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nLab > Latest Changes: Examples of why isomorphism of categories is too strong.

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  1.  
    • CommentRowNumber1.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 3rd 2017

    The next-to-latest revision of equivalence of categories had a “query” to add an “intuitively clear” example why the notion of strict isomorphism of categories is too strong to be useful. I cannot think of a better example than the category of pointed sets versus the category of partial functions. In particular since even readers who have never learned category theory are likely to have been weaned on partial functions. I have therefore started to anser to this “query” with a condensed exposition of this example. The exposition had to broken off for the time being though. I intend to finish it tomorrow, complete with a proof that the categories are not isomorphic and a brief intuitive argument why they are (to be considered) the same nevertheless.

    Comments on whether you agree to use this example appreciated.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 3rd 2017
    • (edited Jun 3rd 2017)

    We have that equivalence described at partial function.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 3rd 2017

    I’m not sure that giving an example of an equivalence (that is not necessarily an isomorphism) really constitutes an adequate explanation of why strict isomorphism is too strong a notion.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2017

    It might be worth looking at the source for equivalences, which I think is the Tohoku paper of Grothendieck.

    • CommentRowNumber5.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 4th 2017

    @David_Corfield. Thanks for pointing out. Then I will perhaps just briefly summarize this in words and link to the existing treatment. Also, an even more “bell-ringing” example, intuitive for many readers, would be the usual example that the category “of integer dimensions”, with real matrices as the morphisms is equivalent but not isomorphic to the category of finite-dimensional real vector spaces, and I now intend to mention this, too, as an exercise in exposition.

    Something intervened today, though, so continuing this little section in equivalence of categories will have to wait till tomorrow.

    • CommentRowNumber6.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 7th 2017

    Not to leave this unfinished for too long, it was now decided to not use the partial-function-example as the requested intuitive example, but rather the usual trivial example of the category of finitely dimensional vector spaces versus the category of finite dimensions. This will be known to most readers. The partial-functions-example is briefly mentioned after that, with a link.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2017

    Thanks for adding these examples! However, I think the discussion that went along with them is unnecessarily wordy and conveys hardly any information aside from the distinction of size, so I condensed it greatly. Also, I generally dislike subsections that are the only subsection in their section, so I removed the section heading and incorporated them as bullet points in the containing section “Isomorphism”.

    • CommentRowNumber8.
    • CommentAuthorPeter Heinig
    • CommentTimeJul 21st 2017

    Re #7. Thanks for the edits and advice. I was aware that this is almost entirely about size. I also like it better this way.

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