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Stream: learning: questions

Topic: Torsion Times NonTorsion

Sam Tenka (Apr 30 2020 at 21:20):

Emily Riehl's introductory book mentions this neat example of an isomorphism that isn't natural:
working in the category of finitely presentable abelian groups, we always have:
$G \cong T(G) \times G/T(G)$
where $T$ extracts the torsion subgroup from $G$.

It's intuitive that this isn't natural, since the equivalence classes that $G/T(G)$ don't have canonical representatives. Visually, the "cylinder" presented as $\mathbb{Z} \times \mathbb{Z}/42\mathbb{Z}$ may be "sheared" to get just as good a presentation.

That said, $G \cong T(G) \times G/T(G)$ seems like a useful thing to talk about. Is there an adjective that applies to this family of isomorphisms and captures my intuition that it is better than a typical, "random" family of isomorphisms? Perhaps by casting the above "shear" in terms of some bundle-y or semi-direct-product-y language?

John Baez (Apr 30 2020 at 21:23):

I think there's a natural transformation $G \to T(G)$: a map of groups sends torsion elements of the first to torsion elements of the second.

Sam Tenka (Apr 30 2020 at 21:24):

John Baez said:

I think there's a natural transformation $G \to T(G)$: a map of groups sends torsion elements of the first to torsion elements of the second.

Yep! That's natural. I realize this isn't a precisely-formed question, but: is it possible to talk about the full isomorphism $G \cong T(G) \times G/T(G)$ as "almost natural"?

John Baez (Apr 30 2020 at 21:24):

Well, the next question I was going to ask is: is there a natural transformation $G \to G/T(G)$?

John Baez (Apr 30 2020 at 21:26):

More precisely: is the obvious quotient map $G \to G/T(G)$ natural? (There's a natural transformation sending everything in $G$ to the identity in $G/T(G)$, but that's not interesting.)

John Baez (Apr 30 2020 at 21:28):

Sorry, I was being dumb, there's no natural transformation $G \to T(G)$. There's a functor sending $G$ to $T(G)$; that's what my very sketchy argument actually showed. And there's natural transformation $T(G) \to G$.

John Baez (Apr 30 2020 at 21:30):

So right now I'm guessing there are natural transformations $T(G) \to G$ and $G \to G/T(G)$, but we can't piece these together to get a natural isomorphism $G \to T(G) \times G/T(G)$ because they're pointing in opposite directions: one natural transformation goes to $G$ (i.e. the identity functor) while the other goes from it!

Sam Tenka (Apr 30 2020 at 21:33):

John Baez said:

Well, the next question I was going to ask is: is there a natural transformation $G \to G/T(G)$?

Oh! Thanks for this clue! There should be... Let's see...
First, is $G/T(G)$ functorial in $G$? Yes: $\phi: G \to H$ induces $\hat \phi: G/T(G) \to H/T(H)$ by sending $g T(G) \mapsto \phi(g T(G)) T(H) = \phi(g) T(H)$ (we used that $\phi(T(G)) \subseteq T(H)$, as we saw when showing that $T$ is a functor). Now, do we have the naturality square that

$G \to G/T(G) \to H/T(H)$ agrees with $G \to H \to H/T(H)$?

Yes! Both just map $g \mapsto \phi(g) T(H)$!
So the family of maps $G \to G/T(G)$ is indeed natural.

So I guess there is only one "bad" direction: $G \to G/T(G)$ is okay, but $G \leftarrow G/T(G)$ is not.

Sam Tenka (Apr 30 2020 at 21:38):

John Baez said:

Sorry, I was being dumb, there's no natural transformation $G \to T(G)$. There's a functor sending $G$ to $T(G)$; that's what my very sketchy argument actually showed. And there's natural transformation $T(G) \to G$.

Wait... now I'm confused! Isn't there a natural transformation $G \to T(G)$, that is, from the identity functor to the functor $T$? The naturality square just says that $G \to H \to T(H)$ matches $G \to T(G) \to T(H)$ ... oh shoot! you're right, there isn't a map in $G \to T(G)$ in the category. $T(G)$ is a subobject, not a quotient object.

Sam Tenka (Apr 30 2020 at 21:49):

To summarize @John Baez's wisdom (and my bumbling) for onlookers:

We have functors that send $G$ to the sub object $T(G)$ and to the quotient object $G/T(G)$. However, the first functor isn't an arrow in the category; that is, there isn't a natural map from the identity functor to $T$. In fact, the functor $T$ enjoys natural maps to but not from the identity functor. By contrast, the functor $G/T(G)$ enjoys natural maps from but not to the identity functor.

John Baez (Apr 30 2020 at 22:00):

I bumbled around too for a while. It was fun.

So I guess there is only one "bad" direction...

Right, but it's a matter of opinion which direction is the "bad" one. It's like when two cars collide head-on in the middle of a desolate plain, we know that one of them was going the wrong way.

Sam Tenka (Apr 30 2020 at 22:44):

intuitively, the subobject $T(G)$ thinks the $\times$ is a coproduct, while the quotient object $G/T(G)$ thinks the $\times$ is a product!

John Baez (May 01 2020 at 00:32):

For abelian groups the binary product is the binary coproduct. And yet I suspect that doesn't help make $G$ naturally isomorphic to $T(G) \times G/T(G)$.

Sam Tenka (May 01 2020 at 00:34):

Right. I was just saying that the relevant property from T(G)'s perspective is that it is a subobject of a coproduct, and dually for G/T(G)) :slight_smile: :tongue:

John Baez (May 01 2020 at 00:39):

@Sam Tenka (naive student) - you posted your comment in the wrong thread. I do that all the time...

Morgan Rogers (he/him) (May 01 2020 at 09:23):

It's easy to fix :slight_smile: Next time I'll also delete the flagging comment so the flow of the discussion isn't interrupted :heart:

Morgan Rogers (he/him) (May 01 2020 at 09:46):

1. Since the torsion elements always form a normal subgroup, this is a special case of decomposing a group as a semidirect product. I mention this because I saw a talk by Peter Faul at TACL last year about viewing semi-direct products as a special case of Artin gluing, which has featured in the recent conversation in #practice: applied ct. The slides I link aren't the most enlightening, but one can probably find his work on this written up somewhere. This doesn't really answer your original question, though.
2. I had a conversation here a few weeks ago here on parametricity, which in my very limited understanding is a formalism for describing relationships between objects and derived objects which can involve repeated entries with different variance, like the relationship between a group $G$ and $\mathrm{Aut}(G) = \mathrm{Hom}(G,G)$. This seems very much like another instance of that! If it turns out that I am right and you find some useful insights in the material shared by Dan Doel or Gershom, please do share them here or over there :grinning_face_with_smiling_eyes: