being called a "homotopical terrorist" by bondal may well be the high point of my career! https://www.ipmu.jp/sites/default/files/imce/news/41E_Workshop2.pdf
@EricPeterson i suspect that you get a t-structure that isn't complete, like the direct limit of tau_{geq n} X might be the fiber of the map from X to its rationalization. i have not yet figured out handedness for t-structures so I don't remember if this is left complete or right complete
i think there's a nice t-structure you get by patching together t-structures with an arithmetic square, like a spectrum X is >= 0 iff its rationalization is >= 1 and its p-completions are all >= 0. (maybe this is what eric meant by perversions?)
@MikeMiller oh haha i think it was just the overall amount of homotopy theory / higher category theory in my talks -- see here (or here for "handout" version)
more generally, i propose that we adopt the terminology "homotopical terrorism" for when more classical mathematics (e.g. the good old hochschild complex) gets nontrivially repackaged into $\infty$-language
... or maybe that's actually culturally insensitive -- or at the very least, a pretty bad way of selling the field :-/
yeah, from a social perspective were I not already inclined to give you the benefit of the doubt I might not want to read the work of a terrorist! And I think this is already a common feeling before that choice of language
@TylerLawson yeah, that's what I intended by "perversions". i certainly don't know what what the official terminology is, but these shifted-based-on-locus t-structures appear in the theory of perverse sheaves, so
in re this discussion of t-structures, i've never really known -- what are the upsides of having a t-structure, and what further properties might it have that would be desirable?
@AaronMazel-Gee Alexey used some more colourful language related to homotopical terrorists in a talk a few weeks ago: he was going on about understanding the enemy (i.e. you) and its propaganda, as that is the first step in winning the war.
He was speaking to a crowd of algebraic geometers, so I'd guess something about everything being translated and reinterpreted in an infinity-language, devoid of all geometry.
@EricPeterson I don't know much the t-structure, but if you also close up under desuspensions, and take a generalized type n spectrum, the associated colocalization functor is C_{n-1}^f
@Pieter on a similar theme, there's also Remark 7.1 of arxiv.org/abs/1706.03417v1: "For the benefit of those millenials who believe the Godement resolution is one of the founding documents of the United Nations, here is a translation of the above construction into [∞-language]..."
@AaronMazel-Gee for starters, just like with spectra it connects your category C to an ordinary abelian category A and gives cohomology theories with coefficients in the objects of A, as well as k-invariants that classify objects of C via their Postnikov tower
if you're into K-theory then t-structures give objects such a canonical decomposition that a devissage theorem applies, relating K-theory of (bounded objects of) C with that of A
@TylerLawson those are both great answers, thanks! for K-theory though, the word "bounded" is going to mean different things when using different t-structures, right?
haha i suppose in a way those are both the same answer -- closely related, anyways
