nLab atom category

Idea

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.

Definition

Definition

(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.

Remark

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

Definition

(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;

Remark

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:

Lemma

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.

Definition

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

Properties

Theorem

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;

Remark

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

Proposition

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

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

Proof

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

Lemma

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

as well as the Cisinski-Eilenberg-Zilber axiom which state that

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

Stability of operations

Proposition

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

Proposition

$(\odot, \otimes, \mathbf{1})$ 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.

Definition

(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.

Theorem

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

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

Test category

Theorem

The category $\odot$ is a test category.

Proof

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.

Related concepts

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

References

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:

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

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)