nLab connective cover





Generally, given a stable \infty -category 𝒜\mathcal{A} equipped with a t-structure 𝒜 0,𝒜 0𝒜\mathcal{A}_{\geq 0}, \mathcal{A}_{\leq 0} \hookrightarrow \mathcal{A}, the coreflection 𝒜𝒜 0\mathcal{A} \to \mathcal{A}_{\geq 0} onto the connective objects is called the connective cover-construction.

For chain complexes

Given an unbounded chain complex V Ch (𝒜)V_\bullet \,\in\, Ch_\bullet(\mathcal{A}) in some abelian category 𝒜\mathcal{A},

V =(V 2 2V 1 1V 0 0V 1 1V 2) V_\bullet \;=\; \big( \cdots \to V_2 \overset{\partial_2}{\to} V_1 \overset{\partial_1}{\to} V_0 \overset{\partial_0}{\to} V_{-1} \overset{\partial_{-1}}{\to} V_{-2} \to \cdots \big)

its connective cover is obtained by

  • retaining its entries in positive degrees,

  • replacing its entries in negative degree by zero objects,

  • replacing its entry in degree 0 by the kernel of the differential in this degree:

cn 0V =(V 2 2V 1 1ker( 0)00). cn_{\geq 0} V_\bullet \;=\; \big( \cdots \to V_2 \overset{\partial_2}{\to} V_1 \overset{\partial_1}{\to} \ker(\partial_0) \to 0 \to 0 \to \cdots \big) \,.

(e.g. Lurie, Rem.

For plain spectra

Connective spectra form a coreflective sub-(∞,1)-category of the (∞,1)-category of spectra (as part of the canonical t-structure in spectra, see there).

The right adjoint (∞,1)-functor from spectra to connective spectra EE0E \mapsto E \langle 0 \rangle is called the connective cover construction. This comes with the coreflection morphism of spectra

(1)E0ϵ EE E\langle 0 \rangle \overset{ \epsilon_E }{\longrightarrow} E

which induces an isomorphism on homotopy groups of spectra in non-negative degrees:

π 0(E0)π 0(ϵ E)π 0(E). \pi_{\bullet \geq 0}\big( E\langle 0\rangle\big) \underoverset {\simeq} { \pi_{\bullet \geq 0} (\epsilon_E) }{\longrightarrow} \pi_{\bullet \geq 0}(E) \,.

For ring spectra

The connective cover functor extends from plain spectra to E-∞ ring spectra (May 77, Prop. VII 4.3, Lurie, Prop., such that the coreflection E0EE \langle0\rangle \longrightarrow E (1) is a homomorphism of E-∞ rings.

Besides a canonically inherited ring structure, the connective cover may sometimes carry further ring structures, but in many examples of interest it is unique (Baker-Richter 05).


  • ku is the connective cover of KU


For plain spectra:

For ring spectra:

Last revised on April 21, 2023 at 13:28:49. See the history of this page for a list of all contributions to it.