homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
An orbital -category is an (∞,1)-category satisfying a weak axiomatic framework generalizing the properties of the orbit category of a finite group allowing for the construction of the Burnside category.
Given an (∞,1)-category, we write for its finite coproduct completion.
An -category is orbital if admits finite (co)products.
An orbital -category is atomic if every retract in is an equivalence.
If is orbital, then is disjunctive, so we may form the Burnside category
the latter denoting the (∞,1)-category of correspondences constructed by Barwick.
The following are all examples of atomic orbital -categories.
orbit categories of finite groups; more generally, orbit categories of profinite groups (where the stabilizers are required to be open);
locally finite groups (where the stabilizers are required to be finite);
any space;
the cyclonic orbit 2-category (see at cyclotomic spectrum);
the 2-category of connected finite groupoids and covering maps (i.e. the global orbit category);
the category of finite sets of cardinality and surjective functions;
the topological categories of finite-dimensional inner product spaces (over and ) of dimension and orthogonal projections.
For many cases of these atomic orbital -categories there is a conservative (∞,1)-functor to a poset, and so they are EI (∞,1)-categories. Essentially finite EI orbital discrete categories have been considered by Wilson on work concerning the slice filtration, where they are called inductive orbital categories.
Many of the original papers on this topic:
Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah, Parametrized higher category theory and higher algebra: A general introduction, (arXiv:1608.03654)
Clark Barwick, Parametrized higher category theory and parameterized higher algebra, video
Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah, Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory, (arXiv:1608.03657)
Jay Shah, Parametrized higher category theory, (arXiv:1809.05892)
Jay Shah, Parametrized higher category theory II: Universal constructions, (arXiv:2109.11954)
Denis Nardin, Parametrized higher category theory and higher algebra: Exposé IV - Stability with respect to an orbital ∞-category, (arXiv:1608.07704)
Denis Nardin, Stability and distributivity over orbital -categories, (thesis)
Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem, (arxiv:1507.01976)
Saul Glasman, Goodwillie calculus and Mackey functors, (1507.01976)
On disjunctive (∞,1)-categories
On the slice filtration for inductive orbital categories
Last revised on May 11, 2024 at 23:44:56. See the history of this page for a list of all contributions to it.