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I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain:

"There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category of integral domains and injective ring maps. The functor from $\mathcal{C}$ to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the forgetful functor from the category of fields to $\mathcal{C}$."

It is usually the case that forgetful functors have free functors as left adjoints. Hence I wonder, can we make sense of the phrase "the free field on an integral domain", and if so, is this structure the field of fractions?

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  • $\begingroup$ What is your question? I mean, Wiki already pointed out that the field of fraction is a "unique up to iso" construction which associates to each domain the field which best approximates it (this is just a rough idea, I have been intentionally sloppy). The precise statement is that the canonical map $R \to Frac(R)$ is universal in the sense that it is the unit of the abovementioned adjunction. $\endgroup$ – Edoardo Lanari Jul 1 '14 at 14:31
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    $\begingroup$ Yes, the left adjoint to the functor $\mathbf{Fields}\longrightarrow\mathbf{Domains}$ sends every integral domain on its field of fractions. The proof is not difficult. Actually, once you know that there exists a left adjoint, you can easily show, that the structure of the elements coincides with the one you obtain via the usual construction. $\endgroup$ – Jakob Werner Jul 1 '14 at 14:40
  • $\begingroup$ "Can we make sense of [...]" Yes, we can : the notion you probably looking for is the notion of free object. $\endgroup$ – Pece Jul 1 '14 at 15:50
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I suggest you'd rather see this adjunction as a reflector-inclusion pair. $$Q:\mathbf{Dom_m} \dashv \mathbf{Field}:i$$

The forgetful functor is actually a full inclusion $i$ into the category of integral domains and monomorphisms, while the left adjoint $Q$ is the reflector "field of quotients". As is usual with full subcategories, you can think of the subcategory objects as having some extra property. So fields are integral domains with a special property and the $Q$ functor adds this special property to any integral domain. See also nlab

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