# What exactly is a property?

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...

• So what's narrow enough to be a property for you? – Hayden Apr 1 '14 at 1:13
• Whatever the case, maybe this will be useful to you: en.wikipedia.org/wiki/… – Hayden Apr 1 '14 at 1:14
• Why is that too broad? Is "has two arms" too broad to be a property of people? – Asaf Karagila Apr 1 '14 at 1:16
• 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 – AIM_BLB Apr 1 '14 at 1:23
• 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! – rschwieb Apr 1 '14 at 1:31