additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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).
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
is an equivalence.
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)
Last revised on April 19, 2024 at 17:06:44. See the history of this page for a list of all contributions to it.