nLab diagrammatic set

Idea

Diagrammatic sets are presheaves over the atom category. There are kind of geometric polygraphs, and can serve as a model for higher categories.

Definition

Definition

(diagrammatic set)
A diagrammatic set $X$ is a presheaf over $\odot$, the category of atoms and cartesian maps. Diagrammatic sets and natural transformations form the category $\odot\mathbf{Set}$

Properties

Let $\mathbf{RDCpx}$ be the category of regular directed complexes and cartesian maps. Since atoms are regular directed complexes, there is a full subcategory inclusion $i \colon \odot \hookrightarrow \mathbf{RDCpx}$. Moreover, any regular directed complex $P$ is canonically a colimit of its atoms, in the sense that there is a functor

$$P^* \colon P \to \mathbf{RDCpx}$$

sending $x \in P$ to the atom $\mathrm{cl}(x)$ and $x \le y$ to the inclusion $\mathrm{cl}(x) \hookrightarrow \mathrm{cl}(y)$, such that

$$\colim P^* \cong P.$$

Since $\odot\mathbf{Set}$ is the free cocompletion of $\odot$, one might wonder what happens when we compute the same colimit, but with the functor:

$$P^* \colon P \to \odot\mathbf{Set}.$$

sending $x \in P$ to $y(\mathrm{cl}(x))$, where $y$ is the Yoneda embedding. Unsurprisingly, we obtain the same thing, in the following precise sense. The subcategory inclusion $i$ defines a restricted Yoneda embedding:

$$i^* \colon \mathbf{RDCpx} \to \odot\mathbf{Set}$$

that sends a regular directed complex $P$ to the presheaf $\mathbf{RDCpx}(i^*(-), P)$.

Proposition

The functor $i^*$ is fully faithful.

This means that $i$ is a dense functor, so we see $\mathbf{RDCpx}$ as a full subcategory of $\odot\mathbf{Set}$, factoring the Yoneda embedding as

$$\odot\mathbf{Set} \hookrightarrow \mathbf{RDCpx} \hookrightarrow \odot\mathbf{Set}.$$

Furthermore, the category $\odot$ is monoidal for the Gray product. By Day convolution, the category $\odot\mathbf{Set}$ is also monoidal for (what we also call) the Gray product, with monoidal unit $y\mathbf{1}$, where $\mathbf{1}$ is the (Point). Let $P, Q$ be regular directed complexes, we have two ways of computing $P \otimes Q$:

• via Day convolution when seeing $P, Q$ as presheaves over $\odot\mathbf{Set}$,
• directly via the Gray product of regular directed complexes.

Of course, those two ways give the same result, i.e. $i^*(P \otimes Q) = i^*(P) \otimes i^*(Q)$.

Another feature is the boundary. Similarly to the boundary of a simplex, any atom $U$ has a boundary $\partial U$ which can be computed in two ways:

• as the regular directed complex $U \backslash \{ \top_U \}$ with underlying poset $U$ without its top-element,
• as the diagrammatic set generated from $yU$ minus its unique non-degenerate cell in dimension $\dim U$.

Again, those two ways coincide in the sense that $i^*(\partial U) = \partial yU$.

Diagrammatic sets and higher category theory

The richeness of shapes in the category of atoms, as well as the quite rigid behavior of the category $\odot$ where they live make them a good candidate for modelling higher categories. In this perspective, in a diagrammatic set $X$, a natural transformation $x \colon U \to X$ whose domain is an $n$-dimensional atom, correspond to an $n$-cell in $X$. If $U$ is more generally an $n$-dimensional molecule, we call it an $n$-diagram, while when $U$ is furthermore round, we speak of a round $n$-diagram.

Before the more conjectural case of weak higher categories, we can deal with the case of higher groupoids.

Higher groupoids

Definition

(diagrammatic horn)
Let $U$ be an atom, let $V \sqsubseteq \partial^\alpha U$ be a rewritable submolecule of either the input or output boundary. The diagrammatic horn defined by $U, V$ is the regular directed complex

$$\Lambda^V_U \coloneqq U \backslash (V \backslash \partial V).$$

We write

$$\lambda^V_U \colon \Lambda^V_U \hookrightarrow U$$

for the inclusion of a horn into its atom.

Example

Recall that the category $\odot$ contains as a full subcategory the oriented simplices, which is a category isomorphic to the simplex category. Then, any simplicial horn is also a diagrammatic horn.

Definition

(Kan diagrammatic set)
A Kan diagrammatic set is a diagrammatic set $X$ in which all diagrammatic horns have a filler. That is, for all horn inclusions, and all solid arrows in there exists a diagonal dashed filler.

The paper CH24homotopy settles the Kan diagrammatic sets as model of ∞-groupoids in their following theorem:

Theorem

There exists a cofibrantely generated model structure on diagrammatic sets where

• cofibrations are monomorphisms, and are generated by $\{ \partial U \hookrightarrow U \mid U\; \text{atom} \}$,
• acyclic cofibrations are generated by the diagrammatic horn inclusions,
• fibrant objects are Kan diagrammatic sets,

which is Quillen equivalent to the classical model structure on simplicial sets.

Remark

Note that the existence of the model structure and the Quillen equivalence already follow from the fact that $\odot$ is a test category.

We can realize the Quillen equivalence in two different ways. For the first one, recall that any atom is in particular a poset, of which we can take the nerve, to get a simplicial set, which defines a functor from $\odot$ to $\mathbf{sSet}$. Then we define $\mathrm{Sd}_\odot \colon \odot\mathbf{Set} \to \mathbf{sSet}$ as the Yoneda extension of this functor, and we name its right adjoint $\mathrm{Ex}_\odot$. Then:

Proposition

The adjunction $\mathrm{Sd}_\odot \dashv \mathrm{Ex}_\odot$ realizes the Quillen equivalence of Theorem .

But also, recall that, up to isomorphism, the simplex category is a full subcategory of $\odot$, thus of $\odot\mathbf{Set}$. The Yoneda extension of this full subcategory inclusion defines the functor $i_\Delta \colon \mathbf{sSet} \to \odot\mathbf{Set}$, together with its right adjoint $X \mapsto X_\Delta$. First we observe:

Proposition

The adjunctions $\mathrm{Sd}_\odot \dashv \mathrm{Ex}_\odot$ and $i_\Delta \dashv (-)_\Delta$

$$\mathbf{sSet} \leftrightarrows \odot\mathbf{Set} \leftrightarrows \mathbf{sSet}$$

factorize the adjunction $\mathrm{Sd} \dashv \mathrm{Ex}$ of simplicial sets (see subdivision).

Hence,

Proposition

The adjunction $i_\Delta \dashv (-)_\Delta$ also realizes the Quillen equivalence of Theorem .

Proof

By two-out-of-three for Quillen equivalences.

Higher category

This part requires notion of internal equivalence in diagrammatic set. Briefly, a cell $x$ is an equivalence if it is invertible up to equivalence, meaning that the “equations” $x \# x^* = 1$ and $1 = x^* \# x$ are witnessed by cells $h$ and $z$ one dimension higher, which are coinductively required to be equivalences. If there exists an equivalence $x \colon a \Rightarrow b$, we write $a \simeq b$. All the details can be found in CH2024eq.

Definition

(merge)
Let $U$ be a round molecule. The merge of $U$ is the atom $\partial^- U \Rightarrow \partial^+ U$.

The merge of a round molecule composes all the top dimensional cells of $U$ inside one. For instance, the merge of is Notice that $\langle U \rangle$ shares the same boundaries as $U$. In particular, since $U$ is round, $U \Rightarrow \langle U \rangle$ is well-defined.

Definition

(weak composite)
Let $X$ be a diagrammatic set, let $x \colon U \to X$ be a round $n$-diagram. A weak composite of $x$ is a cell $\langle x \rangle \colon \langle U \rangle \to X$ such that $x \simeq \langle x \rangle$.

The contender for weak $\infty$-category is the notion of diagrammatic set with weak composites.

Definition

(diagrammatic set with weak composites)
A diagrammatic set has weak composites if all round diagrams have a weak composite. This means that for all round molecules $U$, and all solid arrows in there exists a diagonal filler, which is an equivalence.

Definition

(diagrammatic $(\infty, n)$-category) Let $n\in \mathbb{N} \cup \{ \infty \}$. A diagrammatic $(\infty, n)$-category is a diagrammatic set $X$ with weak composite, and such that for all $k \gt n$, the $k$-cells in $X$ are equivalences.

Definition

(functor) Let $X, Y$ be diagrammatic $(\infty, n)$-categories. A functor between $X$ and $Y$ is simply a natural transformation of the underlying diagrammatic sets.

Definition

($\omega$-equivalence) Let $f \colon X \to Y$ be a functor of diagrammatic $(\infty, n)$-categories. We say that $f$ is an $\omega$-equivalence if:

• for every 0-cell $y \in Y$, there exists a 0-cell $x$ such that $y \simeq f(x)$;
• for every $n \gt 0$, pair of parallel $(n - 1)$-cells $x_1, x_2$ in $X$, and $n$-cell $y \colon f(x_1) \Rightarrow f(x_2)$, there exists an $n$-cell $x \colon x_1 \Rightarrow x_2$ such that $y \simeq f(x)$.

From CH24model, we have

Theorem

For each $n\in \mathbb{N} \cup \{ \infty \}$, there exists a cofibrantely generated model structure on diagrammatic sets where

• cofibrations are monomorphisms, and are generated by $\{ \partial U \hookrightarrow U \mid U\; \text{atom} \}$,
• fibrant object are the diagrammatic $(\infty, n)$-categories,
• weak equivalences between fibrant objects are the $\omega$-equivalences.

Theorem

(homotopy hypothesis)
The model structure for diagrammatic $(\infty, 0)$-categories coincide with the Kan model structure of Theorem .

References

The origin of the name is in the following, where diagrammatic sets were presheaves over Johnson’s pasting schemes:

• Kapranov, M. M., and Voevodsky, V. A.. $\infty$-groupoids and homotopy types. Cahiers de Topologie et Géométrie Différentielle Catégoriques 32.1 (1991) (link)

An older treatment of diagrammatic sets:

• Amar Hadzihasanovic, Diagrammatic sets and rewriting in weak higher categories, (arXiv:2007.14505)

For an updated version, together with results for higher groupoids:

• Clémence Chanavat, Amar Hadzihasanovic, Diagrammatic sets as a model of homotopy types, 2024 (arXiv:2407.06285)

For the notion of equivalence in diagrammatic sets:

• Clémence Chanavat, Amar Hadzihasanovic, Equivalences in diagrammatic sets, 2024 (arXiv:2410.00123)

For the model structures on diagrammatic $(\infty, n)$-categories:

• Clémence Chanavat, Amar Hadzihasanovic, Model structures for diagrammatic $(\infty,n)$-categories, 2024 (arXiv:2410.19053)