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I am reading Paolo Aluffi's book: "Algebra: Chapter 0"
I am currently focussed on Chapter 2: "Groups: First Encounter".
On page 41, Aluffi defines a group as follows:
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Definition: A group is a groupoid with a single object.
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Aluffi has already defined a groupoid on page 29 as follows:
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" ... ... there are categories in which every morphism is an isomorphism; such categories are called groupoids."
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I cannot see how the definition of a group as a groupoid with a single object results in what I have come to know as a group: that is a set G with a binary operation \(\displaystyle \bullet \ : \ G \times G \to G \) that is associative and is such that G possesses a unique identity and unique inverses for each element.
Can someone please help me to see how the definition via category theory is equivalent to the usual 'undergraduate' definition of a group. I would especially like to be able to give a rigorous and formal demonstration of how a category in which all morphisms are isomorphisms leads to the properties of a group.
Hope someone can help in this matter.
... ... Further to the above but possibly covered in the above is the following: ... ... On page 43 Aluffi gives the following proposition pertaining to groups:
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Proposition 1.7. The inverse is unique: if \(\displaystyle h_1, h_2 \) are both inverses of g in G, then \(\displaystyle h_1 = h_2 \)
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Aluffi asks readers to construct a stand-alone proof of this based on the category theory definition of a group and the following proposition:
Proposition 4.2. The inverse of an isomorphism is unique.
I would really appreciate help in constructing such a proof.
PeterPeter
I am currently focussed on Chapter 2: "Groups: First Encounter".
On page 41, Aluffi defines a group as follows:
------------------------------------------------------------------------------
Definition: A group is a groupoid with a single object.
------------------------------------------------------------------------------
Aluffi has already defined a groupoid on page 29 as follows:
-----------------------------------------------------------------------------
" ... ... there are categories in which every morphism is an isomorphism; such categories are called groupoids."
----------------------------------------------------------------------------
I cannot see how the definition of a group as a groupoid with a single object results in what I have come to know as a group: that is a set G with a binary operation \(\displaystyle \bullet \ : \ G \times G \to G \) that is associative and is such that G possesses a unique identity and unique inverses for each element.
Can someone please help me to see how the definition via category theory is equivalent to the usual 'undergraduate' definition of a group. I would especially like to be able to give a rigorous and formal demonstration of how a category in which all morphisms are isomorphisms leads to the properties of a group.
Hope someone can help in this matter.
... ... Further to the above but possibly covered in the above is the following: ... ... On page 43 Aluffi gives the following proposition pertaining to groups:
-----------------------------------------------------------------------------------------------
Proposition 1.7. The inverse is unique: if \(\displaystyle h_1, h_2 \) are both inverses of g in G, then \(\displaystyle h_1 = h_2 \)
------------------------------------------------------------------------------------------------
Aluffi asks readers to construct a stand-alone proof of this based on the category theory definition of a group and the following proposition:
Proposition 4.2. The inverse of an isomorphism is unique.
I would really appreciate help in constructing such a proof.
PeterPeter
Last edited: