A Dictionary for Meta-Ontologies

This is a brief dictionary, and a work in progress, of some of the 100+ terms that represent the most basic concepts of the IFF. Although there are thousands of terms in the IFF, most of those are derived from the basic terms listed here. For example, an opspan concept appears in the namespace of logical languages in the IFF Ontology meta-ontology (IFF-ONT). However, this concept is derived from the basic opspan concept and the concept of a language for first order logic. The concepts listed here should be compared to the concepts indexed in the text by Saunders Mac Lane.

Terms

A

1. term: 'abridgment'
2. synonyms: none
3. dual notions: none
4. dictionary definition: A shortened form that is reduced in scope but retains the general sense and unity of the original.
5. axiomatization: This is axiomatized in the relation namespace of the IFF-TCO meta-ontology
6. uses: This is a key relation. It is used in many, many namespaces in the IFF.

• 'antisymmetric'

B

• 'bijection'

• 'binary coproduct'

• 'binary intersection'

• 'binary product'

• 'binary union'

• 'bond'

• 'bonding pair'

C

• 'categorical equivalence'

• 'category'

• 'class'

• 'classification'

• 'cocone'

• 'coequalizer'

• 'colimit'

• 'colimit injection'

• 'collection'

• 'composition'

• 'concept lattice'

• 'cone'

• 'constant function'

• 'coproduct'

• 'currying'

D

• 'diagram'

• 'disjoint'

• 'domain'

E

• 'empty'

• 'epimorphism'

• 'equalizer'

• 'equivalence relation'

• 'evaluation'

• 'exponent'

F

• 'function'

• 'functional relation'

• 'functor'

H

• 'hom'

• 'hypergraph'

I

• 'identity'

• 'inclusion'

• 'infomorphism'

• 'initial'

• 'injection'

• 'institution'

• 'institution morphism'

• 'inverse currying'

• 'isomorphic'

• 'isomorphism'

L

• 'language'

• 'limit'

• 'limit projection'

M

• 'mediator'

• 'monad'

• 'monomorphism'

• 'morphism'

• 'mor2rel'

N

• 'natural transformation'

• 'nothing'

• 'null'

O

• 'object'

1. term: 'opspan'
2. synonyms: diagram for pullbacks
3. dual notions: none
4. dictionary definition: none
5. axiomatization: This is axiomatized in the IFF-UR meta-ontology, and the object namespace of the IFF-LCO meta-ontology.
6. uses: This is a key concept. It is used in many, many namespaces in the IFF.

• 'one'

• 'order'

P

• 'packing'

• 'pair'

• 'parallel pair'

• 'partial order'

• 'partial function'

• 'pfn2ftn'

• 'pfn2rel'

• 'product'

• 'product with'

• 'pullback'

• 'pushout'

R

• 'reflexive'

• 'relation'

• 'restriction'

S

• 'set'

• 'signature'

• 'source'

• 'span'

• 'spangraph'

• 'subclass'

• 'subcollection'

• 'subordination'

• 'subset'

• 'surjection'

• 'symmetric'

T

• 'target'

• 'terminal'

• 'ternary coproduct'

• 'ternary product'

1. term: 'thing'
2. synonyms: universe of discourse, all, everything
   
3. dual notions: nothing, empty
4. dictionary definition: A state of affairs in general or within a specified or implied sphere. A separate and distinct individual quality, fact, idea, or usually entity. An object or entity not precisely designated or capable of being designated. A material or substance of a specified kind.
5. axiomatization: This is axiomatized in the IFF-UR meta-ontology, and in the basic namespace of the IFF-TCO meta-ontology.
6. uses: This represents the universe. However, it has very few uses now, and probably none in a full categorification of the IFF. In fact, there is a strong desire to eliminate the universe of discourse from the IFF, first since it is not needed, and second since the universe and any power/exponent operators must be kept apart in order to eliminate contradictions due to Cantor's diagonal argument (see Russell's paradox).

• 'three'

• 'total order'

• 'total relation'

• 'transitive'

• 'triple'

• 'two'

U

• 'unique function'

• 'unique morphism'

• 'unit'

• 'unpacking'

V

• 'vocabulary'

Z

• 'zero'