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Higher-order algebras and coalgebras from parameterized endofunctors - - PowerPoint PPT Presentation

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Basic Definitions Results Applications Conclusions Higher-order algebras and coalgebras from parameterized endofunctors Jiho Kim Department of Mathematics Indiana University Bloomington, Indiana Coalgebraic Methods in Computer Science 2010

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Basic Definitions Results Applications Conclusions

Higher-order algebras and coalgebras from parameterized endofunctors

Jiho Kim

Department of Mathematics Indiana University Bloomington, Indiana

Coalgebraic Methods in Computer Science 2010

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

1 Basic Definitions

Higher-order & parameterized endofunctors Initial and final suitability

2 Results 3 Applications 4 Conclusions

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category. Functor categories [B, C], e.g.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category. Functor categories [B, C], e.g. Category, C ∼ = [1, C].

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category. Functor categories [B, C], e.g. Category, C ∼ = [1, C]. Arrow category, C→ ∼ = [2, C].

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category. Functor categories [B, C], e.g. Category, C ∼ = [1, C]. Arrow category, C→ ∼ = [2, C]. Endofunctor category, End(C) = [C, C].

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Higher-order endofunctors

Definition A higher-order endofunctor is an endofunctor on a functor category. Functor categories [B, C], e.g. Category, C ∼ = [1, C]. Arrow category, C→ ∼ = [2, C]. Endofunctor category, End(C) = [C, C]. Monad category, Mon(C) (abusing terminology slightly)

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized Endofunctors

Definition A parameterized endofunctor is a bifunctor of the type B × C → C.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized Endofunctors

Definition A parameterized endofunctor is a bifunctor of the type B × C → C. By currying, a parameterized endofunctor has type B → End(C).

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized Endofunctors

Definition A parameterized endofunctor is a bifunctor of the type B × C → C. By currying, a parameterized endofunctor has type B → End(C). There are more constrained notion of parameterized endofunctors, (e.g. structural actions (Blute-Cockett-Seely ’97), parameterized monads (Uustalu ’03, Atkey ’06), etc.)

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized Endofunctors

Definition A parameterized endofunctor is a bifunctor of the type B × C → C. By currying, a parameterized endofunctor has type B → End(C). There are more constrained notion of parameterized endofunctors, (e.g. structural actions (Blute-Cockett-Seely ’97), parameterized monads (Uustalu ’03, Atkey ’06), etc.) We use the unconstrained definition studied by Kurz and Pattinson ’00.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized endofunctors to higher-order endofunctors

Definition For F : B × C → C, let F : [B, C] → [B, C] be given by

  • F(X)(b) = F(b, Xb)

for X : B → C and b ∈ B. F is the higher-order endofunctor generated by the parameterized endofunctor F.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Higher-order & parameterized endofunctors

Parameterized endofunctors to higher-order endofunctors

Definition For F : B × C → C, let F : [B, C] → [B, C] be given by

  • F(X)(b) = F(b, Xb)

for X : B → C and b ∈ B. F is the higher-order endofunctor generated by the parameterized endofunctor F. Goal Characterize initial algebras and final coalgebras of these higher-order endofunctors in terms of lower-order properties.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Initial and final suitability

Suitable Parameterized Endofunctors

Definition A parameterized endofunctor F : B × C → C is

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Initial and final suitability

Suitable Parameterized Endofunctors

Definition A parameterized endofunctor F : B × C → C is initially suitable if F(b, −) admits an initial algebra for any b ∈ B.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions Initial and final suitability

Suitable Parameterized Endofunctors

Definition A parameterized endofunctor F : B × C → C is initially suitable if F(b, −) admits an initial algebra for any b ∈ B. finally suitable if F(b, −) admits a final coalgebra for any b ∈ B.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

F(x, RF x) rx − → RF x is the initial F(x, −)-algebra.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

F(x, RF x) rx − → RF x is the initial F(x, −)-algebra. RF f is induced by initiality.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

F(x, RF x) rx − → RF x is the initial F(x, −)-algebra. RF f is induced by initiality.

  • F(RF ) = F(−, RF −)

r

= ⇒ RF is a natural isomorphism.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

F(x, RF x) rx − → RF x is the initial F(x, −)-algebra. RF f is induced by initiality.

  • F(RF ) = F(−, RF −)

r

= ⇒ RF is a natural isomorphism.

  • FRF

r

= ⇒ RF is an F-algebra!

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Suitability to Higher-order Algebras and Coalgebras

An initially suitable parameterized endofunctor F : B × C → C induces a C-endofunctor RF : F(x, RF x)

rx

  • F(x,RF f)
  • F(f,RF f)=F(−,RF −)f
  • RF x

RF f

  • F(x, RF y)

F(f,RF y)

F(y, RF y)

ry

RF y

F(x, RF x) rx − → RF x is the initial F(x, −)-algebra. RF f is induced by initiality.

  • F(RF ) = F(−, RF −)

r

= ⇒ RF is a natural isomorphism.

  • FRF

r

= ⇒ RF is an F-algebra! Let SF

s

= ⇒ FSF be the F-coalgebra induced by finally suitable parameterized endofunctors.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

The Punchline

Theorem (J.Kim ’09) Let F be a [B, C]-endofunctor generated by a parameterized endofunctor F : B × C → C. The following are equivalent:

1 F is initially (resp. finally) suitable. 2

  • F admits an initial algebra (resp. final coalgebra).

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

The Punchline

Theorem (J.Kim ’09) Let F be a [B, C]-endofunctor generated by a parameterized endofunctor F : B × C → C. The following are equivalent:

1 F is initially (resp. finally) suitable. 2

  • F admits an initial algebra (resp. final coalgebra).

If F is initially suitable, FRF

r

= ⇒ RF is the initial F-algebra.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

The Punchline

Theorem (J.Kim ’09) Let F be a [B, C]-endofunctor generated by a parameterized endofunctor F : B × C → C. The following are equivalent:

1 F is initially (resp. finally) suitable. 2

  • F admits an initial algebra (resp. final coalgebra).

If F is initially suitable, FRF

r

= ⇒ RF is the initial F-algebra. If F is finally suitable, SF

s

= ⇒ FSF is the final F-coalgebra.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

The Punchline

Theorem (J.Kim ’09) Let F be a [B, C]-endofunctor generated by a parameterized endofunctor F : B × C → C. The following are equivalent:

1 F is initially (resp. finally) suitable. 2

  • F admits an initial algebra (resp. final coalgebra).

If F is initially suitable, FRF

r

= ⇒ RF is the initial F-algebra. If F is finally suitable, SF

s

= ⇒ FSF is the final F-coalgebra. The result can be specialized to “parameterized monads” F : B → Mon(C). A monad structure can be imposed on the higher-order [B, C]-endofunctor F.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor I

Example Let G1

θ

= ⇒ G0 be a natural transformation between two C-endofunctors. Let D: 2 × C → C be given by D(i, x) = Gi(x) and D(!, x) = θx for i ∈ 2, x ∈ C. Recall 2 = 0

id0

  • 1

id1

  • !
  • .

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor I

Example Let G1

θ

= ⇒ G0 be a natural transformation between two C-endofunctors. Let D: 2 × C → C be given by D(i, x) = Gi(x) and D(!, x) = θx for i ∈ 2, x ∈ C. Recall 2 = 0

id0

  • 1

id1

  • !
  • .

D is initially suitable if G0 and G1 admit initial algebras.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor I

Example Let G1

θ

= ⇒ G0 be a natural transformation between two C-endofunctors. Let D: 2 × C → C be given by D(i, x) = Gi(x) and D(!, x) = θx for i ∈ 2, x ∈ C. Recall 2 = 0

id0

  • 1

id1

  • !
  • .

D is initially suitable if G0 and G1 admit initial algebras. D is finally suitable if G0 and G1 admit final coalgebras.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor I

Example Let G1

θ

= ⇒ G0 be a natural transformation between two C-endofunctors. Let D: 2 × C → C be given by D(i, x) = Gi(x) and D(!, x) = θx for i ∈ 2, x ∈ C. Recall 2 = 0

id0

  • 1

id1

  • !
  • .

D is initially suitable if G0 and G1 admit initial algebras. D is finally suitable if G0 and G1 admit final coalgebras. The initiality version of the theorem generalizes a result by Chuang and Lin ’06, proved for arrow categories.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor II

Example Let H be a C-endofunctor. Let E : C × C → C be given by E(a, x) = a + Hx for a, x ∈ C.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor II

Example Let H be a C-endofunctor. Let E : C × C → C be given by E(a, x) = a + Hx for a, x ∈ C. RE is the free monad generated by H.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor II

Example Let H be a C-endofunctor. Let E : C × C → C be given by E(a, x) = a + Hx for a, x ∈ C. RE is the free monad generated by H. SE is the completely iterative monad generated by H.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor II

Example Let H be a C-endofunctor. Let E : C × C → C be given by E(a, x) = a + Hx for a, x ∈ C. RE is the free monad generated by H. SE is the completely iterative monad generated by H. E is finally suitable ⇐ ⇒ E is iteratable

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor II

Example Let H be a C-endofunctor. Let E : C × C → C be given by E(a, x) = a + Hx for a, x ∈ C. RE is the free monad generated by H. SE is the completely iterative monad generated by H. E is finally suitable ⇐ ⇒ E is iteratable The finality version of the theorem generalizes a result by Aczel, Ad´ amek, Milius, Velebil ’03, proved for iteratable endofunctors.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor III

Example Let A, B be nonempty sets. Let F : (Setop × Set) × Set → Set be given by F(A, B, C) = (B × C)A.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor III

Example Let A, B be nonempty sets. Let F : (Setop × Set) × Set → Set be given by F(A, B, C) = (B × C)A. F is finally suitable.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor III

Example Let A, B be nonempty sets. Let F : (Setop × Set) × Set → Set be given by F(A, B, C) = (B × C)A. F is finally suitable. SF A, B = ΓA,B = {f : Aω → Bω : f causal}. Let ΓA,B

γA,B

− − − → (B × ΓA,B)A be given by γA,B(f)(a) = hd ◦ f ◦ ca, tl ◦ f ◦ ca for f ∈ ΓA,B, a ∈ A, ca(σ) = a:σ. (Rutten)

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

An Example of a Parameterized Endofunctor III

Example Let A, B be nonempty sets. Let F : (Setop × Set) × Set → Set be given by F(A, B, C) = (B × C)A. F is finally suitable. SF A, B = ΓA,B = {f : Aω → Bω : f causal}. Let ΓA,B

γA,B

− − − → (B × ΓA,B)A be given by γA,B(f)(a) = hd ◦ f ◦ ca, tl ◦ f ◦ ca for f ∈ ΓA,B, a ∈ A, ca(σ) = a:σ. (Rutten) The first coordinate hd ◦ f ◦ ca is a constant in B since f is causal.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Let SF = Γ: Setop × Set → Set be given by Γ(A, B) = ΓA,B.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Let SF = Γ: Setop × Set → Set be given by Γ(A, B) = ΓA,B. Γ

γ

= ⇒ F(Γ) = F(−, Γ−) is the final F-coalgebra.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Let SF = Γ: Setop × Set → Set be given by Γ(A, B) = ΓA,B. Γ

γ

= ⇒ F(Γ) = F(−, Γ−) is the final F-coalgebra. Define another F-coalgebra

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Let SF = Γ: Setop × Set → Set be given by Γ(A, B) = ΓA,B. Γ

γ

= ⇒ F(Γ) = F(−, Γ−) is the final F-coalgebra. Define another F-coalgebra Let Hom: Setop × Set → Set be the usual hom-bifunctor.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism I

Recall the final F-coalgebra

  • F is a [Setop × Set, Set]-endofunctor.

Let SF = Γ: Setop × Set → Set be given by Γ(A, B) = ΓA,B. Γ

γ

= ⇒ F(Γ) = F(−, Γ−) is the final F-coalgebra. Define another F-coalgebra Let Hom: Setop × Set → Set be the usual hom-bifunctor. Let Hom

e

= ⇒ F Hom = F(−, Hom −) be a F-coalgebra where the components Hom(A, B)

eA,B

− − − − → (B × Hom(A, B))A is given by eA,B(f)(a) = f(a), f.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism II

By finality of γ, there is a higher-order coalgebra morphism Hom

m

= ⇒ Γ:

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism II

By finality of γ, there is a higher-order coalgebra morphism Hom

m

= ⇒ Γ: Hom(A, B)

eA,B

  • mA,B
  • (B × Hom(A, B))A

(B×mA,B)A

  • ΓA,B

γA,B

(B × ΓA,B)A

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism II

By finality of γ, there is a higher-order coalgebra morphism Hom

m

= ⇒ Γ: Hom(A, B)

eA,B

  • mA,B
  • (B × Hom(A, B))A

(B×mA,B)A

  • ΓA,B

γA,B

(B × ΓA,B)A

mA,B(f)(σ0, σ1, σ2, . . .) = (f(σ0), f(σ1), f(σ2), . . .)

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

map as a higher-order coalgebra morphism II

By finality of γ, there is a higher-order coalgebra morphism Hom

m

= ⇒ Γ: Hom(A, B)

eA,B

  • mA,B
  • (B × Hom(A, B))A

(B×mA,B)A

  • ΓA,B

γA,B

(B × ΓA,B)A

mA,B(f)(σ0, σ1, σ2, . . .) = (f(σ0), f(σ1), f(σ2), . . .) mA,B is map!

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

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Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

3 Generalization of known results.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

slide-56
SLIDE 56

Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

3 Generalization of known results. 4 Derivation of map as a higher-order coalgebra morphism.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

slide-57
SLIDE 57

Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

3 Generalization of known results. 4 Derivation of map as a higher-order coalgebra morphism.

Conclusions

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

slide-58
SLIDE 58

Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

3 Generalization of known results. 4 Derivation of map as a higher-order coalgebra morphism.

Conclusions

1 Algebraic and coalgebraic properties of higher-order

endofunctors should be studied.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors

slide-59
SLIDE 59

Basic Definitions Results Applications Conclusions

Overview & Conclusions

Overview

1 Higher-order endofunctors & parameterized endofunctors 2 Characterization of initial algebras and final coalgebras for

higher-order endofunctors generated by parameterized endofunctors.

3 Generalization of known results. 4 Derivation of map as a higher-order coalgebra morphism.

Conclusions

1 Algebraic and coalgebraic properties of higher-order

endofunctors should be studied.

2 Particulars of other “constrained” functor categories should be

studied.

Jiho Kim Department of Mathematics, Indiana University Higher-order algebras and coalgebras from parameterized endofunctors