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Constructive Category Theory and Applications in Algebraic Geometry - - PowerPoint PPT Presentation

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Constructive Category Theory and Applications in Algebraic Geometry Sebastian Gutsche Universitt Siegen Siegen, August 31, 2017 Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 1 / 23 Outline Constructive category

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Constructive Category Theory and Applications in Algebraic Geometry

Sebastian Gutsche

Universität Siegen

Siegen, August 31, 2017

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 1 / 23

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Outline

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Constructive category theory

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 2 / 23

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Outline

1

Constructive category theory

2

Applications to Algebraic Geometry

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 2 / 23

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Constructive category theory

Constructive category theory

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 3 / 23

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Constructive category theory

Abstraction of language

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: Data type: int Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: Assembly Data type: int Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: Assembly Data type: int addi: movl %edi, -4(%rsp) movl %esi, -8(%rsp) movl

  • 4(%rsp), %esi

addl

  • 8(%rsp), %esi

movl %esi, %eax ret Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: Assembly Data type: int addi: movl %edi, -4(%rsp) movl %esi, -8(%rsp) movl

  • 4(%rsp), %esi

addl

  • 8(%rsp), %esi

movl %esi, %eax ret Data type: float addf: movss %xmm0, -4(%rsp) movss %xmm1, -8(%rsp) movss -4(%rsp), %xmm0 addss -8(%rsp), %xmm0 ret

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: C Data type: int Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: C Data type: int int addi( int a, int b ) { return a + b; } Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: C Data type: int int addi( int a, int b ) { return a + b; } Data type: float float addf( float a, float b ) { return a + b; }

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: GAP or Julia Data type: int Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: GAP or Julia Data type: int function( a, b ) return a + b; end; Data type: float

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: GAP or Julia Data type: int function( a, b ) return a + b; end; Data type: float function( a, b ) return a + b; end;

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: GAP or Julia Data type: int, float function( a, b ) return a + b; end;

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Addition of two numbers: GAP or Julia Data type: int, float function( a, b ) return a + b; end; High language leads to generic code!

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 4 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V:

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2 Ideals of Z

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2 Ideals of Z x, y ≤ Z :

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2 Ideals of Z x, y ≤ Z : Euclidean algorithm: lcm (x, y)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2 Ideals of Z x, y ≤ Z : Euclidean algorithm: lcm (x, y) Generic algorithm for both cases?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Abstraction of language

Computing the intersection of two subobjects Vector spaces v1, v2 , w1, w2 ≤ V: Solution of x1v1 + x2v2 =y1w1 + y2w2 Ideals of Z x, y ≤ Z : Euclidean algorithm: lcm (x, y) Generic algorithm for both cases? Category theory!

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 5 / 23

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Constructive category theory

Category theory as programming language

Category theory

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 6 / 23

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Constructive category theory

Category theory as programming language

Category theory abstracts mathematical structures

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 6 / 23

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Constructive category theory

Category theory as programming language

Category theory abstracts mathematical structures defines a language to formulate theorems and algorithms for different structures at the same time

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 6 / 23

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Constructive category theory

Category theory as programming language

Category theory abstracts mathematical structures defines a language to formulate theorems and algorithms for different structures at the same time CAP - Categories, Algorithms, and Programming

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 6 / 23

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Constructive category theory

Category theory as programming language

Category theory abstracts mathematical structures defines a language to formulate theorems and algorithms for different structures at the same time CAP - Categories, Algorithms, and Programming CAP implements a categorical programming language

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 6 / 23

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Constructive category theory

Categories

Definition A category A contains the following data:

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B)

  • : HomA(B, C) × HomA(A, B) → HomA(A, C)

(assoz.)

A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B)

  • : HomA(B, C) × HomA(A, B) → HomA(A, C)

(assoz.)

A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B)

  • : HomA(B, C) × HomA(A, B) → HomA(A, C)

(assoz.)

Neutral elements: idA ∈ HomA(A, A) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B)

  • : HomA(B, C) × HomA(A, B) → HomA(A, C)

(assoz.)

Neutral elements: idA ∈ HomA(A, A) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Categories

Definition A category A contains the following data: ObjA HomA(A, B)

  • : HomA(B, C) × HomA(A, B) → HomA(A, C)

(assoz.)

Neutral elements: idA ∈ HomA(A, A) A B C

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 7 / 23

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Constructive category theory

Computable categories

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • Sebastian Gutsche (Siegen)

Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • Sebastian Gutsche (Siegen)

Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • 1

2

  • ·

3 4

  • = (11)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • 1

2

  • ·

3 4

  • = (11)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • 1

2

  • ·

3 4

  • = (11)

(1)

  • 1

1

  • (1)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Computable categories

A category becomes computable through data structures for objects and morphisms algorithms to compute the composition of morphisms and identity morphisms of objects Finitely generated Q-vector spaces (skeletal) 1 2 1

  • 1

2

  • 3

4

  • 1

2

  • ·

3 4

  • = (11)

(1)

  • 1

1

  • (1)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 8 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories Zero morphisms

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories Zero morphisms Addition and subtraction of morphisms

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories Zero morphisms Addition and subtraction of morphisms Direct sums

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories Zero morphisms Addition and subtraction of morphisms Direct sums Kernels and cokernels of morphisms

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Categorical operations

Some categorical operations in abelian categories Zero morphisms Addition and subtraction of morphisms Direct sums Kernels and cokernels of morphisms ...

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 9 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B).

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). A B ϕ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . A B ϕ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . . . . one needs an object ker ϕ, A B ker ϕ ϕ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . . . . one needs an object ker ϕ, its embedding κ = KernelEmbedding(ϕ), A B ker ϕ ϕ κ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . . . . one needs an object ker ϕ, its embedding κ = KernelEmbedding(ϕ), and for every test morphism τ A B ker ϕ T ϕ κ τ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . . . . one needs an object ker ϕ, its embedding κ = KernelEmbedding(ϕ), and for every test morphism τ a unique morphism λ = KernelLift(ϕ, τ) A B ker ϕ T ϕ κ τ λ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel

Let ϕ ∈ Hom(A, B). To fully describe the kernel of ϕ . . . . . . one needs an object ker ϕ, its embedding κ = KernelEmbedding(ϕ), and for every test morphism τ a unique morphism λ = KernelLift(ϕ, τ), such that A B ker ϕ T ϕ κ τ λ

  • Sebastian Gutsche (Siegen)

Constructive Categories August 31, 2017 10 / 23

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Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ϕ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ ϕ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ ϕ Compute ker ϕ as dim(A) − rank(ϕ)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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SLIDE 74

Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ ϕ κ Compute ker ϕ as dim(A) − rank(ϕ)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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SLIDE 75

Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ ϕ κ Compute ker ϕ as dim(A) − rank(ϕ) κ by solving X · ϕ = 0

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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SLIDE 76

Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ T ϕ κ τ Compute ker ϕ as dim(A) − rank(ϕ) κ by solving X · ϕ = 0

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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SLIDE 77

Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ T ϕ κ τ λ Compute ker ϕ as dim(A) − rank(ϕ) κ by solving X · ϕ = 0

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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SLIDE 78

Constructive category theory

Implementation of the kernel: Q-vector spaces

Obj := Z≥0, Hom (m, n) := Qm×n A B ker ϕ T ϕ κ τ λ Compute ker ϕ as dim(A) − rank(ϕ) κ by solving X · ϕ = 0 λ by solving X · κ = τ

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 11 / 23

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Constructive category theory

What is CAP?

CAP - Categories, Algorithms, and Programming

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 12 / 23

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SLIDE 80

Constructive category theory

What is CAP?

CAP - Categories, Algorithms, and Programming CAP is a framework to implement computable categories and provides

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 12 / 23

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SLIDE 81

Constructive category theory

What is CAP?

CAP - Categories, Algorithms, and Programming CAP is a framework to implement computable categories and provides specifications of categorical operations,

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 12 / 23

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SLIDE 82

Constructive category theory

What is CAP?

CAP - Categories, Algorithms, and Programming CAP is a framework to implement computable categories and provides specifications of categorical operations, generic algorithms based on basic categorical operations,

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 12 / 23

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SLIDE 83

Constructive category theory

What is CAP?

CAP - Categories, Algorithms, and Programming CAP is a framework to implement computable categories and provides specifications of categorical operations, generic algorithms based on basic categorical operations, a categorical programming language having categorical

  • perations as syntax elements.

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 12 / 23

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Constructive category theory

Computing the intersection

Let M1 ⊆ N and M2 ⊆ N subobjects.

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects.

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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SLIDE 86

Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N.

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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SLIDE 87

Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 M2 N ι2 ι1

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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SLIDE 88

Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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SLIDE 89

Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1 π1

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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SLIDE 90

Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2 πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 13 / 23

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ∩ M2 M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ∩ M2 M1 ⊕ M2 M1 M2 N ι2 ι1 π1 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ := KernelEmbedding (ϕ)

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 ι2 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ π1 ι1 M1 ∩ M2 N πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ := KernelEmbedding (ϕ)

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Constructive category theory

Computing the intersection

Let M1 ֒ → N and M2 ֒ → N subobjects. Compute their intersection γ : M1 ∩ M2 ֒ → N. M1 ⊕ M2 M1 M2 ι2 π2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ π1 ι1 M1 ∩ M2 N πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ := KernelEmbedding (ϕ) γ := ι1 ◦ π1 ◦ κ

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Constructive category theory

Translation to CAP

πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2 ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ := KernelEmbedding (ϕ) γ := ι1 ◦ π1 ◦ κ

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Constructive category theory

Translation to CAP

πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 );

ϕ := ι1 ◦ π1 − ι2 ◦ π2 κ := KernelEmbedding (ϕ) γ := ι1 ◦ π1 ◦ κ

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Constructive category theory

Translation to CAP

πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 );

ϕ := ι1 ◦ π1 − ι2 ◦ π2

lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 );

κ := KernelEmbedding (ϕ) γ := ι1 ◦ π1 ◦ κ

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Constructive category theory

Translation to CAP

πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 );

ϕ := ι1 ◦ π1 − ι2 ◦ π2

lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 );

κ := KernelEmbedding (ϕ)

kappa := KernelEmbedding( phi );

γ := ι1 ◦ π1 ◦ κ

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Constructive category theory

Translation to CAP

πi := ProjectionInFactorOfDirectSum ((M1, M2) , i), i = 1, 2

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 );

ϕ := ι1 ◦ π1 − ι2 ◦ π2

lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 );

κ := KernelEmbedding (ϕ)

kappa := KernelEmbedding( phi );

γ := ι1 ◦ π1 ◦ κ

gamma := PostCompose( lambda, kappa );

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Constructive category theory

Translation to CAP

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa );

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Constructive category theory

Translation to CAP

pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa );

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Constructive category theory

Translation to CAP

Schnitt := function( iota1, iota2 ) pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa );

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Constructive category theory

Translation to CAP

Schnitt := function( iota1, iota2 ) M1 := Source( iota1 ); M2 := Source( iota2 ); pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa );

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Constructive category theory

Translation to CAP

Schnitt := function( iota1, iota2 ) M1 := Source( iota1 ); M2 := Source( iota2 ); pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa ); return gamma; end;

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Constructive category theory

Translation to CAP

Schnitt := function( iota1, iota2 ) local M1, M2, pi1, pi2, lambda, phi, kappa, gamma; M1 := Source( iota1 ); M2 := Source( iota2 ); pi1 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 1 ); pi2 := ProjectionInFactorOfDirectSum( [ M1, M2 ], 2 ); lambda := PostCompose( iota1, pi1 ); phi := lambda - PostCompose( iota2, pi2 ); kappa := KernelEmbedding( phi ); gamma := PostCompose( lambda, kappa ); return gamma; end;

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Constructive category theory

Computing the intersection: Q-vector space

Compute the intersection of N M1 M2 3 2 2

ι1 := 1 1 1 1

  • ι2 :=

1 1 1 1

  • Sebastian Gutsche (Siegen)

Constructive Categories August 31, 2017 15 / 23

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Constructive category theory

Computing the intersection: Q-vector space

Compute the intersection of N M1 M2 3 2 2

ι1 := 1 1 1 1

  • ι2 :=

1 1 1 1

  • gap> gamma := Schnitt( iota1, iota2 );

<A morphism in the category of matrices over Q>

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Constructive category theory

Computing the intersection: Q-vector space

Compute the intersection of N M1 M2 3 2 2

ι1 := 1 1 1 1

  • ι2 :=

1 1 1 1

  • gap> gamma := Schnitt( iota1, iota2 );

<A morphism in the category of matrices over Q> gap> Display( gamma ); [ [ 1, 1, 0 ] ] A morphism in the category of matrices over Q

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Constructive category theory

CAP packages

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Constructive category theory

CAP packages

AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules MotivesForBiArrangements

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules MotivesForBiArrangements Bialgebroids FunctorCategories BMod

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules MotivesForBiArrangements Bialgebroids FunctorCategories BMod Actions GroupRepresentations InternalExteriorAlgebra

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules MotivesForBiArrangements Bialgebroids FunctorCategories BMod GradedModulePresentations ToricSheaves Actions GroupRepresentations InternalExteriorAlgebra

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Constructive category theory

CAP packages

IntrinsicCategories AttributeCategory LinearAlgebra ComplexesAndFilteredObjects GeneralizedMorphisms ModulePresentations HomologicalAlgebra CategoriesWithAmbientObjects M2 complex StableCategories FrobeniusCategories TriangulatedCategories Bicomplexes HomotopyCategories QPA2 CategoryOfProjectiveGradedObjects PresentationCategory PresentationsByProjectiveGradedModules MotivesForBiArrangements Bialgebroids FunctorCategories BMod GradedModulePresentations ToricSheaves Actions GroupRepresentations InternalExteriorAlgebra GradedModulePresentations ToricSheaves

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Applications to Algebraic Geometry

Applications to Algebraic Geometry

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Applications to Algebraic Geometry

Coherent sheaves: Affine space

Let K be an algebraically closed field.

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Applications to Algebraic Geometry

Coherent sheaves: Affine space

Let K be an algebraically closed field. Affine space Affine space: An = K n

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 18 / 23

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Applications to Algebraic Geometry

Coherent sheaves: Affine space

Let K be an algebraically closed field. Affine space Affine space: An = K n Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn]

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 18 / 23

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Applications to Algebraic Geometry

Coherent sheaves: Affine space

Let K be an algebraically closed field. Affine space Affine space: An = K n Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory:

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 18 / 23

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Applications to Algebraic Geometry

Coherent sheaves: Affine space

Let K be an algebraically closed field. Affine space Affine space: An = K n Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory: Equivalence of categories S-mod

∼

− → Coh (An)

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 18 / 23

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Applications to Algebraic Geometry

Coherent sheaves

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Applications to Algebraic Geometry

Coherent sheaves

Projective space

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory: Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory: Coh

  • Pn−1

Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 135

Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory: Coh

  • Pn−1

S-mod Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] In the language of category theory: Coh

  • Pn−1

S-mod Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading In the language of category theory: Coh

  • Pn−1

S-mod Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading In the language of category theory: Coh

  • Pn−1

S-mod Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading In the language of category theory: Coh

  • Pn−1

S-grmodZ Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading In the language of category theory: Coh

  • Pn−1

S-grmodZ Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Coh

  • Pn−1

S-grmodZ Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼ Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Projective space Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼ Berechenbarkeit von S-grmodG/S-grmod0

G?

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Projective space Pn−1 = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/K ∗) − {0}, K ∗ ∼ = Hom (Z, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − {0}, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a Z-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on {0}. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on Z. In the language of category theory: Equivalence of categories Coh

  • Pn−1

S-grmodZ/S-grmod0

Z

∼

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SLIDE 152

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on Z. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodZ/S-grmod0

Z

∼

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 153

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on Z. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodG/S-grmod0

G

∼

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 154

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety (smooth) Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on Z. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodG/S-grmod0

G

∼

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 155

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that are only supported on Z. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodG/S-grmod0

G

∼

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 156

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that sheafify to zero. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodG/S-grmod0

G

∼

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 19 / 23

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SLIDE 157

Applications to Algebraic Geometry

Coherent sheaves

Normal toric variety Toric variety X = (K n/G′) − Z, G′ ∼ = Hom (G, K ∗) Coherent sheaves correspond to f. g. modules over S := K [x1, . . . , xn] with a G-grading modulo modules that sheafify to zero. In the language of category theory: Equivalence of categories Coh

  • X
  • S-grmodG/S-grmod0

G

∼ Computability of S-grmodG/S-grmod0

G?

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SLIDE 158

Applications to Algebraic Geometry

Serre quotients

Serre quotient

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SLIDE 159

Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory.

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SLIDE 160

Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with

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SLIDE 161

Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with ObjA/C := ObjA

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SLIDE 162

Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with ObjA/C := ObjA HomA/C (A, B) := A B X ψ ϕ

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Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with ObjA/C := ObjA HomA/C (A, B) := A B X ψ ϕ coker (ψ) ∈ C

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Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with ObjA/C := ObjA HomA/C (A, B) := A B X ψ ϕ coker (ψ) ∈ C ϕ (ker (ψ)) ∈ C

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SLIDE 165

Applications to Algebraic Geometry

Serre quotients

Serre quotient Let A be an abelian category and C a thick subcategory. The Serre quotient A/C is an abelian category with ObjA/C := ObjA HomA/C (A, B) := A B X ψ ϕ coker (ψ) ∈ C ϕ (ker (ψ)) ∈ C

∼

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Applications to Algebraic Geometry

Composition in the Serre quotient

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C X Y B

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C X Y B

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SLIDE 170

Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C B X Y

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C B X Y Z

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C B X Y Z FiberProduct: Algorithm for intersection

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SLIDE 173

Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C X Y B Z

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Applications to Algebraic Geometry

Composition in the Serre quotient

Composition in the Serre quotient A/C A C X Y B Z Composition only by computations in A!

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable,

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian.

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian. S-grmodG/S-grmod0

G ∼

= Coh (X) ?

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SLIDE 179

Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian. S-grmodG/S-grmod0

G ∼

= Coh (X) ? Theorem (G.) Let X be a normal toric variety without torus factors.

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian. S-grmodG/S-grmod0

G ∼

= Coh (X) ? Theorem (G.) Let X be a normal toric variety without torus factors. Then the thick subcategory S-grmod0

G of f. g. G-graded modules over S which sheafify

to zero

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian. S-grmodG/S-grmod0

G ∼

= Coh (X) ? Theorem (G.) Let X be a normal toric variety without torus factors. Then the thick subcategory S-grmod0

G of f. g. G-graded modules over S which sheafify

to zero is decidable.

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Applications to Algebraic Geometry

Computability of toric coherent sheaves

Theorem (Barakat, Lange-Hegermann) Is A computable abelian and C decidable, then A/C is computable abelian. S-grmodG/S-grmod0

G ∼

= Coh (X) ? Theorem (G.) Let X be a normal toric variety without torus factors. Then the thick subcategory S-grmod0

G of f. g. G-graded modules over S which sheafify

to zero is decidable. So Coh (X) is computable abelian!

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian!

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection Homology

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection Homology Diagram chases

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection Homology Diagram chases Spectral sequences

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection Homology Diagram chases Spectral sequences Purity filtration

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 23 / 23

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Applications to Algebraic Geometry

Coherent sheaves

So Coh (X) is computable abelian! We can apply algorithms for abelian categories to coherent sheaves

  • ver toric varieties:

Intersection Homology Diagram chases Spectral sequences Purity filtration ...

Sebastian Gutsche (Siegen) Constructive Categories August 31, 2017 23 / 23