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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 $g$. $\endgroup$ – Qiaochu Yuan Mar 19 '15 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 '15 at 6:44
  • $\begingroup$ So that's basically a syntax for some limits. Thanks. $\endgroup$ – Konstantin Solomatov Mar 19 '15 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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