Timeline for Interlocking (weak) factorization systems
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2020 at 20:33 | comment | added | Tim Campion | @MikeShulman: It contains the isomorphisms and is closed under composition, too :) | |
| Aug 21, 2020 at 2:20 | comment | added | Mike Shulman | Trivial observation about question 2: at least it's automatically closed under retracts... (-:O | |
| Aug 18, 2020 at 14:13 | comment | added | Jiří Rosický | Tim Campion. Your last observation can be extended. Dual of an interlocking factorization system is $(R,M,L)$ where $(R,M)$ is a factorization system and $(M,L)$ is a weak factorization system. I met this situation in my arXiv:1702.08684 paper. Examples are on Boolean algebras, Banach spaces, or commutative $C^\ast$-algebras. The last example yields an interlocking factorization system on compact Hausdorff spaces. | |
| Aug 16, 2020 at 12:10 | comment | added | Tim Campion | I suppose another class of examples would be the following. Let $C^{op}$ be a Grothendieck topos or a Grothendieck abelian category. Then the triple $((Mono^\square)^{op},Mono^{op},Epi^{op})$ is an interlocking factorization system on $C$. | |
| Aug 16, 2020 at 12:03 | comment | added | Tim Campion | @Fosco Thanks! I just came across this question of yours related to the first question above, as well as this one which yields, following Joyal's Catlab, a class of examples generalizing the first and third above. Namely, if $C$ is any variety of algebras, then $({}^\square Surj, Surj, Inj)$ is an interlocking factorization system on $C$. Awesome! | |
| Aug 16, 2020 at 9:09 | comment | added | fosco | Long ago I asked what's up with those FS such that $(E,M)$ is a OFS and $(M,E)$ a WFS; this doesn't seem to be understood. (I would call these particular interlocking FS Frobenius, for obvious reasons). Fiorenza, Marchetti and I studied the notion of "stable n-ary OFS" to describe Postnikov towers in arxiv.org/abs/1501.04658: not that this was really unknown. It just come out better when done $\infty$. Hopefully you can adapt that technology? | |
| Aug 15, 2020 at 23:29 | comment | added | varkor | This isn't quite the same phenomenon, but ternary factorisation systems share similar structure in being overlapping factorisation systems in some sense, though they're stronger than the notion you're interested in (at least when $R_1 = L_2$). | |
| Aug 15, 2020 at 22:55 | comment | added | Simone Virili | One example worth understanding could be the following: it is known that t-structures on the homotopy category of a given $(\infty,1)$-category are in bijection with a suitable class of factorization systems (called normal torsion theories). It could be interesting to study which $t$-structures correspond to "normal interlocking torsion theories"... e.g., are they the TTFs studied in the PhD thesis of Pedro Nicolas (arxiv.org/pdf/0801.0507.pdf) or do we need to impose some more conditions to get a bijection? | |
| Aug 15, 2020 at 21:29 | history | asked | Tim Campion | CC BY-SA 4.0 |