Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
In broad generality, given a stable -category equipped with a t-structure , the objects in the full sub--category are the -connective objects, for any .
Here typically one understands that a plain “connective” is short for “0-connective.”
Accordingly, the coreflections are called the connective cover-constructions.
Dually, the objects of may be called the “-coconnective objects”. For non-negative the intersection of sub-(infinity,1)-categories of objects which are both connective and -coconnective are equivalently the -truncted connective object:
For the standard t-structure on an -category of chain complexes, the -connective objects are the -connective chain complexes, namely those which are concentrated in degrees .
For the standard t-structure on an -category of spectra, the -connective objects are the -connective spectra, namely those whose stable homotopy groups are concentrated in degree .
In an -topos one would — following traditional in algebraic topology — instead speak of the -connected for . 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: -connected corresponds to connective. (See there for more.)
For general discussion in the context of stable -categories see the references at t-structure, such as
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]
Created on April 20, 2023 at 07:14:34. See the history of this page for a list of all contributions to it.