Chain complexes supported in a single degree #
We define single V j c : V ⥤ HomologicalComplex V c
,
which constructs complexes in V
of shape c
, supported in degree j
.
Similarly single₀ V : V ⥤ ChainComplex V ℕ
is the special case for
ℕ
-indexed chain complexes, with the object supported in degree 0
,
but with better definitional properties.
In toSingle₀Equiv
we characterize chain maps to an ℕ
-indexed complex concentrated in degree 0;
they are equivalent to { f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 }
.
(This is useful translating between a projective resolution and
an augmented exact complex of projectives.)
The functor V ⥤ HomologicalComplex V c
creating a chain complex supported in a single degree.
See also ChainComplex.single₀ : V ⥤ ChainComplex V ℕ
,
which has better definitional properties,
if you are working with ℕ
-indexed complexes.
Instances For
The object in degree j
of (single V c h).obj A
is just A
.
Instances For
ChainComplex.single₀ V
is the embedding of V
into ChainComplex V ℕ
as chain complexes supported in degree 0.
This is naturally isomorphic to single V _ 0
, but has better definitional properties.
Instances For
Sending objects to chain complexes supported at 0
then taking 0
-th homology
is the same as doing nothing.
Instances For
Sending objects to chain complexes supported at 0
then taking (n+1)
-st homology
is the same as the zero functor.
Instances For
Morphisms from an ℕ
-indexed chain complex C
to a single object chain complex with X
concentrated in degree 0
are the same as morphisms f : C.X 0 ⟶ X
such that C.d 1 0 ≫ f = 0
.
Instances For
Morphisms from a single object chain complex with X
concentrated in degree 0
to an ℕ
-indexed chain complex C
are the same as morphisms f : X → C.X
.
Instances For
single₀
is the same as single V _ 0
.
Instances For
CochainComplex.single₀ V
is the embedding of V
into CochainComplex V ℕ
as cochain complexes supported in degree 0.
This is naturally isomorphic to single V _ 0
, but has better definitional properties.
Instances For
Sending objects to cochain complexes supported at 0
then taking 0
-th homology
is the same as doing nothing.
Instances For
Sending objects to cochain complexes supported at 0
then taking (n+1)
-st homology
is the same as the zero functor.
Instances For
Morphisms from a single object cochain complex with X
concentrated in degree 0
to an ℕ
-indexed cochain complex C
are the same as morphisms f : X ⟶ C.X 0
such that f ≫ C.d 0 1 = 0
.
Instances For
single₀
is the same as single V _ 0
.