Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



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A TT-complex is a higher-dimensional combinatorial structure with a class of designated thin elements. The concept can be made sense of for various shapes.

In general, the requirements are:

  1. Every degenerate element is thin.
  2. Every hollow shape has a unique thin filler.
  3. If a thin element has every face but one also thin, then the last face is thin as well.

The theory describes a version of higher groupoids, rather than higher categories more generally. See however algebraic quasi-categories for more.

Types of TT-complexes

Simplicial TT-complexes


Cubical TT-complexes


Polyhedral sets and polyhedral TT-complexes

There is also a notion of polyhedral TT-complex, defined in (Jones, 1983). There are no degeneracies in this theory, but it does allow for the shapes quite, but not completely, general forms of regular cell decompositions of cells. This gives a solution to the problem of defining general compositions. One has to define:

  1. What are the pieces that might be composable?
  2. When are they composable?
  3. What is their composite?
  4. What are the axioms on the composition?

On the face of it, the last problem seems the hardest. It turns out that the last two TT-complex axioms are sufficient! Thus the geometry determines the algebra.


See the references at simplicial T-complex.

Polyhedral TT-complexes are discussed in

  • David W. Jones, A general theory of polyhedral sets and the corresponding TT-complexes. Dissertationes Math. (Rozprawy Mat.) 266 (1988) 110. Scanned Thesis or from RBPhDsSupervised
Revised on October 25, 2015 21:43:13 by JCMc Keown (