrammatic sets are a nice place to do higher category theory: we constructed for each $0 \le n \le \infty$ a model structure whose fibrant objects are the diagrammatic $(\infty, n)$-categories.

Clémence Chanavat (May 05 2025 at 09:44):

Now diagrammatic sets are presheaves over the category of atoms, whose objects are a general collection of combinatorial objects for the possible shapes of cells in a higher category (including notably the simplices, the cubes, the globes, and the (positive) opetopes). In particular, atoms are closed under Gray product. Hence via Day convolution, diagrammatic sets inherit this monoidal structure.

Clémence Chanavat (May 05 2025 at 09:45):

In this paper, I answer the natural question of whether this Gray product is homotopical with respect to the aforementioned model structures. Unsurprisingly: yes. The critical result this rely on, and the thing I find most interesting about it, is the following: given $u$ a $n$-cell in a higher category $C$ and $v$ also a $n$-cell in another higher category $D$, then we know that the $n$-cell $u \times v$ in $C \times D$ is, in general, invertible only when both $u$ and $v$ are. For the Gray product $- \otimes -$, this works differently: the $(n + n)$-cell $u \otimes v$ is invertible in $C \otimes D$ as soon as $u$ or $v$ is invertible (notice that I am talking about the fully lax Gray product, and not some pseudo-variation, which forces some tensored cells to be invertible by definition).

Clémence Chanavat (May 05 2025 at 09:45):

Even more interesting, given $u'$ and $v'$, inverses of $u$ and $v$, I would very much be in the incapacity to give you the precise formula for the inverse of $u \otimes v$: it is very far from $u' \otimes v'$ as one could expect. If you want the formula for the particular case where $v$ is a $1$-cell, you can check out the big computations of Lemma 4.2 in this paper by Ara and Lucas (which is where I learnt this result in the first place).

Clémence Chanavat (May 05 2025 at 09:46):

So how does one prove that $u \otimes v$ is invertible then? When in the case of diagrammatic sets, I developped with Amar in a previous paper the notion of contexts and natural equivalences between them, the latter being a coinductive notion of natural isomorphism between contexts, aka the "internal hom-functors". There, we proved my favorite result so far, stating roughly that if $u \colon a \Rightarrow b$ is a cell in a diagrammatic set, then the internal hom-functor $hom(-, u) \circ - \colon hom(-, a) \to hom(-, b)$ is invertible up to a natural isomorphism if and only if the cell $u$ is invertible.

Clémence Chanavat (May 05 2025 at 09:46):

Then, instead of proving that $u \otimes v$ is invertible by hand, I prove that the hom-functor represented by $u \otimes v$ can be decomposed into smaller pieces that I already know to be hom-functors represented by invertible cells, hence I can conclude.

Clémence Chanavat (May 05 2025 at 09:47):

Anyway, what I am trying to say is that this way of doing things has a very similar flavor than the usual lower dimensional Yoneda-yoga. With diagrammatic sets, it becomes available uniformly in all dimensions, pleasantly mixing together the geometric and algebraic nature of higher categories: one manipulates equations whose terms have an underlying regular and directed CW complex. The equality is represented by a homotopy, and pasting (=composing) terms together is a pushout of the underliyng space.

James Deikun (May 05 2025 at 12:24):

Diagrammatic sets seem like they are intended to restore some of the flexibility with regard to arity that you lose with opetopes/multitopes while still maintaining the presheafiness that you lose with full-on computads/polygraphs; is this a fair assessment? I'm a little curious as to where exactly they differ from computads, actually.

Clémence Chanavat (May 05 2025 at 14:47):

Well if you have a diagrammatic set $X$, then you have indeed the strict $\omega$-category $Pd(X)$ of all pasting diagrams inside it. The composition is free, in the sense that the composite of two composable arrows $f$ and $g$, both of shape $\bullet \to \bullet$, is the pasting $f \#_0 g$ whose underlying shape is $\bullet \to \bullet \to \bullet$. So in that sense, it is very much like a (regular) polygraph. Here regular is in the sense of Henry's regular polygraph, where the input and output boundaries of any $(n + 1)$-cell is required to be round, i.e. CW models of the $n$-balls

Clémence Chanavat (May 05 2025 at 14:48):

However, $Pd(X)$ is not a (regular) polygraph. Pasting of pasting diagrams satisfies the equation of strict $\omega$-category (associativity, unitality and exchange), but also a bunch more. This is quite a new phenomenon, that we did not really study in depth yet, but you can find some detailed explanation in chapter 8 of Amar's book. This is very much linked to acyclicity conditions of the underlying pasting diagrams, which do not appear, say, with opetopes shapes. Those extra equations only arise starting dimension 4, and are all topologically sound (and I find them very mysterious)

Clémence Chanavat (May 05 2025 at 14:50):

Another way this diverges from polygraph is that diagrammatic sets are closer in spirit to simplicial (or cubical, or...) sets than to polygraphs: the base category of atoms is made of two classes of maps, the cofaces which restrict a cell to a subcell, and the codegeneracies, which models a certain algebra of units and unitors. For instance the simplex category is a full and faitfull subcategory of the atom category. A contrario, there is no such thing in the category of atom that represents the $\omega$-functor from to $\bullet \to \bullet$ to $\bullet \to \bullet \to \bullet$ mapping the identity of the walking 1-cell to the identity of the composite of the two walking 1-cells

Clémence Chanavat (May 05 2025 at 15:01):

Of course, one could devise a notion of "stricter" $\omega$-category where on top of exchange and associativity, those extra axioms would have to be satisfied, and indeed in that case, $Pd(X)$ would be a "stricter" polygraph. But of course, those stricter $\omega$-categories would still not be sound topologically (ie. model homotopy types), since the usual Eckmann-Hilton argument would still apply. Yet diagrammatic sets model homotopy types: the diagrammatic $(\infty, 0)$-categories are equivalent to the usual Kan complexes, which will never happen with any version of strict $\omega$-groupoids

Clémence Chanavat (May 05 2025 at 15:03):

But yeah, other than that it is morally true that whatever one do with strict $\omega$-categories using the symbol $=$, one can do with the symbol $\simeq$ in diagrammatic sets (provided the underlying argument is sound topologically).