nLab Parametrized Higher Category Theory and Higher Algebra

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 $G$ 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 $G$-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 $G$:

1. Orbital: Fiber products of representable presheaves are finite disjoint unions of representable presheaves, a restatement of the fact that the category of finite $G$-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 $(\infty, 1)$-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 $\leq n$ and surjective functions;

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

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

Related concepts

• G-∞-category, G-symmetric monoidal category

• G-operad

References

Along with an introduction, nine exposés were planned, leading to the following material:

• 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)

On isotropy separation:

• 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 $G$-operads and algebras:

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra, (arxiv:2203.00072)

• Shaul Barkan, Rune Haugseng, Jan Steinebrunner, Envelopes for Algebraic Patterns, (arxiv:2203.00072)

On factorization homology, THH and the $\mathbb{E}_V$ operad:

• Asaf Horev, Genuine equivariant factorization homology, (arXiv:1910.07226)

• Jeremy Hahn, Asaf Horev, Inbar Klang, Dylan Wilson, Foling Zou, Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra, (arXiv:2006.13348)

On presentability, semiadditivity and stability:

• Kaif Hilman, Parameterized presentability over orbital categories, (arxiv:2202.02594)

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Parametrized stability and the universal property of global spectra, (arXiv:2301.08240),

• Bastiaan Cnossen, Twisted ambidexterity in equivariant homotopy theory, (arXiv:2303.00736),

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Partial parametrized presentability and the universal property of equivariant spectra, (arxiv:2307.11001)

• Sil Linskens, Globalizing and stabilizing global $\infty$-categories, (arXiv:2401.02264)

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens, Parametrized stability and the universal property of global spectra, (arXiv: arXiv:2403.07676)

On noncommutative motives:

• Kaif Hilman, Parametrised noncommutative motives and equivariant cubical descent in algebraic K-theory, (arXiv:2202.02591)