Atoms are regular directed complexes with a maximal cell. With the correct notion of maps, they are a contender for geometric shape for higher categories.

**(atom)**

An **atom** is a regular directed complex whose underlying poset has a maximal element.

In regular directed complex, we defined a **morphism** $f \colon P \to Q$ between regular directed complex as a function of the underlying posets that respects both input and output faces, that is, $f$ induces a bijection between $\Delta^\a x$ and $\Delta^\alpha f(x)$.

This notion is, albeit well-suited for defining regular directed complexes, is quite restrictive. For instance, the point $\mathbf{1}$ is not a terminal object in this category. In fact, if there is a morphism $f \colon P \to Q$, then $\dim P \le \dim Q$, which prevent using morphisms for modelling degeneracies.

Instead, we define a less restrictive notion of map, that strictly generalizes the morphisms.

In the following definition, we will impose two conditions (finality, and fibration)

**(cartesian map of regular directed complex)**

A **cartesian map** $f \colon P \to Q$ between regular directed complexes is a function of the underlying set such that

- $f$ is a Grothendieck fibration (in particular, $f$ is order-preserving);
- for all $x \in P$, $\alpha \in \{ -, + \}$, $n \geq 0$,$f(\partial^\alpha_n x) = \partial^\alpha_n f(x);$
- for all $x \in P$, the restriction and corestriction of $f \colon \partial^\alpha_n x \to \partial^\alpha_n f(x)$ is final;

In the previous definition, the fibration and finality condition implicitely see the underlying posets as categories.

This new definition of cartesian map generalizes the notion of morphism in the following sense:

Any morphism of regular directed complex is a cartesian map, and a cartesian map is a morphism if and only if it is dimension-preserving.

**(atom category)**

We define the **atom category**, that we write **$\odot$**, to be a skeleton of the category of atoms and cartesian maps.

The category $\odot$ is a strict Cisinski generalized Reedy category (Def 2.4), where

- the degree map $d \colon \odot \to \mathbb{N}$ is given by the dimension $U \mapsto \dim U$;
- the wide subcategory $\odot^+$ is the cartesian maps that are injective;
- the wide subcategory $\odot^-$ is the cartesian maps that are surjective;

We use the prefix “strict” in the previous definition to mean that the factorisation system $(\odot^-, \odot^+)$ is strict rather than merely orthogonal

In particular, any cartesian map of atom factors uniquely as a surjection, which we also call a **collapse**, followed by an inclusion, which we also call an **inclusion**, and any cartesian map between atoms of the same dimension has to be the identity.

The inclusions are very-well behaved, we have

Let $U$ be an atom, there is a one-to-one correspondance between elements of $U$ and inclusions $V \hookrightarrow U$ given by

$x \in U \mapsto (\iota_x \colon \mathrm{cl}(x) \hookrightarrow U)$

If $\iota \colon V \hookrightarrow U$ is an inclusion, there is an isomorphisms $V \cong \iota(V) \subseteq U$, since $V$ is an atom, so is $\iota(V)$, whose maximal element defines an element $x \in U$ which uniquely characterize $\iota$ (since the category is skeletal).

However, the precise behavior of collapses is not as explicit as in the case of simplex or cube categories. Nevertheless, we have

The point $\mathbf{1}$ is the terminal object of $\odot$.

- every degeneracy has a section;
- if two degeneracies have the same set of section, they are equal.

The Gray product, the join, and the suspension are again atoms. Therefore:

$(\odot, \otimes, \mathbf{1})$ is a monoidal category.

Notice that the only reason why the join $\star$ does not make a monoidal category is because its unit $\emptyset$ is only a regular directed complex, but not an atom.

**(oriented simplex)**

The **$n$-th oriented simplex** $\vec{\Delta}_n$ is defined inductively by

- $\vec{\Delta}_0 \coloneqq \mathbf{1}$;
- inductively, $\vec{\Delta}_n \coloneqq \vec{\Delta}_{n - 1} \star \mathbf{1}$;

We call $\vec{\Delta}$ the full subcategory of $\odot$ on oriented simplices.

The category $\vec{\Delta}$ is isomorphic to the simplex category. More in details, the coface map $d_i \colon \vec{\Delta}_{n - 1} \to \vec{\Delta}_{n}$ can be reconstructed as

$\mathrm{id} \star ! \star \mathrm{id} \colon \vec{\Delta}_{i - 1} \star \emptyset \star \vec{\Delta}_{n - i - 1} \hookrightarrow \vec{\Delta}_{i - 1} \star \mathbf{1} \star \vec{\Delta}_{n - i - 1}$

and the codegeneracy $s_i \colon \vec{\Delta}_{n + 1} \to \vec{\Delta}_{n}$ can be reconstructed as

$\mathrm{id} \star ! \star \mathrm{id} \colon \vec{\Delta}_{i - 1} \star \vec{\Delta}_1 \star \vec{\Delta}_{n - i - 1} \hookrightarrow \vec{\Delta}_{i - 1} \star \mathbf{1} \star \vec{\Delta}_{n - i - 1}.$

The category $\odot$ is a test category.

As $\mathbf{1}$ is the terminal object, $\odot$ is aspherical. Therefore, it suffices to prove that $\odot$ is a local test category. This follows from Cisinski06, Corollaire 8.2.16, using the cylinder object $U \mapsto \vec{I} \otimes U$, extend via Day convolution on presheaves. Since atoms are stable under Gray products, the map $I \otimes U \twoheadrightarrow U$ is representable, so is a weak equivalence (in the sense of here).

In fact, $\odot$ is even a strict test category.

- A presheaf over $\odot$ is a diagrammatic set
- simplex category
- regular directed complex
- geometric shape for higher categories

For the basic properties of atoms, and their relations to cube, simplices, and other shapes:

- Amar Hadzihasanovic,
*Combinatorics of higher-categorical diagrams*, 2024 (link)

Statements about Eilenberg-Zilber category and test category are proved here:

In part 8, we find a lot of results to prove that Eilenberg-Zilber related categories are test categories:

- Denis-Charles Cisinski,
*Les préfaisceaux comme modèles des types d’homotopie*, Astérisque 308, 2006 (doi:10.24033/ast.715, NUMDAM)

Last revised on July 12, 2024 at 14:15:00. See the history of this page for a list of all contributions to it.