Triangulated persistence category (TPC) insymplectic geometry
Jun Zhang(based on joint work with P. Biran and O. Cornea)
University of Science and Technology of China- Institute of Geometry and Physics

August 7, 2022

Motivation• Roughly speaking, Kontsevich’s homological mirror symmetry (HMS)(1994) claims that

D Fuk(X ) ' D Coh(X ∨ )where X is a symplectic manifold and X ∨ is its mirror.
• The A-side D Fuk(X ) and the B-side D Coh(X ∨ ) share some common algebraic structures, for instance, the triangulated structure(cones, distinguished triangles, etc.)• Recent work by Biran-Cornea (2014) reinterprets the triangulated
structure on D Fuk(X ) in terms of their Lagrangian cobordism theory.
In particular, one can define the cone decomposition (iterated cones)
and count the number of triangles in this decomposition.
• Recent work by Dimitrov-Haiden-Katzarkov-Kontsevich (2014) defines a complexity in D Coh(X ∨ ) when it is split-generated. Also
Orlov has some work on the dimension of D Coh(·) (2008).

Motivation (cont.)• However, the A-side D Fuk(X ) seems having richer structures thanthe B-side D Coh(X ∨ ). For instance, it can be studied from a quantitative perspective, since the building block

Lagrangian Floer complex CF(L, L0 ) is a filtered vector space
where the filtration comes from the symplectic action functional(defined on some path space).
• Quantitative measurements on CF(L, L0 ) include spectral invariants(Oh, Viterbo), torsion exponents (FOOO), boundary depth (Usher),barcode (Usher-Z.), partially symplectic quasi-state (PolterovichEntov) etc.
• Recent work by Biran-Cornea-Shelukhin (2021) defines the concept“shadow”, a positive real number, for each cone decomposition when
we study D Fuk(X ) from the perspective of Lagrangian cobordism
theory.

Born of TPC
Inspired by Biran-Cornea-Shelukhin’s work (2021), one can combine
the filtration structure and triangulated structure in some way, sothat one can easily extract numerical data from D Fuk(X ). One
approach is called
triangulated persistence category (TPC)
which is, more precisely,
a persistence category + triangulated structure.
Remark
The word “persistence” is from persistent homology or persistence
module, a theoretical foundation of the newly-developed field topological data analysis (TDA). Background on TDA is not
required for the rest of this talk.

Persistence category
Definition (Biran-Cornea-Z. 2021)
A persistence category (PC) C is a (classical) category enrichedby persistence modules. Explicitly, if for any A, B ∈ Ob(C ), there
exists a functor EA,B : (R, ≤) → Vectk satisfying(i) the hom-set in C is HomC (A, B) = {(f , r ) | f ∈ EA,B (r )}. For
later use, denote Morr (A, B) := EA,B (r );(ii) for r ≤ r 0 and s ≤ s 0 , we have the following commutative
diagramMorr (A, B) × Mors (B , C )EA,B (ir ,r 0 )×EB ,C (is ,s 0 )00Morr (A, B) × Mors (B , C )◦(r ,s)◦(r 0 ,s 0 )/ Morr +s (A, C )

EA,C (ir +s ,r 0 +s 0 )/ Morr 0 +s 0 (A, C )The functor EA,B is called a persistence k-module and ir ,s = EA,B (ir ,s )for r ≤ s is called a structure map.

Persistence category (cont.)• Observe that a PC C is not necessarily additive, that is, HomC is
not always abelian. Here are two ways that help us transfer back toabelian categories.

(a) Denote by C0 the restriction category with HomC0 = Mor0 .
(b) Denote by C∞ the limit category with HomC∞ = lim Morr .
−→r• One can define persistence functors between two PCs, persistence
natural transformations, etc. There will be no surprise, only conventions!
• An important but auxiliary data for a PC C is a shift functor
Σ = ({Σr }r ∈R , {θs ,t }s ,t ), an R-parametrized family of persistencefunctors on C with persistence natural transformations satisfying
certain conditions. For instance, evaluated at any object A ∈ Ob(C ),θr ,0 (A) ∈ Mor−r (Σr A, Σ0 A) = Mor−r (Σr A, A).

When r ≥ 0, map i−r ,0 (θr ,0 (A)) ∈ Mor0 (Σr A, A) is important.

Acyclic objects• In a PC with a shift functor, different from the classical category,elements can be considered up to shift/approximation. One views
i−r ,0 (θr ,0 (A)) as an r -approximation of the identity map eA .

Definition
An object A ∈ Ob(C ) is called r -acyclic if i−r ,0 (θr ,0 (A)) = 0.
Recall that an object A is 0 in any category if and only if eA = 0.
• The object A is r -acyclic if and only if ir ,0 (eA ) = 0 (Exercise).
• There are various equivalent names for acyclic objects (dependingon the context) such as torsion element, homological non-essential
element, etc. The most successful applications of the persistence
module theory to symplectic geometry in recent years almost allcome from the study of acyclic objects (very biased opinion!)

Triangulated category
Adding the triangulated structure to a PC (with a shift functor) willlead to something interesting. Let us highlight three concepts thatcan be derived from a classical triangulated structure.
(1) Cone decomposition. Fix a subset of objects X := (X0 , X1 , ..., Xn ).
A cone decomposition of any object A with respect to X consistsof a family of triangles (∆1 , · · · , ∆n ) in C as follows,∆1 :X0 → Y1 → X1 → TX0Y1 → Y2 → X2 → TY1
 ∆2 :
..
.



∆
:
Y
→
Yn−1 → Xn−1 → TYn−2n −2
 n −1∆n :Yn−1 → A → Xn → TYn−1 .
Cone decomposition can be concatenated (or called refinement).

Triangulated category (cont.)(2) Complexity. Suppose C is split-generated by a single object E .
Consider, in a cone decomposition as above,
Xi = T mi E for some mi ∈ Z.
Define (due to DHKK 2014), for A ∈ Ob(C ),¨ n«Xthe cone decomposition of A
−mi t
δt (E , A) = infe.
with respect to translations of E
i=1(3) Grothendieck group. Let C be a triangulated category, its
K -group is defined, as an abelian group, by
A = B + C if and only if
K (C ) = A ∈ Ob(C ).
∃ B → A → C → TB in C
One should view K (C ) as a linear approximation of C .

Main results
Theorem (Biran-Cornea-Z. 2021)
Let (X , ω) be an exact symplectic manifold. Then the derived
Fukaya category D Fuk(X ) is a TPC.
For simplicity, consider dg-categories. Recall the following steps that
produce a triangulated structure (by Bondal-Kapranov 1990):
dg-category

pre-triangulated completion

homotopy category.

We can enhance everything to be filtered:filtered
dg-category

filteredpre-triangulated completion

filtered.
homotopy category

In particular, the hom-set of a filtered dg-category is a filtered chain
complex. The resulting filtered homotopy category is a TPC (BiranCornea-Z. 2021).

Main results (cont.)
Theorem (Biran-Cornea-Z. 2021)
For any TPC C , we can define a non-trivial pseudo-metric calledfragmentation pseudo-metric dfrag , with values in R≥0 , on Ob(C ).
• This dfrag is a purely algebraic generalization of the “fragmentation shadow metric” defined by Biran-Cornea-Shelukhin (2021) via
Lagrangian cobordism theory.
• This dfrag is defined via a quantity in R≥0 associated to any cone
decomposition and the difficulty is to show this quantity behaves ina sub-additivity way with respect to the concatenation/refinement.
• In terms of dfrag , one can study the dynamical property of endomorphisms F on C . For instance, for any “generator” A of C ,d(F n A,A)

consider limn→∞ frag n.

• The estimation of dfrag is possible but computing the exact valueof dfrag is almost never possible (even when C is very concrete).

Main results (cont.)• Denote by FKk the category of filtered chain complexes over k.
Its filtered homotopy category, denoted by H0 (FKk ), is a TPC.
• A result by Usher-Z. (2016) provides a complete classification of
objects in H0 (FKk ). Explicitly,M
MX=
E1 (a)[m] ⊕E2 (b , c)[m0 ]b ≤c ∈R,m0 ∈Z

a∈R,m∈Z

where E1 (a) = (· · · → 〈x 〉 → · · · ) and E2 (b , c) = (· · · → 〈z 〉 −∂→ 〈y 〉k → · · · )where filtrations of x , y , z are a, b , c (spectrum) respectively.
Theorem (Biran-Cornea-Z. 2022)
In the TPC H0 (FKk ), the (modified) DHKK-complexity isXXδt (E1 (0), X ) =e −aki t − e −bki t = magnitude of H∗ (X ).
deg k

i

Here, (persistence) magnitude is from Govc-Hepworth (2021).

Main results (cont.)
Theorem (Biran-Cornea-Z. 2022)
For any TPC C , there exists a pairing, a bilinear (but notnecessarily symmetry) map
κ : K (C ) ⊗ K (C ) → Λ P

where K (C ) is the K -group of the
P restriction category C0 and ΛP
is the Novikov polynomial ring { finite nr t r | nr ∈ Z, r ∈ R}.
This pairing κ is a generalization of the classical Euler pairing, but
the target ΛP is novel (and somewhat mysterious).
Corollary
Let X be a plumbing of two copies of T ∗ S 1 . Then there exists
M ∈ K (D Fuk(X )), as a K -class, that can not be represented byany embedded closed Lagrangian of X .

Main results (cont.)• Basic fact: for a TPC C , the restriction of C on acyclic objects
is also a TPC, denoted by A C .
• For any TPC C , the K -group K (C ) can always be viewed as a
ΛP -module, explicitly by t r · [A] := [Σr A] where Σr is the shift functorfrom the underlying PC structure of C .
• Denote by T PC the category of TPCs and MΛP the category ofΛP -modules. One can consider the following three functors fromT PC to MΛP :K (C ) := K (C0 ), K A (C ) := K (A C 0 ), K∞ (C ) := K (C∞ ).
Theorem (Biran-Cornea-Z. 2022)
There exists a functor TorK : T PC → MΛP such that the following
sequence of functors is exact,
0 → TorK → K A → K → K∞ → 0.

TPC
Definition
A triangulated persistence category (TPC) is a triple (C , Σ, T )satisfying the following conditions.
(i) C is a persistence category endowed with a shift functor Σ.
(ii) C0 is triangulated where T is the translation autoequivalence.
(iii) Functors Σ and T commute.
(iv) For r ≥ 0 and A ∈ Ob(C ), the map i−r ,0 (θr ,0 (A)) : Σr A → A
completes into a triangle in C0 as follows,
i−r ,0 (θr ,0 (A))Σr A −−−−−−−→ A → K → T Σr A

such that K is r -acyclic.
One should view (iv) as an r -approximation of the standard axiomeAA −→ A → 0 → TA.

Keys to proofs• In a TPC, one can define a weighted triangle in the form of

A → B → C → Σ−r TA.
This is a new family of triangles in C0 and it is “r -closed” to a
standard triangle. From this, one can derive weighted octahedral
axiom.
• One can compute K (H0 (FKk )). At present, this is the only casethat we can give a precise answer (Biran-Cornea-Z. 2022).
f.g.

λ : K (H0 (FKk )) ' ΛP

where f.g. means every filtered chain complex is in total of finite
rank, and the isomorphism λ is an isomorphism of rings.
• The isomorphism λ above can be described explicitly by λ :[E1 (a)] → t a (hence, [E2 (b , c)] = t b − t c ).

Keys to proofs (cont.)• The pairing κ can be constructed by the following recipe (first on
the level of Ob(C )). For any A, B ∈ Ob(C ), consider

A, B

HomC (A, T i B)

XA,T i B ∈ Ob(FKk )κ([A], [B]) :=X(−1)i λ(XA,T i B ) ∈ ΛP .

λ(XA,T i B ),then definei

The difficulty is to show κ is well-defined on the level of K -group.
• When C satisfies certain duality (say, Calabi-Yau duality), thenfor any a, b ∈ K (C ), one obtains the following skew-symmetry,κ(a, b)(t) = ±κ(b , a)(t −1 ).

This implies that in D Fuk(X ), we have κ([L], [L]) = χ (L), which isin particular a constant!

Keys to proofs (cont.)• Here is the explicit picture for Corollary above.
S
Y

Z0

y0

x

x
y 00

y 00

Area(shade) = a

where the immersed Lagrangian S is constructed by doing a surgery
at the intersection point y 0 . Denote by the MS in D Fuk(X ) themodule represented by S, then one computes that
κ([MS ], [MS ]) = t −a − t a .

K -theoretical measurements
The quantitative nature of a TPC C allows one to study its K -group
K (C ) from a quantitative perspective.
• When C = H0 (FKk ), we start from the following observation(Biran-Cornea-Z. 2022),θ : ΛP ' LCB
P
where LCB =finite nI 1I | I ⊂ R interval, nI ∈ Z and 1I is the indicator of I. Importantly θ , defined by t a 7→ 1[a,∞) , is an isomorphismof rings, but the ring structure on LCB is peculiar. Then one canapply various operations on functions to elements σ in LCB . For
instance,Z(σ − σ(∞)1[0,∞) )d µ.

σ 7→

R

This is called the length of σ and σ(∞) is called the Euler characteristic of σ.

K -theoretical measurements (cont.)• For a general TPC C where Ob(C ) is endowed with a pseudometric dC , one can definedK (C ) (a, b) := inf dC (A, B) [A] = a, [B] = b .

So far, this approach is embarrassing since for most examples that weknow, for instance dfrag , the induced pseudo-metric dK (C ) is trivial.
• The deep reason is that when passing to the K -group, one can notdistinguish “small torsion” with “big torsion”. Here is an example.
Again, for C = H0 (FKk ), we have

[E2 (0, 100)] = [E2 (0, 1)] + · · · + [E2 (99, 100)]in its K -group.
• One can use filtration shifts of autoequivalences of C to measure
the distance between objects in C and then descent to K (C ).

Outlook• Categorical dynamics on K (C ) (TPC + DHKK 2014).
• Quantitative refinement of homological mirror symmetry (??)• Filtered derived functors/derived category (part of these has been
written up)• Other algebraic structures that combine the filtration structure
and the triangulated structure (PTC?).

THANK YOU!