Being universal with respect to a property

Question

I have always concerned universal properties under a statement of the form

Universal Property of the Quotient Group: Let $G$ be a group and $H$ a subgroup of $G$. If $f: G \to G'$ is a group homomorphism such that $H \subset ker \ f$, there exists a unique group homomorphism $\bar{f}: G/H \to G'$ such that $f= \bar{f} \circ \pi$.

Reciently, I've come across that this universal property can be rephrased as

Universal Property of the Quotient Group: The quotient group $G/H$ is another object together with a map $\pi: G \to G/H$ such that $H \subset ker \ \pi$ and it is universal with respect this property.

I know that the last sentence means exactly the same as the previous formulation, ie, that any other map satisfying the mentioned property factorizes through the object.

For instance, the universal property of the localization of a ring can be also stated in this previous way, namely "it is an object $S^{-1}A$ together with a morphism $j : A \to S^{-1}A$ such that $j(S) \subset (S^{-1}A)^*$ and it is universal wrt this property"

But my question is: can all universal properties be stated in this form, claiming that it is an object together with a (or some) map(s) such that some property if satisfied and it is universal wrt it?

For instance, I am unable to express in this form

--. Universal property of the free R-module.

--. Universal property of the direct sum or product.

--. Universal property of the localization of modules.

With respect what properties are these objects (with the obvious maps in every case) universal?

Answer 1

The direct sum $\oplus A_i$ is the universal object equipped with maps from each $A_i$, while the product $\prod A_i$ is the universal object with maps to each $A_i$. We see already there are two different kinds of universal properties: the first kind determines maps out of the universal object, while the second determines maps in.

All your other examples fit into this mode. The free $R$-module on a set $A$ is the universal $R$-module with a map from $A$ into its underlying set. The localization of a module $B$ at $S$ is the universal module with a map from $B$ on which $S$ acts invertibly. So both of these objects have "mapping in" universal properties, more formally, they're colimits or values of left adjoints. More examples of objects with mapping-in universal properties, that is limits or values of right adjoints, include pullbacks and kernels, indiscrete topological spaces, and values of forgetful functors (for instance the underlying set $UB$ of a module $B$ is the universal set equipped with a map $FUB\to B$, where $F$ is the free module functor.)

The answer to your general question is, essentially, yes, when defined by the observation that there are two ways of being universal: universal properties fall into classes of objects such that maps out or maps in are canonically determined by the starting data. I say "essentially" because there are, arguably, certain counterexamples, depending on what objects you consider as defined by a universal property-consider for instance the homology of a chain complex. All or most such examples can be squeezed into the framework just described, though, and in any case people generally don't seem too concerned with giving a general abstract definition of universal property. I do think Mac Lane defines a universal mapping problem in the manner you ask about, to give one reference.