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(i) Types of mathematical objects: In the following discussion, we shall often
speak of “types of mathematical objects”, i.e., such as groups, rings, topological
spaces equipped with some additional structure, schemes, etc. This notion of a “type
of mathematical object” is formalized in [IUTchIV], §3, by introducing the notion of
a “species”. On the other hand, the details of this formalization are not so important
for the following discussion of the notion of multiradiality. A “type of mathematical
object” determines an associated category consisting of mathematical objects of this
type ? i.e., in a given universe, or model of set theory ? and morphisms between such
mathematical objects. On the other hand, in general, the structure of this associated
category [i.e., as an abstract category!] contains considerably less information than the
information that determines the “type of mathematical object” that one started with.
For instance, if p is a prime number, then the “type of mathematical object” given by
rings isomorphic to Z/pZ [and ring homomorphisms] yields a category whose equivalence
class as an abstract category is manifestly independent of the prime number p.

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