nLab n-connective object

Idea

In stable $\infty$-categories

Definition

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

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

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

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

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

(Lurie, Warning 1.2.1.9)

Examples

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

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

In $\infty$-toposes

In an $\infty$-topos one would — following traditional in algebraic topology — instead speak of the $k$-connected for $k \in \mathbb{N}$. 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: $n$-connected corresponds to $n+1$ connective. (See there for more.)

Related concepts

• 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]