t-Structures are Normal Torsion Theories

We characterize t-structures in stable∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure t on a stable∞-category C is equivalent to a normal torsion theory F on C, i.e. to a factorization system F = (E,M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts. Introduction. The ideal reader of this note is acquainted with the basic theory of factorization systems, here treated in their ∞-categorical counterparts presented in [Joy04] and [Lur09], and t-structures in triangulated categories, for which the main references will be the classical [BBD82] and section 1.2 of Lurie’s Higher Algebra, [Lur11], of which this note is intended to be a modest paralipomenon. There seem to be no (or better to say, too many) comprehesive reference about the first topic, since every author seems to rebuild the basic theory from scratch each time they prove a new result. Nevertheless, having to choose once and for all a reference for the interested reader, we couldn’t help but mention the seminal paper by Freyd and Kelly [FK72], the refined notion of "algebraic" factorization system proposed in Garner’s [Gar09], and Emily Riehl’s thesis [Rie11], whose first and second chapters, albeit being mainly interested on weak factorization systems, constitute the best-approximation to a complete compendium about the basic theory, and finally the short, elementary note [Rie08]. Again, we must mention the paper [CHK85] by Cassidy, Hébert, and Kelly, which together with [RT07] and the first section of [BR07] constitute our main references for the connections between factorization systems and torsion theories in (pointed additive) categories. In particular, we would like to address the interested reader to [CHK85] for a crystal-clear treatment of what we called “fundamental connection” in our Section 1.1 and several adaptions of this notion in various particular contexts (pointed, well-complete and additive categories above all), and to [BR07] for making clear that t-structures can be regarded as the triangulated counterpart of torsion theories in abelian categories. Date: Thursday 4th September, 2014. 2010 Mathematics Subject Classification. 18E30, 18E35, 18A40.