mathlib documentation

category_theory.limits.shapes.zero_objects

Zero objects #

A category "has a zero object" if it has an object which is both initial and terminal. Having a zero object provides zero morphisms, as the unique morphisms factoring through the zero object; see category_theory.limits.shapes.zero_morphisms.

References #

structure category_theory.limits.is_zero {C : Type u} [category_theory.category C] (X : C) :
Prop

An object X in a category is a zero object if for every object Y there is a unique morphism to : X → Y and a unique morphism from : Y → X.

This is a characteristic predicate for has_zero_object.

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noncomputable def category_theory.limits.is_zero.to {C : Type u} [category_theory.category C] {X : C} (h : category_theory.limits.is_zero X) (Y : C) :
X Y

If h : is_zero X, then h.to Y is a choice of unique morphism X → Y.

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noncomputable def category_theory.limits.is_zero.from {C : Type u} [category_theory.category C] {X : C} (h : category_theory.limits.is_zero X) (Y : C) :
Y X

If h : is_zero X, then h.from Y is a choice of unique morphism Y → X.

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Any two zero objects are isomorphic.

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A zero object is in particular initial.

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A zero object is in particular terminal.

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The (unique) isomorphism between any initial object and the zero object.

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The (unique) isomorphism between any terminal object and the zero object.

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Construct a has_zero C for a category with a zero object. This can not be a global instance as it will trigger for every has_zero C typeclass search.

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Every zero object is isomorphic to the zero object.

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There is a unique morphism from the zero object to any object X.

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There is a unique morphism from any object X to the zero object.

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