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I am currently reading Steve Awodey's book on category theory. On pg. 101 he uses set notation for generalized elements, namely $\{ a \mid f(a) = g(a) \}$ What does it mean in an arbitrary category?

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  • $\begingroup$ It means the pullback of $f$ and $g$. $\endgroup$ Mar 19, 2015 at 6:07
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    $\begingroup$ The pullback would be $\{ (a, b) : f (a) = g ( b) \}$. This is the equaliser. $\endgroup$
    – Zhen Lin
    Mar 19, 2015 at 6:44
  • $\begingroup$ So that's basically a syntax for some limits. Thanks. $\endgroup$ Mar 19, 2015 at 17:44

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The use of set notation to describe different types of limits is discussed briefly in Toposes, Triples and Theories in Section 1.6 and 1.7, where it is referred to as the "objectification" of the limit.

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