Contents

# Contents

## 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.

A definition and further discussion can be found in

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

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