Is there a standard notation for the k-connected cover of a spectrum E?

@ChrisSchommer-Pries Some people use E<k>.

I have also seen E[k,\infty] for the k-connected cover and E[0,k] for the k-truncation.

I've been going with $E_{\geq k}$ for connected cover.

so, one thing that's a bit strange about the product monoidal structure on Set is that there are ways of thinking about it that make no reference to the usual universal property of products. one is to think about it as left adjoint to taking sets of functions. another is to think of it as determined by the fact that it's cocontinuous in both variables and has unit 1. and a third, related to the second, is as follows:

the 2-category of presentable cocomplete categories and cocontinuous functors has a symmetric monoidal product representing functors cocontinuous in both variables, and its unit object is Set. the unit object of any symmetric monoidal higher category itself naturally inherits a commutative monoid structure from the unit maps; in this case, Set inherits the structure of a symmetric monoidal cocomplete category where the symmetric monoidal structure is cocontinuous in both variables, which...

is just the usual cartesian monoidal structure. but you can tell exactly the same story with "cocomplete Ab-enriched categories" (and note that this is a property, not a structure, which does not require a general theory of enrichment to describe), whose unit is Ab, and this reproduces the tensor product of abelian groups

a somewhat more interesting example is to repeat the same story with presentable cocomplete stable infty-categories, which gives you back the smash product of spectra. anyway, vague question: has anyone seen a story like this in the literature?

(the motivation here is that i wonder whether it's possible to recover tannaka reconstruction over Vect from "tannaka reconstruction over Set," which is the statement that geometric functors into G-sets are the same thing as G-torsors in, say, a grothendieck topos, although weaker assumptions are possible; morally the idea is to think of G-reps as G-sets tensor vect, but this operation takes the product monoidal structure to tensor product)

Any chance I could get some help with a homological algebra problem? I work in differential geometry but I've run into this in my research. Unfortunately my homological algebra background is entirely self-taught so I've gotten stuck on this issue.