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    • CommentRowNumber1.
    • CommentAuthorjesuslop
    • CommentTimeJul 25th 2018
    • (edited Jul 25th 2018)

    Hi, in Category of elements, it says:

    It is the category whose objects are pairs (c,x)(c,x) where cc is an object in CC and xx is an element in P(c)P(c) and morphisms (c,x)(c,x)(c,x) \to (c',x') are morphisms u:ccu:c \to c' such that P(u)(x)=xP(u)(x) = x'.

    S. Awodey in his notes here in his definition (actually in proposition 8.10) defines the concept in the same terms, but he adds:

    actually, the arrows are triples of the form (u,(x,C),(x,C))(u,(x,C),(x',C')) satisfying…

    And would be nice to have that here also (that the arrows are those triples). If not technically the same arrow could belong to two hom-sets.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2018

    My own feeling is that this is merely a slight and forgivable abuse of language; for me it is implicit that the domain and codomain are always part of the data of a morphism, and the uu names the only extra datum needed to complete the description.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 25th 2018

    In other words, we generally use this definition of a category in which it doesn’t matter whether or not the hom-sets are disjoint. If you want a structured way to make the hom-sets disjoint, see protocategory.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 25th 2018

    However, this is a good example of that point, so perhaps it should be brought out more explicitly.

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