nLab indexed coproduct

Idea

Indexed coproducts are to coproducts as equivariant symmetric monoidal categories are to symmetric monoidal categories.

Definition

Definition

If $\mathcal{C}$ is a G-∞-category, $H \subset G$ a subgroup, and $S$ a finite H-set, by composing the diagonal functor with restriction, we have a functor

$$\Delta^S:\mathcal{C}_H \xrightarrow{\Delta} \prod_{H/K_i \in \mathrm{Orb}(S)} \mathcal{C}_{H} \xrightarrow{\Res} \prod_{H/K_i \in \mathrm{Orb}(S)} \mathcal{C}_{K_i}.$$

If a pointwise left adjoint to $\Delta^S$ exists at a tuple $(X_{K_i})$, then the resulting object is denoted $\coprod_{K_i}^H X_{K_i}$ and called the $S$-indexed coproduct of $(X_{K_i})$.

Dually, if a pointwise right adjoint to $\Delta^S$ exists at a tuple $(X_{K_i})$, then the resulting object is denoted $\prod_{K_i}^H X_{K_i}$ and called the $S$-indexed product of $(X_{K_i})$.

Related concepts

• Parametrized Higher Category Theory and Higher Algebra, G-∞-category, equivariant symmetric monoidal category

• equivariant homotopy theory

References

Originally,

• Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]

In higher category theory,

• 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)