Q:

Do journals have legal copyrights to the online supplemental information?

I have a paper in an biology journal (Wiley is the publisher) for which I would like to put a much more mathematical version on ArXiv, mainly the theorems and proofs located in the online supplement (that biologists will not read but that perhaps a mathematician who does not read this journal might be interested in), with no figures or sentences copied from the biology paper. I could perhaps call it a pre-submission version of the paper (which would be somewhat untrue, but Wiley does allow pre-peer-review preprint publishing on ArXiV; none of the peer-review process affected the supplement; so in spirit this would be sort of true).
Does Wiley’s copyright agreement prevent me from doing this? Wiley’s information on supplemental information doesn’t seem to address the issue.

A:

Wiley
Taking a look at Wiley’s general copyright agreement, they only “the Contribution”, which in turn is defined in the header as “Manuscript entitled ___”.
Now it’s debatable whether the supplemental material is part of the manuscript. However Wiley needs at least some permission from you to publish the supplemental material and I could not find any information on separate copyright agreements for the supplemental material, so I assume that at least Wiley considers the supplemental material to be included in the manuscript.

Wiley-Blackwell licenses back the following rights to the Contributor in the version of the Contribution as originally submitted for publication:
a. The right to […] place in a subject matter archive, […]. This right extends to […] the Internet. The Contributor may not update the submission version or replace it with the published Contribution. […]

Again, it’s not specified whether this applies to parts of “the Contribution”, but given that a partial publication is less than a full publication in terms of copyright, I would argue that it’s included.
Finally note that all of the above only applies to the general copyright agreement. Wiley mentions that the individual journals may have differing agreements:

Author re-use rights vary between journals. Please refer to the copyright form you have signed or are required to sign to review the applicable re-use rights.

As there are some questions of exegesis in the above, the easiest safe way is to ask Wiley whether they agree with your endeavour.
In General
Addressing the titular question:

Do journals have legal copyrights to the online supplemental information?

As journals want to disseminate the supplemental material, they need you to at least transfer at least the rights needed for this. Since those rights are a subset of the usual copyright agreement, it makes sense from the journal’s perspective to plainly treat the supplementary materials the same as the main paper:

It’s less complicated for the reader.
It’s less work.
Most people do not care much about supplementary materials anyway.

Therefore I do not find it surprising that most journals (in my experience) do this. However, at the end of the day, you have to look into the journal’s copyright agreement to be sure, or even talk to the journal itself, if the copyright agreement is ambiguous and you do not want to risk any bad blood.
Finally a sidenote from personal experience: I once wrote a paper presenting a new method and wanted to attach the source code for an implementation. Since the journal’s copyright agreement did not allow me to disseminate the source code on GitHub with a free license simultaneously and the copyright department did not want to relax the copyright agreement in that respect, I ended up removing the supplementary material and posting it on GitHub alone.

