Pretriangulated A ∞-categories

483 Pages·2008·3.126 MB·English

by  Yu. Bespalov, V. Lyubashenko, O. Manzyuk

Pretriangulated A -categories
∞
Yuri Bespalov1, Volodymyr Lyubashenko2 and Oleksandr Manzyuk3
February 18, 2012
1Bogolyubov Institute for Theoretical Physics, Metrologichna str., 14-b, Kyiv 143, 03143 Ukraine; be-
[email protected]
2Institute of Mathematics, NASU, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine;
[email protected]
3Institute of Mathematics, NASU, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine;
[email protected]
2
Preface
Mudrec(cid:126) (cid:25)e Fiziku provadiv,
I tolkovav (cid:31)kihs(cid:126) monadiv,
I dumav, vidkil(cid:126) vz(cid:31)vs(cid:31) svit?
Ivan Kotl(cid:31)revs(cid:126)kii˘,
Vergili(cid:13)va Ene¨ıda, qastina tret(cid:31), 1798.1
Monoids associative up to homotopy appeared first in topology in works of Stasheff in early
sixties. Simultaneously he started to study their differential graded algebraic analogues,
called A -algebras. In nineties mirror symmetry phenomenon discovered by physicists
∞
was spelled by Fukaya in a new language of A -categories. They combine features of
∞
categories and of A -algebras. The binary composition in A -categories is associative
∞ ∞
up to a homotopy which satisfies an equation that holds up to another homotopy, etc.
If these homotopies are trivial, we deal with a differential graded category. Bondal and
Kapranov gave a notion of a pretriangulated envelope of a differential graded category.
They indicated that pretriangulated differential graded categories are in the origin of the
truncated notion – triangulated categories of Grothendieck and Verdier. Inspired by all
these developments Kontsevich formulated the homological mirror conjecture: equiva-
lence of pretriangulated A -categories associated with a complex manifold and with its
∞
symplectic partner. Jointly with Soibelman he proved recently this conjecture for certain
type of manifolds.
Summing up, the framework of differential graded categories and functors is too narrow
for many problems, and it is preferable to consider wider class of A -functors even dealing
∞
with differential graded categories. We have noticed that many features of A -categories
∞
and A -functors come from the fact that they form a symmetric closed multicategory.
∞
This structure is revealed in the language of comonads.
In the first part of this book the theory of multicategories is presented including its new
ingredients: closed multicategories and multicategories enriched in symmetric multicate-
gories. In the second part we apply this theory to (differential) graded k-linear quivers.
We deduce from it various properties of A -categories, A -functors and A -transforma-
∞ ∞ ∞
tions such as unitality – existence of unit or identity morphisms. Then we construct two
1
A wise man there Physics honored,
He was explaining certain monads,
And thinking: where the world came from?
Ivan Kotlyarevsky, Eneida, part III, 1798.
3
4 Preface
ingredients of pretriangulated categories: the monad of shifts and the Maurer–Cartan
monad. First some predecessors are defined, which are monads in the category of graded
k-linear quivers. Then we deduce commutation rules between these monads and the ten-
sor comonad that defines A -categories. This leads to monads on the multicategory of
∞
A -categories. Commutation between the monad of shifts and the Maurer–Cartan monad
∞
is also important to us, since it turns their product into another monad, the monad of
pretriangulated A -categories. The latter A -categories constitute the main subject of
∞ ∞
the book.
We prove that zeroth homology of a pretriangulated A -category is a strongly trian-
∞
gulated category. As the name suggests, the latter notion due to Maltsiniotis is a trian-
gulated category with stronger axioms. In appendices we do some algebra in 2-categories
necessary for taking tensor products of A -categories with differential graded categories.
∞
The results proven in this book open up new possibilities for researchers in algebraic
and symplectic geometry, homological algebra, algebraic homotopy theory and category
theory. Graduate courses on mirror symmetry or representation theory would benefit
from the considered topics in A -category theory as well. We hope that the reader will
∞
find the developed techniques useful for his own research.
Acknowledgements. We were able to give the manuscript its final shape due to hos-
pitality and excellent conditions of work in Technische Universit¨at Kaiserslautern and
Max-Planck-Institut fu¨r Mathematik during visits of the second author. Commutative
diagrams in this book were typeset with the diagrams package created by Paul Taylor.
Kyiv, Yuri Bespalov
April 2008 Volodymyr Lyubashenko
Oleksandr Manzyuk
Chapter 1
Introduction
A -algebras were introduced by Stasheff in 1963 as algebraic counterpart of his the-
∞
ory of H-spaces, topological monoids associative up to homotopies which are in turn
coherent up to higher homotopies, etc. [Sta63]. Since then the theory of A -algebras
∞
underwent quiet development motivated by topological applications in works of Smirnov
[Smi80, Smi85, Smi93, Smi01], Kadeishvili [Kad80, Kad82, Kad85, Kad88] and some other
mathematicians. The interest to the subject grew significantly, when Fukaya suggested
a construction of an A -category arising from symplectic geometry data [Fuk93]. This
∞
idea was supported and developed further by Kontsevich [Kon95], who studied the phe-
nomenon of mirror symmetry, discovered by physicists. He came up with the homological
version of the mirror symmetry conjecture. It stated equivalence of two A -categories,
∞
one coming from the symplectic structure of a manifold, another from the complex struc-
ture of its mirror manifold. The attempts to face this conjecture required significant
efforts on the symplectic and complex sides of the story [Fuk03, KS09, BS04]. It already
has led Kontsevich and Soibelman to the proof of homological mirror symmetry conjec-
ture for certain type of manifolds [KS01]. These findings are expected to be summarized
in the books by Fukaya, Oh, Ohta and Ono [FOOO09], by Kontsevich and Soibelman
[KS07] and by Seidel [Sei08]. On the other hand, the basic notions of A -categories and
∞
of A -functors have been studied in works of Fukaya [Fuk02], Keller [Kel01], the second
∞
author [Lyu03] and Soibelman [Soi04].
In the present book the A -category theory is elaborated aiming at another appli-
∞
cation. In 1963 Verdier has materialized the ideas of Grothendieck about homological
algebra in the notion of triangulated category [Ver77]. It was, however, clear that the
new concept was a truncation of something underlying it. An insight came from Bon-
dal and Kapranov [BK90], who suggested to look for pretriangulated differential graded
(dg) categories, whose 0-th homology would give the triangulated category in question.
Drinfeld has succeeded in doing this for derived categories [Dri04] by relating quotient con-
structions for pretriangulated dg-categories and quotients (localizations) of triangulated
categories.
Clearly, the class of differential graded functors between pretriangulated dg-categories
is too small to provide a sufficient supply of morphisms. We propose to extend this class
to unital A -functors. It is advantageous to extend simultaneously the class of objects to
∞
pretriangulated A -categories. The aim of this book is to define such notions and to study
∞
their properties. Taking 0-th homology we get from these properties some known results
in the theory of triangulated (derived) categories. However, work with pretriangulated
5
6 1. Introduction
A -categories instead of triangulated categories opens up new possibilities. For instance,
∞
the quotient pretriangulated dg-category of complexes of quasicoherent sheaves is expected
to encode more information about the variety than the corresponding derived category,
cf. [BO02, Pol03].
1.1 Conventions and basic notions. We work within set theory in which all sets
are elements of some universes [GV73]. In particular, a universe is an element of another
universe. One of them, U (containing an element which is an infinite set) is considered
basic. It is an element of some universe U (cid:48) (cid:51) U .
A structure is called U -small if it consists of sets which are in bijection with elements
of the universe U [GV73, Expos´e I.1]. For instance, sets, rings, modules, categories, etc.
can be U -small. A category V is a U -category if the sets of morphisms V(X,Y ) are
U -small for all objects X, Y of V. A U -category V is U -small if and only if ObV is a
U -small set.
Let k be a U -small commutative associative ring with unity. Let gr denote the
category of graded k-modules. Let C = dg denote the abelian category of complexes of
k
k-modules with chain maps as morphisms. It is a closed monoidal category. A graded
quiver C always means for us a U -small set of objects ObC together with U -small
Z-graded k-modules of morphisms C(X,Y ), given for each pair X,Y ∈ ObC. A differential
graded category is a graded quiver C equipped with differentials m : C(X,Y ) → C(X,Y )
1
of degree 1 and a category structure such that the binary compositions m : C(X,Y ) ⊗
2
C(Y,Z) → C(X,Z) and units 1 : k → C(X,X) are chain maps. In other terms, a
X
differential graded category is a category enriched in dg, or a dg-category. The first
example of a differential graded category is the dg-category C = dg of complexes of
k
k-modules.
For any graded k-module M there is another graded k-module sM = M[1], its sus-
pension, with the shifted grading (sM)k = M[1]k = Mk+1. The mapping s : M → sM
given by the identity maps Mk===(cid:42)(cid:43)M[1]k−1 has degree −1.
Tensor product of homogeneous mappings f : X → Y , g : U → V between graded
k-modules means the following mapping X ⊗ U → Y ⊗ V of the degree degf + degg:
(x ⊗ u).(f ⊗ g) = (−1)degu·degfx.f ⊗ u.g = (−1)ufx.f ⊗ u.g.
As a rule, we shorten up the usual notation (−1)degx to (−)x. In the same spirit, (−)x+y
might mean (−1)degx+degy, and (−)xy might mean (−1)degx·degy, etc.
An A -category means for us a graded quiver C with n-ary compositions
∞
b : sC(X ,X ) ⊗ ··· ⊗ sC(X ,X ) → sC(X ,X ),
n 0 1 n−1 n 0 n
of degree 1 given for all n (cid:62) 1 (we assume for simplicity that b = 0) such that b2 = 0 for
0
1.1. Conventions and basic notions. 7
the k-linear map b : TsC → TsC, whose matrix components are maps
(cid:88)
b = b = 1⊗r ⊗ b ⊗ 1⊗t : TksC → TlsC (1.1.1)
kl n
r+n+t=k
r+1+t=l
of degree 1, where the tensor quiver TsC = ⊕ TnsC = ⊕ (sC)⊗n is defined as
n(cid:62)0 n(cid:62)0
TsC(X,Y ) = ⊕ TnsC(X,Y ) = ⊕n(cid:62)0 sC(X,X ) ⊗ ··· ⊗ sC(X ,Y ).
n(cid:62)0 X ,...,X ∈ObC 1 n−1
1 n−1
In particular, T1sC = sC, T0sC(X,X) = k and T0sC(X,Y ) = 0 if X (cid:54)= Y .
The equation b2 = 0 is equivalent to the system of equations for n > 0:
(cid:88)
(1⊗r ⊗ b ⊗ 1⊗t)b = 0 : TnsC(X,Y ) → sC(X,Y ).
k r+1+t
r+k+t=n
In particular, b : sC(X,Y ) → sC(X,Y ) satisfying the equation b2 = 0 is a differential.
1 1
Notice that maps here are composed from the left to the right. In general, the composition
f g
of maps, morphisms, functors, etc. is denoted in this book by fg = f ·g = → → =
g◦f. A function (or a functor) f : X → Y applied to an element is denoted f(x) = xf =
x.f = x f and occasionally fx. Preimage is denoted f−1(y) = f−1y.
•
The compositions b determine also operations
n
m = (cid:0)C⊗n s⊗→n (sC)⊗n bn→sC s−→1 C(cid:1)
n
of degree 2−n. A differential graded category is an example of an A -category for which
∞
b = 0 and m = 0 if n > 2. It has only the differential m and an associative composition
n n 1
m , which is a chain map with respect to differentials 1 ⊗ m + m ⊗ 1 and m . For a
2 1 1 1
general A -category m remains a chain map, but fails to be associative. Instead it is
∞ 2
associative up to boundary of the homotopy m . Jointly m and m satisfy an equation,
3 2 3
which holds up to higher homotopy m and so on.
4
The tensor quiver TC becomes a counital coalgebra when equipped with the cut co-
multiplication ∆ : TC(X,Y ) → ⊕ TC(X,Z)(cid:78) TC(Z,Y ), h ⊗ h ⊗ ··· ⊗ h (cid:55)→
0 Z∈ObC k 1 2 n
(cid:80)n (cid:78)
h ⊗ ··· ⊗ h h ⊗ ··· ⊗ h . The map b given by (1.1.1) is a coderivation
k=0 1 k k+1 n
with respect to this comultiplication. Thus b is a codifferential. Any coderivation b :
TsC(X,Y ) → TsC(X,Y ) (thus, b∆ = ∆(1 ⊗ b + b ⊗ 1)) has form (1.1.1) for b = b · pr :
n 1
TnsC(X,Y ) → sC(X,Y ) (in general, b might be non-zero, but we do not consider such
0
structures). Thus one of the approaches to A -categories is to view them as differential
∞
graded coalgebras. This suggests to use morphisms of dg-coalgebras as A -functors.
∞
An A -functor f : A → B is a map of objects f = Obf : ObA → ObB, X (cid:55)→ Xf
∞
and k-linear maps f : TsA(X,Y ) → TsB(Xf,Y f) of degree 0 which agree with the cut
comultiplication and commute with the codifferentials b. Such f is determined in a unique
8 1. Introduction
way by its components f = f pr : TnsA(X,Y ) → sB(Xf,Y f), n (cid:62) 1 (we require that
n 1
f = 0):
0
(cid:88)
f = f = f ⊗ f ⊗ ··· ⊗ f : TnsA(X,Y ) → TmsB(Xf,Y f).
nm i i i
1 2 k
i +···+i =n
1 k
As shown e.g. in [Lyu03] 0-th homology of an A -category C is a non-unital k-linear
∞
category. To ensure that H0(C) were unital we may ask C to be unital. An A -category
∞
C is unital if for any object X of C there is a cycle iC ∈ C(X,X)[1]−1 such that the
X 0
chain maps −( iC ⊗ 1)b : sC(X,Y ) → sC(X,Y ), (1 ⊗ iC)b : sC(X,Y ) → sC(X,Y ) are
X 0 2 Y 0 2
homotopic to 1. An A -functor f : A → B is unital if iAf − iB ∈ Imb for all objects
∞ X 0 1 Xf 0 1
X of C.
A -transformations r : f → g : A → B (not necessarily natural) can be also intro-
∞
duced using coalgebra structure as (f,g)-coderivations, that is, maps r : TsA(X,Y ) →
TsB(Xf,Y g) of certain degree degr such that r∆ = ∆(f ⊗r+r⊗g). They are restored
from its components r = r · pr : TnsA(X,Y ) → sB(Xf,Y g) via its matrix entries:
n 1
(cid:88)
r = f ⊗ r ⊗ g : TnsA(X,Y ) → TmsB(Xf,Y g).
nm kp a lq
k+a+l=n
p+1+q=m
Given two A -categories A, B one can form a third one A (A,B) (“the category of
∞ ∞
functors”), whose objects are A -functors f : A → B and graded k-modules of morphisms
∞
A (A,B)(f,g)[1] are formed by A -transformations r : f → g. The n-ary compositions
∞ ∞
B in A (A,B) are given by explicit formulae [Fuk02, Lyu03, LH03]. In this book we are
n ∞
going to explain this phenomenon. We show that A -categories form a closed category
∞
(not a monoidal one!), and A (A,B) are inner homomorphism objects in this category.
∞
Actually, we show more: A -categories form a closed symmetric multicategory, and we
∞
explore consequences of this fact.
Before describing these new features we recall some already known results about
A -categories. Objects X, Y of a unital A -category C are said isomorphic if they are iso-
∞ ∞
morphic in the ordinary category H0(C). If B is a unital A -category then C = A (A,B)
∞ ∞
is unital as well. According to general terminology, A -functors f,g : A → B are called
∞
isomorphic if they are isomorphic in H0(A (A,B)). Thus, there are natural A -isomor-
∞ ∞
phisms r : f → g, p : g → f. Naturality of r, p means that they are cycles of degree
−1 in A (A,B)[1], which correspond to cycles of degree 0 in A (A,B). These cycles
∞ ∞
are mutually inverse in the sense of H0(A (A,B)). This gives meaning to the notions
∞
of A -equivalences f : A → B (if both A and B are unital). The full subcategory
∞
of A (A,B) consisting of unital A -functors is denoted Au (A,B). Unital A -cate-
∞ ∞ ∞ ∞
gories are objects of a Cat-category (a 2-category) Au with the categories of morphisms
∞
H0(Au (A,B)). Natural A -isomorphisms (resp. A -equivalences) are precisely invert-
∞ ∞ ∞
ible 2-morphisms (resp. quasi-invertible 1-morphisms) in this 2-category.
1.1. Conventions and basic notions. 9
The following theorem gives a criterion of quasi-invertibility for an A -functor φ with
∞
values in a unital A -category: φ is an A -equivalence iff H0(φ) is essentially surjective on
∞ ∞
objects, and φ consists of homotopy isomorphisms. Notice that the source A -category
1 ∞
is not assumed unital, but is proven to be such.
1.2 Theorem ([Lyu03, Theorem 8.8]). Let C be an A -category and let B be a unital
∞
A -category. Let φ : C → B be an A -functor such that for all objects X, Y of C the chain
∞ ∞
map φ : (sC(X,Y ),b ) → (sB(Xφ,Y φ),b ) is homotopy invertible. Let h : ObB → ObC
1 1 1
be a mapping. Assume that each object U of B is isomorphic to Uhφ in H0(B). Then C
is unital, φ is an A -equivalence, and there is an A -functor ψ : B → C quasi-inverse to
∞ ∞
φ such that Obψ = h.
The theorem was originally proven in a more refined form, which allows to draw
inductive conclusions. If the A -functor ψ and the natural A -isomorphism r : id →
∞ ∞ B
ψφ : B → B are constructed up to k-th component, then the inductive procedure implies
existence of the remaining components with the required properties.
The following proposition was proven in various forms by Gugenheim and Stasheff
[GS86], Merkulov [Mer99], Kontsevich and Soibelman [KS01, Section 6.4], see also the
survey [LM08a, Proposition 2.5].
1.3 Proposition. Let (sC,d) be a differential graded quiver, let B be an A -category,
∞
and let f : (sC,d) → (sB,b ) be a chain quiver morphism such that chain maps f :
1 1 1
sC(X,Y ) → sB(Xf ,Y f ) are homotopy invertible for all pairs X, Y of objects of C. Then
1 1
there is an A -category structure on C such that bC = d and an A -functor f : C → B,
∞ 1 ∞
whose first component is the given morphism f .
1
Note that the proof is constructive. Since one knows the contracting homotopy of the
cone of a certain homotopy isomorphism, one can write down recursive formulas for the
components b , f , and to express them in terms of trees.
n n
If in the above proposition the A -category B is unital, the statement combines with
∞
the previous theorem. We find out that C is unital and f : C → B gives an A -equivalence
∞
of C with a full A -subcategory of B.
∞
The above applies, in particular, if the graded k-modules C(X,Y ) = H(B(X,Y ),m )
1
with zero differential are homotopy isomorphic to the complexes (B(X,Y ),m ) (e.g. if k
1
is a field). In this case it is possible to transfer the A -category structure from B to its
∞
homology C = H(B), cf. Kadeishvili [Kad82]. Notice that bC = 0. A -categories with
1 ∞
such property are called minimal. Composition m in minimal A -categories is strictly
2 ∞
associative.
This construction has consequences for the derived category of a k-linear abelian cat-
egory, when k is a field. The derived category can be presented as zeroth homology
of a certain differential graded category B, the quotient of all complexes over acyclic
complexes, constructed by Drinfeld [Dri04, Section 3]. Massey products in the derived
10 1. Introduction
category are related to the operations m for H•(B), cf. Lu, Palmieri, Wu, and Zhang
n
[LPWZ06].
Another application of A -equivalences is related to the Yoneda Lemma. Recall that
∞
C denotes the differential graded category of complexes of k-modules. For any unital
k
A -category A there is the Yoneda A -functor Y : Aop → Au (A,C ), constructed by
∞ ∞ ∞ k
Fukaya [Fuk02, Section 9]. It takes an object X of A to the A -functor HX = A(X, ) :
∞
A → C , Z (cid:55)→ A(X,Z). Higher components of HX and Y do not vanish, they are
k
given by explicit formulae, which we do not reproduce here. The first result about the
Yoneda A -functor is that Y is an A -equivalence with its image, the full subcategory
∞ ∞
of the dg-category Au (A,C ) consisting of A -functors HX. This theorem of Fukaya
∞ k ∞
[Fuk02, Theorem 9.1] was extended to arbitrary A -categories, unital in the above sense,
∞
in [LM08c, Proposition A.9]. It has an important corollary:
1.4 Corollary. Let A be a unital A -category. Then there is a differential graded cate-
∞
gory D with ObD = ObA and an A -equivalence f : A → D such that Obf = id.
∞
More general A -form of the Yoneda Lemma is the following:
∞
1.5 Proposition ([LM08b, Proposition A.3]). Let A be a unital A -category, let X be
∞
an object of A, and let f : A → C be a unital A -functor. Then the natural chain map
k ∞
Xf → Au (A,C )(HX,f), defined by explicit formulae, is homotopy invertible.
∞ k
For the case of a ground field k, this was also proved by Seidel [Sei08, Lemma 2.12].
The following proposition is proven in [LO06]. Several variations of hypotheses and
conclusions are listed there.
1.6 Proposition ([LO06, Proposition 8.6]). Let A, B be unital A -categories, and let
∞
f : A → B be a unital A -functor. Let g : ObA → ObB be a mapping. Let cycles
∞
r s−1 ∈ Z0B(Xf,Xg) be invertible in H0B for all objects X of A. Then the map g
X 0
extends to a unital A -functor g : A → B and the given r extend to an invertible
∞ X 0
natural A -transformation r : f → g : A → B. Furthermore, such extensions up to level
∞
n can be continued to the full A -notion. That is, given A -functor (Obg,g ,...,g ) and
∞ n 1 n
A -transformation (r ,r ,...,r ) with invertible r (such that the m-th components of
n 0 1 n 0
gb − bg and rb + br vanish for all 0 (cid:54) m (cid:54) n) extend to a unital A -functor g and to an
∞
invertible natural A -transformation r.
∞
The latter statement is due to inductive nature of the proof of Proposition 8.1 of
[LO06].
There are five notions of quotient relevant to this book:
– Verdier’s localization of triangulated categories from [Ver77, Ver96];
– Keller–Drinfeld’s quotient of dg-categories from [Dri04]. If B is a full dg-subcategory
of C, then by Drinfeld’s result there is an equivalence
H0(“Keller–Drinfeld” Ctr/Btr) (cid:39) “Verdier” H0(Ctr)/H0(Btr);