I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I am doing and as such references is of outmost importans, and I found this note document which gave an interesting definition of free objects which I'll cite

A free object is the mother of all objects of that type, in the sense that every object of that type is a quotient of the free object

This is an extremely good definition for what I am after, especially when you comapare it with this definition which I have seen many times before. I don't include the diagram or that one becuase I do not know how to draw a diagram on here. However I curious, does anyone know a good source for the first definition? Or how to prove it from the more common definition?


This is not a definition, it is a property of free objects. Every object surjecting onto a free object satisfies this too, and these do not have to be free. (Consider the group $S_n \times \mathbb{Z}$ for instance.) I advise you to read any text / book on universal algebra. There, you can find concise definitions and constructions of free objects.

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  • $\begingroup$ They state it as a definition. Would you know any recommendation of books where I could find it? $\endgroup$ – Zelos Malum Feb 14 '16 at 11:22
  • $\begingroup$ Well done on giving constructive comments to question :) Well done! $\endgroup$ – Zelos Malum Feb 14 '16 at 11:25
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    $\begingroup$ No, they don't mean this to be a definition. They write: "We begin with a somewhat cryptic definition, whose meaning will soon become clear." -- in other words, you will have to read further. It is a preliminary "definition". $\endgroup$ – Martin Brandenburg Feb 14 '16 at 11:25

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