semiorthogonal decompositions

Stream: maths

[Matthew Ballard]

variable [pretriangulated T]

def cone {X Y: T} (f : X ⟶ Y) : T :=
Exists.some(distinguished_cocone_triangle X Y f)

structure triang_subcategory :=
(P : T → Prop)
(cone_mem : ∀ {X Y : T} (f : X ⟶ Y), P X → P Y → P (cone T f))
(shift_mem: ∀ (X : T), P X → P (X ⟦1⟧))

structure semiorthogonal_decomposition :=
(A : triang_subcategory T)
(B : triang_subcategory T)
(semiorthogonal : ∀ {X Y: T} (f : X ⟶ Y), A.P Y → B.P X → f = 0)
(triangle : ∀ (X : T), ∃ (XA XB : T) (f : XB ⟶ X) (g : X ⟶ XA) (h : XA ⟶ (XB)⟦1⟧), (A.P XA) ∧ (B.P XB) ∧ triangle.mk _ f g h ∈ dist_triang T)

A semi-orthogonal decomposition $\langle A, B \rangle$ of triangulated category $T$ is a pair of triangulated subcategories $A, B$ such that

$$\operatorname{Hom}(b,a) = 0$$

when $a \in A$ and $b \in B$ and

• for any $x \in T$ there is a triangle

$$b \to x \to a \to b[1]$$

with $a \in A$ and $b \in B$. A canonical reference is Bondal and Orlov's paper.

What do people think about this design choice?

[Reid Barton]

I assume P is also meant to be invariant under isomorphism, and probably P 0 should hold, and maybe you want to assume something about direct sums too?
I tend to assume it's better to state things like cone_mem in the form "for every distinguished triangle $X \to Y \to Z$, P X -> P Y -> P Z" rather than making some arbitrary (but useless) choice of "the" cone, and then having to invoke isomorphism invariance

[Reid Barton]

(e.g. compare "a pushout of a cofibration is a cofibration" https://github.com/rwbarton/lean-homotopy-theory/blob/lean-3.4.2/src/homotopy_theory/formal/cofibrations/cofibration_category.lean#L55-L56)

[Matthew Ballard]

Since

$$X \overset{\operatorname{id}_X}{\to} X \to 0 \to X[1]$$

is a triangle, you get P 0 from P (cone T f).

Reid Barton said:

I assume P is also meant to be invariant under isomorphism,

Should it? It seems to fall in the same space as full vs strictly full subcategories.

Reid Barton said:

I tend to assume it's better to state things like cone_mem in the form "for every distinguished triangle $X \to Y \to Z$, P X -> P Y -> P Z" rather than making some arbitrary (but useless) choice of "the" cone, and then having to invoke isomorphism invariance

Yes, this is very good advice. Thanks.

[Reid Barton]

Matthew Ballard said:

Since

$$X \overset{\operatorname{id}_X}{\to} X \to 0 \to X[1]$$

is a triangle, you get P 0 from P (cone T f).

but P might not hold for any object to start with!

[Matthew Ballard]

Ah yes :embarrassed:

[Reid Barton]

If P is closed under cofibers then it is also closed under replacing objects by isomorphic ones (since $0 \to X \to X'$ is distinguished if $X \to X'$ is an iso)

[Reid Barton]

In other words if you rephrase cone_mem in the style I suggested then isomorphism invariance is a consequence (once you have P 0). If you ask for it to be closed under forming "the" cone then it's a weird concept if you did not already assume it to be isomorphism invariant.

[Matthew Ballard]

I agree

[Matthew Ballard]

About direct sums, you want to prove some results about (co)limits and left/right orthogonals. Is it valuable to package some assumptions directly?

[Johan Commelin]

@Matthew Ballard It will be very interesting to hear if you find the current setup of pretriangulated categories pleasant to work with. My gut feeling is that there might be too much bundling of triangles, and that this might be annoying to work with in practice.

[Johan Commelin]

But I haven't used it yet, so I don't have more than a gut feeling