nLab semiadditive (∞,1)-category

Idea

An (∞,1)-category is $n$-semiadditive for $n \in \mathbb{N}$ if (∞,1)-colimits over m-finite spaces coincide with their (∞,1)-limits in a canonical way (see Hopkins-Lurie 14, def. 4.4.2). Here for $n=0$ the condition is that coproducts coincide with products, hence that bilimits exist.

So an (∞,1)-category that happens to be an ordinary category is 0-semiadditive precisely if it is a semiadditive category in the traditional sense. Generally, an (∞,1)-category is 0-semiadditive precisely if its homotopy category is semiadditive (Hopkins-Lurie 14, remark 4.4.13).

Examples

• For $R$ an A-∞ ring such that its connected component ring $\pi_0(A)$ is a vector space over the rational numbers, then the (∞,1)-category of modules $A Mod$ is $n$-semiadditive for all $n \in \mathbb{N}$ (Hopkins-Lurie 14, example 4.4.22).

• By Carmeli-Schlank-Yanovski, the $\infty$-category of m-commutative monoids is a mode classifying $m$-semiadditivity; in particular, for all $\mathcal{C}$, the $\infty$-category $\mathrm{CMon}_m(\mathcal{C})$ is $m$-semiadditive, and $\mathcal{C}$ is $m$-semiadditive if and only if the free functor

  $$\mathcal{C} \rightarrow \mathrm{CMon}_m(\mathcal{C})$$

  is an equivalence.

Related concepts

• additive (∞,1)-category

• stable (∞,1)-category

• m-finite space, m-commutative monoid

References

• Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory (2014)

• Shachar Carmeli?, Tomer Schlank, Lior Yanovski, Ambidexterity and Height (2020) (arXiv:2007.13089)