nLab manifold-diagrammatic n-category



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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.


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

  • Associative n n -categories are instances of manifold diagrammatic nn-categories (and, in fact, their combinatorics precedes some of the stratified-geometric ideas of manifold-diagrammatic nn-categories). The relation of manifold-diagrammatic composition with (higher) associativity derives from the observation that geometric gluings do not remember the order in which they occur.


  • Coherences in manifold-diagrammatic categories derive from isotopies of manifold strata. 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.

  • While in some instances (such as the ‘free’ case of geometric computads) definitions of manifold-diagrammatic nn-categories have been given, the theory of manifold diagrammatic higher categories overall remains under-developed.


An introduction to and overview of some relevant ideas in the area can be found in:

with further details spelled out in:

Last revised on March 19, 2023 at 13:02:14. See the history of this page for a list of all contributions to it.