CategoryTheory.Limits.has_colimits_of_hasCoequalizers_and_coproducts
theorem CategoryTheory.Limits.has_colimits_of_hasCoequalizers_and_coproducts :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasCoproducts C]
[inst_2 : CategoryTheory.Limits.HasCoequalizers C], CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, u} C
The theorem `CategoryTheory.Limits.has_colimits_of_hasCoequalizers_and_coproducts` states that for any category `C`, if `C` has coproducts and coequalizers, then `C` has all colimits.
In more detail, a coproduct in a category is a generalized version of the notion of disjoint union or sum, while a coequalizer is a specific type of colimit used to describe the effect of identifying or "gluing" together certain elements of a set. The theorem asserts that the existence of these two types of colimits in a category ensures the existence of all possible colimits in that category. This result is a fundamental theorem in category theory.
More concisely: In any category with coproducts and coequalizers, all colimits exist.
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CategoryTheory.Limits.hasColimit_of_coequalizer_and_coproduct
theorem CategoryTheory.Limits.hasColimit_of_coequalizer_and_coproduct :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} [inst_1 : CategoryTheory.SmallCategory J]
(F : CategoryTheory.Functor J C)
[inst_2 : CategoryTheory.Limits.HasColimit (CategoryTheory.Discrete.functor F.obj)]
[inst_3 : CategoryTheory.Limits.HasColimit (CategoryTheory.Discrete.functor fun f => F.obj f.fst.1)]
[inst_4 : CategoryTheory.Limits.HasCoequalizers C], CategoryTheory.Limits.HasColimit F
This theorem states that given a category `C` and a functor `F` from a small category `J` to `C`, if `C` has the necessary coproducts (possibly finite) and coequalizers, then `F` has a colimit. Specifically, `C` must have a colimit of the functor which maps each object of `J` to an object of `C` using `F`, and `C` must also have a colimit for the functor that maps each pair of objects in `J` to an object of `C` using `F`. Also, every pair of morphisms in `C` needs to have a coequalizer. The requirement for all coequalizers is technically more than necessary for the existence of the colimit of `F`.
More concisely: Given a functor `F` from a small category `J` to a category `C` with finite coproducts and coequalizers, if `C` has colimits of functors that map each object and each pair of objects in `J` to `C` using `F`, and every pair of morphisms in `C` has a coequalizer, then `F` has a colimit in `C`.
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CategoryTheory.Limits.hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
theorem CategoryTheory.Limits.hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasFiniteCoproducts C]
[inst_2 : CategoryTheory.Limits.HasCoequalizers C], CategoryTheory.Limits.HasFiniteColimits C
The theorem `CategoryTheory.Limits.hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts` states that if a category `C` has finite coproducts and coequalizers, then it also has all finite colimits. In category theory, colimits are a generalization of several concepts in algebra and set theory, such as the codomain or target of a function, the union of sets, and the direct sum of modules. The condition of having finite coproducts and coequalizers ensures the existence of these general constructs in the category. You can find more details about this theorem [here](https://stacks.math.columbia.edu/tag/002Q).
More concisely: A category with finite coproducts and coequalizers has all finite colimits.
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CategoryTheory.Limits.hasFiniteLimits_of_hasEqualizers_and_finite_products
theorem CategoryTheory.Limits.hasFiniteLimits_of_hasEqualizers_and_finite_products :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasFiniteProducts C]
[inst_2 : CategoryTheory.Limits.HasEqualizers C], CategoryTheory.Limits.HasFiniteLimits C
This theorem states that for any category `C`, if `C` has finite products and equalizers, then `C` has all finite limits. In other words, the presence of finite products and equalizers is sufficient to guarantee the existence of all finite limits in the category. This is a result in category theory, a branch of mathematics that deals with abstract algebraic structures. For more details, you can refer to the link provided, which leads to an entry in the Stacks Math Project.
More concisely: In any category with finite products and equalizers, all finite limits exist.
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CategoryTheory.Limits.hasLimit_of_equalizer_and_product
theorem CategoryTheory.Limits.hasLimit_of_equalizer_and_product :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} [inst_1 : CategoryTheory.SmallCategory J]
(F : CategoryTheory.Functor J C) [inst_2 : CategoryTheory.Limits.HasLimit (CategoryTheory.Discrete.functor F.obj)]
[inst_3 : CategoryTheory.Limits.HasLimit (CategoryTheory.Discrete.functor fun f => F.obj f.fst.2)]
[inst_4 : CategoryTheory.Limits.HasEqualizers C], CategoryTheory.Limits.HasLimit F
This theorem states that, given a category `C` and a functor `F` from a small category `J` to `C`, if `C` has all equalizers (which is technically a stronger condition than necessary) and `C` has the limits of the functors from the discrete categories generated by the objects and pairs of objects of `J` to `C`, then `C` has a limit of the functor `F`. In simpler terms, if we have sufficient conditions in terms of equalizers and products, then we can guarantee the existence of a limit for a functor.
More concisely: If category `C` has all equalizers and limits of functors from discrete categories generated by objects and pairs of objects in small category `J`, then `F` from `J` to `C` has a limit in `C`.
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CategoryTheory.Limits.has_limits_of_hasEqualizers_and_products
theorem CategoryTheory.Limits.has_limits_of_hasEqualizers_and_products :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasProducts C]
[inst_2 : CategoryTheory.Limits.HasEqualizers C], CategoryTheory.Limits.HasLimitsOfSize.{w, w, v, u} C
This theorem states that, for any category `C`, if this category has products and equalizers, then the category has all limits. The "products" in this context refer to the existence of a product for every collection of objects in the category, and "equalizers" refer to the existence of an equalizer for every pair of morphisms in the category. Therefore, this theorem is essentially stating a sufficient condition for a category to have all limits: having both products and equalizers. The link provided references a proof of this theorem in the Stacks Project.
More concisely: In any category with products and equalizers, all limits exist.
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