n-connective object in nLab

Contents

Contents

Idea

In stable ∞\infty-categories

Definition

In broad generality, given a stable ∞ \infty -category 𝒜\mathcal{A} equipped with a t-structure 𝒜 ≥n,𝒜 ≤n↪𝒜\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 n∈ℤn \in \mathbb{Z}.

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

Accordingly, the coreflections 𝒜→𝒜 ≥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 k∈ℕk \in \mathbb{N} the intersection 𝒜 ≥0∩𝒜 ≤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 ≥0≃𝒜 ≥0∩𝒜 ≤k. \tau_{\leq k} \mathcal{A}_{\geq 0} \;\simeq\; \mathcal{A}_{\geq 0} \cap \mathcal{A}_{\leq k} \,.

(Lurie, Warning 1.2.1.9)

Examples

• For the standard t-structure on an ∞ \infty -category of chain complexes, the nn-connective objects are the nn-connective chain complexes, namely those which are concentrated in degrees ≥n\geq n.

• For the standard t-structure on an ∞ \infty -category of spectra, the nn-connective objects are the nn-connective spectra, namely those whose stable homotopy groups are concentrated in degree ≥n\geq n.

In ∞\infty-toposes

In an ∞ \infty -topos one would — following traditional in algebraic topology — instead speak of the k k -connected for k∈ℕk \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.)

• t-structure

• connective chain complex

• connective spectrum

• connective spectrum object

References

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

• Jacob Lurie, Section 1.2.1 in: Higher Algebra

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]