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.

References

Along with an introduction, nine exposés are planned:

• 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, 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 and higher algebra: Exposé II - Indexed homotopy limits and colimits, (arXiv:1809.05892)

• Denis Nardin, Parametrized higher category theory and higher algebra: Exposé IV - Stability with respect to an orbital ∞-category, (arXiv:1608.07704)

A survey talk is

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