Augmented simplicial objects with an extra degeneracy #
In simplicial homotopy theory, in order to prove that the connected components
of a simplicial set X
are contractible, it suffices to construct an extra
degeneracy as it is defined in Simplicial Homotopy Theory by Goerss-Jardine p. 190.
It consists of a series of maps π₀ X → X _[0]
and X _[n] → X _[n+1]
which
behave formally like an extra degeneracy σ (-1)
. It can be thought as a datum
associated to the augmented simplicial set X → π₀ X
.
In this file, we adapt this definition to the case of augmented simplicial objects in any category.
Main definitions #
- the structure
ExtraDegeneracy X
for anyX : SimplicialObject.Augmented C
ExtraDegeneracy.map
: extra degeneracies are preserved by the application of any functorC ⥤ D
SSet.Augmented.StandardSimplex.extraDegeneracy
: the standardn
-simplex has an extra degeneracyArrow.AugmentedCechNerve.extraDegeneracy
: the Čech nerve of a split epimorphism has an extra degeneracyExtraDegeneracy.homotopyEquiv
: in the case the categoryC
is preadditive, if we have an extra degeneracy onX : SimplicialObject.Augmented C
, then the augmentation on the alternating face map complex ofX
is a homotopy equivalence.
References #
- [Paul G. Goerss, John F. Jardine, Simplicial Homotopy Theory][goerss-jardine-2009]
The datum of an extra degeneracy is a technical condition on
augmented simplicial objects. The morphisms s'
and s n
of the
structure formally behave like extra degeneracies σ (-1)
.
Instances For
If ed
is an extra degeneracy for X : SimplicialObject.Augmented C
and
F : C ⥤ D
is a functor, then ed.map F
is an extra degeneracy for the
augmented simplicial object in D
obtained by applying F
to X
.
Equations
Instances For
If X
and Y
are isomorphic augmented simplicial objects, then an extra
degeneracy for X
gives also an extra degeneracy for Y
Equations
- One or more equations did not get rendered due to their size.
Instances For
The shift of a morphism f : [n] → Δ
in SimplexCategory
corresponds to
the monotone map which sends 0
to 0
and i.succ
to f.toOrderHom i
.
Equations
Instances For
The obvious extra degeneracy on the standard simplex.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The extra degeneracy map on the Čech nerve of a split epi. It is
given on the 0
-projection by the given section of the split epi,
and by shifting the indices on the other projections.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The augmented Čech nerve associated to a split epimorphism has an extra degeneracy.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If C
is a preadditive category and X
is an augmented simplicial object
in C
that has an extra degeneracy, then the augmentation on the alternating
face map complex of X
is a homotopy equivalence.
Equations
- One or more equations did not get rendered due to their size.