nLab manifold-diagrammatic n-category

Idea

The notion of manifold-diagrammatic higher categories (in some places also referred to as a ‘geometric’ or ‘geometrical’ higher categories) broadly refers to higher categories whose composition operation is modelled on the stratified-geometric notion of manifold diagrams.

Overview

Models of higher categories are usually defined as certain collections of k-morphisms that are parametrized by (that is, presheafs on) some category of ‘combinatorial shapes’: examples include presheafs of globular, simplicial or opetopic shapes. Manifold diagrams, while defined in geometric terms, admit a combinatorialization in terms of trusses (or, more precisely, combinatorial manifold diagrams) and can therefore be used as combinatorial shapes for defining higher categories as presheafs on them. Importantly, combinatorial manifold diagrams come with a rich class of combinatorial mappings which (in contrast to most other classes of shapes) include internal representations of both ‘embeddings’ (=‘face and degeneracy’ maps, up to geometric dualization) as well as ‘quotients’ (=‘subdivisions’, up to geometric dualization). We give geometric insight into these classes below, but ultimately only care about their combinatorial definitions.

Embeddings of manifold diagrams

Geometric idea

Embeddings of manifold $n$-diagrams are framed stratified maps whose underlying maps are embeddings of spaces (this is different from the notion of substratifications). The next figure illustrates mappings in this class (in dimension $n = 2$).

Figure 1: Embeddings of manifold diagrams (for which substratifications and coarsenings form a factorization system).

Geometric dualization unveils that embeddings of manifold diagrams are precisely ‘cellular’ maps of their dual framed cell complexes (‘cellular’ is usually meant to imply that maps map closed cells to closed cells), which can be seen to have an (epi,mono) factorization system by degeneracy and face maps. The next figure illustrates this: the shown maps are precisely the the geometric duals of the maps in the previous figure.

Figure 2: Geometric duals of embeddings are cellular maps (for which face and degeneracy maps form a factorization system)

Combinatorial definition

Combinatorially, we define the following.

Definition

Embeddings of combinatorial manifold diagrams $T,S$ are described by spans of the form

where $T' \xrightarrow{\mathsf{norm}} T$ is a normalizing map and $T' \hookrightarrow S$ is map of labeled trusses whose underlying map is regular and injective.

Such spans can be composed by pullback.

Quotients of manifold diagrams

Geometric idea

Quotients of manifold diagrams are framed stratified maps whose underlying maps are quotient maps. The next figure illustrates mappings in this class (in dimension $n = 2$).

Figure 3: Quotients of manifold diagrams are the categorically dual notion to embeddings.

Geometric dualization unveils that quotients of manifold diagrams correspond to subdivisions of framed cell complexes. The next figure illustrates the geometric duals of the quotient maps in the preceding figure.

Figure 4: The geometric duals of quotients are subdivisions.

Combinatorial definition

Combinatorially, we define the following.

Definition

Quotients of combinatorial manifold diagrams $T, S$ are maps of labeled trusses

$$T \twoheadrightarrow S$$

whose underlying map of trusses is singular and surjective.

Remark

To summarize: while the notion embeddings of manifold diagrams recovers (after geometric dualization) face and degeneracy maps often found in the combinatorics of shapes, in contrast, the notion of quotients dual to that of embeddings captures (after geometric dualization) something new: namely, subdivisions of cells. This is quite powerful e.g. for describing compositions as we will exploit below.(Moreover, it turns out that classifying subdivisions is generally a difficult (undecidable) problem, see e.g. Dorn-Douglas 2021, so it is convenient that it can be easily solved in the setting of manifold diagrams!)

Sketch Definition

We first organize embeddings and quotients (in the sense of Def. resp. Def. ) into categories as follows.

Terminology

Together, embeddings and quotients organize into the double category $\mathbb{M}\mathsf{Diag}_n$ of combinatorial manifold $n$-diagrams, with horizonal morphisms being embeddings, vertical morphisms being quotients, squares being commuting diagrams of the following form:

(note that the dashed arrow is unique if it exists.) The category of manifold $n$-diagrams $\mathsf{MDiag}_n$ will refer to just the horizontal part of this double category.

When defining higher categories parametrized by manifold $n$-diagrams, we would want higher categories to be compatible both with embeddings (‘morphism attachment’) and quotients (‘morphism composition’) in the following sense.

Sketch definition. A manifold diagrammatic $n$-category $\mathcal{C}$ should be a presheaf on $\mathsf{MDiag}_n$ subject to the following two conditions.

1. The collection of morphisms parametrized by a given diagram should be determined by subdiagrams that cover it. Formally, this can be enforced by an appropriate sheaf condition.
2. The composition of a given diagram should be compatible with the composition of its subdiagrams. Formally, this can be enforced by requiring that $\mathcal{C}$ extends to a double-(co)presheaf $\mathbb{M}\mathsf{Diag}^{\mathrm{op},\mathrm{co}}_n \to \mathbb{S}\mathbf{et}$ (covariant in the vertical part, where $\mathbb{S}\mathbf{et}$ is the double category of squares $\mathrm{Sq}(\mathbf{Set})$ in Set)).

Remark

(Coherence conditions). What’s worth pointing out about the above definition sketch is that there are no explicit higher-categorical coherence conditions (like filler or Segal conditions) beyond ordinary sheaf conditions. This is because coherences can be expressed internally to the category of manifold diagrams (namely, as so-called ‘manifold-diagram isotopies’ including, for instance, a braiding).

Remark

(The $n = \infty$ case). So far we worked with $n$-diagrams for fixed finite $n \in \mathbb{N}$ which work well as parametrizing objects for $n$-categories. However, we can also address the case $n = \infty$ as follows. Given a combinatorial manifold $n$-diagram $T$, note its cylinder $\mathbb{I} \times T$ is a combinatorial manifold $(n+1)$-diagram. This defines a chain of inclusions of (ordinary resp. double) categories, the colimit of which is a category of manifold diagrams $\mathsf{MDiag}$ (resp. the double category $\mathbb{M}\mathsf{Diag}$) which may then be used to parametrized $(\infty,\infty)$-categories via the sketch definition above.

A formal definition

To make the preceding sketch precise, one would need to formalize the sheaf condition. We now mention one way of how this could be achieved.

Terminology

The category $\mathrm{MDiag}_n$ (and similarly, $\mathrm{MDiag}$ from Rmk. ) contains wide subcategories $\mathrm{MDiag}^L_n$ resp. $\mathrm{MDiag}^R_n$ consisting of spans

We define a coverage for $\mathrm{MDiag}^R_n$. (Note that $\mathrm{MDiag}^R_n$ does not have all pullbacks since, e.g., subdiagrams can intersect in non-diagrams.)

Definition

The neighborhood coverage $J_n$ for $\mathrm{MDiag}^R_n$ is the coverage that assigns to $T \in \mathrm{MDiag}$ the single family $\{f_x : T^{\leq x} \to T \}_{x \in T}$ comprised of the stratified neighborhoods of $T$.

We may now formalize our earlier sketch definition as follows.

Definition

A manifold-diagrammatic $n$-category $C$ is a presheaf on $\mathrm{MDiag}_n$ such that:

1. $C$ is a sheaf on $(\mathrm{MDiag}^R_n,J_n)$ and locally constant on (i.e. constant on the connected components of) $\mathrm{MDiag}^L_n$,
2. $C$ extends to a double-(co)presheaf $\mathbb{M}\mathsf{Diag}^{\mathrm{op},\mathrm{co}}_n \to \mathbb{S}\mathbf{et}$.

(Note, however, that the practicality of this definition has not yet been established, and several variants are under consideration, cf. next section.)

Variants

Instead of considering presheafs on combinatorial manifold diagrams, one may also consider

• presheafs on (unlabeled) trusses
• presheafs on (unlabeled) atomic trusses

with various sheaf conditions in place. In both cases, faces, degeneracies and compositions may be understood via mappings as outlined above. The potential advantage of dropping labelings from our parametrizing objects in this way (i.e. considering trusses in place of labeled trusses) is simplicity. For instance, an immediate simplification is that we can work with truss maps instead of spans of truss maps when combinatorially modelling embeddings. This could simplify the comparison to other presheaf models of higher categories.

Another related, potentially useful observation is that the spans occurring in the definition of the category of manifold diagrams may be expressed concisely in terms of bundles (slightly generalizing the notion of labeled truss bordism).

Several other approaches to and variations of the idea of manifold-diagrammatic higher categories are currently under consideration. The categories of combinatorial shapes considered here (whether combinatorial manifold diagrams, or (atomic) trusses, or similar) often have peculiarities that one doesn’t encounter in other settings (e.g. failure to be a Reedy category due to the novel concept of normalizing maps). The precise nature of these shape categories remains to be fully understood.

Examples

• Geometric computads are free instances of manifold-diagrammatic $n$-categories.

• Associative $n$-categories are, in an extended sense, instances of manifold diagrammatic $n$-categories (and, in fact, their combinatorics precedes some of the stratified-geometric ideas of manifold-diagrammatic $n$-categories).

Remark

(Globular sets in associative $n$-categories). Note that associative $n$-categories use globular shapes as parametrizing objects rather than manifold diagrams. This can be related to the above discussion as follows. Using Rmk. , denote by $\mathsf{MDiag^{st}}$ the category of (‘strictly globular’) manifold diagrams with source/target inclusions as maps, and by $\mathsf{Glob}$ the globe category. Then there is a retraction $\mathsf{Glob} \hookrightarrow \mathsf{MDiag^{st}}$ (which maps $n$ as the ‘terminal’ manifold $n$-diagram, namely, the geometric dual of the $n$-globe) and $\mathsf{MDiag^{st}} \to \mathsf{Glob}$ (which maps an $n$-manifold diagram to the $n$-globe). This retraction can be used to ‘work with globular sets whose composition is defined via manifold diagrams’.

Remarks

• As mentioned above, coherences in manifold-diagrammatic categories derive from isotopies of manifold diagrams. This provides a different mechanism for coherence construction than the mechanism found in many existing, spatially-inspired models of higher categories (which create coherences by enforcing contractibility of ‘spaces of composites’). The conceptual difference between the two mechanisms is discussed in more detail in this blog post.

• Manifold-diagrammatic categories have also been referred to ‘geometric higher categories’ or ‘geometrical higher categories’, based on a distinction of ‘geometric’, ‘topological’ and ‘combinatorial’ models of higher structures as explained here. See also here for a relevant discussion about terminology.

Related concepts

• manifold $n$-diagrams

• meshes

• trusses

• directed topological spaces

• stratified spaces

• higher category theory

References

A condensed write-up of the mathematics needed for stating the above formal definition of manifold-diagrammatic higher categories was compiled in

• From zero to manifold-diagrammatric categories, pdf.

A general introduction to some of the relevant ideas in the area can be found in

• An Invitation to Geometric Higher Categories, $n$-Category Café post,

with further details spelled out in

• Christoph Dorn, Nine short stories about geometric higher categories, 2023 (pdf).

The type of ‘directed combinatorial topology’ that governs the stratified topology of manifold diagrams was described in

• Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021 (arXiv, latest)