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Stream: learning: questions

Topic: The other universal property of free groups

Jencel Panic (Aug 18 2026 at 11:39):

The free group is defined by and adjunction, and every adjunction defines two universal properties.

(Initial morphism) ηY:Y→GFY which describes how to build the free group of a set Y.
(terminal morphism) εX:FGX→X which describes that the free group can be converted in a unique way to any other group, which has the generators of the free group as elements.

The initial property is known as the universal property of free groups. But what about the terminal one?

It seems to me that it is equally important: the initial one tells you how to build a free group from set, but the terminal property actually tells you what is defining property that the free group has.

It also seems to me that the terminal property also, as the initial one, defines the free group uniquely.

So does it have a name?

And of course the same question stands about every construct that is defined by universal property...

Ruby Khondaker (she/her) (Aug 18 2026 at 17:12):

The initial morphism gives you the universal property for the free group on a set $X$ - given a morphism $X \to U(K)$ where $U$ is the underlying set functor, it extends uniquely to a morphism $FX \to K$ which fits into a commutative triangle involving the initial morphism $X \to UFX$ . Or, in terms of representability, it gives you a description for $\text{Hom}(FX, -)$ , namely that it's naturally isomorphic to $\text{Hom}(X, U(-))$ .

The terminal morphism instead gives you the universal property for the underlying set of a group $K$ . Given a morphism $FX \to K$ where $F$ is the free group functor, there is a unique morphsim $X \to U(K)$ which fits into a commutative triangle involving the terminal morphism $FU(K) \to K$ . Or, in terms of representability, it gives you a description for $\text{Hom}(-, UK)$ , namely that it's naturally isomorphic to $\text{Hom}(F(-), K)$ .

So the initial and terminal parts are how you go back-and-forth along the adjunction. The initial one says "any function can be uniquely extended to a group homomorphism", while the terminal one says "any group homomorphism comes from a unique function". I hope that makes sense?

Björn Gohla (Aug 18 2026 at 18:12):

I would just call that a co-universal property.

Matteo Capucci (he/him) (Aug 19 2026 at 11:30):

I think (2) is used every time we say a homomorphism out of $FX$ is just a function out of $X$

Sridhar Ramesh (Aug 19 2026 at 17:07):

We have a natural correspondence between $\mathrm{Hom}(FY, X)$ and $\mathrm{Hom}(Y, GX)$ , which you can think of as a property about a specific $Y$ and a varying $X$ , or a property about a specific $X$ and a varying $Y$ , as you like.

If you look at a specific $Y$ and varying $X$ , then the correspondence amounts to that the specific comma category $(Y / G)$ has an initial object (which is a particular group $FY$ and map $\eta_Y : Y \to GFY$ ). This tells us, if we already know what $G$ is in general, how $FY$ is defined. So you might say this is the universal property of $FY$ as a "free" construction left adjoint to $G$ (presuming you already know what $G$ is).

If you look at a specific $X$ and varying $Y$ , then the correspondence amounts to that the specific comma category $(F / X)$ has a terminal object (which is a particular set $GX$ and map $\epsilon_X : FGX \to X$ ). This tells us, if we already know what $F$ is in general, how $GX$ is defined. So you might say this is the universal property of $GX$ as a "co-free" construction right adjoint to $F$ (presuming you already know what $F$ is).

In the case of free groups and underlying sets, you are not very likely to be in a situation where you already know what free groups are but don't know what underlying sets are. You are (or at some point in your life were) much more plausibly in the reverse situation of already knowing what underlying sets are and now wanting to define free groups. So there is correspondingly more emphasis on defining $FY$ via the initial object of $(Y / G)$ than on defining $GX$ via the terminal object of $(F / X)$ .

Regardless, it seems to me most natural to think of the salient property as the symmetric fact of the $\mathrm{Hom}(FY, X) \leftrightarrow \mathrm{Hom}(Y, GX)$ correspondence itself, rather than viewing it asymmetrically. It just happens to be a fact that this is equivalent to "Each $(Y / G)$ has an initial object" and also equivalent to "Each $(F / X)$ has a terminal object".

Sridhar Ramesh (Aug 19 2026 at 17:16):

Matteo Capucci (he/him) said:

I think (2) is used every time we say a homomorphism out of $FX$ is just a function out of $X$

I think (1) is being used there (or rather, the fact that $\eta_X : X \to GFX$ is initial in the comma category $(X / G)$ ). Note that this just requires $F$ to be defined at the object $X$ (and for $G$ to be defined at all objects), while the corresponding terminality fact about (2) requires $F$ to be defined at all objects.

Granted, one can derive (1) from (2) holding at all objects (and vice versa). But I think (1) is more directly salient in this case, thinking of $F$ as defined at one particular input and $G$ as defined everywhere.

Jencel Panic (Aug 20 2026 at 11:28):

Sridhar Ramesh said:

lient property as the symmetric fact of the $\mathrm{Hom}(FY, X) \leftrightarrow \mathrm{Hom}(Y, GX)$ correspondence itself, rather than viewing it asymmetrically. It just happens to be a fact that this is equivalent to "Each $(Y / G)$ has an initial object" and also equivalent to "Each $(F / X)$ has a terminal object".

Yes, that is my intuition as well, don't know why the initial property gets all the credit :)

John Baez (Aug 20 2026 at 15:09):

It's just that mapping out of a free group on a set is more exciting than mapping into the underlying set of a group. The free functor is exciting because it takes a set and puffs it up into a group through a rather nontrivial process. The forgetful functor just takes a group, forgets the group structure and leaves a set: ordinary mathematicians tend to do that almost without noticing.

"Freedom is more fun than forgetting".

John Baez (Aug 20 2026 at 15:10):

But this does not mean that forgetful functors are unimportant.

Mike Shulman (Aug 20 2026 at 15:37):

Also, the forgetful functor comes first. We start with the category of sets, then we define a group to be a set with structure, and so the forgetful functor is staring us in the face. It's less obvious that there is an object with the universal property of the free group, and so we consider that universal property to define that object. In order to use the universal property of the forgetful functor to define it, we'd have to know about the free group functor but not the forgetful functor, which is hard to imagine.

John Baez (Aug 20 2026 at 15:40):

Good point: that's why the forgetful functor is "boring": we define groups and many other things to be sets with structure, so we completely expect that they'll have underlying sets.

Jencel Panic (Aug 20 2026 at 19:23):

When I learned about the free group (well, it was the free monoid) I was most excited that you can convert it to any other group in a canonical way, I wasn't all that impressed by the fact that you can construct a group from a set.

This fact seems (at least from first sight) to correspond to the terminal universal property, than to the initial one.

Jencel Panic (Aug 20 2026 at 19:26):

But, when I read a bit more, most sources tend to cite the hom isomorphism, as the universal property of the free group, this is from Wikipedia:

The free group F S is the universal group generated by the set S. This can be formalized by the following universal property: given any function f from S to a group G, there exists a unique homomorphism φ : F S → G making the following diagram commute (where the unnamed mapping denotes the inclusion from S into F S:

That is, homomorphisms F S → G are in one-to-one correspondence with functions S → G.

https://en.wikipedia.org/wiki/Free_group

Mike Shulman (Aug 20 2026 at 20:17):

Well, fundamentally they are the same property, namely the adjunction between the free and forgetful functors. But we generally think of the initial universal property as a property of the free group, since it tells you how to construct a homomorphism whose domain is the free group and whose codomain is arbitrary, and the terminal universal property as a property of the underlying set, since it tells you how to construct a function whose codomain is the underlying set and whose domain is arbitrary.

Jencel Panic (Aug 26 2026 at 08:23):

Thanks, this is helpful, there was one other mention of "universal property of the underlying set".