nLab G-semiadditivity

Definition

Definition

A G-∞-category $\mathcal{C}$ is G-semiadditive if, for all finite H-sets $S$ and tuples $(X_{H_i} \in \mathcal{C}_{H_i})_{G/H_i \in \mathrm{Orb}(S)}$, the canonical map from the indexed coproduct to the indexed product

$$\coprod_{H_i}^G X_{H_i} \rightarrow \prod_{H_i}^G X_{H_i}$$

is an equivalence.

More generally, if $I$ is a weak indexing category, we say that $\mathcal{C}$ is $I$-semiadditive if for all $I$-admissible H-sets $S$ and tuples $(X_{H_i} \in \mathcal{C}_{H_i})_{G/H_i \in \mathrm{Orb}(S)}$, the canonical map

$$\coprod_{H_i}^G X_{H_i} \rightarrow \prod_{H_i}^G X_{H_i}$$

is an equivalence.

Semiadditive closure

(…)

Related concepts

• Mackey functor

• equivariant symmetric monoidal category

• G-stability

References

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

• Bastiaan Cnossen, Tobias Lenz, Sil Linskens: Parameterized higher semiadditivity and the universality of spans, [arXiv:2403.07676]