nLab connective cover

Idea

General

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

For chain complexes

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

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

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:

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

(e.g. Lurie, Rem. 1.2.3.4)

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 $E \mapsto E \langle 0 \rangle$ is called the connective cover construction. This comes with the coreflection morphism of spectra

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

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

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

For ring spectra

The connective cover functor extends from plain spectra to E-∞ ring spectra (May 77, Prop. VII 4.3, Lurie, Prop. 7.1.3.13), such that the coreflection $E \langle0\rangle \longrightarrow E$ (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).

Examples

• ku is the connective cover of KU

Related concepts

• n-connective object

• n-connected object of an (infinity,1)-topos

References

For plain spectra:

• Stefan Schwede, chapter II, section 8, and chapter III, section 7 of Symmetric spectra (2012)

For ring spectra:

• Peter May, Prop. VII 4.3 in: $E_\infty$-Ring spaces and $E_\infty$ ring spectra, Lecture Notes in Mathematics 577, Springer 1977 (pdf, cds:1690879)

• Andrew Baker, Birgit Richter, Uniqueness of $E_\infty$-structures for connective covers, Proc. Amer. Math. Soc. 136 (2008), 707-714 (arXiv:math/0506422, doi:10.1090/S0002-9939-07-08984-8)

• Jacob Lurie, 7.1.3.11-13 in: Higher Algebra