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There is a lot > of potential for this idea contributing to future works in category > theory; for example, Mike Shulman has extended B=C3=A9nabou's definabilit= y > 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 =3D Spec(Z[x]). This is an internal ring object in the fibre and so we can form its subobject N <=3D 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 =3D x.g^-1.y, and another with x * y =3D y.g^-1.x. Richard ---------- You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. Leave group: https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.as= hx?source=3DEscalatedMessage&action=3Dleave&GuestId=3D4eb9b40c-9b3a-48a5-97= 81-836e5a171e8b