nLab geometric computad

Idea

Geometric $n$-computads are free geometric n-categories, whose notion of composition is modelled on manifold diagrams. Geometric $n$-computads are a model for a semistrict flavor of n-categories.

Definition

Terminology

We will say a manifold $k$-diagram $(\mathbb{R}^k,f)$ is a stratum $k$-type if the diagram is of the form $(\mathrm{Cone}(S^{k-1}), \mathrm{cone}(\partial f))$. We call $(S^{k-1},\partial f)$ the type boundary of $f$.

Let $\mathcal{E} : \mathbf{Strat} \to \mathbf{Pos}$ denote the entrance path poset functor, that maps stratified spaces to their entrance path posets.

Definition

A geometric $n$-computad is an $[n]$-graded set $C$ together with, for each $c \in C_i$, a stratum type $(\mathbb{R}^i, f_c)$ and a ‘labeling’ function $l_c : \mathcal{E}_0 f_c \to C_{\leq i}$ with $l_c\{0\} = c$, such that $k$-strata $s$ in $f_c$ have stratum type $f_{l_c(s)}$ and $l_{l_c(s)}$ factors through $l_c$ by the induced map $\mathcal{E}_0 f_{l_c(s)} \to \mathcal{E}_0 f_c$.

In words, a computad $C$ is a bunch of ‘$i$-morphisms’ (indexed by the sets $C_i$) for $i \leq n$, and each morphism has a stratum type whose boundary is consistently labeled in lower-dimensional morphisms.

Remarks

• The remarkable observation about geometric computads is the following. Classically, building computads is an inductive two-step process: namely, usually one constructs computads inductively by, in the $n$th step, adding generating $n$-morphisms with boundary in existing $(n-1)$-morphisms, and then passing to the closure under composition and coherence conditions (which, passing back and forth between “new composites” and “new coherences” is often a pretty infinite process). In geometric computads, this is process becomes unnecessary: boundaries of $n$-morphisms are manifold $(n-1)$-diagrams (or, if you work dually, cell $(n-1)$-diagrams) which already can express all the composites and isotopies you could possibly need. This makes geometric computads really easy to work with.

• While it is straight-forward to define geometric computads (and related notions, such as their functors and certain transformations), a comprehensive theory of computads is still in development.

Related pages

• manifold diagram

• geometric n-category

• homotopy.io

References

A definition and further discussion can be found in

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

The notion has been the topic of several talks and blog posts.

• Christoph Dorn, From trusses to geometric computads to combinatorial manifold diagrams (2022)

• Christoph Dorn, The categorical Pontryagin-Thom construction (2022) contains definitions of functors and certain transformations of geometric computads.

The finitely generated case can be efficiently manipulated using the proof-assistant homotopy.io

• Jamie Vicary et. al, Introducing homotopy.io: A proof assistant for geometrical higher category theory (2022)