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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeMay 6th 2014
   • PermaLink
   Author: Fosco
   Format: MarkdownItexI was undecided whether to post my question here on the nforum or rather turn it into a private email. As I’m looking forward to test the reaction of a possible audience, I finally decided to put it here. I need a long introduction to give you the background. Be patient! And thank you for reading. During the last months I extensively worked with factorization systems. The reason is a resemblance between "classical" t-structures in homological algebra, defined on a triangulated category $\mathbf{A}$, and suitable factorization systems defined on the base category of the t-structure. This resemblance, albeit known (see for example Rosicki and Tholen [paper](http://arxiv.org/abs/0801.0063) "Factorization, Fibration and Torsion", and the previous work by Cassidy, Hebert and Kelly "Reflexive subcategories, localisations and factorisation systems" which suggest -without telling it explicitly- that t-structures happen to be *torsion theories* in triangulated categories), seems utterly neglected in the literature, even if factorization systems happen to be one of the most ancient notions in CT (and even if several authors [build](http://www.math.uoi.gr/~abeligia/torsion.pdf) [the link](http://www.math.jussieu.fr/~keller/publ/aisles.pdf) between the two notions without explicitly mentioning it, again). I would like to extend this result to the realm of stable $\infty$-categories, using the classical characterization plus (and this is fundamental) the fact that in a stable $\infty$-category suspensions (loops) are true colimits (limits). Far from being vacuous, this point of view sheds some light on parts of Lurie’s HA1 in which, for example, t-structures on a stable $\infty$-category $\mathcal{C}$ are defined as *t-structures on the homotopy category* of $\mathcal{C}$, rather than as genuinely-higher-categorical obejcts. I think I'm able to prove by simple $\infty$-categorical means that the heart $\mathbf{D}_\ge\cap \mathbf{D}_\le$ of any $\infty$-categorical t-structure is "$\infty$-abelian" (read as: its homotopy cat is abelian), exploiting only universal constructions. (Sorry for this long introduction! I simply wanted to make clear what I'm doing, and clearly present my aims.) Now, the next step goes as follows: people working in Algebraic Geometry call a "t-stability" something which should somehow correspond to a [k-fold FS](http://ncatlab.org/nlab/show/k-ary+factorization+system) in a stable infinity category: in AG you say that a "t-stability" is a way to factor any terminal morphism $X \to 0$ into a finite sequence of arrows, $X\to E_1\to\dots \to E_n\to 0$ whose (homotopy) fibers have "definite phase", i.e. each of them lie in a distinguished subcategory of "objects of type $\alpha$" for $\alpha$ the element of a poset. These decompositions are called *Harder-Narashiman filtrations*, and this is the first step to construct stability conditions on triangulated categories *a la* Bridgeland. Now, [Stigler's law](http://en.wikipedia.org/wiki/Stigler's_law_of_eponymy) entails that Harder-Narashiman filtrations are no more than *Postnikov towers* in a triangulated category! If $\alpha$ is a natural number $n$, then the distinguished categories correspond to shifts of the heart of the canonical t-structure in the stable category of spectra (this heart is [nothing more than the category $\mathbf{Ab}$](http://mathoverflow.net/a/112418/7952)), and "having definite phase" corresponds to "being a EM space". Add to this that Postnikov decompositions are deduced in the $\infty$-setting from a multiple factorization system on that category, and you'll obtain the main reason I began investigating this problem from that point of view. Finally, my questions: I'm converging to some results, but I'm stuck in a series of annoying problems I'm not able to solve alone: exactly the point where I need the feedback of the community. 1. Why does this point of view seem to be so neglected? 2. To what extent all this material is known to somebody-out-there? 3. There's the famous joke about the student proving several high-sounding theorems about "$\mathcal{L}$-groups", until he discovers that the class of $\mathcal{L}$-groups contains only $\mathbb Z/2$ and $0$. This is why I asked the [question](http://mathoverflow.net/questions/164894/factorization-systems-in-a-triangulated-stable-category) on MO about FS in a triangulated category: given that the "obvious" FS which rely on monics tend to be extremely trivial in a triangulated/stable context, how can I be sure that I'm working in a nontrivial setting? 4. Having to work with $k$-fold factorization systems, to understand Postnikov decompositions, I want to better understand the $n$Lab page which defines them. In the case of $k=3$ it's [a matter of computations](http://ncatlab.org/nlab/show/ternary+factorization+system) to obtain a factorization of type $(L_1, L_2\cap R_1, R_2)$; what about the general case? Drawing a couple of diagrams I can make a guess that every arrow factors as $(L_1, R_1\cap L_3 , L_3\cap R_2, R_2\cap L_3, R_3)$, but for the moment I'm not 100% sure about this. To obtain this factorization I simply reproduced the argument of the ternary case, applied to the FS $(L_i, R_i), (L_j, R_j)$ for $i \lneq j$, and then recursively applying orthogonality and cancellation properties of the various classes.

   I was undecided whether to post my question here on the nforum or rather turn it into a private email. As I’m looking forward to test the reaction of a possible audience, I finally decided to put it here. I need a long introduction to give you the background. Be patient! And thank you for reading.

   During the last months I extensively worked with factorization systems. The reason is a resemblance between “classical” t-structures in homological algebra, defined on a triangulated category $\mathbf{A}$, and suitable factorization systems defined on the base category of the t-structure. This resemblance, albeit known (see for example Rosicki and Tholen paper “Factorization, Fibration and Torsion”, and the previous work by Cassidy, Hebert and Kelly “Reflexive subcategories, localisations and factorisation systems” which suggest -without telling it explicitly- that t-structures happen to be torsion theories in triangulated categories), seems utterly neglected in the literature, even if factorization systems happen to be one of the most ancient notions in CT (and even if several authors build the link between the two notions without explicitly mentioning it, again).

   I would like to extend this result to the realm of stable $\infty$-categories, using the classical characterization plus (and this is fundamental) the fact that in a stable $\infty$-category suspensions (loops) are true colimits (limits).

   Far from being vacuous, this point of view sheds some light on parts of Lurie’s HA1 in which, for example, t-structures on a stable $\infty$-category $\mathcal{C}$ are defined as t-structures on the homotopy category of $\mathcal{C}$, rather than as genuinely-higher-categorical obejcts. I think I’m able to prove by simple $\infty$-categorical means that the heart $\mathbf{D}_\ge\cap \mathbf{D}_\le$ of any $\infty$-categorical t-structure is “$\infty$-abelian” (read as: its homotopy cat is abelian), exploiting only universal constructions.

   (Sorry for this long introduction! I simply wanted to make clear what I’m doing, and clearly present my aims.)

   Now, the next step goes as follows: people working in Algebraic Geometry call a “t-stability” something which should somehow correspond to a k-fold FS in a stable infinity category: in AG you say that a “t-stability” is a way to factor any terminal morphism $X \to 0$ into a finite sequence of arrows, $X\to E_1\to\dots \to E_n\to 0$ whose (homotopy) fibers have “definite phase”, i.e. each of them lie in a distinguished subcategory of “objects of type $\alpha$” for $\alpha$ the element of a poset. These decompositions are called Harder-Narashiman filtrations, and this is the first step to construct stability conditions on triangulated categories a la Bridgeland.

   Now, Stigler’s law entails that Harder-Narashiman filtrations are no more than Postnikov towers in a triangulated category! If $\alpha$ is a natural number $n$, then the distinguished categories correspond to shifts of the heart of the canonical t-structure in the stable category of spectra (this heart is nothing more than the category $\mathbf{Ab}$), and “having definite phase” corresponds to “being a EM space”. Add to this that Postnikov decompositions are deduced in the $\infty$-setting from a multiple factorization system on that category, and you’ll obtain the main reason I began investigating this problem from that point of view.

   Finally, my questions: I’m converging to some results, but I’m stuck in a series of annoying problems I’m not able to solve alone: exactly the point where I need the feedback of the community.

   1. Why does this point of view seem to be so neglected?
   2. To what extent all this material is known to somebody-out-there?
   3. There’s the famous joke about the student proving several high-sounding theorems about “$\mathcal{L}$-groups”, until he discovers that the class of $\mathcal{L}$-groups contains only and $\mathbb Z0$. This is why I asked the question on MO about FS in a triangulated category: given that the “obvious” FS which rely on monics tend to be extremely trivial in a triangulated/stable context, how can I be sure that I’m working in a nontrivial setting?
   4. Having to work with $k$-fold factorization systems, to understand Postnikov decompositions, I want to better understand the $n$Lab page which defines them. In the case of $k=3$ it’s a matter of computations to obtain a factorization of type $(L_1, L_2\cap R_1, R_2)$; what about the general case? Drawing a couple of diagrams I can make a guess that every arrow factors as $(L_1, R_1\cap L_3 , L_3\cap R_2, R_2\cap L_3, R_3)$, but for the moment I’m not 100% sure about this.

   To obtain this factorization I simply reproduced the argument of the ternary case, applied to the FS $(L_i, R_i), (L_j, R_j)$ for $i \lneq j$, and then recursively applying orthogonality and cancellation properties of the various classes.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexRe #4, I think in the quaternary case it should be $(L_1, L_2 \cap R_1, L_3 \cap R_2, R_3)$, and in general $(L_1, L_2 \cap R_1, \dots, L_n \cap R_{n-1}, R_n)$.

   Re #4, I think in the quaternary case it should be $(L_1, L_2 \cap R_1, L_3 \cap R_2, R_3)$, and in general $(L_1, L_2 \cap R_1, \dots, L_n \cap R_{n-1}, R_n)$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexFosco, quick general comment on a rushed day: this story looks really interesting, I wasn't aware of it this way.

   Fosco, quick general comment on a rushed day: this story looks really interesting, I wasn’t aware of it this way.

4. • CommentRowNumber4.
   • CommentAuthorFosco
   • CommentTimeMay 7th 2014
   • (edited May 7th 2014)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexUrs: nobody seems to be, and I begin wondering why. I am extremely happy to have your positive feedback! The proof of the abelianity of the heart (which is, to be honest, due to Domenico, like almost all the other insights in the above exposition) is one of the most clear. And yet nobody seemed to worry about making in explicit. Mr. Shulman, about #4: thanks! Can you give (a sketch of) a proof? Or maybe I can post mine... Are you aware of a reference which exposes the basic theory of multiple OFS? Is the nLab page the only one?

   Urs: nobody seems to be, and I begin wondering why. I am extremely happy to have your positive feedback! The proof of the abelianity of the heart (which is, to be honest, due to Domenico, like almost all the other insights in the above exposition) is one of the most clear. And yet nobody seemed to worry about making in explicit.

   Mr. Shulman, about #4: thanks! Can you give (a sketch of) a proof? Or maybe I can post mine… Are you aware of a reference which exposes the basic theory of multiple OFS? Is the nLab page the only one?