Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
Some examples in Category Theory SET⊥
PAR
Proc
AUTO
REACH
D. Gift Samuel THEOLTL
CLOSURE
Functors
SET – MON
April 25, 2007
Outline Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
◮ SET, SET⊥ , Proc PAR
Proc
◮ AUTO, REACH AUTO
REACH
◮ THEO, PRES, SPRES THEOLTL
◮ POSET,Poset,GRAPH, PROOF, LOGI CLOSURE
Functors
SET – MON
Definitions Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
◮ A category C is a triple SET⊥
PAR
◮ G = (G0 , G1 , src, trg ) is a graph Proc
AUTO
◮ ; is a map from G2 into G1 REACH
◮ id is a map from G0 into G1 THEOLTL
◮ ; satisfies associative properties. CLOSURE
◮ (f ; g ); h = f ; (g ; h) Functors
SET – MON
◮ idx satisfies identity morphism.
◮ If f : x → y , idx ; f = f ; idy = f
Category: POSET Some examples in
Category Theory
D. Gift Samuel
Outline
◮ Objects are posets (A, ≤) Category
◮ Morphisms are monotonic functions POSET
SET
◮ Composition is well defined and it is closed. SET⊥
PAR
Proc
1. Let (P, ≤P ) and (Q, ≤Q ) be posets.
AUTO
f : P → Q and g : Q → R, (f ; g )(x) = g (f (x)) REACH
2. x ≤P y ⇒ f (x) ≤Q f (y ) by f is monotonic THEOLTL
⇒ g (f (x)) ≤R g (f (y )) by g is monotonic CLOSURE
⇒ (f ; g )(x) ≤R (f ; g )(y ) by definition of Functors
composition SET – MON
3. f ; (g ; h) = (f ; g ); h is true as f,g,h are functions
◮ for each poset (P, ≤P ), indentity morphism is
identity function
1. idP : P → P is monotonic
2. it satisfies the identity axioms; f : P → Q ,
idP ; f = f and f ; idQ = f
Category: SET Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
◮ Objects are sets PAR
Proc
◮ Morphisms are total functions AUTO
◮ Composition is functional composition. REACH
THEOLTL
◮ If f : A → B and g : B → c are total, then so is.
CLOSURE
◮ Functional compositional is associative
Functors
◮ Identity morphisms are identity functions SET – MON
◮ Identity function is total
◮ for any function f : A → B, idA ; f = f ; idB = f
Category: SET⊥ Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ Objects are pair where ⊥A ∈ A
SET
◮ morphism between and are SET⊥
PAR
total functions s.t. f (⊥A ) = ⊥B Proc
AUTO
◮ morphism for SET⊥ are all morphism of SET s.t. it REACH
satisfies the above condition. THEOLTL
PROOF CLOSURE
◮ composition is defined by functional composition Functors
SET – MON
which are inherited from SET
◮ composition law is closed for SET⊥
◮ identity map assigns to every set the identity
function which are also inherited from SET
◮ Identities are morphism in SET⊥
Comma Category Some examples in
Category Theory
D. Gift Samuel
Outline
◮ Given a category C and an object c : C , we define Category
a↓C POSET
SET
◮ Objects are all the pairs where f is a SET⊥
PAR
morphism f : a → x in C . Proc
AUTO
◮ Morphism between f : a → x and g : a → y s.t. REACH
f ;h = g THEOLTL
a CLOSURE
a Functors
SET – MON
f g h
g
f
x z
i y j
x y
h
◮ Category isomorphism between the 1 ↓ SET and
SET⊥
Categories:Co-reflective sub-categories Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ D be a co-reflective sub-category of a category C iff SET
every C-object c, co-reflection for c is C-Morphism SET⊥
PAR
i : d → c s.t. for any C-morphism f : d ′ → c where Proc
AUTO
d’ is a D-Morphism , there is a unique D-morphism REACH
f ′ : d ′ → d s.t. f=f’;i THEOLTL
d i c
CLOSURE
Functors
SET – MON
f’
f
d’
SET is co-reflective sub-category of PAR Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ PAR is a category where objects are sets and POSET
morphisms are partial functions SET
SET⊥
◮ SET is subcategory of PAR, but it is not a full PAR
Proc
subcategory of PAR AUTO
REACH
◮ PAR is a Co-reflective sub-categories of SET THEOLTL
◮ Proof CLOSURE
elevation of A
A⊥ ∈ SET i Functors
A SET – MON
f ⊥ ∈ SET
f ∈ PAR
B
Category: FSET⊥ Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
◮ Objects are pair where ⊥A ∈ A and A is PAR
Proc
finite AUTO
REACH
◮ morphism between and are
THEOLTL
total functions s.t. f (⊥A ) = ⊥B CLOSURE
◮ composition is functional composition and identity Functors
map assigns to every set the identity function SET – MON
◮ FSET⊥ is a full subcategory of SET⊥
Category: Proc Some examples in
Category Theory
D. Gift Samuel
◮ Objects are process behaviour(or process) Outline
P = where A⊥ is a finite pointed set and Category
Λ ⊆ Aω .
⊥ POSET
◮ λ ∈ Λ is a function from λ : ω → A⊥ . an finite SET
sequence of elements of A⊥ SET⊥
PAR
◮ Given a process , A⊥ is called events of Proc
P,and A⊥ \ ⊥A is called alphabet of P(denoted as AUTO
REACH
Pα ). Λ is called behaviours of P. THEOLTL
◮ ⊥ is called the environment event of P
CLOSURE
◮ process morphism Functors
h : P =→ Q = is a SET – MON
morphism h : A⊥ → B⊥ s.t. hω (ΛA ) ⊆ ΛB . where
hω (λ) = λ; h
◮ processes and process morphism constitute a
category Proc
◮ Forget functor U⊥ : Proc → FSet⊥ that sends each
process to its alphabets and each morphism
f : (A1 , Λ1 ) → (A2 , Λ2 ) to f : A1 → A2 is faithful.
Category: Proc Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ Morphism h : P =→ Q = is SET
SET⊥
a embedding of the process Q within P, making Q a PAR
Proc
component-of P. AUTO
◮ h :→ is a morphism REACH
h : A⊥ → B⊥ s.t. hω (ΛA ) ⊆ ΛB THEOLTL
CLOSURE
◮ environment of P are also in the environment of Q
Functors
◮ any alphabet of P can be mapped onto ⊥B : P SET – MON
identifies part of the environment of Q: P doesn’t
participate on the event.
◮ the behaviour of P be compatible with Q: the life
cycle of Q is mapped to life cycle of P.
Definition: Cone, Limit Some examples in
Category Theory
D. Gift Samuel
◮ A communtative cone over diagram δ consists of an Outline
object C together with morphism p : C → δ s.t. for Category
POSET
every f : Ai → Aj we have f ; δi = δj SET
SET⊥
◮ A limit for the diagram D is a commutative cone PAR
Proc
p : C → δ s.t. for every commutative cone AUTO
′ ′
p ′ : C ′ → δ there is a unique morphism f : C → C REACH
′ ′
such that p; f = p ( pa ; f = pa for every edge ) THEOLTL
CLOSURE
◮ Terminal, products,equalizers and pushback are
Functors
speciallization of limits SET – MON
X C
′ f C
qb
qa
pb
pa pb
pa
b B
a f A LIMIT
CONE
Definition: Universal properties Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ Limit of two object without any morphism is a POSET
product. The limit of empty diagram is the terminal SET
object SET⊥
PAR
Proc
◮ Limit of two parallel morphism with the same AUTO
domain and co-domain is the equalizer. Pullback is REACH
the limit of two morphism with the same co-domain. THEOLTL
CLOSURE
B
Functors
A B f g
SET – MON
∅ A g
B
A f C
P
P
p1 p2 g
A C
A B f
A B$ C
Terminal Object C Product P Equaliser pullback
Proc: Initial and Terminal Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ Terminal process is . Its alphabet
∅ SET
contains only the witness for action of the SET⊥
PAR
environment. It is a model for idle process. Proc
AUTO
◮ Terminal process is the innermost component REACH
THEOLTL
◮ Initial process . It does nothing. It
CLOSURE
models a deadlock process. deadlock any process to
Functors
which it is connected SET – MON
◮ Initial process is the outmost component.
◮ Initial objects and terminals objects are in SET⊥ are
singleton sets
Universal property: Product Some examples in
Category Theory
D. Gift Samuel
◮ The object z is the product of x and y with Outline
projection πx : z → x and πy : z → y iff for any v Category
POSET
and pair of morphism fx : v → x fy : v → y , then
SET
there is a unique morphism k : v → z SET⊥
PAR
◮ Every poset(P, ≤) is a category. Objects are Proc
AUTO
elements in P. Morphisms are given by the relation REACH
≤. THEOLTL
◮ Composition is defined by transitivity law. Identity CLOSURE
morphism are defined by reflexitivity laws Functors
SET – MON
◮ Universal Properties
◮ The least element is the initial object, the greatest
element is the terminal object
◮ In a category of poset (P, ≤P ), greatest lower bound
z is the product of p and q
◮ glb(p, q) ≤ p, glb(p, q) ≤ q, are projections
◮ if c ≤ p and c ≤ q, then c ≤ glb(p, q), which is
unique
Universal properties of Proc Some examples in
Category Theory
D. Gift Samuel
◮ Product of two processes alphabets Outline
{a, ⊥A }, ⊥A Category
{b, ⊥B }, ⊥B
POSET
SET
SET⊥
PAR
Proc
AUTO
REACH
THEOLTL
{a|b, ⊥A |b, a|⊥B , ⊥A |⊥B }, ⊥A |⊥B
CLOSURE
DENOTED AS
{a} Functors
SET – MON
{b}
{a|b, b, a}
◮ Parallel composition without interaction
Universal properties of Proc Some examples in
Category Theory
D. Gift Samuel
◮ product represents traditional trace-based semantics Outline
of parallel compositions without synchronisation Category
◮ Product of two processes and POSET
SET
is obtained SET⊥
◮ by computing the product of the alphabets (A1 × A2 ) PAR
Proc
◮ consist all lifecycles of the products, once projected AUTO
into the components alphabets, give life cycles of REACH
the component process THEOLTL
◮ consists λ, λ : ω → (A1 × A2 ) such that g1 (λ) ∈ Λ1
ω CLOSURE
Functors
and g2 (λ) ∈ Λ2
ω
SET – MON
c0 , S0
f1 f2
c1 , S1
c2 , S2
g1 g2
−1 −1
c, g1 (S) ∩ g2 (S2 )
◮ Parallel composition without interaction
Idle and deadlock process Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
◮ When terminal process put in parallel with another PAR
Proc
process, the result of the parallel composition is the AUTO
other process REACH
THEOLTL
◮ Initial process(blocking process) absorbs any other CLOSURE
process when put in parallel Functors
SET – MON
◮ product with other process returns an empty
behaviour
Universal properties of Proc Some examples in
Category Theory
D. Gift Samuel
◮ Pullback represents traditional trace-based semantics Outline
of parallel compositions with synchronisation Category
◮ Product of two morphism f1 and f2 is obtained by POSET
SET
computing the product of the alphabets and by SET⊥
PAR
taking as set of behaviours the intersection of the Proc
inverse images of the set of behaviours of the AUTO
REACH
components and equal THEOLTL
◮ consists λ, λ : ω → (A1 ×A A2 ) such that CLOSURE
g1 (λ) ∈ Λ1 and g2 (λ) ∈ Λ2
ω ω
Functors
c0 , S0 SET – MON
f1 f2
c1 , S1
c2 , S2
g1 g2
−1 −1
c, g1 (S) ∩ g2 (S2 )
◮ Parallel composition without interaction
Universal properties of Proc Some examples in
Category Theory
D. Gift Samuel
◮ A1 = {⊥, a, c, d}, A2 = {⊥, b, e, k} and A = {⊥, x} Outline
with functions f = {⊥ → ⊥, a → ⊥, c → x, d → x} Category
and g = {⊥ → ⊥, b → ⊥, e → x, k → x} POSET
SET
◮ product A1 × A2 = SET⊥
PAR
{⊥, a, c, d, b, e, k, a|b, a|e, a|k, c|b, c|e, c|k, d|b, d|e, d|k}Proc
◮ pullback is obtained by keeping only the events that AUTO
REACH
after being projected to A1 and A2 are mapped THEOLTL
through f and g to the same element of A. CLOSURE
◮ A1 ×A A2 = {⊥, a, b, a|b, c|e, c|k, d|e, d|k} Functors
{x} SET – MON
c→x e→x
d →x k →x
{a, c, d } Pullback {b, e, k}
{a, b, a|b, c|e, c|k, d |e, d |k}
An Example Some examples in
Category Theory
D. Gift Samuel
◮ ΛP1 contains all the life cycle of the form Outline
⊥∗ a⊥∗ c⊥∗ a⊥∗ c⊥∗ · · · and ΛP2 contains Category
POSET
⊥∗ e⊥∗ b⊥∗ e⊥∗ b⊥∗ · · · SET
◮ Pullback is SET⊥
PAR
⊥∗ a⊥∗ c|e⊥∗ {a⊥∗ b, b⊥∗ a, a|b}⊥∗ c|e⊥∗ Proc
AUTO
{a⊥∗ b, b⊥∗ a, a|b} · · · REACH
THEOLTL
◮ e can synchronised either with c or d, so e has to
CLOSURE
wait until c or d appears
Functors
{x} SET – MON
c→x e→x
d →x k →x
{a, c, d } Pullback {b, e, k}
{a, b, a|b, c|e, c|k, d |e, d |k}
Reference Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
PAR
Proc
AUTO
◮ Mirror, Mirror in my Hand: a duality between REACH
specifications and models of process behaviour THEOLTL
J.L. Fiadeiro and J.F. Costa CLOSURE
Functors
SET – MON
Category of Automatas(AUTO) Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
◮ Objects in AUTO are automata (X , S, Y , f , s0 , g ) SET⊥
PAR
◮ Morphisms in AUTO are simulations: B simulates A Proc
AUTO
if A → B REACH
◮ Morphism from A = (X , S, Y , f , s0 , g ) to THEOLTL
′ ′ ′ ′ ′ ′
B = (X , S , Y , f , s0 , g ) is a tuple CLOSURE
′ ′ ′ Functors
such that SET – MON
′
◮ i(s0 ) = s0
′
◮ f ; i = h × i; f
′
◮ g ; j = i; g
Proof for AUTO Some examples in
Category Theory
D. Gift Samuel
◮ Composition of two morphisms is a morphism
Outline
◮ (h, i , j) : A = (X , S, Y , f , s0 , g ) to Category
′ ′ ′ ′ ′ ′
B = (X , S , Y , f , s0 , g ) POSET
′ ′ ′ ′ ′ ′ ′ ′ ′
(h , i .j ): B = (X , S , Y , f , s0 , g ) to SET
SET⊥
′′ ′′ ′′ ′′ ′′ ′′
C = (X , S , Y , f , s0 , g ) and PAR
Proc
′ ′ ′
◮ Composition is AUTO
◮ (i ; i ′ )(s0 ) REACH
= i (i ′ (s0 ))
THEOLTL
by composition definition
′ CLOSURE
= i (s0 ) by is morphism
′′ ′ ′ ′ Functors
= s0 by is morphism SET – MON
′
◮ f ; (i ; i )
′
= (f ; i ); i by associativity
′ ′
= (h × i ; f ); i by morphism
′ ′
= h × i ; (f ; i ) by associativity
′ ′ ′′
= (h × i ; (h × i ); f by morphism
′ ′ ′′
= (h × i ; h × i ); f by associativity
′ ′ ′′
= (h; h ) × (i ; i ); f by composition
Initials and Terminals Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
PAR
◮ ∅ is the initial object in SET Proc
AUTO
◮ {a} is a terminal object in SET. REACH
THEOLTL
◮ (∅, ∅, ∅, ∅, ∅, ∅) is a initial objects in AUTOM
CLOSURE
◮ ({i }, {s}, {o}, {i × s → s}, s, {s → o}) is a terminal Functors
object in AUTOM SET – MON
Co-reflective sub-categories of AUTO Some examples in
Category Theory
D. Gift Samuel
◮ D be a co-reflective sub-category of a category C iff Outline
every C-object c, co-reflection for c is C-Morphism Category
POSET
i : d → c such that for any C-morphism f : d ′ → c SET
where d’ is a D-Morphism , there is a unique SET⊥
PAR
D-morphism f ′ : d ′ → d such that f=f’;i Proc
AUTO
d i c REACH
THEOLTL
CLOSURE
f’ Functors
f SET – MON
d’
◮ Reachable Automata: automate that is obtained by
removing all non-reachable states.
◮ In REACH, objects are reachable automata.
◮ Morphisms are simulations.
REACH is a Co-reflective sub-category of Some examples in
Category Theory
AUTO D. Gift Samuel
Outline
◮ A is related to canonical reachable automata R by Category
POSET
c :R→A SET
R c SET⊥
A
PAR
Proc
AUTO
h’ h REACH
THEOLTL
CLOSURE
R’ Functors
SET – MON
◮ A = (X , S, Y , s0 , f , g ) and R = (X , SR , Y , s0 , fR , gR )
where SR ⊆ S, X,Y are identities
◮ Given any reachable automata R ′ and simulation
h : R ′ → A, there is a unique morphism of reachable
automata h′ : R ′ → R such that h = h′ ; c
◮ Co-reflector for an object is a morphism through
which all communication must go.
Temporal propositions Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ A signature of LTL is a set of actions symbols. SET
SET⊥
◮ The action symbols provide atomic propositions in PAR
Proc
the LTL formula. AUTO
◮ The set of temporal propositions prop(Σ) for a REACH
THEOLTL
signature is inductively defined as
CLOSURE
◮ Every action symbol is a temporal propositions
Functors
◮ beg is a temporal propositions (denotating the initial SET – MON
state)
◮ if φ is a temporal proposition so is ¬φ
◮ if φ1 and φ2 is a temporal proposition so are
φ1 ⊃ φ2 , φ1 Uφ2 and φ1 W φ2
Semantics Some examples in
Category Theory
D. Gift Samuel
◮ An interpretation structure for a signature Σ is a Outline
sequence λ ∈ (2Σ )ω Category
POSET
◮ λ ∈ (2Σ )ω is true at state i which we write λ |=Σ,i φ SET
◮ if φ ∈ Σ, λ |=Σ,i φ iff φ ∈ λ(i) SET⊥
PAR
◮ φ ∈ Σ, λ |=Σ,i beg iff i = 0 Proc
◮ φ ∈ Σ, λ |=Σ,i ¬φ iff it is not the case λ |=Σ,i φ AUTO
REACH
◮ φ ∈ Σ, λ |=Σ,i φ1 ⊃ φ2 iff λ |=Σ,i φ1 implies THEOLTL
λ |=Σ,i φ2 CLOSURE
◮ φ ∈ Σ, λ |=Σ,i φ1 Uφ2 iff for some j > i, λ |=Σ,j φ2
Functors
and λ |=Σ,k φ1 for every i ≤ k ≤ j SET – MON
◮ (weak until) (φ1 W φ2 ) holds (φ1 Uφ2 ), or φ2 will
forever be false and φ1 true.
◮ φ is true in φ for λ, written, λ |=Σ φ iff λ |=Σ,i φ for
every state i
◮ Φ ⊢Σ φ iff φ is true in every sequence that makes all
the propositions in Φ true
Examples Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
◮ specification vending machine is PAR
Proc
signature coin,cake,cigar
AUTO
axioms: beg ⊃ REACH
((¬cake ∧ ¬cigar ) ∧ (coin ∨ (¬cake ∧ ¬cigar )Wcoin)) THEOLTL
coin ⊃ (¬coin)W (cake ∨ cigar ) CLOSURE
(cake ∨ cigar ) ⊃ (¬cake ∧ ¬cigar )Wcoin Functors
SET – MON
cake ⊃ (¬cigar )
◮ accepts coins, delivers cakes and cigars
Interpretations between theories Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ Let Σ be a signature and Ψ a subset of prop(Σ) is POSET
SET
said to be closed iff for every φ ∈ prop(Σ), Ψ ⊢Σ φ SET⊥
PAR
implies φ ∈ Ψ. Proc
AUTO
◮ cΣ (Ψ) denotes the least closed set that contains Ψ. REACH
′
◮ Let Σ andΣ be signatures. Every functions THEOLTL
′
f : Σ → Σ extends to a ′
CLOSURE
prop(f ) : prop(Σ) → prop(Σ ) as follows Functors
SET – MON
◮ prop(f)(beg)=beg
◮ if a ∈ Σ then prop(f)(a)=f(a)
◮ prop(f )(¬φ) = (not prop(f )(φ))
◮ prop(f )(φ1 ⊃ φ2 ) = (prop(f )(φ1 ) ⊃ prop(f )(φ2 )
◮ prop(f )(φ1 ⊃ φ2 ) = (prop(f )(φ1 )Uprop(f )(φ2 )
More Categories Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ THEOLTL is the category of theories with: POSET
◮ objects: , where Φ = cΣ (Φ). SET
SET⊥
◮ morphism f :→ is a sig. morph. PAR
Proc
f : Σ → Σ′ such that prop(f )(Φ) ⊆ Φ′ .
AUTO
◮ ◮ PRESLTL is the category of presentation with: REACH
◮ objects: , THEOLTL
◮ morphism f :→ is a sig. CLOSURE
morph. f : Σ → Σ′ such that Functors
prop(f )(cΣ (Φ)) ⊆ cΣ′ (Φ′ ). SET – MON
◮ THEOLTL is a category
◮ prop(f ; g )(Φ) = prop(g )prop(f )(Φ)
◮ (f ; g ) is a theory morphism
◮ PRESLTL is a category
THEOLTL is a subcategory of PRESLTL Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
SET
SET⊥
PAR
Proc
◮ Every theory is a presentation. AUTO
′ ′
◮ Given a theory f : (Σ, Φ) → (Σ , Φ ), we need to REACH
′ THEOLTL
prove prop(f )(cΣ (Φ) ⊆cΣ (Φ )
CLOSURE
′
◮ As f is a theory morphism, prop(f )(Φ) ⊆ Φ and Φ Functors
′ ′ ′
and Φ are closed, c(Φ) = Φ and c(Φ ) = Φ SET – MON
Category:CLOSURE Some examples in
Category Theory
D. Gift Samuel
Outline
Category
POSET
◮ A closure system is a pair where L is a set SET
and c : 2L → 2L is a total function satisfying the SET⊥
PAR
following properties: Proc
◮ Reflexivity: for every Φ ⊆ L, Φ ⊆ c(Φ). AUTO
REACH
◮ Monotonicity: for every Φ, Γ ⊆ L, Φ ⊆ Γ implies THEOLTL
c(Φ) ⊆ c(Γ).
CLOSURE
◮ Idempotence: for every Φ ⊆ L, c(c(Φ)) ⊆ c(Φ).
Functors
◮ Objects in CLOSURE are closure systems SET – MON
◮ morphisms f :→ are the maps
f :L→L ′ such that f (c(Φ)) ⊆ c ′ (f (Φ)) for all
Φ ⊆ L.
Reflective & Co-Reflective sub categories Some examples in
Category Theory
D. Gift Samuel
Outline
Category
Consider a closure systems (L, c). POSET
SET
Φ ⊆ L is closed iff Φ = c(Φ). SET⊥
PAR
◮ THEO is the category of theories with: Proc
AUTO
◮ objects: closed subset of L REACH
◮ morphism: Inclusions. THEOLTL
◮ SPRES is the category of strict presentation with: CLOSURE
◮ objects: subsets of L Functors
SET – MON
◮ morphism: Inclusions.
◮ PRES is the category of presentation with:
◮ objects: subsets of L
◮ morphism: by the preorder Φ ≤ Γ iff c(Φ) ⊆ c(Γ)
Categories: Reflective sub-categories Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ D be a Reflective sub-category of a category C iff POSET
every C-object c, reflection for c is C-Morphism SET
SET⊥
o : c → d such that for any C-morphism f : c → d ′ PAR
Proc
where d’ is a D-Morphism , there is a unique AUTO
D-morphism f ′ : d → d ′ such that f=o;f’ REACH
THEOLTL
o
C d
CLOSURE
Functors
SET – MON
f f’
d’
Reflective & Co-Reflective sub categories Some examples in
Category Theory
D. Gift Samuel
Outline
◮ THEO is a full subcategory of PRES and SPRES.
Category
◮ SPRES is a subcategory of PRES. POSET
It is an immediate consequence of the monotonicity SET
SET⊥
PAR
◮ THEO is a reflective subcategory of PRES and Proc
SPRES. AUTO
REACH
◮ SPRES is not co-reflective subcat of PRES. THEOLTL
Φ c(Φ) c(Φ) Φ
CLOSURE
Functors
SET – MON
′
′ Φ
Φ
◮ THEO is a coreflective subcategory of PRES.
◮ THEO is not a coreflective subcategory of SPRES.
◮ SPRES is a coreflective subcategory of PRES.
Functor between SET and MON Some examples in
Category Theory
D. Gift Samuel
◮ SET is a category. Outline
Category
◮ Objects in SET are sets POSET
◮ Morphisms in SET are total functions SET
◮ MON is a category. SET⊥
PAR
Proc
◮ Objects in MON are (List(S), ⋄, [])
AUTO
◮ List(S)is free monoid generated by S. REACH
◮ Morphism in MON are monoid homomorphism. THEOLTL
◮ functor LIST maps SET → List (which is object CLOSURE
part of functor) Functors
SET – MON
′
◮ functor LIST maps f : S → S to a function
′
LIST (f ) : List(S) → List(S ) (which forms
morphism part of functor )
◮ Given a list L =[s1 , s2 , · · · , sn ] maps f over the
elements of list :
LIST (f )(L) = f ∗ (L) = [f (s1 ), f (s2 ), · · · , f (sn )]
Functor between SET and MON [conti...] Some examples in
Category Theory
D. Gift Samuel
Outline
Category
◮ f ∗ is a homomorphism POSET
SET
f ∗ ([]) = [] SET⊥
f ∗ (L ⋄ L′ ) = f ∗ (L) ⋄ f ∗ (L′ )
PAR
Proc
f ∗ ([s]) = [f (s)] AUTO
REACH
◮ Any total function between sets induces the monoid THEOLTL
homomorphism between the corresponding monoids. CLOSURE
PROOF Functors
◮ Preservation of Identities SET – MON
◮ List(ids )(L) = [ids (s1 ), · · · , ids (s1 )]
◮ = [s1 , · · · sn ] = L
◮ = idList(s) (L)
◮ Preservation of composition