> One thing I learned from Thomas Streicher's paper on universes in
> toposes is that definability is related to descent — for instance, if
> you restrict the codomain fibration to a stable class of maps, you get
> a full subfibration, and definability in the sense of Bénabou is the
> gap between this subfibration and its stack completion. There is a lot
> of potential for this idea contributing to future works in category
> theory; for example, Mike Shulman has extended Bénabou's definability
> from "classes" of things in a fibration (i.e. properties) to a notion
> of definability that makes sense for structures; Andrew Swan has given
> a very interesting and thorough investigation of this generalised
> definability and its practical implications here:
> https://arxiv.org/abs/2206.13643.

Definability is quite a fascinating thing. Peter Freyd's work on the
core of a topos and the subsequent work on isotropy groups of toposes I
find very pretty. One related thing that I am reminded of is the
following cute fact. Possibly it is well known; I do not know the SGAs
and EGAs very well.

Suppose I have a fibration E ---> B and some X in E(b). I can consider
the existence of an "object G(X) of group structures on X". What this
means is a map f: G(X) ---> b in B, together with a group structure on
f^*(X) in E(G(X)), which is universal among such in the expected way.
Clearly this is an instance of (the more general?) definability.

Let us consider this for the following fibration. The base B is the
category of affine schemes, CRng^op. The fibre over a ring k is the
category of formal affine k-schemes, i.e., the Ind-completion of the
k-Alg^op. This is a full subfibration of the codomain fibration of
Ind(k-Alg^op).

In the terminal fibre of this fibration we have the affine line L =
Spec(Z[x]). This is an internal ring object in the fibre and so we can
form its subobject N <= L of nilpotent elements ("Spec of Z[[x]]").

Now for any commutative ring k, to give a group structure on k^*(N)
("Spec of k[[x]]") in the category of formal affine k-schemes is to give
a formal group law with coefficients in k. It follows that the "object
of group structures on N" is Lazard's universal coefficient ring for a
formal group law.

What is also quite fun is to compute the object of group structures on
the generic group O in the (codomain fibration of) the group classifier
[Grp_fp, Set]; this turns out to be O+O. This is because, given any
group G, there are G+G ways of making it into a group. Indeed, each g in
G yields two group structures on G, one with x * y = x.g^-1.y, and
another with x * y = y.g^-1.x.

Richard


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