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IFF Category Theory Ontology

- IJCAI 2001
- Workshop on the IEEE Standard Upper Ontology

http//suo.ieee.org/Kent-IFF.pdf

Sections

- Introduction (6 slides)
- The Core Namespace (6 slides)
- The Category Theory Ontology (11 slides)
- Examples some with logical code (16 slides)
- Summary Future Work

Table of Contents Introduction

- Origins and Influences
- Possible outcomes of the SUO Effort
- The Truth Classification Truth Concept Lattice
- Components of Logic
- Architecture
- The Category Theory Principle

Origins and Influences

Possible outcomes of the SUO Effort

- Hayes
- A single, usable, well-designed, universal upper

ontology is created . - A standard upper ontology is created which is a

shambles . - The whole enterprise is eventually abandoned .
- Sowa
- A framework is created which can support an

open-ended number of theories (potentially

infinite) organized in a lattice together with

systematic metalevel techniques for moving from

one to another, for testing their adequacy for

any given problem, and for mixing, matching,

combining, and transforming them to whatever form

is appropriate for whatever problem anyone is

trying to solve. - Kent (the IFF Foundation Ontology)
- A framework is created for the truth concept

lattice of a first-order language L generated

from the truth classification of the language. - Replace of the theories of outcome 4 with the

concepts in the truth concept lattice. - Suitable set-theoretic foundational concerns are

covered in the IFF Category Theory Ontology

(discussed here), a subontology of the IFF

Foundation Ontology. - Proposal Use the IFF Foundation Ontology as a

structuring methodology for the SUO metalevel. - This will provide a principled approach to John

Sowas framework.

The Truth Classification Truth Concept Lattice

- Truth classification of 1st-order language L
- Instances are L-structures.
- Types are L-sentences.
- Incidence is satisfaction M ?L ? when ? is true

in M. - Truth concept lattice of L
- In the IFF approach, this is the appropriate
- lattice of ontological theories.
- Formal concept is a pair c ?extent, intent(c)?

where - The intent(c) is a closed theory (set of

sentences). - The extent(c) is the collection of all models for

that theory. - Lattice order is the opposite of theory

inclusion. - The join or supremum of two theories is the

intersection of the theories. - The meet or infimum of two theories is the theory

of the common models. - Both L-structures and L-sentences generate formal

truth concepts (theories). - An object concept is the theory of a model.
- An attribute concept is the models of a sentence.

Components of Logic

Architecture

- Lower metalevel
- Declare, define, axiomatize and reason about

categories, functors, natural transformations,

adjunctions, colimits, monads, etc. - Categories include Set, Classification, IF-Logic,

Top, Rel, Gph, Lang, etc. - Functors include inst, typ, inst-pow, typ-pow,

etc. - Object level
- Declare, define, populate and reason about

objects and morphisms. - Objects may be sets, partial orders, topological

spaces, classifications, hypergraphs, languages,

models, etc. - Morphisms may be functions, continuous maps,

infomorphisms, 1st-order interpretations, model

morphisms, etc.

The Lower Metalevel

- The lower metalevel makes heavy use of the upper

metalevel for both representation and reasoning. - The following module will be placed on the lower

metalevel in a later version of the IFF

Foundation Ontology (version 2.0 or 3.0) - IF Model Theory Ontology
- Other possible modules on the lower metalevel

include the following - Module for categorical model theory
- Modules for modal, tense and linear logic
- Modules for rough and fuzzy sets
- Module for semiotics
- etcetera

The Categorical Design Principle

- Principle A central goal in modeling the lower

metalevel is to abide by the following

categorical property. - strictly category-theoretic all axioms are

expressed in terms of category-theoretic notions,

such as the composition and identity of

large/small functions or the pullback of diagrams

of large/small functions. - no KIF no axioms use explicit KIF connectives

or quantification. - no basic KIF ontology no axioms use terms from

the basic KIF ontology, other than pair

bracketing '-' or pair projection '(- 1)', '(-

2)'. - This principle is an ideal that has proven very

useful in the design of the IF Model Theory

Ontology. All modules that satisfy this property

should (i) be easier to design and (ii) provide

the basis for simpler proof techniques. - This design principle would seem to extend to all

ontologies for true categories (not

quasi-categories) those categories whose object

and morphism collections are classes (not

conglomerates). All ontologies that reside at the

lower metalevel will be centered on true

categories.

Table of Contents The Core Namespace

- Collection Hierarchy
- Kinds of Specific Collections
- Core Conglomerates and Functions
- Principal Terms at the Core
- Sub-namespaces in the Core Namespace
- Terms introduced in the Core Namespace

Collection Hierarchy

Kinds of Specific Collections

- Conglomerates
- The collections of classes, functions, opspans

and binary cones - The collections of large categories and functors

between large categories - The collections of natural transformations and

adjunctions - Classes
- The object and morphism collections in any large

category - The collections of algebras and homomorphisms of

a monad (e.g. groups) - The collections of diagrams and cocones for

finite colimits in a category - Sets
- Sets as collections and the extent of their

functions - The instance and type sets of classifications and

the instance and type functions of infomorphisms

Core Conglomerates and Functions

Principal Terms at the Core

- CNG
- conglomerate
- (possibly n-ary) function
- (binary) relation
- SET
- class
- (binary) function
- source
- target
- (binary) relation
- set
- (small) set
- (binary) function
- source
- target

- (binary) relation
- subcollection
- subclass
- (function) restriction

Sub-namespaces in the Core Namespace

- The Core Namespace (SET)
- Classes
- Functions (SET.FTN)
- Finite Completeness (SET.LIM)
- The Terminal Class
- Binary Products (SET.LIM.PRD)
- Equalizers (SET.LIM.EQU)
- Subequalizers (SET.LIM.SEQU)
- Pullbacks (SET.LIM.PBK)
- Topos Structure (SET.TOP)

Terms introduced in the Core Namespace

Currently there are 11 terms in the conglomerate

namespace and about 130 terms in the core

namespace. About 100 of these 130 terms are for

finite limits, especially pullbacks. Eventually

the core namespace will grow to contain several

hundred terms.

Table of Contents Category Theory Ontology

- Category Theory
- Sub-namespaces in the Category Theory Ontology
- Terms introduced in the Category Theory Ontology
- Formal definition of a category C
- KIF Formalism for Category
- KIF Formalism for Graph Multiplication
- Formal definition of a functor F
- Monad Theory
- Formal Definition of a Monad M
- Formal Definition of an M-algebra
- Limits/Colimits

Category Theory

- Started in 1945 with Eilenberg Mac Lanes paper

entitled "General Theory of Natural

Equivalences." - It is a general mathematical theory of structures

and systems of structures. - Reveals how structures of different kinds are

related to one another (morphisms), as well as

the universal components of a family of

structures of a given kind (limits/colimits). - It is considered by many as being an alternative

to set theory as a foundation for mathematics.

Sub-namespaces in the Category Theory Ontology

- The Namespace of Graphs (GPH)
- Graphs
- Graph Morphisms (GPH.MOR)
- The Namespace of Categories (CAT)
- The Namespace of Functors (FUNC)
- The Namespace of Natural Transformations (NAT)
- The Namespace of Adjunctions (ADJ)
- The Namespace of Monads (MND)
- Monads and Monad Morphisms
- Algebras and Freeness (MND.ALG)

- The Namespace of Colimits (COL)
- Initial Objects
- Binary Coproducts (COL.PRD)
- Coequalizers (COL.COEQU)
- Pushouts (COL.PSH)
- The Namespace of Kan Extensions (KAN)
- The Namespace of Classifications (CLS)
- IF Theories (CLS.TH)
- IF Logics (CLS.LOG)
- The Namespace of Concept Lattices (LAT)
- The Namespace of Topoi (TOP)

Terms introduced in the Category Theory Ontology

Currently there about 170 terms in the category

theory subontology. When the base-line

terminology is complete, there may be 250 terms.

Eventually it will grow to contain thousands of

terms.

Formal definition of a category C

- C is a graph.
- There are two classes the objects obj(C) and the

morphisms mor(C). - Any morphism f ? mor(C) has a source and target

object f a ? b. - C has monoidal structure.
- For any composable pair of morphisms f a ? b

and g a ? b, there is a composition morphism

(f C g) a ? c. - For every object a ? obj(C), there is a morphism

idC(a) a ? a called the identity on a. - Associativity and identity laws hold.
- If f a ? b, g b ? c and h c ? d, then f C

(g C h) (f C g) C h. - If f a ? b, then (idC(a) C f) f and (f C

idC(b)) f.

KIF Formalism for Category

(1) (CNGconglomerate category) (2)

(CNGfunction underlying) (CNGsignature

underlying category GPHgraph) (3)

(CNGfunction mu) (CNGsignature mu category

GPH.MOR2-cell) (forall (?c (category ?c))

(and ( (GPH.MORsource (mu ?c))

(GPHmultiplication (underlying ?c)

(underlying ?c))) ( (GPH.MORtarget

(mu ?c)) (underlying ?c)))) (4)

(CNGfunction composition) (CNGsignature

composition category SET.FTNfunction)

(forall (?c (category ?c)) (

(composition ?c) (GPH.MORmorphism (mu

?c)))) (5) (CNGfunction eta)

(CNGsignature eta category GPH.MOR2-cell)

(forall (?c (category ?c)) (and (

(GPH.MORsource (eta ?c))

(GPHunit (GPHobject (underlying ?c))))

( (GPH.MORtarget (eta ?c))

(underlying ?c)))) (6) (CNGfunction identity)

(CNGsignature identity category

SETfunction) (forall (?c (category ?c))

( (identity ?c) (GPH.MORmorphism

(eta ?c))))

KIF Formalism for Graph Multiplication

(1) (CNGfunction multiplication-opspan)

(CNGsignature multiplication-opspan graph graph

SET.LIM.PBKopspan) (forall (?g0 (graph ?g0)

?g1 (graph ?g1)) (ltgt (exists (?s

(SET.LIM.PBKopspan ?s)) (

(multiplication-opspan ?g0 ?g1) ?s))

( (object ?g0) (object ?g1)))) (forall

(?g0 (graph ?g0) ?g1 (graph ?g1)) (gt

( (object ?g0) (object ?g1)) (and

( (SET.LIM.PBKopvertex (multiplication-opspan

?g0 ?g1)) (object

?g0)) ( (SET.LIM.PBKopfirst

(multiplication-opspan ?g0 ?g1))

(target ?g0)) (

(SET.LIM.PBKopsecond (multiplication-opspan ?g0

?g1)) (source

?g1))))) (2) (CNGfunction multiplication)

(CNGsignature multiplication graph graph graph)

(forall (?g0 (graph ?g0) ?g1 (graph ?g1))

(ltgt (exists (?g (graph ?g))

( (multiplication ?g0 ?g1) ?g)) (

(object ?g0) (object ?g1)))) (forall (?g0

(graph ?g0) ?g1 (graph ?g1)) (gt (

(object ?g0) (object ?g1)) (and (

(object (multiplication ?g0 ?g1))

(object ?g0)) ( (morphism

(multiplication ?g0 ?g1))

(SET.LIM.PBKpullback (multiplication-opspan ?g0

?g1))) ( (source

(multiplication ?g0 ?g1))

(SETcomposition

(SET.LIM.PBKprojection1 (multiplication-opspan

?g0 ?g1)) (source

?g0))) ( (target

(multiplication ?g0 ?g1))

(SETcomposition

(SET.LIM.PBKprojection2 (multiplication-opspan

?g0 ?g1)) (target

?g1))))))

Formal definition of a functor F

- F is a graph morphism.
- There is a source and a target category src(F)

A and tgt(F) B, F A ? B. - There are two functions the object function

obj(F) obj(A) ? obj(B) and the morphism

function mor(F) mor(A) ? mor(B). - These are compatible w.r.t. source/target for

any A-morphism f a ? b, - src(B)(mor(F)(f)) obj(F)(src(A)(f)) and
- tgt(B)(mor(F)(f)) obj(F)(tgt(A)(f)).
- F preserves monoidal structure.
- For any composable pair of A-morphisms f a ? b

and g a ? b, the images are composable and

mor(F)(f A g) mor(F)(f) B mor(F)(g). - For every object a ? obj(C), there is a morphism

idC(a) a ? a called the identity on a. - Associativity and identity laws hold for functor

composition. - If F A ? B, G B ? C and H C ? D, then F ?

(G ? H) (F ? G) ? H. - If F A ? B, then IdA ? F F and F ? IdB F.

Monad Theory

- Relates universal algebra with adjunctions
- For each type of algebra or equational

presentation t ??? ?? with - category Algt of t-algebras and t-homomorphisms.
- underlying functor Ut Algt ? Set
- free functor Ft Set ? Algt
- adjunction Ft ? Ut
- with insertion of variables natural

transformation ?t IdSet ? Ft Ut - with evaluation natural transformation

et Ut Ft ? IdAlgt - there is a monad ?Tt? ?t? µt? over the category

Set, where - Tt Ft ? Ut Set ? Set equationally

equivalent terms - µt ?F ? e ??U multiplication (substitution) of

terms into terms.

Formal Definition of a Monad M

- M ?T? ?? µ?
- underlying endofunctor T A ? A
- natural transformation ? IdA ? T
- natural transformation µ T ? T ? T

Formal Definition of an M-algebra

- Assume M ?T? ?? µ? is a monad over category A.
- An M-algebra
- ?a? ??
- consists of
- an underlying object a ? obj(A),
- a structure map ? T?a? ? a
- that satisfies the associative and unit laws.
- An M-homomorphism
- h ?a? ?? ? ?a?? ???
- consists of
- an A-morphism h a ? a? that preserves

structure.

Limits/Colimits

- Limits
- Notions of producting, and-ing and matching
- A large part of the core namespace (100 of the

130 terms) - Finite kinds
- Terminal object (unit class in core)
- Binary product
- Equalizer
- Subequalizer
- Pullback (core pullback is heavily used in

Category Theory Ontology)

- Colimits
- Notions of summing, or-ing, and fusion.
- No terms in the core namespace
- Axiomatized in the colimit namespace of the

Category Theory Ontology - Finite kinds
- Initial object
- Binary coproduct
- Coequalizer
- Pushout

Table of Contents Examples

- Categories
- The ordinal category ord3
- The category Set
- The category Classification
- Monads and Algebras
- The instance and instance power functors
- The inst ? pow adjunction
- Data Structures as Initial Algebras (stacks)
- Data Structures as Initial Algebras

(s-expressions binary trees) - Data Structures as Initial Algebras

(trees-forests) - Object level logical code Domain Ontology ieee

namespace - Lower metalevel logical code Model Theory

(sub)Ontology classification namespace - Upper metalevel logical code core namespace

Category Theory (sub)Ontology graph category

namespace - Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology pullback - Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology equalizer - Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology subequalizer

Categories

- Almost every known example of a mathematical

structure with the appropriate structure

preserving map yields a category. - Sets with functions between them.
- Groups with group homomorphisms.
- Topological spaces with continuous maps.
- Vector spaces and linear transformations.
- Any class itself is a category with only identity

morphisms. - Any monoid is a one-object category with elements

being morphisms. - Any preordered class is a category with morphisms

being pair orderings. - Classifications and infomorphisms (or bonds, or

bonding pairs). - Hypergraphs and their morphisms.
- Models and their morphisms.
- Concept lattices and concept lattice morphisms.
- Complete lattices and adjoint morphisms (or

complete homomorphisms).

The ordinal category ord3

((SET.TOPelement SET.TOPnatural-numbers)

ord30) ((SET.TOPelement SET.TOPnatural-numbers)

ord31) ((SET.TOPelement SET.TOPnatural-numbers

) ord32) ( ord30 SET.TOPzero) ( ord31

(SET.FTNcomposition ord30 SET.TOPsuccessor)) (

ord32 (SET.FTNcomposition ord31

SET.TOPsuccessor)) (CATcategory

ord3)((CATobject ord3) ord30) ((CATobject

ord3) ord31) ((CATobject ord3)

ord32) ((CATmorphism ord3) ord300) ((CATmorphi

sm ord3) ord311) ((CATmorphism ord3)

ord322) ((CATmorphism ord3) ord301) ((CATmorph

ism ord3) ord312) ((CATmorphism ord3)

ord302) ( ((CATsource ord3) ord301)

ord30) ( ((CATtarget ord3) ord301) ord31) (

((CATsource ord3) ord312) ord31) (

((CATtarget ord3) ord312) ord32) (

((CATsource ord3) ord302) ord30) (

((CATtarget ord3) ord302) ord32) (

((CATidentity ord3) ord30) ord300) (

((CATidentity ord3) ord31) ord311) (

((CATidentity ord3) ord32) ord322) (

((CATcomposition ord3) ord301 ord312) ord302)

- Obj(ord3) 0, 1, 2
- Mor(ord3) 00, 11, 22, 01, 12, 02
- one nontrivial composition
- 01 12 02.
- The code on the right defines the category ord3

by populating it using terminology from the

Category Theory Ontology.

The category Set

- Objects in Obj(Set) are (small) sets.
- Morphisms in Mor(Set) are (unary) functions

between (small) sets. - The code below makes the following assertion

using terminology from both the Category Theory

Ontology and the (Small) Set Ontology in an

external namespace - There is a KIF axiomatization for the Set

category in the set namespace. - Some theorem-proving is involved here.

(CATcategory set)( (CATunderlying set)

set.ftnset-graph) ( (CATobject set)

setset) ( (CATmorphism set)

set.ftnfunction) ( (CATsource set)

set.ftnsource) ( (CATtarget set)

set.ftntarget) ( (CATcomposable set)

set.ftncomposable) ( (CATcomposition set)

set.ftncomposition) ( (CATidentity set)

set.ftnidentity)

The category Classification

- Objects in Obj(Classification) are

classifications. - Morphisms in Mor(Classification) are

infomorphisms. - The code below (front diagram) makes the

following assertion using terminology from both

the Category Theory Ontology and the (Small)

Classification Ontology in an external namespace - There is a KIF axiomatization for the

Classification category in the if.cls

namespace. - Some theorem-proving is involved here.

The Webster Classification

(if.clsClassification Webster)((if.clstype

Webster) Noun)((if.clstype Webster)

Intransitive-Verb)((if.clstype Webster)

Transitive-Verb)((if.clstype Webster)

Adjective) ((if.clsinstance Webster)

bet)((if.clsinstance Webster)

eat)((if.clsinstance Webster)

fit)((if.clsinstance Webster)

friend)((if.clsinstance Webster)

square)...((if.clsincidence Webster) bet

Noun)((if.clsincidence Webster) bet

Intransitive-Verb)((if.clsincidence Webster)

bet Transitive-Verb)(not ((if.clsincidence

Webster) bet Adjective))(not

((if.clsincidence Webster) eat

Noun))((if.clsincidence Webster) eat

Intransitive-Verb)((if.clsincidence Webster)

eat Transitive-Verb)(not ((if.clsincidence

Webster) fit Adjective))((if.clsincidence

Webster) fit Noun)((if.clsincidence Webster)

fit Intransitive-Verb)((if.clsincidence

Webster) fit Transitive-Verb)((if.clsincidence

Webster) fit Adjective)((if.clsincidence

Webster) friend Noun)(not ((if.clsincidence

Webster) friend Intransitive-Verb))((if.clsinc

idence Webster) friend Transitive-Verb)(not

((if.clsincidence Webster) friend

Adjective))((if.clsincidence Webster) square

Noun)(not ((if.clsincidence Webster) square

Intransitive-Verb))((if.clsincidence Webster)

square Transitive-Verb)((if.clsincidence

Webster) square Adjective)...

- This is the Webster classification on page 70 of

the text Information Flow The Logic of

Distributed Systems by Barwise and Seligman. - This classification is (a small part of) the

classification of English words according to

parts of speech as given in Websters dictionary.

Monads and Algebras

- Universal Algebra Each type (equational

presentation) t ??? E? induces a monad Mt

?Tt? ?t? µt? over Set with ??? E?-algebras

equivalent to Mt-algebras. - Closure If the underlying category is a

preorder, then a monad is a closure operator and

an algebra is a closed element. - Monoid action Any monoid ?M? e? ?? defines a

monad on Set - TM(A) A?M
- ?M A ? A?M a ? (a, e)
- µM A?M?M ? A?M (a, m?, m?) ? (a, m? ? m?)
- and an algebra ?A? d A?M ? A? is called an

M-action (A-automaton) - d(a, m? ? m?) d(d(a, m?), m?)
- d(a, e) a
- Module Any ring R defines a monad on Ab (abelian

groups) - TR(A) A?R
- ?A A ? A?R a ? (a, 1)
- µA A?R?R ? A?R (a, r?, r?) ? (a, r? ? r?)
- and an algebra is a right R-module

The instance and instance power functors

- The instance functor inst ? Classification ? Setop

forgets type information. - The power functor pow ? Setop ? Classification

adds subsets and inverse image. - The code below makes the following assertion

using terminology from both the Category Theory

Ontology and the (Small) Classification Ontology

in an external namespace There is a KIF

axiomatization for the instance and

instance-power functors in the if.cls

namespace. Some theorem-proving is involved here.

(FUNCfunctor instance) ( (FUNCsource

instance) classification) (

(FUNCtarget instance) (CATopposite set))

( (FUNCunderlying instance)

if.cls.infoinstance-graph-morphism) (

(FUNCobject instance) if.clsinstance)

( (FUNCmorphism instance) if.cls.infoinstance)

(FUNCfunctor instance-power) (

(FUNCsource instance-power) (CATopposite

set)) ( (FUNCtarget instance-power)

classification) ( (FUNCunderlying

instance-power) if.cls.infoinstance-power-graph

-morphism) ( (FUNCobject

instance-power) if.clsinstance-power)

( (FUNCmorphism instance-power)

if.cls.infoinstance-power)

The inst ? pow adjunction

- There is a natural transformation

? IdClassification ? inst pow whose component

at any classification is the extent infomorphism

associated with that classification. - The code below makes the following assertion in

an external namespace using terminology from both

the Category Theory Ontology and the (Small)

Classification Ontology Appropriate code in

the if.cls namespace represents the fact that

the underlying instance functor is left adjoint

to the instance power functor with ? being the

unit of this adjunction and the counit being the

identity.

(NATnatural-transformation eta) ( (NATsource

eta) (FUNCidentity Classification)) (

(NATtarget eta) (FUNCcomposition instance

instance-power)) ( (NATcomponent eta)

if.clsextent) (FUNCcomposable instance

instance-power) ( (FUNCcomposition

instance-power instance) (FUNCidentity

set)) (ADJadjunction inst-pow) (

(ADJunderlying-category inst-pow)

Classification) ( (ADJfree-category inst-pow)

Set) ( (ADJleft-adjoint inst-pow)

instance) ( (ADJright-adjoint inst-pow)

instance-power) ( (ADJunit inst-pow)

eta) ( (ADJcounit inst-pow)

(NATidentity (FUNCidentity Set)))

Data Structures as Initial Algebras (stacks)

- Stacks over atoms A
- Solution to recursive equation X ? 1 X?A
- Endofunctor F Set ? Set Q ?1 Q?A
- F-algebras are A-automata ?Q? ? Q?A ? Q?
- ?Stack(A)? ??? is the initial F-algebra over A

with ?? unfolding to empty-stack and push.

Also, ?? 1 Stack(A)?A ? Stack(A) is an

isomorphism (bijection), with inverse being

(is-empty, pop-top). - Initial algebra ?Stack(A)? ??? defined via the

?-colimit of the following iteration diagram in

Set - ? ? F??? ? ? Fn??? ?
- F??? 1 ??A 1
- F2??? F?F???? F?1? 1 1?A A?2
- F3??? F?F2???? 1 ?11?A??A 1 1?A

1?A?A A?3 - Unique homomorphism to ?Q? ?? is the run map for

the automaton - ?_at_ ?Stack(A)? ??? ? Q

Data Structures as Initial Algebras

(s-expressions binary trees)

- S-expressions over atoms A
- Solution to recursive equation S ? 1 A S?S
- Endofunctor F Set ? Set Q ?1 A Q?Q
- ?Sexp(A)? ??? is the initial F-algebra over A

with ?? unfolding to empty-s-expression,

atom-as-sexp and cons. Also, ?? 1 A

Sexp(A)?Sexp(A) ? Sexp(A) is an isomorphism with

inverse being (is-null, is-atom, car-cdr). - Initial algebra ?Sexp(A)? ??? defined via the

?-colimit of the following iteration diagram in

Set - ? ? F??? ? ? Fn??? ?
- Binary trees over atoms A
- Solution to recursive equation B ? 1 B?B?A
- Endofunctor F Set ? Set Q ?1 Q?Q?A
- ?BinTree(A)? ??? is the initial F-algebra over A

with ?? unfolding to empty-bintree and

make-bintree. Also, ?? 1 BinTree(A)?BinTree(

A)?A ? BinTree(A) is an isomorphism (bijection),

with inverse being (is-empty,

left-right-atom). - Initial algebra ?BinTree(A)? ??? defined via the

?-colimit of the following iteration diagram in

Set - ? ? F??? ? ? Fn??? ?

Data Structures as Initial Algebras

(trees-forests)

- Trees and Forests
- Solution to pair of recursive equations
- T ? F?A, F ? 1 F?T
- Endofunctor TF Set?Set ? Set?Set ?P, Q? ?

?Q?A, 1 Q?P? - TF-algebras are function pairs ?P, Q? ? Q?A ?

P? ? 1 Q?P ? Q? - ?Tree(A)? Forest(A)? ??? ??? is the initial

TF-algebra over A with - ?? Forest(A)?A ? Tree(A) being make-tree,

and - ?? 1 Forest(A)?Tree(A) ? Forest(A)

unfolding to empty-forest and push-tree.

Also, ?? Forest(A)?A ? Tree(A) and ?? 1

Forest(A)?Tree(A) ? Forest(A) are isomorphisms

(bijections), with inverse being (data,

forest) and (is-empty, pop-top),

respectively. - Isomorphism Forests ? Binary trees
- Substitute T into F-equation
- F ? 1 F?T ? 1 F?(F?A) ? 1 F?F?A

Object level logical codeDomain Ontology ieee

namespace

- Lower metalevel terms
- if.loglogic, if.logtheory,

if.logclassification, if.thsubtype,

if.clstype, if.clsobject,

if.clsincidence, - Object level terms
- ieeelogic, ieeetheory, ieeeclassificatio

n, ieeeorganization, ieeetechnical-society,

ieeeregion,

(if.loglogic ieeelogic) ( (if.logtheory

ieeelogic) ieeetheory) ( (if.logclassification

ieeelogic) ieeeclassification) ((if.clstype

ieeeclassification) ieeeorganization) ((if.clst

ype ieeeclassification) ieeetechnical-society) (

(if.clstype ieeeclassification)

ieeeregion) ((if.thsubtype ieeetheory)

ieeeregion ieeeorganization) ((if.thsubtype

ieeetheory) ieeetechnical-society

ieeeorganization) ((if.clsobject

ieeeclassification) ieeecircuits-and-systems-soc

iety) ((if.clsobject ieeeclassification)

ieeecomputer-society) ((if.clsincidence

ieeeclassification) ieeecircuits-and-system

s-society ieeetechnical-society) ((if.clsincide

nce ieeeclassification) ieeecomputer-societ

y ieeetechnical-society)

Lower metalevel logical code Model Theory

(sub)Ontology classification namespace

- Upper metalevel terms
- Core namespace
- SETclass, SET.FTNfunction,

SET.FTNsource, SET.FTNtarget, - Category Theory (sub)Ontology graph category

namespaces - GPHgraph, GPHobject, GPHmorphism,

GPHsource , GPHtarget, CATcategory, - CATunderlying , CATcomposable ,

CATcomposition , CATidentity - Lower metalevel terms
- if.clsclassification, if.clstype,

if.clsobject, if.clsincidence

if.infoinfomorphism , if.infosource ,

if.infotarget , if.infocomposable ,

if.infocomposition , if.infoidentity,

Upper metalevel logical code core namespace

Category Theory (sub)Ontology graph category

namespace

- Upper metalevel terms
- Conglomerate Core namespaces
- CNGconglomerate, CNGfunction ,

CNGsignature , SETclass, SET.FTNfunction,

- Category Theory (sub)Ontology graph category

namespaces - GPHgraph, GPHmultiplication,

GPH.MOR2-cell, GPH.MORsource,,

GPH.MORtarget, CATcategory, - CATunderlying , CATmu , CATcomposition

,

Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology pullback

- SET.LIM.PBKpullback
- Two morphisms of a category are composable when

the target of the first is identical to the

source of the second.

(CNGfunction composable-opspan) (CNGsignature

composable-opspan category SET.LIM.PBKopspan) (fo

rall (?c (category ?c)) ( (composable-opspan

?c) (GPHmultiplication-opspan (underlying

?c) (underlying ?c)))) (CNGfunction

composable) (CNGsignature composable category

SETclass) (forall (?c (category ?c)) (

(composable ?c) (GPHmorphism

(GPHmultiplication (underlying ?c) (underlying

?c)))))

Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology equalizer

- SET.LIM.EQUequalizer
- A classification is extensional when any two

coextensive types are identical.

(SET.FTNfunction coextensive) ( (SET.FTNsource

coextensive) classification) ( (SET.FTNtarget

coextensive) relrelation) ( (SET.FTNcomposition

coextensive relobject) type) ( coextensive

relequivalence2) (SET.LIM.EQUparallel-pair

extensional-parallel-pair) ( (SET.LIM.EQUsource

extensional-parallel-pair) classification) (

(SET.LIM.EQUtarget extensional-parallel-pair)

relrelation) ( (SET.LIM.EQUfunction1

extensional-parallel-pair) coextensive) (

(SET.LIM.EQUfunction2 extensional-parallel-pair)

(SET.FTNcomposition type relidentity)) (SET

class extensional) (SETsubclass extensional

classification) ( extensional (SET.LIM.EQUequali

zer extensional-parallel-pair))

Use of SET finite limits by namespaces in the

Category Theory (sub)Ontology subequalizer

- SET.LIM.SEQUsubequalizer
- An endorelation is reflexive when it contains the

identity relation on its class.

Summary

- I propose that the IFF Foundation Ontology be

used to represent the metalevel structure of the

SUO. - The IFF Foundation Ontology is founded on

category theory more strongly, the upper level

of the IFF Foundation Ontology represents

Category Theory. - The two levels of the IFF Foundation Ontology

represent the large/small distinction. - The upper metalevel consists of the Category

Theory Ontology anchored at the core namespace. - It is strongly recommended that the object level

and the metalevel have a distinct and obvious

boundary.

Future Work

- Future versions
- Version 2.0 add terminology/axioms for the IFF

Model Theory (sub)Ontology. - Namespaces small sets, small relations, small

classifications, spans and hypergraphs, models,

and 1st-order interpretations. - Version 3.0 etc.
- Namespaces limits, Kan extensions, large

classifications concept lattices, topoi and

fibrations. - Applications
- Kestrel Institutes Specware
- Semantics for DAML-OIL
- Category Theory Review
- Initial reviewers
- Ross Street, a category-theorist who has interest

in applications see his link to Category Theory

at The Boeing Company, which refers to the

Specware software mentioned in Michael Uscholds

message Composing Ontologies using morphisms and

colimits. - Colin McLarty, who gave the challenge topos

axioms, and has written a topos-theory book

"Elementary Categories, Elementary Toposes". - Bill Lawvere, who is one of the primary creators

of topos theory, and whose interest in

applications (although a theorist) is indicated

in the very detailed response that he gave to a

question of Mike Healy's on the category list. - Submit to the category list at large for review,

comments and criticisms.

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