Well-structured collections are used in the Cantorian expansion that underlies the IFF metastack architecture.

Overview

Abstract sets are the object of disussion here. By a set we intuitively mean an object depleted of all of its structure. What remains is only a bunch of elements whose connection, if any, is not mentioned. A set is just a bag of points. For convenience in our discussion, we use standard set-theoretic notation.

We would normally expect to discuss sets of related elements, such as a subset {red, green, blue} of colors, or a subset {love, excitement, anger} of emotions, or a subset {0, 1, 2 …} of numbers. However, sets of unconnected, unrelated elements are also possible. Consider the set {red, teal, excitement, disgust, 2}, which is the union of a subset of colors, a subset of emotions and a subset of numbers. This set is isomorphic to the set {color:red, color:teal, emotion:excitement, emotion:disgust, number:2} with type labels, which is clearly a subset of the disjoint union (coproduct) of the sets of colors, emotions and numbers.

As stated by Saunders Mac Lane in a section on foundations in his book Categories for the Working Mathematician (1971), “One of the main objectives of category theory is to discuss properties of totalities of mathematical objects such as the ‘set’ of all groups or the ‘set’ of all homomorphisms between any two groups.” Since the IFF not only uses but also axiomatizes category theory, it too must consider such totalities of mathematical objects. These include, following Mac Lane, the set of all groups, the set of all topological spaces, and the set of all vector spaces. Of course, each group, topological or vector space has an underlying set. Consider the set of all sets, as used in everyday mathematics. Such totalities of abstract sets are collections of some kind. For the IFF, such collections are required to be well-structured. A collection of abstract sets is said to be well-structured when it satisfies the core assumptions: subobject/image closure and existence of power/union operators. All of the collections of sets {setn | 0 ≤ n} that are used in the Cantorian expansion are assumed to be well-structured by convention.

Core Assumptions

Let set be a collection of sets, and let ftn denote the collection of all functions with source and target in set. The collection set is closed under subobjects when it satisfies the assumption: if A ∈ set and B → A is any injection, then B ∈ set. For example, if {yellow, teal, magenta} ∈ set, then {yellow, magenta} ∈ set and also Ø = {} ∈ set. The collection set is closed under images when it satisfies the assumption: if A ∈ set and A → C is any surjection, then C ∈ set. These subobject/image closure assumptions are useful in image factorization: if f ∈ ftn factors as f : A → I → B, then I ∈ set.

We assume that we can form the set of all subsets of any abstract set. For example, if {city, state, country} ∈ set, then also {{}, {city}, {state}, {country}, {city,state}, {city,country}, {state,country}, {city,state,country}} ∈ set. We use the notation PA = {B | B ⊆ A} for the set of all subsets of A, and we call PA the powerset of A. In a nutshell, we assume that set is closed under the powerset operator. Hence, if A ∈ set, then PA ∈ set; that is, we assume existence of the power operator

P : set → set.

The isomorphism PA ≅ 2A, of the power of A to the collection of all functions with source A and target 2 ≅ {0,1} (unique up to isomorphism), illustrates the name of the power operator and reveals a connection to general homsets of the form BA. Indeed, in the generic core IFF axiomatization, we assume that the well-structure collection of sets setn is the object component of a cartesion-closed category Setn. This implies that setn possesses a powerset operator, and thus illustrates the interdependence between the core and structural components of the IFF.

We assume that we can form the union of any set of sets in the collection set; that is, if A ∈ set and B ∈ set for all B ∈ A, then ⊔A = {x | x ∈ B some B ∈ A} ∈ set. We call ⊔A the (unbounded) union of A. In a nutshell, we say that set is closed under the (unbounded) union operator. Define C set = {A ∈ set | A ⊂ set} = set ∩ Pset to be the collection of sets in set that are collections of sets in set. Hence, if A ∈ C set, then we assume ⊔A ∈ set; that is, we assume existence of the (unbounded) union operator

⊔ : C set → set.

Any set A in C set corresponds to a bijection ≎A : A → set, where ≎A(B) = B for all B ∈ A. Any function B : A → set is essentially an A-indexed family {Ba ∈ set | a ∈ A} of sets in the collection set. This reveals a connection to discrete diagrams of the form setA. Unions of such discrete diagrams are known as “indexed unions”. These notions of unbounded or indexed unions are stronger than the notion of a bounded “powerset union” operators ∪A: PPA → PA for sets A ∈ set. In the generic core IFF axiomatization, we assume that the cartesion-closed category Set is cocomplete, and hence has all coproducts ∐A = ∐{B ∈ A} ∈ set. This, along with the image closure axiom, implies that set is closed under the (unbounded) union operator, since there is a canoncial surjection ∐A → ⊔A : (B,b) ↦ b.

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