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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Definitions
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A -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:
The theory describes a version of higher groupoids, rather than higher categories more generally. See however algebraic quasi-categories for more.
See
See
There is also a notion of polyhedral -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:
On the face of it, the last problem seems the hardest. It turns out that the last two -complex axioms are sufficient! Thus the geometry determines the algebra.
See the references at simplicial T-complex.
Polyhedral -complexes are discussed in
Last revised on October 26, 2015 at 01:43:13. See the history of this page for a list of all contributions to it.