A limit cone c
in a simplicial category A
is a simplicially enriched limit if for every
X : C
, the cone obtained by applying the simplicial coyoneda functor (X ⟶[A] -)
to c
is a
limit cone in SSet
.
- isLimit : CategoryTheory.Limits.IsLimit c
- isSLimit : (X : C) → CategoryTheory.Limits.IsLimit (((CategoryTheory.SimplicialCategory.sHomFunctor C).obj (Opposite.op X)).mapCone c)
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Conical simplicial limits are also limits in the unenriched sense.
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- CategoryTheory.IsSLimit_islimit c slim = slim.isLimit
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Characterization in terms of the comparison map. #
There is a canonical comparison map with the limit in SSet
, the following proves that a limit
cone in A
is a simplicially enriched limit if and only if the comparison map is an isomorphism
for every X : A
.
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- One or more equations did not get rendered due to their size.
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- CategoryTheory.SimplicialCategory.isSLimitOfIsIsoLimitComparison c hc = { isLimit := hc, isSLimit := fun (X : C) => let_fun this := ⋯; Classical.choice ⋯ }
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ConicalLimitCone F
contains a cone over F
together with the information that it is a conical
limit.
- cone : CategoryTheory.Limits.Cone F
The cone itself
- isSLimit : CategoryTheory.IsSLimit self.cone
The proof that is the limit cone
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A conical limit cone is a limit cone.
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- CategoryTheory.ConicalLimitCone_isLimitCone cone slim = slim.isLimit
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HasConicalLimit F
represents the mere existence of a limit for F
.
- mk' :: (
- exists_limit : Nonempty (CategoryTheory.ConicalLimitCone F)
There is some limit cone for
F
- )
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Use the axiom of choice to extract explicit ConicalLimitCone F
from HasConicalLimit F
.
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- ⋯ = ⋯
An arbitrary choice of limit cone for a functor.
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An arbitrary choice of conical limit object of a functor.
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The projection from the conical limit object to a value of the functor.
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- CategoryTheory.conicalLimit.π F j = (CategoryTheory.conicalLimit.cone F).π.app j
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Evidence that the arbitrary choice of cone provided by conicalLimit.cone F
is a conical
limit cone.
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The morphism from the cone point of any other cone to the limit object.
Equations
- CategoryTheory.conicalLimit.lift F c = (CategoryTheory.conicalLimit.isConicalLimit F).isLimit.lift c
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C
has conical limits of shape J
if there exists a conical limit for every functor
F : J ⥤ C
.
- has_conical_limit : ∀ (F : CategoryTheory.Functor J C), CategoryTheory.HasConicalLimit F
All functors
F : J ⥤ C
fromJ
have limits
Instances
Equations
- ⋯ = ⋯
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- ⋯ = ⋯
If a functor F
has a conical limit, so does any naturally isomorphic functor.
Equations
- ⋯ = ⋯
If a E ⋙ F
has a limit, and E
is an equivalence, we can construct a limit of F
.
We can transport conical limits of shape J
along an equivalence J ≌ J'
.
C
has all conical limits of size v₁ u₁
(HasLimitsOfSize.{v₁ u₁} C
)
if it has conical limits of every shape J : Type u₁
with [Category.{v₁} J]
.
- has_conical_limits_of_shape : ∀ (J : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} J], CategoryTheory.HasConicalLimitsOfShape J C
All functors
F : J ⥤ C
from all smallJ
have conical limits
Instances
Equations
- ⋯ = ⋯
A category that has larger conical limits also has smaller conical limits.
hasConicalLimitsOfSizeShrink.{v u} C
tries to obtain HasLimitsOfSize.{v u} C
from some other HasLimitsOfSize C
.
C
has all (small) conical limits if it has limits of every shape that is as big as its
hom-sets.
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- ⋯ = ⋯
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- ⋯ = ⋯
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- ⋯ = ⋯
An abbreviation for HasSLimit (Discrete.functor f)
.
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- ⋯ = ⋯
An abbreviation for HasSLimit (Discrete.functor f)
.
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- ⋯ = ⋯
- has_conical_limits_of_shape : ∀ (J : Type w), CategoryTheory.HasConicalLimitsOfShape (CategoryTheory.Discrete J) C
All discrete diagrams of bounded size have conical products.
Instances
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- ⋯ = ⋯
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- ⋯ = ⋯
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- ⋯ = ⋯
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- ⋯ = ⋯
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- ⋯ = ⋯