Definition Triangulated category

1 definition

1.1 tr 1
1.2 tr 2
1.3 tr 3
1.4 tr 4: octahedral axiom

definition

a translation functor on category d automorphism (or authors, auto-equivalence) t d d. 1 uses notation



x
[
n
]
=

t

n


x


{\displaystyle x[n]=t^{n}x}

, likewise morphisms x y.

a triangle (x, y, z, u, v, w) consists of 3 objects x, y, , z, morphisms u : x → y, v : y → z , w : z → x[1]. triangles written in unravelled form:

x


→





u




y


→





v




z


→





w




x
[
1
]


{\displaystyle x{\xrightarrow {{} \atop u}}y{\xrightarrow {{} \atop v}}z{\xrightarrow {{} \atop w}}x[1]}

or

x


→





u




y


→





v




z


→





w






{\displaystyle x{\xrightarrow {{} \atop u}}y{\xrightarrow {{} \atop v}}z{\xrightarrow {{} \atop w}}}

for short.

a triangulated category additive category d translation functor , class of triangles, called distinguished triangles, satisfying following properties (tr 1), (tr 2), (tr 3) , (tr 4). (these axioms not entirely independent, since (tr 3) can derived others.)

tr 1

for object x, following triangle distinguished:








x


→
id


x
→
0
→
x
[
1
]


{\displaystyle x{\overset {\text{id}}{\to }}x\to 0\to x[1]}






for morphism u : x → y, there object z (called mapping cone of morphism u) fitting distinguished triangle








x


→





u




y
→
z
→
x
[
1
]


{\displaystyle x{\xrightarrow {{} \atop u}}y\to z\to x[1]}






any triangle isomorphic distinguished triangle distinguished. means if








x


→





u




y


→





v




z


→





w




x
[
1
]


{\displaystyle x{\xrightarrow {{} \atop u}}y{\xrightarrow {{} \atop v}}z{\xrightarrow {{} \atop w}}x[1]}




is distinguished triangle, , f : x → x′, g : y → y′, , h : z → z′ isomorphisms, then






x
′



→

g
u

f

−
1






y
′



→

h
v

g

−
1






z
′



→

f
[
1
]
w

h

−
1






x
′

[
1
]


{\displaystyle x {\xrightarrow {guf^{-1}}}y {\xrightarrow {hvg^{-1}}}z {\xrightarrow {f[1]wh^{-1}}}x [1]}




is distinguished triangle.

tr 2

if

x


→





u




y


→





v




z


→





w




x
[
1
]


{\displaystyle x{\xrightarrow {{} \atop u}}y{\xrightarrow {{} \atop v}}z{\xrightarrow {{} \atop w}}x[1]}

is distinguished triangle, 2 rotated triangles

y


→





v




z


→





w




x
[
1
]


→

−
u
[
1
]



y
[
1
]


{\displaystyle y{\xrightarrow {{} \atop v}}z{\xrightarrow {{} \atop w}}x[1]{\xrightarrow {-u[1]}}y[1]}

and

z
[
−
1
]


→

−
w
[
−
1
]



x


→





u




y


→





v




z
.
 


{\displaystyle z[-1]{\xrightarrow {-w[-1]}}x{\xrightarrow {{} \atop u}}y{\xrightarrow {{} \atop v}}z.\ }

the second rotated triangle has more complex form when



[
1
]


{\displaystyle [1]}

,



[
−
1
]


{\displaystyle [-1]}

not isomorphisms mutually inverse equivalences since



−
w
[
−
1
]


{\displaystyle -w[-1]}

morphism



z
[
−
1
]


{\displaystyle z[-1]}





(
x
[
1
]
)
[
−
1
]


{\displaystyle (x[1])[-1]}

, obtain morphism



[
x
]


{\displaystyle [x]}

1 must compose component of natural transformation



(
x
[
1
]
)
[
−
1
]


→



x


{\displaystyle (x[1])[-1]{\xrightarrow {}}x}

. leads complex questions possible axioms 1 has impose on natural transformations making



[
1
]


{\displaystyle [1]}

,



[
−
1
]


{\displaystyle [-1]}

pair of inverse equivalences. due issue assumption



[
1
]


{\displaystyle [1]}

,



[
−
1
]


{\displaystyle [-1]}

mutually inverse isomorphisms usual choice in definition of triangulated structure.

tr 3

given 2 distinguished triangles , map between first morphisms in each triangle, there exists morphism between third objects in each of 2 triangles makes commute. means in following diagram (where 2 rows distinguished triangles , f , g form map of morphisms such gu = u′f) there exists map h (not unique) making squares commute:

tr 4: octahedral axiom

suppose have morphisms u : x → y , v : y → z, have composed morphism vu : x → z. form distinguished triangles each of these 3 morphisms according tr 1. octahedral axiom states (roughly) 3 mapping cones can made vertices of distinguished triangle commutes .

more formally, given distinguished triangles

x


→

u




y


→

j




z
′



→

k





{\displaystyle x{\xrightarrow {u\,}}y{\xrightarrow {j}}z {\xrightarrow {k}}}






y


→

v




z


→

l




x
′



→

i





{\displaystyle y{\xrightarrow {v\,}}z{\xrightarrow {l}}x {\xrightarrow {i}}}






x


→






v
u





z


→

m




y
′



→

n





{\displaystyle x{\xrightarrow {{} \atop vu}}z{\xrightarrow {m}}y {\xrightarrow {n}}}

there exists distinguished triangle

z
′



→

f




y
′



→

g




x
′



→

h





{\displaystyle z {\xrightarrow {f}}y {\xrightarrow {g}}x {\xrightarrow {h}}}

such that

l
=
g
m
,

k
=
n
f
,

h
=
j
[
1
]
i
,

i
g
=
u
[
1
]
n
,

f
j
=
m
v
.


{\displaystyle l=gm,\quad k=nf,\quad h=j[1]i,\quad ig=u[1]n,\quad fj=mv.}

this axiom called octahedral axiom because drawing objects , morphisms gives skeleton of octahedron, 4 of faces distinguished triangles. presentation here verdier s own, , appears, complete octahedral diagram, in (hartshorne 1966). in following diagram, u , v given morphisms, , primed letters cones of various maps (chosen every distinguished triangle has x, y, , z letter). various arrows have been marked [1] indicate of degree 1 ; e.g. map z′ x in fact z′ t(x). octahedral axiom asserts existence of maps f , g forming distinguished triangle, , f , g form commutative triangles in other faces contain them:

two different pictures appear in (beilinson, bernstein & deligne 1982) (gelfand , manin (2006) present first one). first presents upper , lower pyramids of above octahedron , asserts given lower pyramid, can fill in upper pyramid 2 paths y y′, , y′ y, equal (this condition omitted, perhaps erroneously, hartshorne s presentation). triangles marked + commutative , marked d distinguished:

the second diagram more innovative presentation. distinguished triangles presented linearly, , diagram emphasizes fact 4 triangles in octahedron connected series of maps of triangles, 3 triangles (namely, completing morphisms x y, y z, , x z) given , existence of fourth claimed. pass between first 2 pivoting x, third pivoting z, , fourth pivoting x′. enclosures in diagram commutative (both trigons , square) other commutative square, expressing equality of 2 paths y′ y, not evident. arrows pointing off edge degree 1:

this last diagram illustrates useful intuitive interpretation of octahedral axiom. since in triangulated categories, triangles play role of exact sequences, can pretend




z
′

=
y

/

x
,

y
′

=
z

/

x
 


{\displaystyle z =y/x,y =z/x\ }

in case existence of last triangle expresses on 1 hand

x
′

=
z

/

y
 


{\displaystyle x =z/y\ }

(looking @ triangle



y
→
z
→

x
′

→


{\displaystyle y\to z\to x \to }

 ), and





x
′

=

y
′


/


z
′

 


{\displaystyle x =y /z \ }

(looking @ triangle




z
′

→

y
′

→

x
′

→


{\displaystyle z \to y \to x \to }

 ).

putting these together, octahedral axiom asserts third isomorphism theorem :

(
z

/

x
)

/

(
y

/

x
)
=
z

/

y
.


{\displaystyle (z/x)/(y/x)=z/y.}

when triangulated category



k
(
a
)


{\displaystyle k(a)}

abelian category a, , when x, y, z objects of placed in degree 0 in eponymous complexes, , when maps x → y, y → z injections in a, cones literally above quotients, , pretense becomes truth.

finally, neeman (2001) gives way of expressing octahedral axiom using 2 dimensional commutative diagram 4 rows , 4 columns. beilinson, bernstein, , deligne (1982) give generalizations of octahedral axiom.