nLab geometric computad




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



We will say a manifold k k -diagram ( k,f)(\mathbb{R}^k,f) is a stratum kk-type if the diagram is of the form (Cone(S k1),cone(f))(\mathrm{Cone}(S^{k-1}), \mathrm{cone}(\partial f)). We call (S k1,f)(S^{k-1},\partial f) the type boundary of ff.

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


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

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


  • 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 nnth step, adding generating nn-morphisms with boundary in existing (n1)(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 nn-morphisms are manifold (n1)(n-1)-diagrams (or, if you work dually, cell (n1)(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

Created on March 14, 2023 at 11:24:24. See the history of this page for a list of all contributions to it.