Natural isomorphism of Left adjoints

Since we have the natural isomorphisms \varphi and \psi, we can use them to define the morphisms \alpha_x and \alpha_y for any objects x and y in \mathcal{C}. And since the composition of isomorphisms is an isomorphism, we can show that for any morphism f:x\to y, the equation H(f)\circ\alpha_x = \alpha_y\circ F(f) holds, thus proving that F and H are naturally isomorphic. In summary, to show that two left adjoints F and H are naturally isomorphic, we need to construct a natural isomorphism \alpha using the natural isomorphisms \varphi and \psi, and then
  • #1
dmuthuk
41
1
Given two left adjoints [tex]F,H:\mathcal{C}\to\mathcal{D}[/tex] of a functor [tex]G:\mathcal{D}\to\mathcal{C}[/tex], how do we show that [tex]F[/tex] and [tex]H[/tex] are naturally isomorphic? This is my idea so far (I am working with the Hom-set defenition of adjunction):

We need to construct a natural isomorphism [tex]\alpha[/tex]. So, for each [tex]x\in\mathcal{C}[/tex], I need a morphism [tex]\alpha_x:F(x)\to H(x)[/tex]. Suppose we are given the natural isomorphisms [tex]\varphi:\mbox{Hom}(F-,-)\to\mbox{Hom}(-,G-)[/tex] and [tex]\psi:\mbox{Hom}(H-,-)\to\mbox{Hom}(-,G-)[/tex]. Then, I can simply let [tex]\alpha_x := \varphi_{x,Hx}^{-1}\circ\psi_{x,Hx}(1_{Hx})[/tex]. But, I am stuck here. I don't know how to show that for a given morphism [tex]f:x\to y[/tex] in [tex]\mathcal{C}[/tex], [tex]H(f)\circ\alpha_x = \alpha_y\circ F(f)[/tex].
 
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  • #2
You have (natural) isos

(F?,?)-->(?,G?)-->(H?,?)

is the composition of (natural) isos an iso?
 
  • #3
n_bourbaki said:
You have (natural) isos

(F?,?)-->(?,G?)-->(H?,?)

is the composition of (natural) isos an iso?

Yes, I believe so.
 

1. What is a natural isomorphism of left adjoints?

A natural isomorphism of left adjoints is a type of isomorphism that exists between two functors, where one functor is a left adjoint of the other. It is a special case of natural transformations, which are mappings between functors that preserve the structure of the underlying category.

2. How is a natural isomorphism of left adjoints different from a regular isomorphism?

A natural isomorphism of left adjoints differs from a regular isomorphism in that it is a specific type of isomorphism that exists between two functors, while a regular isomorphism is a general concept in mathematics that refers to a bijection between two objects.

3. What are the properties of a natural isomorphism of left adjoints?

A natural isomorphism of left adjoints has three main properties: it is a natural transformation, it is an isomorphism, and it is a left adjoint. This means that it is a mapping between two functors that preserves the structure of the underlying category, it is a bijective mapping, and it is a left adjoint of the other functor.

4. When do we use natural isomorphisms of left adjoints?

Natural isomorphisms of left adjoints are used in many areas of mathematics, including category theory, algebra, and topology. They are particularly useful in proving the equivalence of categories, which is a fundamental concept in category theory.

5. How do we prove the existence of a natural isomorphism of left adjoints?

The existence of a natural isomorphism of left adjoints can be proven using the Yoneda lemma, which states that every natural transformation between two functors can be represented as the image of a unique morphism under the Yoneda embedding. This lemma is a powerful tool in category theory and is often used to prove the existence of natural isomorphisms of left adjoints.

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