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Stream: theory: category theory

Topic: complementary subcategories

Patrick Nicodemus (Apr 23 2022 at 02:51):

I'd like to know if the following concept is studied in the literature.

Let $C$ be an $\infty$ category.

Let $D$, $E$ be two subcategories of objects in $C$. Not necessarily subcategories, maybe just functors $F : D\to C$, $G : E\to C$. Assume for all $d \in D$, $e \in E$, the hom-$\infty$-category $Hom(F(d), G(e))$ is contractible to a discrete set. What do we call this phenomenon? I think of these two as kind of "complementary" subcategories. If we can replace an arbitrary object in $C$ by an object in the image of $D$ or $E$, in some kind of fibrant/cofibrant replacement, studying maps into/out of it becomes easier, it's essentially a set of morphisms rather than a space.

fosco (Apr 23 2022 at 08:08):

Yes, this phenomenon is largely studied when $F,G$ are fully faithful, and related to the theory of orthogonality and factorization systems. 1-categorically, there is a fairly large amount of literature; $\infty$-categorically, it depends what you want to study: for stable $\infty$-categories I said something in my thesis, although in a sort of hybrid form 1-/$\infty$.

Patrick Nicodemus (Apr 23 2022 at 08:13):

Ok, great. If you have any other references than your thesis I will appreciate them

fosco (Apr 23 2022 at 08:47):

These are excellent starting points:

C. Cassidy, M. Hébert, and G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 287–329.

J. Rosický and W. Tholen, Factorization, fibration and torsion, J. Homotopy Relat. Struct. 2 (2007), no. 2, 295–314.

the stable/abelian/triangulated case is explored mainly in relation to the notions of t-structure, semiorthogonal decomposition and similar:

B. Keller and D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239–253, Deuxième Contact Franco-Belge en Algèbre (Faulxles-Tombes, 1987).

Nicolás, Pedro. "On torsion torsionfree triples." arXiv preprint arXiv:0801.0507 (2008).

Alexei Bondal and Dmitri Orlov, Semiorthogonal decomposition for algebraic varieties, arXiv preprint alg-geom/9506012 (1995).

A. Kuznetsov, Base change for semiorthogonal decompositions, Compos. Math. 147 (2011), no. 3, 852–876.

None of these references makes the connection with factorization systems; as a rule of thumb the connection exists for triangulated categories that are homotopy categories of stable higher categories (you can choose pretty much any of your preferred models..); I was able to devise a notion of "triangulated" factorization system, but I wouldn't advise anyone to use it for a concrete purpose:

Loregian, Fosco; Virili, Simone (2020). Triangulated factorization systems and t-structures. Journal of Algebra, 550, 219−241.

An important application of this machinery is to the theory of "recollements", related to tilting/representation theory and suchlike topics: classical references are easy to find by just googling "triangulated category recollement".

Modern ones, a chapter of my thesis: https://arxiv.org/abs/1507.03913 which never became a paper because there was nothing new, and this short "étude" by Glasman and Barwick, who sadly decided I wasn't worth of being mentioned :grinning: https://arxiv.org/abs/1607.02064