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.
Given an unbounded chain complex $V_\bullet \,\in\, Ch_\bullet(\mathcal{A})$ in some abelian category $\mathcal{A}$,
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:
(e.g. Lurie, Rem. 1.2.3.4)
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
which induces an isomorphism on homotopy groups of spectra in non-negative degrees:
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).
For plain spectra:
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
Last revised on April 21, 2023 at 13:28:49. See the history of this page for a list of all contributions to it.