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How is a property $P$ formally defined in mathematics?

I mean for example if $f$ is a morphism from an object $X$ to $Y$ in some category, then somehow I feel that "has codomain $Y$" is too broad to be considered a property...

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  • $\begingroup$ So what's narrow enough to be a property for you? $\endgroup$ – Hayden Apr 1 '14 at 1:13
  • $\begingroup$ Whatever the case, maybe this will be useful to you: en.wikipedia.org/wiki/… $\endgroup$ – Hayden Apr 1 '14 at 1:14
  • $\begingroup$ Why is that too broad? Is "has two arms" too broad to be a property of people? $\endgroup$ – Asaf Karagila Apr 1 '14 at 1:16
  • $\begingroup$ I was thinking a property should be something that is independent of the category axioms, like does not make referene to domain codomain, "is a morphism" "is an object" etc.. but this is difficult to express clearly in mathematics $\endgroup$ – AIM_BLB Apr 1 '14 at 1:23
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    $\begingroup$ Hmm, well in my personal experience, a formal mathematical definition of "property" has never been necessary, nor do I think one would have been useful. The natural language meaning has always been enough. I bet logicians have a formal definition, though: good for them! $\endgroup$ – rschwieb Apr 1 '14 at 1:31
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The wiki says:

In mathematical terminology, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x| p(x) = true}; p is its indicator function.

This is plausible, but strictly speaking one would like to allow X to be a proper class so that one could talk about a property of groups, topologies, etc.

Really, the natural language meaning is enough to do most mathematics.

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