Category Theory .... Isomorphisms and Inverses .... Awodey, Section 1.5 .... ....

In summary, Steenis explains that in order to prove that two arrows in a category are inverse, one must first prove that they are the same arrow in different positions. Then, using equation (1) and (2), one can conclude that the arrows are inverse.
  • #1
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I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ...

I need some further help in order to fully understand some further aspects of Definition 1.3, Page 12, including some remarks Awodey makes after the text of the definition ... ...

The start of Section 1.5, including Definition 1.3 ... reads as follows:View attachment 8354In the text of Definition 1.3 we read the following:

" ... ... Since inverses are unique (proof!), we write \(\displaystyle g = f^{-1}\). ... ... "Can someone please demonstrate a rigorous proof that in a category, inverses are unique ... ?Help will be appreciated ...

Peter
 
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  • #2
You have a isomorphism $f:A \rightarrow B$ in a category $C$

Suppose, there are two arrows $g,h:B \rightarrow A$ such that

$fg=1_B$, $gf=1_A$, $fh=1_B$, and $hf=1_A$

in other word, each is an inverse of $f$, we have to prove that $g=h$

Consider $hfg:B \rightarrow A \rightarrow B \rightarrow A$

On one hand, we have $hfg=h(fg)=h1_B=h$

On the other hand, we have $hfg= \cdots$, can you finish this ?
 
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  • #3
steenis said:
You have a isomorphism $f:A \rightarrow B$ in a category $C$

Suppose, there are two arrows $g,h:B \rightarrow A$ such that

$fg=1_B$, $gf=1_A$, $fh=1_B$, and $hf=1_A$

in other word, each is an inverse of $f$, we have to prove that $g=h$

Consider $hfg:B \rightarrow A\rightarrow B$

On one hand, we have $hfg=h(fg)=h1_B=h$

On the other hand, we have $hfg= \cdots$, can you finish this ?
Thanks Steenis ...

Hmm ... easy when you see how ... :) ...

We have ...

\(\displaystyle hfg = h(fg) = h 1_B = h\) ... ... ... ... ... (1)

and

\(\displaystyle hfg = (hf)g = 1_A g = g\) ... ... ... ... ... (2) ... so it follows that ...

(1) (2) \(\displaystyle \Longrightarrow g = h\)Hope that is correct ...Thanks again Steenis ...

Peter
 
  • #4
Yes that is correct

You did these things before, for instance, in Group Theory

In post #2, I meant $hfg:B \rightarrow A \rightarrow B \rightarrow A$
Already corrected
 
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Related to Category Theory .... Isomorphisms and Inverses .... Awodey, Section 1.5 .... ....

1. What is the definition of an isomorphism in category theory?

An isomorphism in category theory is a morphism that has an inverse, meaning there exists another morphism that, when composed with the original morphism, yields the identity morphism. Isomorphisms preserve all categorical structures, including objects, arrows, and composition.

2. What is the significance of isomorphisms in category theory?

Isomorphisms are significant because they represent a way for objects in one category to be equivalent to objects in another category. This allows for the comparison and translation of concepts between different categories, making it easier to understand and analyze complex systems.

3. How do you prove that two objects are isomorphic?

To prove that two objects are isomorphic, you must show that there exists an isomorphism between them. This can be done by demonstrating the existence of an inverse morphism, as well as showing that the composition of the two morphisms yields the identity morphism. In some cases, it may also be necessary to show that the objects have the same categorical properties.

4. What is the difference between an isomorphism and an equivalence in category theory?

An isomorphism is a specific type of equivalence, where the objects in two categories are considered equivalent if there exists an isomorphism between them. Equivalences are more general and can include other types of mappings, such as functors and natural transformations.

5. How are isomorphisms used in mathematical proofs?

Isomorphisms are often used in mathematical proofs to show that two seemingly different structures are equivalent. By demonstrating the existence of an isomorphism between the structures, the properties of one structure can be applied to the other, making it easier to prove certain theorems or statements.

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