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An initial and terminal object of the category 'Set'

  1. Sep 23, 2008 #1

    According to the book 'Categories for the Working Mathematician' (p20), the empty set is an initial object and any one-point set is a terminal object for the category 'Set'.

    My question is,
    "Why an empty set cannot be the terminal object for the category 'Set' as well?".

    Is this because there is no isomorphism between one-point set and empty set, so we just discard empty set as a terminal object for the category 'Set'?

    Thanks in advance.
    Last edited: Sep 24, 2008
  2. jcsd
  3. Sep 26, 2008 #2


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    if T={t} is a set containing just the element t, then for every set A there is a unique function f:A->T. This is given by f(a)=t for all a in A. So, T is a terminal object.

    If A is any non-empty set then there is no function f:A->{} from A to the empty set. There is no possible value that can be assigned to f(a) for any a in A. So, by definition, {} is not a terminal object.
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