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nLab > Latest Changes: free category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2017
    • (edited May 5th 2017)

    I have given free category its own little entry. This used to be (and still is) the topic of one subsection at path category, but I feel that for pointing people to just the plain concept of a free category, the entry “path category” is not focused enough. But both entries remain cross-linked, so the reader may still explore further, if desired.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeNov 6th 2021

    small adjustments of wording.

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeApr 28th 2023
    I think this should be edited as this entry is problematic. There isn't really a "1-category of categories"; there's either a bicategory of categories or a 1-category of strict categories.

    There is indeed a full adjunction between the path category construction and the underlying graph functor for strict categories. There isn't for normal (weak) categories as equivalence of categories is not respected.

    But I'm not entirely sure quite what I should say. The above is perhaps a little too stark?
  4. (That comment was supposed to be signed - I thought I had signed in!)
    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 28th 2023

    Changed the wording to fix the size issues:

    More formally, there is a forgetful functor from the category of small categories to that of directed graphs, and the free category construction is the left adjoint to this functor.

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023

    Just to highlight that the issue raised in #3 is not the size issue, but the strictness issue.

    I have adjusted wording further to account for this (here).

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023

    added also (here) a sentence and adjunction diagram for the further free groupoid construction

    diff, v5, current

  8. Thanks, these fixes are perfect @Urs.
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023

    While we are at it, I have added section-pointers to the first couple of references.

    (Didn’t actually find free categories discussed in Borceux’s vol 1, so I removed that item.)

    diff, v6, current

1 to 9 of 9

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