# mathlibdocumentation

category_theory.limits.connected

# Connected limits #

A connected limit is a limit whose shape is a connected category.

We give examples of connected categories, and prove that the functor given by (X × -) preserves any connected limit. That is, any limit of shape J where J is a connected category is preserved by the functor (X × -).

@[instance]
@[instance]
@[instance]
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def category_theory.prod_preserves_connected_limits.γ₂ {C : Type u₂} {J : Type v₂} {K : J C} (X : C) :

(Impl). The obvious natural transformation from (X × K -) to K.

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@[simp]
theorem category_theory.prod_preserves_connected_limits.γ₂_app {C : Type u₂} {J : Type v₂} {K : J C} (X : C) (Y : J) :
def category_theory.prod_preserves_connected_limits.γ₁ {C : Type u₂} {J : Type v₂} {K : J C} (X : C) :

(Impl). The obvious natural transformation from (X × K -) to X

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@[simp]
theorem category_theory.prod_preserves_connected_limits.γ₁_app {C : Type u₂} {J : Type v₂} {K : J C} (X : C) (Y : J) :
@[simp]
theorem category_theory.prod_preserves_connected_limits.forget_cone_X {C : Type u₂} {J : Type v₂} {X : C} {K : J C}  :
@[simp]
theorem category_theory.prod_preserves_connected_limits.forget_cone_π {C : Type u₂} {J : Type v₂} {X : C} {K : J C}  :
def category_theory.prod_preserves_connected_limits.forget_cone {C : Type u₂} {J : Type v₂} {X : C} {K : J C}  :

(Impl). Given a cone for (X × K -), produce a cone for K using the natural transformation γ₂

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The functor (X × -) preserves any connected limit. Note that this functor does not preserve the two most obvious disconnected limits - that is, (X × -) does not preserve products or terminal object, eg (X ⨯ A) ⨯ (X ⨯ B) is not isomorphic to X ⨯ (A ⨯ B) and X ⨯ 1 is not isomorphic to 1.

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