The Meta Namespace

Overview

The meta namespace of the IFF metashell (IFF-META) is a large-sized namespace (~ 2000 terms, with ~ 500 terms in the kernel) located directly below the IFF-TYPE namespace and directly above the natural part of the IFF. It enables declarations of all natural things. Below we list the complete terminology for the IFF-META namespace (emphasized terms were previously introduced in the IFF-TYPE namspace). Look into the linked PDF files for more detailed comments and the axiomatization. The IFF-META namespace consists of five nested namespaces: kernel, finite diagrams, finite limits, finite colimits and exponent namespaces.

kernel

The kernel nested namespace continues the axiomatization for the basic collections of sets and functions, and initiates the axiomatization for the other core concepts. By introducing basic terminology, it defines at the meta level a finitely-complete category Set of Cantorian featureless sets and functions enriched with factorization and subobjects.

• There is an outer namespace for terms relating to the category theory of the kernel. The namespace terminology is listed in Table 1. This consists of 8 terms and concepts representing Set the category of sets and functions, Rel the category of sets and (binary) relations, connecting adjoint functors (f2r, r2f, η = {-}, ε)  : Set → Rel, and the generated power monad P = (P = f2r ∘ r2f, {-}, ∪).

			Natural
	Category	Functor	Transformation	Adjunction	Monad
	set	ftn2rel
	rel	rel2ftn
		power	singleton union		power
	Technical Prefix: meta Recommended Prefix: meta
	Table 3: The Meta Outer Namespace

• The set namespace terminology is listed in Table 3. This consists of 42 terms and 36 concepts (6 synonyms) representing the set basic collection and its components, and subobject structure.

	Type Set	Type Function
.set
basics	set isomorphic-relation [in]finite-set	binary-(union | intersection) difference
instances	zero = initial = empty one = terminal two three	constant-zero constant-one counique unique
subobject	subset-relation	smaller larger inclusion predicate power power-pair leq bottom top binary-(join | meet) minus complement implication singleton union = join intersection = meet
		exists = power inverse-image forall
	Technical Prefix: meta.krnl Recommended Prefix: meta
	Table 3: The Meta Set Namespace

• The function namespace terminology is listed in Table 4. This consists of 57 terms and concepts representing the concepts of function, function (optimal-)restriction order, composition functionality, element, application functionality, factorization, conversion to other kinds, function morphism and belonging order.

	Type Set	Type Function
.ftn
basics	function [optimal-]restriction-relation	source target [optimal-](smaller | larger)
	element constant-function	set constant-element construction
category theory	composable-pair	factor0 factor1 composition identity
	appliable-pair	element-factor function-factor application
factorization	injection surjection bijection	range kernel coimage injective-factor coapplication surjective-factor coequalizer inverse
conversion		predicate span relation fiber unit
instances		initial terminal counique unique
.ftn.mor
basics	2-cell belonging equivalence isomorphism	source target function element0 element1
category theory	composable-pair	factor0 factor1 composition identity
	Technical Prefix: meta.krnl Recommended Prefix: meta
	Table 4: The Meta Function Namespace

finite diagrams

The finite diagram nested namespace essentially defines four categories of diagrams (Figure 4): the category Set² of set pairs and function pairs (diagrams for binary products), the category Set³ of set triples and function triples (diagrams for ternary products), the (comma) category Ppr = (Δ, Δ) of parallel pairs and parallel pair morphisms (diagrams for equalizers), and the (comma) category Ospn = (1, Δ) of opspans and opspan morphisms (diagrams for pullbacks).

The diagram namespace terminology is listed in Table 8. This consists of 91 terms and 85 concepts (6 synonyms) representing the four basic kinds of things (Set², Set³, Ppr, Ospn) and their components, category theory, and conversion maps.

Table 8: The Meta Finite Diagram Namespace

finite limits

The limit namespace terminology of this namespace, which is listed in Table 9, consists of 131 terms and 131 concepts (with no synonyms). There are six kinds of limit according to diagram shape: binary and ternary products and their power variants, equalizers and pullbacks.

Table 9: The Meta Finite Limit Namespace

In the meta namespace, a large part of the kernel is the specialization (using subset, delimitation, (optimal-)restriction and abridgment) of the type namespace kernel. The meta namespace initiates axiomatization for exponents.

Figure 1: The IFF Meta Namespace Architecture

In the current phase of IFF development, the old IFF Upper Core (meta) Ontology (IFF-UCO) has moved up and become the kernel of the metashell namespace (IFF-META). Like the axiomatization for finite type limits and the axiomatization for finite limits of categories and functors, this axiomatization is in adjunctive form. The IFF-META namespace plays a central role in the IFF axiomatization — it is the most referenced namespace at the metalevel. The architecture of the IFF-META namespace is illustrated in Figure 1. The IFF-META kernel consists of the axiomatization for metasets, metapredicates and metafunctions. At the metashell meta level the axiomatization for relations, which includes order-theoretic concepts, has branched off into a namespace in its on right. The kernel and limit namespace axiomatizations are essentially copies of those in the type level namespace (IFF-TYPE). These axiomatizations rely heavily upon the subset, delimitation, restriction and abridgement relations for sets, predicates, functions and relations, respectively. The colimit namespace is a categorical dual to the limit namespace. The exponent namespace axiomatization is new.