CategoryTheory.Abelian.coimageImageComparison.proof_4
theorem CategoryTheory.Abelian.coimageImageComparison.proof_4 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C] {P Q : C}
(f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f)
(CategoryTheory.Limits.kernel.lift (CategoryTheory.Limits.cokernel.π f) f ⋯) =
0
This theorem in the category theory of abelian categories, named `CategoryTheory.Abelian.coimageImageComparison.proof_4`, asserts that for any morphism `f : P ⟶ Q` in an abelian category `C` with zero morphisms, kernels, and cokernels, the composition of the morphism from the kernel of `f` to the source of `f`, with a certain lifted morphism related to the morphism from the target of `f` to the cokernel of `f`, is the zero morphism. In more mathematical terms, it states that `kernel.ι(f) ∘ kernel.lift(cokernel.π(f)) = 0`.
More concisely: In an abelian category with zero morphisms, kernels, and cokernels, the composition of the inclusion morphism from the kernel of a morphism to its source, and the lifted morphism from the cokernel of the morphism to its target, is the zero morphism.
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CategoryTheory.Abelian.coimageImageComparison.proof_3
theorem CategoryTheory.Abelian.coimageImageComparison.proof_3 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasCokernels C] {P Q : C} (f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.cokernel.π f) = 0
The theorem `CategoryTheory.Abelian.coimageImageComparison.proof_3` states that for any Category `C` that has zero morphisms and cokernels, and for any morphism `f` from object `P` to `Q` in `C`, the composition of `f` and the map from the target of `f` to its cokernel is the zero morphism. In other words, given `f : P ⟶ Q`, we have `f ≫ cokernel.π f = 0`. This is a characteristic property of cokernels in category theory, which in essence captures the idea of 'quotienting out' by the image of a morphism.
More concisely: For any morphism `f` in an Abelian category with zero morphisms and cokernels, `f ≫ cokernel.π f = 0`.
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CategoryTheory.Abelian.coimage.fac
theorem CategoryTheory.Abelian.coimage.fac :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C] {P Q : C}
(f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.coimage.π f)
(CategoryTheory.Abelian.factorThruCoimage f) =
f
The theorem `CategoryTheory.Abelian.coimage.fac` asserts that for any categories `C` with zero morphisms, kernels, and cokernels, and for any morphism `f` from `P` to `Q` in `C`, the composition of the canonical morphism `π` (the projection onto the coimage of `f`) and the factorization of `f` through its coimage is equal to `f` itself. Essentially, it encapsulates the property that any morphism in an Abelian category factors through its coimage, with the factorization being mediated by the canonical projection onto the coimage.
More concisely: In an Abelian category with zero morphisms, for any morphism `f`, the composition of `f` with the canonical projection onto the coimage of `f` is equal to the factorization of `f` through its coimage.
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CategoryTheory.Abelian.image.fac
theorem CategoryTheory.Abelian.image.fac :
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C] {P Q : C}
(f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.factorThruImage f) (CategoryTheory.Abelian.image.ι f) =
f
This theorem, named `CategoryTheory.Abelian.image.fac`, states that for any category `C` with zero morphisms, kernels, and cokernels, and for any two objects `P` and `Q` in this category, any morphism `f` from `P` to `Q` factors through its image via the canonical morphism `p`. In other words, the composition of `f` with the factorization through its image, followed by the canonical morphism `p`, is equal to `f` itself. This is essential in the Abelian category theory, which studies categories that behave like the category of abelian groups, where morphisms can be "factored" in this way.
More concisely: For any morphism $f$ in an Abelian category with zero morphisms, for every object $P$, there exists a commutative diagram $P \xrightarrow{f} Q \xrightarrow{p} I \xleftarrow{i} Im(f)$ such that $fi = pi$.
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CategoryTheory.Abelian.factorThruCoimage.proof_2
theorem CategoryTheory.Abelian.factorThruCoimage.proof_2 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] {P Q : C} (f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f) f = 0
The theorem `CategoryTheory.Abelian.factorThruCoimage.proof_2` states that for any category `C` with zero morphisms and kernels, and for any two objects `P` and `Q` within `C`, the composition of the morphism from the kernel of a given morphism `f` (from `P` to `Q`) and `f` itself is the zero morphism. This is a property of abelian categories, where the kernel of a morphism is the source of a zero morphism when composed with the original morphism.
More concisely: In an Abelian category with zero morphisms and kernels, for any morphism `f` in the category, the composition of `f` and the kernel of `f` is the zero morphism.
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CategoryTheory.Abelian.image.proof_1
theorem CategoryTheory.Abelian.image.proof_1 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C] {P Q : C}
(f : P ⟶ Q), CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.cokernel.π f)
This theorem states that for any morphism `f` from object `P` to object `Q` in category `C` (which has zero morphisms, kernels, and cokernels), the map from the target of `f` to the cokernel of `f` has a kernel. In the language of category theory, this means that there exists an object and a morphism, such that the morphism is a kernel of the cokernel map of `f`. This is a property of abelian categories.
More concisely: In an abelian category with zero morphisms, kernels, and cokernels, for any morphism `f` from object `P` to object `Q`, the cokernel of `f` has a kernel, i.e., there exists an object `K` and a morphism `i` such that `fi = 0` and for all `h`, `hfi = h` implies `h = 0`.
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CategoryTheory.Abelian.factorThruImage.proof_2
theorem CategoryTheory.Abelian.factorThruImage.proof_2 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasCokernels C] {P Q : C} (f : P ⟶ Q),
CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.cokernel.π f) = 0
This theorem states that for any category `C` with zero morphisms and cokernels, and any two objects `P` and `Q` in `C`, the composition of any morphism `f` from `P` to `Q` and the cokernel of `f` is equal to the zero morphism.
In mathematical terms, if we have a category with zero morphisms and cokernels i.e., the category has all cokernels (which are a kind of colimit), then for any two objects `P` and `Q` in the category, and a morphism `f: P -> Q`, the composition of `f` with the cokernel of `f`, denoted as `f ≫ cokernel.π f`, is the zero morphism.
The cokernel of a morphism is a universal object that essentially captures the 'quotient' or 'residual' structure when we mod out by the image of the morphism.
More concisely: For any morphism `f` in a category with zero morphisms and cokernels, the composition of `f` with its cokernel's projection is the zero morphism.
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CategoryTheory.Abelian.coimage.proof_1
theorem CategoryTheory.Abelian.coimage.proof_1 :
∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasKernels C] [inst_3 : CategoryTheory.Limits.HasCokernels C] {P Q : C}
(f : P ⟶ Q), CategoryTheory.Limits.HasCokernel (CategoryTheory.Limits.kernel.ι f)
This theorem states that, in the setting of an abelian category (which contains zero morphisms, kernels, and cokernels), given any two objects `P` and `Q` in this category and a morphism `f` from `P` to `Q`, the kernel of `f` has a cokernel. In other words, you can form the cokernel of the morphism from the kernel of `f` to `P`.
In the context of category theory, a kernel is a limit of the parallel pair `f, 0`, and a cokernel is a colimit of the parallel pair `f, 0`. This theorem ensures the dual property that if `f` has a kernel, then there exists a cokernel of the morphism from this kernel to the source of `f`.
More concisely: In an abelian category, if a morphism has a kernel, then it has a cokernel and the cokernel of the morphism can be formed from its kernel and the source object.
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