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Parametrized Higher Category Theory and Higher Algebra

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

This entry relates to a series of papers, beginning with Barwick-Dotto-Glasman-Nardin-Shah 16 which aim to provide common foundations for several different parts of homotopy theory, among which equivariant homotopy theory, parametrized homotopy theory, global homotopy theory and Goodwillie calculus.

For GG a finite group, various concepts in equivariant homotopy theory are constructed as indexed over an orbit category, such as the homotopy theory of topological G-spaces or of GG-equivariant spectra, when regarded via Elmendorf's theorem.

An important ingredient of this program is the concept of an atomic orbital \infty-category which is defined in terms of two important properties of the orbit category of GG:

  1. Orbital: Fiber products of representable presheaves are finite disjoint unions of representable presheaves, a restatement of the fact that the category of finite GG-sets has pullbacks (Mackey decomposition), so a version of the Beck-Chevalley condition;

  2. Atomic: The triviality of retracts (that is every retraction is an equivalence).

Examples of such (,1)(\infty, 1)-categories satisfying these two properties include:

  1. orbit categories of finite groups;

  2. more generally, orbit categories of profinite groups (where the stabilizers are required to be open);

  3. locally finite groups (where the stabilizers are required to be finite);

  4. any ∞-groupoid;

  5. the cyclonic orbit 2-category (see at cyclotomic spectrum);

  6. the 2-category of connected finite groupoids and covering maps;

  7. the category of finite sets of cardinality n\leq n and surjective functions;

  8. the topological categories of finite-dimensional inner product spaces (over \mathbb{R} and \mathbb{C}) of dimension n\leq n and orthogonal projections.

The program looks to generate results which hold for all atomic orbital \infty-categories, for any instance of which, TT, there are the associated concepts of TT-\infty-category, TT-space and TT-spectrum.

For many cases of these atomic orbital \infty-categories there is a conservative (∞,1)-functor to a poset, and so they are EI (∞,1)-categories.

References

Along with an introduction, nine exposés are planned:

A survey talk is

  • Clark Barwick, Parametrized higher category theory and parameterized higher algebra, video

Last revised on September 28, 2018 at 03:43:34. See the history of this page for a list of all contributions to it.