nLab n-connective object



Stable Homotopy theory

(,1)(\infty,1)-Category theory



In stable \infty-categories


In broad generality, given a stable \infty -category 𝒜\mathcal{A} equipped with a t-structure 𝒜 n,𝒜 n𝒜\mathcal{A}_{\geq n}, \mathcal{A}_{\leq n} \hookrightarrow \mathcal{A}, the objects in the full sub- \infty -category 𝒜 n\mathcal{A}_{\geq n} are the nn-connective objects, for any nn \in \mathbb{Z}.

Here typically one understands that a plain “connective” is short for “0-connective.”

Accordingly, the coreflections 𝒜𝒜 n\mathcal{A} \to \mathcal{A}_{\geq n} are called the connective cover-constructions.

Dually, the objects of 𝒜 n\mathcal{A}_{\leq n} may be called the “nn-coconnective objects”. For non-negative kk \in \mathbb{N} the intersection 𝒜 0𝒜 k\mathcal{A}_{\geq 0} \cap \mathcal{A}_{\leq k} of sub-(infinity,1)-categories of objects which are both connective and kk-coconnective are equivalently the k k -truncted connective object:

τ k𝒜 0𝒜 0𝒜 k. \tau_{\leq k} \mathcal{A}_{\geq 0} \;\simeq\; \mathcal{A}_{\geq 0} \cap \mathcal{A}_{\leq k} \,.

(Lurie, Warning


In \infty-toposes

In an \infty -topos one would — following traditional in algebraic topology — instead speak of the k k -connected for kk \in \mathbb{N}. If one insists on saying “connective” also in this case (as is the convention in Lurie‘s Higher Topos Theory) then there is a shift in degree: nn-connected corresponds to n+1n+1 connective. (See there for more.)


For general discussion in the context of stable \infty -categories see the references at t-structure, such as

For the terminology “connective”/“coconnective” in this context, see for instance:

  • Harry Smith, Def. 2.1 in: Bounded t-structures on the category of perfect complexes over a Noetherian ring of finite Krull dimension, Advances in Mathematics 399 (2022) 108241 [doi:10.1016/j.aim.2022.108241, pdf]

  • Emanuele Pavia, p. 4 of: t-structures on ∞-categories (2021) [pdf]

Created on April 20, 2023 at 07:14:34. See the history of this page for a list of all contributions to it.